**20**

votes

**5**answers

2k views

### The Matrix-Tree Theorem without the matrix

I'm teaching an introductory graph theory course in the Fall, which I'm excited about because it gives me the chance to improve my understanding of graphs (my work is in topology). A highlight for me ...

**-3**

votes

**0**answers

83 views

### Have any author ever distinguish between “good” and “bad” “indeterminate forms”? [closed]

In the context of two-point compactification of real numbers (affinely extended real line) some expressions with infinite elements are usually left undefined. Yet it seems they can be defined to ...

**10**

votes

**4**answers

1k views

### Integrating Powers without much Calculus

I'll jump into the question and then back off into qualifications and context
Using the definition of a definite integral as the limit of Riemann sums, what is the best way (or the very good ways) ...

**9**

votes

**2**answers

1k views

### Good examples of random variables whose image is not a measurable set?

Are their simple/natural examples of real-valued Borel-measurable random variables whose image is not a Borel set? Something that occurs "naturally"?
I am teaching Doob's lemma (for two real-valued ...

**186**

votes

**97**answers

31k views

### Not especially famous, long-open problems which anyone can understand

Question: I'm asking for a big list of not especially famous, long open problems that anyone can understand. Community wiki, so one problem per answer, please.
Motivation: I plan to use this list ...

**22**

votes

**7**answers

2k views

### Why are two notions of Gaussian curvature are the same - what is the simplest & most didactic proof?

This question is still wide open - all of the answers so far rely on magical calculations. I've only accepted an answer because, by bounty rules, otherwise one would be accepted automatically. I can't ...

**120**

votes

**6**answers

41k views

### Where to buy premium white chalk in the U.S., like they have at RIMS?

While not a research-level math question, I'm sure this is a question of interest to many research-level mathematicians, whose expertise I seek.
At RIMS (in Kyoto) in 2005, they had the best white ...

**26**

votes

**13**answers

9k views

### How to draw knots with Latex?

I am writing an exam for my students, and the topic is intro knots theory. I have no idea how to put knots into the file, but I know many MO users who can draw amazing diagrams in their papers.
Can ...

**46**

votes

**45**answers

16k views

### An example of a beautiful proof that would be accessible at the high school level?

The background of my question comes from an observation that what we teach in schools does not always reflect what we practice. Beauty is part of what drives mathematicians, but we rarely talk about ...

**9**

votes

**9**answers

2k views

### Not especially famous, long-open problems which higher mathematics beginners can understand

This is a pair to
Not especially famous, long-open problems which anyone can understand
So this time I'm asking for open questions so easy to state for students of subjects such as undergraduate ...

**31**

votes

**19**answers

6k views

### Interesting applications (in pure mathematics) of first-year calculus

What interesting applications are there for theorems or other results studied in first-year calculus courses?
A good example for such an application would be using a calculus theorem to prove a ...

**9**

votes

**11**answers

3k views

### Textbook for undergraduate course in geometry

I've been assigned to teach our undergraduate course in geometry next semester. This course originally was intended for future high-school teachers and focused on axiomatic, Euclid-style geometry ...

**71**

votes

**25**answers

26k views

### What are the most misleading alternate definitions in taught mathematics?

I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in ...

**5**

votes

**2**answers

1k views

### What is the best *general triangle*?

During courses on geometry it is sometimes necessary to draw a triangle on the blackboard that can easily be recognized as a general triangle. It must not be rectangular and must not have two or more ...

**21**

votes

**7**answers

2k views

### Conceptual algebraic proof that Grassmannian is closed in Plucker embedding

I'm planning lectures for my intro algebraic geometry course, and I noted something awkward that is coming up. We're starting projective varieties soon. Of course, we'll prove that projective maps are ...

**4**

votes

**1**answer

169 views

### How to teach generalizing the induction hypothesis? [closed]

I just finished teaching a class on using proof assistants (in this case, Agda) to write provably correct programs. Reflecting on how it went, the biggest difficulty I noticed the students having was ...

**18**

votes

**12**answers

7k views

### How seriously should a graduate student take teaching evaluations?

Pretty much the question in the title. If a grad student gets bad reviews as a TA, how much does that hurt them later? How much do good reviews help? What if the situation is more complex? (For ...

**28**

votes

**7**answers

9k views

### Collecting proofs that finite multiplicative subgroups of fields are cyclic.

I teach elementary number theory and discrete mathematics to students who come with no abstract algebra. I have found proving the key theorem that finite multiplicative subgroups of fields are cyclic ...

**27**

votes

**12**answers

2k views

### Historical (personal) examples of teaching-based research

The phrase "teaching-based research" brings to mind research about teaching, though important, it is not what I mean. Unfortunately, I couldn't come up with a better phrase, thus please bear with me ...

**25**

votes

**3**answers

2k views

### Is “problem solving” a subject to be taught?

I am witnessing a new curriculum change in my country (Iran). It includes the change of all the mathematics textbooks at all grades. The peoples involved has sent me the textbook for seven graders (13 ...

**6**

votes

**3**answers

846 views

### An application of Maschke's theorem

I've been teaching some elementary representation theory to undergraduates, and want to provide applications of Maschke's theorem to complex group algebras to present in class. In particular, I'd like ...

**34**

votes

**20**answers

6k views

### Fun applications of representations of finite groups

Are there some fun applications of the theory of representations of finite groups? I would like to have some examples that could be explained to a student who knows what is a finite group but does not ...

**42**

votes

**20**answers

11k views

### Interesting Calculus Questions/Exercises

I am in the process of redesigning the calculus course that I have taught five or six times. What I would like to know is if anyone has some really good examples or exercises that I could either do ...

**7**

votes

**4**answers

2k views

### Help me find good math questions for my students

I am a teacher at 西铁一中。 I teach mathematics in English for students going abroad.
Now this is my problem, there are few mathematics books written in English that are at the level of high school, ...

**14**

votes

**5**answers

1k views

### Permission to use Online Notes

I am a new professor in Mathematics and I am running an independent study on Diophantine equations with a student of mine. Online I have found a wealth of very helpful expository notes written by ...

**26**

votes

**6**answers

2k views

### Means of Promoting Mathematics in Young Countries!

We all know mathematics is life, this question is for Mankind. It's mathoverflow here when some parts of the world we have mathunderflow! I think we can do something through ideas. A similar ...

**15**

votes

**17**answers

12k views

### Undergraduate Differential Geometry Texts

Can anyone suggest any basic undergraduate differential geometry texts on the same level as Manfredo do Carmo's Differential Geometry of Curves and Surfaces other than that particular one?
(I know a ...

**17**

votes

**19**answers

5k views

### Math books for advanced high school students

I'm working in a program for teaching a group of students selected in a Olympiad competition. The program is aimed to acquaint the students with the diverse aspects of higher mathematics in a way ...

**8**

votes

**5**answers

1k views

### Applications of Liouville's theorem

I'm looking for "nice" applications of Liouville's theorem (every bounded entire map is constant) outside the area of complex analysis.
An example of what I'm not looking for : a non-constant entire ...

**58**

votes

**32**answers

39k views

### Why do we teach calculus students the derivative as a limit?

I'm not teaching calculus right now, but I talk to someone who does, and the question that came up is why emphasize the $h \to 0$ definition of a derivative to calculus students?
Something a teacher ...

**50**

votes

**2**answers

1k views

### History of $\frac d{dt}\tan^{-1}(t)=\frac 1{1+t^2}$

Let $\theta = \tan^{-1}(t)$. Nowadays it is taught:
1º that
$$
\frac{d\theta}{dt} = \frac 1{dt\,/\,d\theta} = \frac 1{1+t^2},
\tag1
$$
2º that, via the fundamental theorem of calculus, this is ...

**31**

votes

**10**answers

2k views

### effective teaching

Eric Mazur has a wonderful video describing how physics is taught at many universities and his description applies word for word to the way I learned mathematics and the way it is still being taught, ...

**11**

votes

**3**answers

457 views

### Why does inconstructibility of $\sqrt[3]{2}$ imply impossibility of cube doubling? [closed]

In this question "constructing" and "doubling" is meant in the compass-and-straightedge sense.
On my desk I have five Basic Algebra texts treating constructability in the plane $\mathbb{C}$ or ...

**21**

votes

**14**answers

6k views

### “Homotopy-first” courses in algebraic topology

A first course in algebraic topology, at least the ones I'm familiar with, generally gets students to a point where they can calculate homology right away. Building the theory behind it is generally ...

**20**

votes

**6**answers

4k views

### an engineering Ph.D. teaching math in college

I have a friend who has been teaching college-level math (e.g., all levels of calculus)
for about 4 years, although all of his education, including his Ph.D., was in engineering.
Now he is ...

**51**

votes

**6**answers

3k views

### What does it take to run a good learning seminar?

I'm thinking about running a graduate student seminar in the summer. Having both organized and participated in such seminars in the past, I have witnessed first-hand that, contrary to what one might ...

**48**

votes

**12**answers

9k views

### How misleading is it to regard $\frac{dy}{dx}$ as a fraction?

I am teaching Calc I, for the first time, and I haven't seriously revisited the subject in quite some time. An interesting pedagogy question came up: How misleading is it to regard $\frac{dy}{dx}$ as ...

**18**

votes

**2**answers

954 views

### Teaching the fundamental group via everyday examples

This question is a "prequel" to a similar question about homology. Both questions were inspired by seeing a talk, by Tadashi Tokieda, about the interesting physics that appears in toys.
What ...

**12**

votes

**2**answers

1k views

### Bitcoin Research

I have recently been assigned to advise a student on a senior thesis. She has taken linear algebra, introductory real analysis, and abstract algebra. Her interest is in cryptography. And she has a ...

**6**

votes

**3**answers

1k views

### The etale fundamental group of a field

Background and motivation:
I am teaching the "covering space" section in an introductory algebraic topology course. I thought that, in the last five minutes of my last lecture, I might briefly sketch ...

**6**

votes

**2**answers

1k views

### Which universities teach true infinitesimal calculus? [closed]

My colleague and I are currently teaching "true infinitesimal calculus" (TIC), in the sense of calculus with infinitesimals, to a class of about 120 freshmen at our university, based on the book by ...

**12**

votes

**2**answers

763 views

### Teaching stochastic calculus to students who know no measure theory (or PDE, or…)

I've got quite a challenge as my teaching assignment for the next Fall (not that I want to get rid of it, quite the contrary, but I still feel like asking for advice won't hurt :-)).
I'm to teach the ...

**12**

votes

**1**answer

436 views

### A funny factorization of the Jacobian coming from the lines on the Fermat cubic

Here is something which came up in my algebraic geometry class, and I'm wondering if it has a deeper explanation. Let $F(w,x,y,z) = w^3+x^3+y^3+z^3$ and let $X$ be the cubic surface in $\mathbb{P}^3$ ...

**30**

votes

**21**answers

9k views

### Why linear algebra is fun!(or ?)

Edit: the original poster is Menny, but the question is CW; the first-person pronoun refers to Menny, not to the most recent editor.
I'm doing an introductory talk on linear algebra with the ...

**6**

votes

**0**answers

278 views

### Total spaces of tangent/cotangent bundles in a course where all varieties are quasi-projective

$\def\PP{\mathbb{P}}$In a course where all varieties are quasi-projective (as in Shafarevich Volume I), I am trying to figure out whether I can justify talking about the total spaces of the tangent ...

**22**

votes

**17**answers

3k views

### What are your favorite puzzles/toys for introducing new mathematical concepts to students?

We all know that the Rubik's Cube provides a nice concrete introduction to group theory. I'm wondering what other similar gadgets are out there that you've found useful for introducing new math to ...

**36**

votes

**5**answers

7k views

### Are there any “related rates” calculus problems that don't feel contrived?

I just finished teaching a freshman calculus course (at an American state university), and one standard topic in the curriculum is related rates. I taught my students to answer questions such as the ...

**6**

votes

**0**answers

252 views

### Lower semicontinuity of naive fiber size

I would like to present the following result in my algebraic geometry class, but it is seeming much harder than I would expect. Since my class is working with closed points over an algebraically ...

**5**

votes

**3**answers

572 views

### Higher dimensional Bezout via Hilbert polynomials: a reference

For the purposes of teaching my elementary course in algebraic geometry I am looking for a reference (or notes) that contains a complete proof of a higher-dimensional weak Bezout theorem. I only want ...

**15**

votes

**13**answers

4k views

### Category theory sans (much) motivation?

So I have a friend (no, really) who's taking algebra and is struggling to gain intuition for it. My story is as follows: I used to hate abstract algebra, with pretty much a burning passion, until I ...