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Results tagged with teaching
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user 3106
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 …
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 …
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 …
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 …
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 …
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 …