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Results tagged with mathematics-education
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user 290
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/
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 …
7
votes
How do you motivate a precise definition to a student without much proof experience?
Sipser's Introduction to the Theory of Computation spends some time motivating the need for precise definitions, since it's aimed towards a computer science audience who may not have experience with p …
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 …
6
votes
3
answers
4k
views
(How) should I take notes on a subject for self-study? [closed]
Suppose I am interested in really learning / thoroughly reviewing some subject (e.g. the basic theorems of infinite Galois theory, or the classification of compact Lie groups). One approach I might c …
8
votes
Do you find your students are less competent in basic algebra and arithmetic, and, if so, do...
For what it's worth, there is a fairly specific villain to blame for this problem in the school district where I attended high school. In this district - which is not the district I grew up in - the …
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
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 …
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
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 …
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 …
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 …
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 …
3
votes
Accepted
An "Elementary" Math Question Generalized (Ring Theory Perhaps)
I don't think this is a good problem for metacognition. Solving it is too contingent on what people have taught you about irreducibility.
Anyway, as for your general question, I am sure you can find …
- 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 …
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