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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/.
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 …
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 …
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 …
10
votes
Category theory sans (much) motivation?
The Unapologetic Mathematician covers category theory before it covers group theory, so after a little more experience with abstraction you might want to expose your friend to the beginning posts. I …
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 …
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
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 …
11
votes
Math History Question about the exponential function
Short answer: Most likely undefined.
Long answer: The "naive" definition of $f(x) = a^x$ where $a, x \in \mathbb{R}$ and $a > 0$ is as follows. You know how to define $f(n)$ where $n$ is an intege …
- 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
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 …
33
votes
Demystifying complex numbers
If the students have had a first course in differential equations, tell them to solve the system
$$x'(t) = -y(t)$$
$$y'(t) = x(t).$$
This is the equation of motion for a particle whose velocity vect …
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
13
votes
Math books for advanced high school students
The list I give undergraduates and strong high schoolers is here.
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} …
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