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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/
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 …
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
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