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Results tagged with teaching
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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/.
9
votes
Elementary applications of linear algebra over finite fields
Suppose you want to compute the period of the Fibonacci sequence $\bmod p$. This reduces to examining the powers of the matrix $\left[ \begin{array}{cc} 1 & 1 \\\ 1 & 0 \end{array} \right]$ over $\mat …
- 33.8k
22
votes
Why should one still teach Riemann integration?
Paul Siegel mentioned that to teach undergrads the Lebesgue integral you have to spend half the semester on measure theory. This is actually not true. There is a self-contained definition of the Leb …
- 4,713
5
votes
How to motivate the skein relations?
Here is a sketch of how the skein relations appear in the approach to knot invariants based on braided monoidal categories coming e.g. from representations of quantum groups.
Suppose $V$ is a dualiz …
- 118k
9
votes
What (fun) results in graph theory should undergraduates learn?
The (finite, simple) graphs with the property that their adjacency matrices have spectral radius less than $2$ are precisely the simply laced Dynkin diagrams $A_n, D_n, E_6, E_7, E_8$. Similarly, the …
- 118k
13
votes
Math books for advanced high school students
The list I give undergraduates and strong high schoolers is here.
- 118k
5
votes
Conceptual algebraic proof that Grassmannian is closed in Plucker embedding
Here are some details for what I think is a variant of Lev Borisov's proposal involving the Plücker relations in the comments. I think if your goal is just to show that the Grassmannian is closed then …
- 118k
24
votes
Is Euclid dead?
As long as this question is open I might as well throw in my two cents. I think it is not useful to teach Euclidean geometry to high school students. Here are some reasons I can think of for people to …
- 118k
16
votes
Accepted
Can this informal argument (for the fact that almost all reals in the unit interval are irra...
You can make sense of the uniform probability distribution on lots of infinite sets, notably any compact topological group $G$, where "uniform probability distribution" should mean "normalized Haar me …
- 118k
22
votes
Integrating powers without much calculus
This may not be in the spirit of what you want, but... by scaling arguments it suffices to establish that $\int_0^1 x^p dx = \frac{1}{p + 1}$. Consider the following probabilistic argument (not entire …
- 118k
20
votes
How to mentor an exceptional high school student?
Tell him about his other opportunities (although perhaps being on AoPS he is already aware of them). Summer programs like
Ross
PROMYS
the Canada/USA Mathcamp
HCSSiM
and others come highly recomme …
- 118k
3
votes
Short Course Suggestions For High School Students
I think a course about homogeneous linear recurrence relations with constant coefficients should be manageable. The simplest nontrivial example is probably the Fibonacci recurrence
$$F_{n+2} = F_{n+1} …
- 118k
5
votes
Suggestions for teaching advanced high school students
The site mentioned in Greg's answer, Art of Problem Solving, was my old haunt in high school (the pre-MO days!) so perhaps I should say a word about it. First, there is a large forum there with lots o …
- 118k
18
votes
Collecting proofs that finite multiplicative subgroups of fields are cyclic
Let $G$ be a finite subgroup of $F^{\ast}$ of order $n$. Then all the elements of $G$ satisfy $x^n = 1$ in $F$. Since polynomials of degree $n$ over a field have at most $n$ roots, it follows that the …
- 118k
33
votes
Why should one still teach Riemann integration?
I haven't really thought this through, but how does one actually compute integrals? For the Riemann integral one can either prove the fundamental theorem and have a large table of derivatives handy, …
- 118k
14
votes
Applications of knot theory
Colin Adams' The knot book discusses the following applications:
Knotting in DNA,
Molecular knots,
Statistical mechanics (e.g. the Potts model).
Constructing invariants of knots is also related t …
- 118k