**40**

votes

**14**answers

4k 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 ...

**26**

votes

**6**answers

2k 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

2k 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

692 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 ...

**43**

votes

**9**answers

2k 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 ...

**3**

votes

**2**answers

886 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 ...

**5**

votes

**0**answers

602 views

### Mathematics outside of university

Hello,
this is my first post on mathoverflow so excuse me if this is not the right kind of question.
My situation is the following, without delving into reasons and *should've*s *would've*s I got ...

**40**

votes

**44**answers

12k 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 ...

**37**

votes

**10**answers

5k 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

3k 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

487 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 ...

**22**

votes

**18**answers

5k 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 ...

**15**

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

890 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 ...

**80**

votes

**7**answers

7k views

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

Hope this question is fine. Nakayama's lemma http://en.wikipedia.org/wiki/Nakayama_lemma#Statement is mentioned in the majority of books on algebraic geometry that treat varieties. So I think, I red ...

**12**

votes

**4**answers

857 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 ...

**55**

votes

**73**answers

11k 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

2k 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 ...

**8**

votes

**4**answers

1k 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 ...

**35**

votes

**8**answers

4k 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

**6**answers

2k 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

784 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 ...

**6**

votes

**1**answer

4k 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) ...

**11**

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 ...

**174**

votes

**71**answers

73k 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 ...

**21**

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 ...

**6**

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

873 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, ...

**30**

votes

**12**answers

7k 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 ...

**10**

votes

**6**answers

870 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 ...

**12**

votes

**2**answers

531 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 ...

**55**

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

813 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
...

**3**

votes

**0**answers

2k views

### What is the geometric meaning of third derivative of a function at a point? [closed]

What is the geometrical meaning of third derivative of a function at a point?
The question is now open in the sister site ...

**32**

votes

**6**answers

2k 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 ...

**1**

vote

**1**answer

847 views

### Best examples of physics providing insight into math [duplicate]

Possible Duplicates:
Examples where physical heuristics led to incorrect answers?
Examples of using physical intuition to solve math problems
V. I. Arnold argues ...

**76**

votes

**26**answers

10k views

### How To Present Mathematics To Non-Mathematicians?

(Added an epilogue)
I started a job as a TA, and it requires me to take a five sessions workshop about better teaching in which we have to present a 10 minutes lecture (micro-teaching).
In the last ...

**9**

votes

**4**answers

2k views

### A learning roadmap for Additive combinatorics.

Hello,
I'd love to learn more about the field of additive combinatorics. From what I've understand, there's a book by Tao and Vu out on the subject, and it looks fun, but I think I lack the ...

**3**

votes

**2**answers

2k views

### Mathematics Graduate Student Summer Opportunities

I am currently a mathematics graduate student at Western Kentucky University in Bowling Green, KY. I am looking for some kind of summer opportunity to participate in during summer 2011.
Does anyone ...

**7**

votes

**2**answers

432 views

### How quickly will billiard trajectories cluster?

Suppose you launch $n$ point-particles on
distinct reflecting nonperiodic billiard trajectories
inside a convex polygon. Assume they all have the same speed.
Define an $\epsilon$-cluster as a ...

**33**

votes

**5**answers

4k views

### Is the boundary $\partial S$ analogous to a derivative?

Without prethought, I mentioned in class once that the reason the symbol $\partial$
is used to represent the boundary operator in topology is
that its behavior is akin to a derivative.
But after ...

**4**

votes

**1**answer

681 views

### What topics should be included in a calculus-for-the-liberal arts course?

I have in mind a course taken by liberal-arts students who will probably never take another math course. I would like such a course to convey some of the way mathematical thinking is done (i.e. not a ...

**9**

votes

**4**answers

1k views

### Name for a basic principle of calculus?

$$
[\text{size of boundary}] \times [\text{rate of motion of boundary}] = [\text{rate of change of size of bounded region}]
$$
This differs from the fundamental theorem of calculus in that it does not ...

**13**

votes

**2**answers

2k views

### Does any textbook take this approach to the isomorphism theorems?

Below, I present an outline of a proof of the first isomorphism theorem for groups. This is how I usually think of the first isomorphism theorem for ______, but groups will get the points across. My ...

**5**

votes

**3**answers

2k views

### (How) should I take notes on a subject for self-study?

Suppose I am interested in really learning / thoroughly reviewing some subject (e.g. the basic theorems of infinite Galois theory, or the classification of compact Lie groups). One approach I might ...

**9**

votes

**12**answers

3k views

### How do I explain the number e to a ten year old? [closed]

Hardly a research level question, but interesting nonetheless, I hope. Pi is easy, but not e. Where could I start?

**53**

votes

**10**answers

5k views

### Teaching proofs in the era of Google

Dear members,
Way back in the stone age when I was an undergraduate (the mid 90's), the internet was a germinal thing and that consisted of not much more than e-mail, ftp and the unix "talk" command ...

**20**

votes

**11**answers

5k views

### Categories First Or Categories Last In Basic Algebra?

Recently, I was reminded in Melvyn Nathason's first year graduate algebra course of a debate I've been having both within myself and externally for some time. For better or worse, the course most ...

**0**

votes

**5**answers

1k views

### If you could redesign a high school mathematics curriculum from the ground up, what would you include? [closed]

Let's assume that we also get to redesign our high school mathematics teachers as well, in the sense that we can assume that they know and can teach whatever material we choose to cover.
This is ...

**11**

votes

**3**answers

1k views

### Analysis from a categorical perspective

I have not studied category theory in extreme depth, so perhaps this question is a little naive, but I have always wondered if analysis could be taught naturally using categories. I ask this because ...