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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/
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 …
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 …
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 …
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 …
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} …
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
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 …
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 …
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 …
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 …
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 …
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 …
9
votes
Do rational numbers admit a categorification which respects the following "duality"?
Let me expand on the answer I gave in meta. In my mind the appropriate "categorification" begins with the observation that "cups" is a unit, and in the first approach you endow only the numerator wit …
- 118k
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