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I am finding it difficult to motivate students on why they should how to prove mathematical results. They learn them just to pass examinations but show no real interest or enthusiasm for this. How can I inspire them to love essential kind of mathematics? They love doing mathematical techniques. Any resources or any answers would really help me.

Additional information: 1st year undergraduates who dislike proofs. The university mostly thinks the maths courses are there for supplying other subjects. The students don't like proof based courses.

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    $\begingroup$ 1ST year undergraduates who dislike proofs. THe university mostly thinks the maths courses are there for supplying other subjects. THe students don't like proof based courses. $\endgroup$ – matqkks Apr 2 at 13:59
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    $\begingroup$ Proofs train the brain. To learn how to prove something logically (deduction for "soft proof", induction for hard proof, reductio ad absurdum, counting all the outcomes, etc.) No one who does not do math after school — and even many of those who do — remember all the proofs from their geometry course. But they gained something in the process. They look at politicians and policy makers with different eyes. They easily spot holes in software specs. They write coherent articles. Obviously, the powers that be prefer to keep their lemmings compliant and stupid. Logic is for pencil-necks. $\endgroup$ – Rusty Core Apr 2 at 16:49
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    $\begingroup$ If they're first-year undergraduates, does that mean that this is in the context of first-semester calculus? If so, is it some kind of special proof-based calc course for math majors, or is it the one that is mainly for engineering and biology majors? $\endgroup$ – Ben Crowell Apr 2 at 20:45
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    $\begingroup$ Maybe they need a course or unit on basic logic and methods of proof using the simplest possible ("trivial") examples -- not jumping right into the deep end with deltas and epsilons in calculus. If the university won't go for that, you should maybe resign yourself to teaching your students a useful bag of mathematical tricks for common problems in whatever subject is their specialty. $\endgroup$ – Dan Christensen Apr 3 at 13:44
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    $\begingroup$ Thanks for all the detailed replies.This is a module to understand proofs so that they can use them for the other modules later on such as number theory, linear algebra, analysis, abstract algebra etc. $\endgroup$ – matqkks Apr 5 at 20:00
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Surprise them.

Especially in a (mathematical) culture where "getting the right answers" is prized and excellence on such examinations is valued in general, it is reasonable that students will value this above other expressions of mathematics. (I have a little teaching experience in that context, though most of my experience is in the "just get through it" American culture of math.) So one way to get students interested in proof is to express something where the mathematical techniques are irrelevant and do not help in solving an interesting problem.

Ideally, you would be able to have students in a course like number theory or graph theory - both of which can be quite applied - and then give them a problem which simply doesn't have a (purely) computational answer. Like "is there an integer point on this elliptic curve", or "can I embed this graph on a computer chip with no crossings of circuits", or something. Maybe those are too advanced, but I hope you take my meaning. The only way to know for sure is to prove.

The difficulty is that most course content in this situation is more in the precalculus-through-differential equations trajectory, which doesn't always lend itself well to this. You can prove things like the intermediate value theorem or existence of solutions to equations, but students will not care about these. Because they seem obvious, like the fact there are no integers between $0$ and $1$ - you can prove it from some axioms, but students will not see the point.

But (perhaps ironically) in numerical methods in these areas, proofs are perhaps much more useful. How accurate is Simpson's rule? How much computer power do you need to get within $0.0001$ for your integral? Proofs are key to this - especially the somewhat (to students) bizarre requirements about absolute values of various derivatives. Show an example that shouldn't work, but it does. Now we need a proof.

Another idea is to talk about integration techniques. Give them a really hard integration by partial fractions or nasty multiple parts/substitution, then another. When does it work, when not? You can prove it - or that it isn't, Liouville's Theorem.

I can't guarantee even a little success; it's hard to find resources at times (my examples are probably half-baked), and you have a lot of headwinds, as is evident from your post. But in my experience, showing people what mathematics cannot do is a first direction that will motivate them to love math for its own sake, not just for the satisfaction of a good successful computation. We need both.

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    $\begingroup$ One additional point about the obviousness of the intermediate value theorem: It even took mathematicians some time after Newton's calculus to get to a point where our epistemology required a proof this. i.e. it was about 150 years after Newton's Calculus that Bolzano provided the first proof. $\endgroup$ – Adam Apr 4 at 13:19
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Maybe you can motivate the value of proofs by showing seemingly true claims which are in fact false, justifying the need for a proof of even an "obvious" claim.

For example, this appears to be a dissection of an $8 \times 8$ square to a $13 \times 5$ rectangle: $64$ vs. $65$ unit squares. It takes some effort to uncover the flaw.


          fake-dissection
          (Image from David Eppstein.)


Perhaps better is this MSE question, 'Obvious' theorems that are actually false. For example,

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    $\begingroup$ Thanks for the link, and please update if you find others. Of course, one would want to mine such lists of 'obvious' theorems for ones the particular students care about; sounds like the particular cohort in question may not find this one compelling (though I do!). $\endgroup$ – kcrisman Apr 4 at 13:33
  • $\begingroup$ @kcrisman: Good point to match the cohort's interests. $\endgroup$ – Joseph O'Rourke Apr 4 at 14:17
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As usual, my advice is contrarian, but you need to consider alternate viewpoints.

I think you would be better off not pushing so hard. It's like pestering a girl who doesn't like you. Not a good approach. Slightly pathetic.

Make proofs the spice or garnish or relish or what have you to the main course. Don't try to convince people they are better than the meat and potatoes. They are entitled to their point of view of enjoying technique more. (For that matter they are CORRECT in saying calculus and diffyQs, for most students, are taught as service courses for engineering and science.)

I would also communicate some sympathy and exceptionalism. "We've spared most of the proofs, but this is a really important one, so pay attention."

If you are forcing them to learn/test more proofs than are in the text (or drilling them more than the homework percentage of proofs), stop. Just stop. The students are seeing you pushing MORE peas at them than even in the balanced diet.

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    $\begingroup$ "As usual"? Why don't you register, so member can recognize you as a repeat visitor? $\endgroup$ – JoeTaxpayer Apr 3 at 23:24
  • $\begingroup$ tanquam ex ungue leonem $\endgroup$ – kcrisman Apr 4 at 13:34
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    $\begingroup$ "They are entitled to their point of view of enjoying technique more." - I'll push back on this a little. Not that one isn't "entitled" to a variety of things, but it's not clear to me that one needs to buy into a consumerist viewpoint of pedagogy. Of course it's getting more popular, and one can see these debates about "useful" throughout the centuries, but I think one can let people enjoy something while exposing them to something else worthwhile. From a strict viewpoint nearly every course is "useless" to most people in the long run, but in fact in sum total they are very useful. $\endgroup$ – kcrisman Apr 4 at 13:38
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    $\begingroup$ Nonetheless up vote for the third paragraph, which is actually the truly student-centered one and worth thinking about. $\endgroup$ – kcrisman Apr 4 at 13:43
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    $\begingroup$ @JoeTaxpayer: It's so he can dodge the standard reputation and moderation systems. $\endgroup$ – Daniel R. Collins Apr 10 at 3:39

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