**6**

votes

**13**answers

6k views

### Useless math that became useful

I'm writing an article on Lychrel numbers and some people pointed out that this is completely useless.
My idea is to amend my article with some theories that seemed useless when they are created but ...

**5**

votes

**0**answers

2k views

### What can we do to raise awareness of reciprocity laws? [closed]

The study of reciprocity laws is a centerpiece of modern mathematics. Of the last ten Fields Medalists, two of them (Ngô Bảo Châu and Laurent Lafforgue) were awarded Fields Medals for their work on ...

**9**

votes

**4**answers

757 views

### Seeking a Geometric Proof of a Generalized Alternating Series' Convergence

Let $z \in \mathbb{C} \backslash \lbrace 1 \rbrace$ with $|z| = 1$. We consider the following infinite series, which necessarily converges:
$$S(z) := \sum_{n = 1}^{\infty}\frac{z^n}{n}$$
Note that ...

**16**

votes

**4**answers

3k views

### Is $x \, \tan(x)$ integrable in elementary functions?

I'm teaching Calculus and my students asked me to calculate the integral of $x \, \tan(x)$.
I spent quite a lot of effort to do this, but I'm now even not sure if the integral could be presented in ...

**70**

votes

**20**answers

9k views

### “Mathematics talk” for five year olds

I am trying to prepare a "mathematics talk" for five year olds from my daughter's elementary school. I have given many mathematics talks in my life but this one feels
very tough to prepare. Could the ...

**2**

votes

**2**answers

502 views

### Function with all but mixed second partial derivatives twice differentiable?

Let $f(x,y)$ be a a real valued function on an open subset of $\mathbf{R}^2$ with continuous partial derivatives $\frac{\partial^2 f}{\partial x^2}$ and $\frac{\partial^2}{\partial y^2}$. Is $f$ twice ...

**7**

votes

**5**answers

2k views

### Advantages of the sequence definition of limits

I will be teaching an introductory analysis course in the coming semester. In it the students will learn about limits of real sequences, and then will learn about limits of functions in terms of ...

**5**

votes

**0**answers

243 views

### MathJax (or something like it) as a classroom “blackboard”

(I tried this first at http://math.stackexchange.com/questions/187265/mathjax-or-something-like-it-as-a-classroom-blackboard , but didn't get satisfactory responses.)
What is the best desktop ...

**13**

votes

**10**answers

3k views

### Undergraduate Topology

I am developing an introductory topology course for undergraduates, and I am wondering what topics to cover. At my institution, real analysis is not a prerequisite for the course, so it is more than ...

**5**

votes

**3**answers

2k views

### Battle of the brains; cultural mathematics

Firstly, I apologize if my question is long.
Three years ago, I watched a video with the name Battle of the Brains. It was a wonderful video about challenging some famous peoples to solve some ...

**13**

votes

**1**answer

717 views

### Classroom platonism

I'd like to know whether any form a certain hypothesis about the
learning of higher mathematics has entered the mathematical or
educational literature. I'll frame the hypothesis here but not defend
...

**1**

vote

**0**answers

474 views

### Arguments against Reductio ad Absurdum [closed]

Could Reductio ad Absurdum not be consireded a valid proof method? Are there any compelling arguments against it, or at it's favor?
I feel like I am assuming some metamathematical hypothesis about my ...

**6**

votes

**2**answers

1k views

### Advice on doing physics under the umbrella of mathematics and the converse

Note: This is a question directly copied from Theoretical Physics SE primarily to get the advice of people indulged in mathematics.
In the current scenario of research in QFT and string theory (and ...

**9**

votes

**4**answers

2k views

### Differential Equation Examples for Calculus Students

I've been teaching calculus courses for a while now, and something always bothers me each time I teach it. Students always seem to have trouble connecting with the differential equation material for ...

**16**

votes

**7**answers

1k views

### Unexpected applications of the fact that nth degree polynomials are determined by n+1 points

I had a funny idea for proving an identity in Euclidean geometry. While it didn't end up being a very nice proof strategy in my case, I would still like to collect nice examples of where the proof ...

**2**

votes

**2**answers

229 views

### learning sources about Ihara Coefficient

Do we have any good sources(lecture notes or books) for learning about $Ihara$ Coefficient?
Is there any relation between $Ihara$ Coefficient and the eigenvalues of graphs?
Thanks for any help.

**13**

votes

**17**answers

2k views

### Short Course Suggestions For High School Students

I am planning to teach a course for talented high school students at a summer camp and I need suggestions for possible topics. The students usually have different backgrounds but most of them are ...

**62**

votes

**14**answers

6k views

### How to write popular mathematics well?

Recently, some classmates and I were lamenting the fact that our classmates in other disciplines had almost no conception of what we did, despite the large mathematics population at Waterloo. Instead ...

**29**

votes

**6**answers

3k views

### Taylor's theorem and the symmetric group

Anytime I see an $n!$ in some formula, my instinct is to look for the symmetric group on $n$ letters coming in somewhere. I have never done this seriously with the $n!$ in Taylor's theorem.
Question: ...

**14**

votes

**12**answers

3k views

### Motivating Algebra and Analysis for Average Undergraduates

I work at a small liberal arts college, where many of our mathematics majors will not attend graduate school in mathematics. My hope in asking the following question is to gather innovative ideas for ...

**7**

votes

**2**answers

757 views

### Virtual algebraic calculation within proofs

It seems to me that the undergraduates I teach have particular difficulty with proofs that involve reasoning about algebraic calculations that arise only theoretically. Since I have in mind doing ...

**52**

votes

**9**answers

3k views

### Taking “Zooming in on a point of a graph” seriously

In calculus classes it is sometimes said that the tangent line to a curve at a point is the line that we get by "zooming in" on that point with an infinitely powerful microscope. This explanation ...

**5**

votes

**2**answers

1k views

### Is beauty at the high school level even possible? [closed]

This question is a follow up to 74841, and follows from a suggestion by Gian-Carlo Rota that beauty as judged by the educated public differs from that experienced by mathematicians (he gives Euclidean ...

**48**

votes

**45**answers

17k 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 ...

**52**

votes

**12**answers

10k 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 ...

**4**

votes

**2**answers

6k views

### An image of the hierarchy of algebraic structures

Hello! Does anybody know an image of a graph featuring the hierarchy of algebraic structures? Something rather complete.
So far I've found similar images describing the hierarchies of ...

**4**

votes

**3**answers

543 views

### Looking for ideas concerning the teaching of lower-division differential equation courses…

I'm looking for problems/lessons plans that could be used in a lower-division differential equations course that involve discerning properties of solutions of an equation, IVP, or BVP, without looking ...

**29**

votes

**18**answers

10k views

### Interesting and Accessible Topics in Graph Theory

This summer, I will be teaching an introductory course in graph theory to talented high school seniors. The intent of the course is not to establish proficiency in graph theory, per se. Rather, I hope ...

**16**

votes

**1**answer

1k views

### Looking for an appealing counterexample in probability

There is a commonly-encountered-but-wrong rule of thumb that says something like
If a probability distribution is positively skewed, its mean is greater than its median.
(You sometimes also see ...

**7**

votes

**6**answers

980 views

### Seemingly emergent structures in mathematics

I rather suspect that this must have come up here on MO already, but my handful of searches didn't turn up the thread, so...
I'm curious about examples of mathematical structure that seems to arise ...

**114**

votes

**7**answers

11k views

### How to memorise (understand) Nakayama's lemma and its corollaries?

Nakayama's lemma is mentioned in the majority of books on algebraic geometry that treat varieties. So I think Ihave read the formulation of this lemma at least 20 times (and read the proof maybe ...

**12**

votes

**4**answers

913 views

### Simple groups with the same cardinality as PSL_2(Z/p)

In an undergrad honors algebra course it's sometimes shown that when $p$ is prime and $>3$ then
$PSL_2(Z/p)$ is simple of of order $p(p-1)(p+1)/2$. But that this is the "only" simple group
having ...

**67**

votes

**77**answers

12k views

### Elementary+Short+Useful

Imagine your-self in front of a class with very good undergraduates
who plan to do mathematics (professionally) in the future.
You have 30 minutes after that you do not see these students again.
You ...

**9**

votes

**4**answers

3k views

### Place of Analytic geometry in modern undergraduate curriculum

Hello.
I am a freshmen student in mathematics at Moscow State University (in Russia) and I'm confused with placing the subject called "analytic geometry" into the system of mathematical knowledge (if ...

**9**

votes

**4**answers

2k views

### Which topics/problems could you show to a bright first year mathematics student?

I am teaching a one semester course (January to June) to first year students pursuing various different degrees. Because there are students studying actuarial science, physics, other sciences, other ...

**42**

votes

**10**answers

6k views

### Possibility of an Elementary Differential Geometry Course

I have to admit I'm not sure if this is an appropriate question. It's related to research in math education, but not directly to math.
I've found that in talking to professional physicists and ...

**10**

votes

**8**answers

3k views

### Leibnizian calculus textbook

Where can I find a calculus textbook that emphasizes differentials?
Is there such a book that I could realistically require my calculus students to use?
I want a textbook that supports me when I tell ...

**9**

votes

**2**answers

1k views

### Can formally differentiating give a derivative of a discrete function?

When I teach calculus, I really try to stress the importance of knowing the domain of a function.
One example that I sometimes like to use to show students the importance of inspecting the domain is ...

**7**

votes

**1**answer

8k views

### Self-taught undergrad math: ordering of topics?

After some initial research on math topics, it seems there are about 4 main streams as follows:
1) calculus -> analysis -> complex variables
2) linear algebra -> abstract algebra -> topology
3) ...

**12**

votes

**1**answer

2k views

### Is there evidence whether undergraduate math courses improve problem-solving?

The most commonly stated reason for why mathematics should be a required condition for graduating is }to improve problem-solving skills". Usually it's taken for granted that taking a mathematics ...

**246**

votes

**72**answers

92k views

### Video lectures of mathematics courses available online for free

It can be difficult to learn mathematics on your own from textbooks, and I often wish universities videotaped their mathematics courses and distributed them for free online. Fortunately, some ...

**22**

votes

**5**answers

3k views

### Simple but serious problems for the edification of non-mathematicians

When people graduate with honors from prestigious universities thinking everything in math is already known and the field consists of memorizing algorithms, then the educational system has failed in ...

**7**

votes

**8**answers

3k views

### Graduate School

How does one apply to graduate school when he has been working for sometime? I am interested in pursuing a PhD in math and making a career switch. Would my work experience benefit my application (I am ...

**2**

votes

**1**answer

964 views

### Best Practices for Learning Mathematics (especially in the classroom) [closed]

I am an undergraduate CS major with strong interests in applied math and theoretical computer science. In the past, I've done reasonably well grade-wise in all math-related (that is, pure math, ...

**32**

votes

**12**answers

12k views

### Teaching undergraduate students to write proofs

In my experience, there are roughly two approaches to teaching (North American) undergraduates to write proofs:
Students see proofs in lecture and in the textbooks, and proofs are explained when ...

**11**

votes

**6**answers

1k views

### Reasons for the importance of planarity and colorability?

Could it have been foreseen that - exemplarily - planarity and colorability would turn out to be such important concepts in graph theory (there's almost no textbook on graphs without two chapters ...

**16**

votes

**2**answers

659 views

### Where and when did “transition to abstraction” courses start?

I often find myself debating the content and structure of such courses and I would find it useful to know the basic history.
I don't remember any such offerings during my own undergraduate days in ...

**63**

votes

**6**answers

2k views

### Good ways to engage in mathematics outreach?

Greetings all, I have often heard that it would be good if we as a community did more in the way of mathematics outreach: more to explain what it is we do to the community at large, more to expose ...

**3**

votes

**3**answers

854 views

### Pedagogical question concerning $\Gamma(z)$

Pedagogically speaking, I see two problems with defining
$\Gamma(z)$ (at least for real $z$) by the limit
$$\Gamma(z)=\lim_{m\to\infty}\frac{m! m^z}{\prod_{i=0}^m (z+i)}$$
as compared with the formula
...

**35**

votes

**6**answers

3k views

### What is the simplest, most elementary proof that a particular number is transcendental?

I teach, among many other things, a class of wonderful and inquisitive 7th graders. We've recently been studying and discussing various number systems (N, Z, Q, R, C, algebraic numbers, and even ...