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Once my professor told me that sometimes questions that she gets from her students turn out to be precious inputs for deepening her own mathematical understanding or for doing some new research in mathematics.

Did this happen to some of you too? That is, did questions of students help you to understand some mathematical concept in a better or more profound way, or did they even give you input for your own mathematical research?

If so, please, mention these questions and explain in what way they helped you.

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    $\begingroup$ Generically speaking, when students ask about generalization in nontraditional places it leads to new thinking for me. It's very rare. $\endgroup$ – James S. Cook Oct 18 '14 at 23:46
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    $\begingroup$ There is a meta discussion about this question and discussion-type questions in general. I think discussing this type of questions at large would be more appropriate in meta. $\endgroup$ – Joonas Ilmavirta Oct 19 '14 at 8:49
  • $\begingroup$ I edited the question slightly in view of some meta discussion and removed some (in part now obsolete comments); they are on meta for reference and could be restored if needed (please ask on the meta in case). $\endgroup$ – quid Oct 20 '14 at 6:09
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At my university, the usual multivariable calculus class is split into two semesters: the first does geometry in $\mathbb{R}^3$ and then derivatives (gradients, Lagrange multipliers, etc.) while the second focuses on integration (Green's theorem, Stokes' theorem etc.) In the second semester there are usually about three free weeks or so at the end of the course for supplementary topics, and I decided to give my students a primer on the calculus of variations.

Two of the examples that I discussed were the isoperimetric inequality (the curve of a given length which encloses the maximal area is a circle) and geodesics on a sphere. One of my students tried to make a connection between these two examples and asked a good question: can one use the calculus of variations to solve the isoperimetric problem on the sphere?

The standard variational proof of the planar isoperimetric inequality uses Green's theorem, so it is natural to try to use Stokes' theorem on the sphere. I quickly realized that this can't work, and after a little research learned that one should use the Gauss-Bonnet theorem instead. This led me into the literature on isoperimetric inequalities on Riemannian manifolds and got me thinking about research questions involving connections between isoperimetric problems and Dirac operators. I never got any results, but the questions are still in the back of my mind.

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I am not a research-level mathematician, but this absolutely happens to me. There is so much in math that we can think more deeply about.

Just last week, a student came into my office to get help on a derivative problem that used the quotient 'rule'. He was getting zero in the numerator. I showed him his mistake, and said "You won't get zero in the numerator." He said he thought it could happen on some problems. I said "Well, let's prove it. Here's my conjecture: If y = f(x)/g(x), where f(x) does not equal g(x), and neither is a constant, then y' will not equal zero."

Embarrassingly, I was stuck then. I was trying to think of it algebraically, and was getting nowhere. The way to think about it (which I realized while discussing it with another teacher) is conceptually. If y is not a constant, then y' cannot equal zero. That's all there is to it. Is my understanding deeper now? My understanding of derivative was fine, but my approach to proof was too constrained. I hope I've learned something.

Pam Sorooshian, an unschooling advocate, has said that, "Once people memorize a technique or a 'fact' they have the feeling that they 'know it' and they stop questioning it or wondering about it." I think this probably happened to even the best mathematicians among us, when we were young. How much of elementary math do you take for granted? Can you get a deeper understanding of it, and still stay connected to a child's perspective? What bits of math do you 'know' to be true even though you've never personally proved them?

For years in my algebra courses, I had students using the Pythagorean theorem. It finally occurred to me that I didn't really know why it was true. (And I only knew that it was true because of all the history that references it, not because of any logical necessity.) This was not one of the times that a student question provoked me to dig deeper, but it could have been. Ever since then, I always get my students to work through the most visual proof I know.

I am not remembering all of the student questions that have made me think, but I know it has happened repeatedly. It's always a delight to me when I have to say that I don't know and I'll think about it. I never would have imagined that I could keep learning more about basic algebra, just about every time I teach it. Calculus too. And that is part of the joy of teaching for me.

I hope others (with better memories) will offer more questions students have asked, that got them thinking more deeply or more carefully.

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    $\begingroup$ This is a very nice answer. It reflects my experience as well, and is put more eloquently than I probably could muster on an answer of my own. There is no "big discovery", but thousands of small improvements in depth of knowledge. This is a big part of what keeps me going as a teacher. $\endgroup$ – Steven Gubkin Oct 20 '14 at 15:08
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    $\begingroup$ $f(x) = 2 e^x$, $g(x) = e^x$. $\endgroup$ – Pete L. Clark Oct 20 '14 at 18:20
  • $\begingroup$ Ahh, good call, Pete. Any quotient which would simplify to a constant. Perhaps there are some where it is not easy to see that you have a constant, and the derivative shows more clearly as being equal to zero? $\endgroup$ – Sue VanHattum Oct 20 '14 at 20:16
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    $\begingroup$ @SueVanHattum I think you were not able to prove it only because you were trying to prove an incorrect statement. As Pete pointed out, it's enough for the numerator to be a constant multiple of the denominator, for the derivative of the quotient to be zero. If you try to prove this, you do get the proof: $u'v - v'u = 0 \Rightarrow \\ \dfrac{u'} u = \dfrac{v'} v \Rightarrow\\ \log(u) = \log(v) + c \Rightarrow\\ u = kv \Rightarrow\\ \dfrac u v = k$ $\endgroup$ – M. Vinay Oct 21 '14 at 8:43
  • $\begingroup$ @M. Vinay, thank you! What a lovely proof. I should have seen the anti-derivatives there. (Which my student wouldn't see.) It makes me happy to see this. My algebraic attempt would have been fruitful if I had thought this way. $\endgroup$ – Sue VanHattum Oct 21 '14 at 19:16

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