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1500 questions
21
votes
7 answers
Is $e^{i\pi}+1=0$ a good motivation for introducing $e$ or $i$? Why (not)?
Most mathematicians would agree that
$$e^{i\pi}+1=0$$
is one of the most impressive formulas.
Imagine your students have just learned about the definition of $e$ or $i$ (just assume it's $e$, normally $i$ comes later in curricula); $\pi$ should…
Markus Klein
- 9,438
- 3
- 41
- 96
21
votes
3 answers
Pedagogical challenge: Homeomorphic vs. Homotopy equivalent vs. Homologous?
I believe it is the case that, between spaces,
homeomorphism
is stronger than
homotopy equivalence
which is stronger than having
isomorphic homology groups.
For example,
the annulus and the circle are not homeomorphic
but they have the same…
Joseph O'Rourke
- 29,827
- 6
- 61
- 140
21
votes
2 answers
Handing out $2 calculators for tests?
In a store last weekend, I saw a simple solar-powered calculator for $2. It had four functions plus a square root key. I'm considering the possibility of buying 50 such calculators and handing them out for use during exams in my freshman calc…
user507
21
votes
6 answers
Becoming a better instructor: where to start?
I just finished a PhD in math at a top department, but not one that placed much emphasis on graduate student teaching. Grad students here teach only as TAs, and the training is minimal. I got great student evaluations, but I realize this was due…
Mark
- 311
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- 4
21
votes
7 answers
Is there a canonical name for a polynomial-like expression allowing for negative powers?
When introducing the techniques of differentiation, polynomials come up all the time as great examples to familiarize students with the "power rule" and the linearity of differentiation.
A common extension is to then work with expressions involving…
Kelvin Soh
- 313
- 1
- 5
21
votes
6 answers
Educators, what resources have you built to better serve your students?
What resources (Desmos, Geogebra, math3d, etc.) have you built to help your students better understand or visualize topics in mathematics?
Cameron Williams
- 694
- 4
- 13
21
votes
3 answers
When did the American school system's progression of math classes take its current form?
In the United States, secondary education students generally progress through pre-algebra courses, then algebra, Euclidean geometry, more algebra/trigonometry, then calculus or statistics.
I am particularly interested in the place that geometry…
Brian Rushton
- 11,680
- 14
- 67
- 137
21
votes
2 answers
Example "bad proofs"?
As a sidetrack in this question it came up that it is important to have students read texts (in particular proofs) critically. As examples it is nice to have correct proofs at hand (presumably in the textbook/lecture notes), but also a variety of…
vonbrand
- 12,306
- 3
- 27
- 59
21
votes
4 answers
Evaluating the reception of (epsilon, delta) definitions
Both education researchers and mathematicians discuss the challenge of (epsilon, delta) type definitions in real analysis and the student reception of them. My impression has been that mathematicians often hold an upbeat opinion on the success of…
Mikhail Katz
- 2,242
- 15
- 23
21
votes
6 answers
Physical applications of higher terms of Taylor series
Depressingly many of the physical "applications" of Taylor series that I can find in textbooks and online are actually just applications of linear approximation, since they only take the constant and linear term of a Taylor series. For instance,…
Mike Shulman
- 6,560
- 21
- 53
21
votes
6 answers
Why is the concept of injective functions difficult for my students?
I was aware that students find the definition of function too abstract and thus find it difficult. However, I thought, once you understand functions, the concept of injective and surjective functions are easy. True to my belief students were able…
Divakaran Divakaran
- 701
- 6
- 16
21
votes
2 answers
Adult Mathematical Literacy
In short, my question is:
What percentage of American adults know what a prime number is?
Since this question is very specific, and my interests are a bit broader, I'd also be happy with:
Is there a good source for statistics on adult…
Daniel Litt
- 161
- 8
21
votes
8 answers
How do I learn advanced mathematics without forgetting?
I am pursuing mathematics through distance education and I find that it takes me a long time to understand the concepts (e.g. sigma fields, measure theory, connected topological spaces, etc.).
After I have understood the theorems, I practiced them…
user14243
21
votes
2 answers
Pressure vs. Laissez-faire: Literature dealing with balance in university-classes
I am seeking for some pedagogical literature dealing with the following question: Imaging you have an average class in college/university: What is a good balance between Laissez-faire and pressure?
i.e., a balance between "I'm offering you some…
Markus Klein
- 9,438
- 3
- 41
- 96
21
votes
1 answer
How do I track down the sources of solutions which students have used to cheat on exams?
I recently taught an introduction to real analysis. I assigned a (covid-induced) take-home final, which included the question:
Define the set S by
$$
\bigcup_{n=1}^\infty \left\{
\frac{a}{2^n}\colon 0\le a\le 4^n, a\in\mathbb{Z} \right\}.
$$
…
Anthony Quas
- 311
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- 5