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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/
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 …
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 …
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 …
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 …
55
votes
Cool problems to impress students with group theory
An obvious choice is the enumeration of orbits of finite group actions, which show up everywhere in middle- and high-school competitions in disguise. The "cute" example here is coloring a cube or a r …
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 …
35
votes
What should be offered in undergraduate mathematics that's currently not (or isn't usually)?
I think undergraduates should take problem-solving classes. I don't think such classes are widely available, but bright students who didn't do a lot of problem-solving in high school would definitely …
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 …
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 …
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 …
27
votes
How should one present curl and divergence in an undergraduate multivariable calculus class?
As far as explaining the formulas for div and curl, you should be able to do this starting with the definitions given in the Wikpedia articles by taking the corresponding integrals on rectangles and b …
- 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
25
votes
Why is a topology made up of 'open' sets?
This is an attempt to synthesize ideas that have appeared in other answers, for example sigfpe's and Tim Perutz's. Feel free to edit if you think the ideas can be better expressed.
The idea I want t …
24
votes
Interesting results in algebraic geometry accessible to 3rd year undergraduates
This isn't a result so much as a perspective, but it is one of the main reasons I first got interested in algebraic geometry.
In basic algebraic number theory you learn that some extensions of the in …
- 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 …