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Results tagged with mathematics-education
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For questions in Mathematics Education as a scientific discipline. For more hands-on questions on teaching Mathematics, please use the tag teaching. There is also a Stack Exchange community http://matheducators.stackexchange.com/
32
votes
Real-world applications of mathematics, by arxiv subject area?
math.CO Combinatorics
Combinatorics finds applications in computer science, especially in the run-time analysis of algorithms. It has also in recent years found applications in physics, at least in …
- 4,713
73
votes
Sophisticated treatments of topics in school mathematics
It's common in calculus classes and textbooks to state that the antiderivative of $\frac{1}{x}$ is $\log |x| + C$, where $C$ is a constant. This is incorrect: $C$ need only be a locally constant funct …
- 18.7k
86
votes
Sophisticated treatments of topics in school mathematics
The angle addition formula $\tan(\alpha + \beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha) \tan(\beta)}$ for tangent gives one of the simplest nontrivial examples of a formal group law, nam …
- 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
38
votes
Are there proofs that you feel you did not "understand" for a long time?
The first proof of Tychonoff's theorem I learned, from the Alexander subbase theorem, was completely mysterious to me. I didn't understand it at all. In particular I didn't really understand what the …
- 118k
3
votes
Seeking a Geometric Proof of a Generalized Alternating Series' Convergence
Here's an idea. Group the series into blocks
$$\sum_{n=dk}^{d(k+1) - 1} \frac{z^n}{n}$$
where $d$ is fixed and large enough that the complex numbers $1, z, z^2, ... z^{d-1}$ are approximately unifor …
- 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
25
votes
Taylor's theorem and the symmetric group
One way is to use a combinatorial definition of the derivative. Let $A(z) = \sum a_n z^n$ be a power series. In combinatorics, where $A$ is likely to be an ordinary generating function, $a_n$ is likel …
- 118k
115
votes
What are your favorite instructional counterexamples?
A polynomial $p(x) \in \mathbb{Z}[x]$ is irreducible if it is irreducible $\bmod l$ for some prime $l$. This is an important and useful enough sufficient criterion for irreducibility that one might w …
- 18.7k
17
votes
Teaching undergraduate students to write proofs
Regarding different flavors of approach 1, here are some words from Halmos.
I have taught courses whose entire content was problems solved by students (and then presented to the class). The number of …
- 118k
32
votes
How to present mathematics to non-mathematicians?
There is this nice quote whose wording I can't quite recall. It is something like "physics is the study of the laws of God. Mathematics is the study of the laws even God must follow."
I think there …
- 35.5k
5
votes
How do I explain the number e to a ten year old?
Here is one way which I learned from Clio Cresswell's Mathematics and Sex, although unfortunately I'm not sure how to prove it. Suppose you are sure that you will meet exactly $n$ suitable marriage p …
- 118k
34
votes
Examples of common false beliefs in mathematics
The quotient $G/Z(G)$ of a group by its center is centerless. I definitely thought this until it was pointed out to me in a Lie theory textbook that this wasn't true in general, but is true for (edit …
- 118k
3
votes
What are your experiences of handouts in mathematics lectures?
One basic observation, as a student. A big reason for providing notes is if the class works out of more than one textbook (or none at all!) and you want to keep the narrative straight. The professor …
- 118k
278
votes
Examples of common false beliefs in mathematics
I don't know if this is common or not, but I spent a very long time believing that a group $G$ with a normal subgroup $N$ is always a semidirect product of $N$ and $G/N$. I don't think I was ever sho …
- 118k