**7**

votes

**2**answers

450 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 ...

**59**

votes

**7**answers

9k 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

784 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 ...

**14**

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 ...

**6**

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

4k 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?

**57**

votes

**10**answers

6k 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 ...

**25**

votes

**11**answers

6k 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 ...

**12**

votes

**3**answers

2k 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 ...

**1**

vote

**0**answers

293 views

### Who should teach the mathematics subjects in the engineering curriculum? [closed]

Should mathematics courses in Engineering Degree Programs, particularly those which are covered in licensure examinations, be taught by someone whose background is mathematics, or by someone whose ...

**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 ...

**6**

votes

**2**answers

545 views

### To what extent can algorithms in undergraduate linear algebra be made continuous/polynomial/etc.?

I feel like many of the algorithms that I learned — indeed, that I have taught — in undergraduate linear algebra classes depend sensitively on whether certain numbers are $0$. For ...

**13**

votes

**17**answers

3k views

### What is your favorite isomorphism? [closed]

The other day I was trying to figure out how to explain why isomorphisms are important. I pulled Boyer's A History of Mathematics off the bookshelf and was surprised to find that isomorphism isn't ...

**18**

votes

**16**answers

5k views

### Journals for undergraduates

Are there math journals that are aimed for undergraduates? I don't mean here journals where students can publish their papers, but journals that publish introductory articles that an undergraduate can ...

**22**

votes

**1**answer

2k views

### Community experiences writing Lamport's structured proofs

About two years ago, I came across this paper by Lamport
http://research.microsoft.com/en-us/um/people/lamport/pubs/lamport-how-to-write.pdf
on writing proofs hierarchically. It changed how I wrote ...

**5**

votes

**3**answers

1k views

### Graphical representation of mathematical structures (in the spirit of unified modeling language)

In software engineering the unified modeling language ("UML") is a well established technique for providing overview of complex systems and an efficient means of communicating about them. There are ...

**5**

votes

**2**answers

2k views

### How to study a math text [closed]

Hello,
recently I've been trying various attempts regarding how to approach a math book to learn in the best way. Should one memorize the theorems and proofs so that one can recite them? I tend to ...

**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 ...

**32**

votes

**7**answers

4k views

### Why the Killing form?

I'm teaching a short summer course on algebraic groups and it's time to talk about the Killing form on the Lie algebra. The students are all undergrads of varying levels of inexperience, and I try to ...

**47**

votes

**7**answers

7k views

### What is Lagrange Inversion good for?

I am planning an introductory combinatorics course (mixed grad-undergrad) and am trying to decide whether it is worth budgeting a day for Lagrange inversion. The reason I hesitate is that I know of ...

**11**

votes

**2**answers

2k views

### The probabilistic method - reference to less challenging questions

I am teaching a course in combinatorics and large part of it is dedicated to the probabilistic method especially in the case of graphs. The course is an undergraduate level (almost none of the ...

**19**

votes

**11**answers

5k views

### The role of the mean value theorem (MVT) in first-year calculus.

Should the mean value theorem be taught in first-year calculus?
Most calculus textbooks present the MVT just before the section that says that if $f'>0$ on an interval then $f$ increases on that ...

**1**

vote

**0**answers

485 views

### Good sources for linear algebra for convex optimization and graph analysis?

What are some good sources for linear algebra for convex optimization and graph analysis?
In Particular, is Gilbert Strang's MIT course suitable, or some other online course? I prefer online courses ...

**11**

votes

**11**answers

6k views

### Theorems in Euclidean geometry with attractive proofs using more advanced methods

The butterfly theorem is notoriously tricky to prove using only "high-school geometry" but it can be proved elegantly once you think in terms of projective geometry, as explained in Ruelle's book The ...

**9**

votes

**4**answers

1k views

### Problem suggestions for polymath for undergraduates research

I'm inspired by the polymath project. It might be great for few undergraduates to work together on a research topic.
What are some research problems with the following properties(Experimental ...

**17**

votes

**6**answers

3k views

### Euclid with Birkhoff

I'm looking for an short and elementary book which does Euclidean geomety with Birkhoff's axioms.
It would be best if it would also include some topics in projective (and/or) hyperbolic geometry.
...

**18**

votes

**6**answers

2k views

### Yet another 'roadmap' style request- a second bite of the cherry

Okay, so I know MO has had a recent proliferation of this kind of question, and I know MO is not really for this type of question (though I suspect perhaps this is a phenomenon that is likely to ...

**8**

votes

**7**answers

6k views

### Undergraduate approach to learning math [closed]

I am going into my sophomore year as an undergraduate and I would like to ask the more experienced folks a couple questions about learning math and related things. What are your experiences and advice ...

**17**

votes

**19**answers

13k views

### Good combinatorics textbooks for teaching undergraduates?

Hello, can anyone recommend good combinatorics textbooks for undergraduates? I will be teaching a 10-week course on the subject at Stanford, and I assume that the students will be strong and motivated ...

**2**

votes

**5**answers

2k views

### Learning Roadmap for Software Engineer

Hi,
I did some browsing around here on MathOverflow, but I feel that my question is different enough from others that it warrants its own question.
Background:
I'm a computer science major, with a ...

**18**

votes

**6**answers

1k views

### Uppercase Point Labels in High-School Diagrams: from Euclid?

I wonder if the convention of labeling points in geometric
diagrams with uppercase symbols ultimately derives from
Greek mathematics, which was originally written in
"majuscule" (uppercase) Greek ...

**54**

votes

**20**answers

8k views

### What should we teach to liberal arts students who will take only one math course?

Even professors in academic departments other than mathematics---never mind other educated people---do not know that such a field as mathematics exists. Once a professor of medicine asked me whether ...

**6**

votes

**4**answers

1k views

### What would be good to know before starting my undergraduate studies to become a good mathematician?

First of all, I'm sorry if this isn't the kind of question that should be made in MathOverflow. I read the FAQ and I didn't consider this (that) inappropriate. I couldn't resist! People here are ...

**8**

votes

**10**answers

6k views

### Learning Algebra & Group Theory on my own [closed]

I'm learning Algebra & Group Theory, casually, on my own. Professionally, I'm a computer consultant, with a growing interest in the mathematical and theoretical aspects. I've been amazed with ...

**15**

votes

**9**answers

4k views

### Mathematics and autodidactism

Mathematics is not typically considered (by mathematicians) to be a solo sport; on the contrary, some amount of mathematical interaction with others is often deemed crucial. Courses are the student's ...

**71**

votes

**14**answers

6k views

### Why do we care about L^p spaces besides p = 1, p = 2, and p = infinity?

I was helping a student study for a functional analysis exam and the question came up as to when, in practice, one needs to consider the Banach space $L^p$ for some value of $p$ other than the obvious ...

**18**

votes

**16**answers

4k views

### What are your experiences of handouts in mathematics lectures?

There are many different styles of lecturing, and many different aspects that are blended together to give a whole "lecturing style". That said, I'm particularly interested in hearing people's ...

**25**

votes

**7**answers

5k views

### On starting graduate school and common pitfalls…

Hi,
I'll be starting graduate school soon, and when I look back at my college career, there are certain things I wish I could have done differently. In hindsight, I wished I wasn't in such a rush to ...

**4**

votes

**4**answers

1k views

### Less-known conjectures of significant influence and the contrary

In mathematics, it is common that theorems/results and problems appearing dull in one generation get revitalized and become the center of research in another one.
Sometimes conjectures that are ...

**10**

votes

**11**answers

17k views

### Why does undergraduate discrete math require calculus?

Often undergraduate discrete math classes in the US have a calculus prerequisite.
Here is the description of the discrete math course from my undergrad:
A general introduction to basic
...

**1**

vote

**1**answer

4k views

### Websites hosting free math ebooks. [duplicate]

Possible Duplicates:
Free, high quality mathematical writing online?
Most helpful math resources on the web
A lot has been said about different kinds of math resources here in MO.
To ...

**25**

votes

**14**answers

3k views

### Making sure that you have comprehended a concept

Hi,
I have a question that I've been thinking about for a long time.
How can you assure yourself that you've fully comprehended a concept or the true meaning of a theorem in mathematics?
I mean how ...

**7**

votes

**3**answers

5k views

### Applications of Group Theory Which Motivate Theoretical Questions?

I'm going to be a teaching assistant for an undergraduate class in abstract algebra next semester, for students who have not taken abstract algebra before. It will deal with group theory and linear ...

**5**

votes

**3**answers

1k views

### How do undergrad students write papers by themselves?

Can a student write a paper and send it to a professor review?

**35**

votes

**12**answers

7k views

### Examples of undergraduate mathematics separation from what mathematicians should know

I'm looking for examples of four kinds of things:
Material that is usually covered in standard undergraduate mathematics courses and/or in first-year graduate work (or tested in qualifying ...

**9**

votes

**10**answers

1k views

### A place to find original papers

I currently use scholar.google.com to find papers in cases like Sophus Lie's original papers on "Transformation Groups". Does anyone know of other places that collect original works like this, i.e. ...

**5**

votes

**2**answers

647 views

### Does A “Connections” Blog/Podcast exist for Math?

What I mean is this:
Does there exist a mathematics podcast where a mathematician of some sort looks at undergraduate/graduate mathematical topics and look into the history (how those objects came ...

**4**

votes

**5**answers

666 views

### Resources for learning domain theory?

I'm a computer programmer who's caught on to denotational semantics. I mostly work with Ruby, JavaScript and C, but I know a little Haskell and ML. I've taken my first steps towards reasoning about ...