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Results tagged with teaching
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For questions related to teaching mathematics. For questions in Mathematics Education as a scientific discipline there is also the tag mathematics-education. Note you may also ask your question on http://matheducators.stackexchange.com/.
21
votes
Zorn's lemma: old friend or historical relic?
Your choice (ha) to prove the existence of a basis by using a well-ordering in place of Zorn’s Lemma turns things around historically: the first proof of the existence of algebraic bases (using only f …
- 50.6k
18
votes
Teaching polarisation formula
The polarization formula carries over to algebraic settings not directly involving real and complex numbers. Let $L/K$ be a quadratic Galois extension, where the nontrivial element of ${\rm Gal}(L/K)$ …
- 50.6k
15
votes
Historical (personal) examples of teaching-based research
While thinking about teaching finite fields for students who knew some group theory and what a field is, but had not seen abstract linear algebra, I wondered if there might be a way to show them a finite …
5
votes
Interesting applications (in pure mathematics) of first-year calculus
In number theory, here are four applications of techniques or results in first-year calculus.
(1) Finding equations of tangent lines by first-semester calculus methods lets us add points on elliptic …
16
votes
Are there any "related rates" calculus problems that don't feel contrived?
Here are two examples I think are interesting. (Update: I have added a third example at the end, more intricate than the other two.)
A ladder that is leaning against a wall starts slipping down. If t …
5
votes
Suggestions for teaching advanced high school students
Show them the rational parametrization of the unit circle. Then generalize the method to other conics with an obvious rational point like $x^2 + y^2 = 5$ and $x^2 - 2y^2 = 1$ and discuss why it doesn' …
48
votes
Collecting proofs that finite multiplicative subgroups of fields are cyclic
I once collected six [edit: now seven [edit: now eight [edit:now nine [edit: now ten]]]] proofs of this theorem, for the field $\mathbf Z/(p)$, and they can be found here. While $\mathbf Z/(p)$ is not …
114
votes
Demystifying complex numbers
The nicest elementary illustration I know of the relevance of complex numbers to calculus
is its link to radius of convergence, which student learn how to compute by various tests, but more mechanical …
12
votes
Applications of connectedness
Do you want an example where you can give the proof, not just the statement, in the lecture too? Lots of other suggestions are great uses of connectedness, but in one lecture where you first introduc …
5
votes
Accepted
Text/structure for an analysis course for students with pre-existing understanding of some a...
If you want a reference that will not bore them, supplement the main text with "Metric Spaces: Iteration and Application" by Victor Bryant. The book is short and it shows in several contexts how the c …
15
votes
Pedagogical question about linear algebra
To understand a definition, show the students (a) lots of examples, (b) lots of non-examples and why they don't work, (c) misconceptions related to the definition
(e.g., coordinates and real/imag. par …