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For questions related to teaching mathematics. For questions in Mathematics Education as a scientific discipline there is also the tag mathematics-education. Note you may also ask your question on http://matheducators.stackexchange.com/.

7 votes

Hard problems with an easy-to-understand answer

Lomonosov's theorem that a bounded operator on a complex (infinite-dimensional) Banach space that commutes with a nontrivial compact operator has a nontrivial invariant subspace had a surprisingly sim …
2 votes

Interesting examples of systems of linear differential equations with constant coefficients

There has been some discussion of rigid-body dynamics in the comments. It's a little hard for me to imagine that there was a newly (1967+) "discovered" system of linear differential equations with con …
Timothy Chow's user avatar
  • 82.7k
14 votes

Interesting examples of systems of linear differential equations with constant coefficients

There is some indication of what Rota had in mind in the book Ordinary Differential Equations by Birkhoff and Rota. I don't have a copy handy, but the preview on Amazon has this to say in the Introduc …
Timothy Chow's user avatar
  • 82.7k
5 votes

Alternate algorithms for Chinese remainder theorem

If you're using the (extended) Euclidean algorithm to compute modular inverses mod $m$, then the time complexity is roughly $O((\log m)^2)$. So if $m = abc$ then you'd prefer to do $(\log a)^2 + (\lo …
Timothy Chow's user avatar
  • 82.7k
3 votes

Demystifying complex numbers

I noticed this old question because it got bumped recently, and am surprised that the original, historical motivation for complex numbers—namely, a formula for solving a cubic equation—does not seem t …
26 votes

What kid-friendly math riddles are too often spoiled for mathematicians?

To make this suitable for MO rather than math.SE, perhaps we can define a "too often spoiled" puzzle to be one that can be recognized instantly by a mathematician even with what looks like far too lit …
16 votes

Math talk for all ages

My inclination would be to convey that it's fun to be a professional mathematician. How many people in the world have a fun job that they love doing? Only a small percentage. I feel privileged to be i …
10 votes

Important open exposition problems?

I'd nominate the theory of Macdonald polynomials (and associated topics). This is an extremely important area of algebraic combinatorics. Even if we restrict to type A, there are certain features of …
13 votes

Why do we need random variables?

Let me try to address the vague question of "why random variables." The short answer is that probability theory without random variables is like language without nouns. When I think about probabilit …
Timothy Chow's user avatar
  • 82.7k
2 votes

How to teach generalizing the induction hypothesis?

The following observation about your question 2 might be too obvious, but one simple way to recognize when you need to strengthen the induction hypothesis is to notice that when you try to prove your …
Timothy Chow's user avatar
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3 votes

Teaching homology via everyday examples

The following may not quite count as "everyday life," but in the 2013 MIT Mystery Hunt, one of the most elaborate team puzzle-solving contests in the world, one of the puzzles involved solving a cross …
23 votes
Accepted

Does seeing beyond the course you teach matter? The case of linear algebra and matrices

These examples may not translate directly into useful material for your teaching. However, I do believe that they give a good taste of how mathematicians think about linear algebra. …
2 votes

Elementary applications of linear algebra over finite fields

There is a Martin Gardner problem, reprinted in his Unexpected Hanging collection, that goes like this: Miranda beat Rosemary in a set of tennis, winning 6–3. There were five service breaks. Who …
1 vote

Elementary applications of linear algebra over finite fields

Rubik's Clock and Its Solution by Dénes and Mullen (Math. Mag. 68 (1995), 378–381) uses linear algebra modulo 12 to solve the Rubik's clock puzzle.
131 votes

Not especially famous, long-open problems which anyone can understand

The lonely runner conjecture. As Wikipedia puts it: Consider $k + 1$ runners on a circular track of unit length. At $t = 0$, all runners are at the same position and start to run; the runners' sp …

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