**3**

votes

**2**answers

3k views

### Which area of mathematics is the hardest to learn ? [closed]

I was asked this question by a friend's daughter a few days back, and I was interested to see what people here think.
I realize of course that this is quite subjective, but still, I get the impression ...

**25**

votes

**12**answers

2k views

### Where have you used computer programming in your career as an (applied/pure) mathematician?

For background: I'm working on a book to help mathematicians learn how to program. However, I need to see some examples from people in the field that have done different kinds of things than I have.
...

**11**

votes

**10**answers

3k views

### Math History books

I'm teaching a course over the summer (it's a sort of make-your-own course for non-majors) and I'm planning on organizing it as a math history course, hitting on major threads through about 1900, and ...

**477**

votes

**197**answers

123k views

### Examples of common false beliefs in mathematics

The first thing to say is that this is not the same as the question about interesting mathematical mistakes. I am interested about the type of false beliefs that many intelligent people have while ...

**18**

votes

**3**answers

901 views

### Do rational numbers admit a categorification which respects the following “duality”?

I need to give a lot of quite basic background to this question because I think (at least from conversing with fellow graduate students) that most mathematicians have not really thought about ...

**4**

votes

**0**answers

1k views

### Good textbooks on probability and/or stochastic processes, emphasizing simulation

Any recommendations for textbooks on probability and/or stochastic processes that emphasize simulation? I'll be teaching this course in the Fall.

**43**

votes

**18**answers

13k views

### How should one present curl and divergence in an undergraduate multivariable calculus class?

I am a TA for a multivariable calculus class this semester. I have also TA'd this course a few times in the past. Every time I teach this course, I am never quite sure how I should present curl and ...

**0**

votes

**7**answers

1k views

### Good/Economical textbook for undergraduate intro to diff.eq. for engineers?

In the fall I will be teaching an intro to diff.eq.s course for undergrad engineers. The usual textbook is $150 with solution manual and it's not that great. There must be a cheaper alternative that's ...

**20**

votes

**4**answers

3k views

### The Interrelationship Problem Of Modern Mathematics- How To Deal With it In First Year Graduate Courses?

I was reading recently online Peter May's complaints (I'm a fan, you can tell, I'm sure) about teaching the third quarter of the graduate algebra sequence at the University of Chicago. This course ...

**13**

votes

**5**answers

12k views

### Reference letters for graduate school after a couple years in the industry

How does one return to graduate school after spending a couple years in the industry? In particular, what are ways of getting good recommendations? I'm not concerned about the "adjustment" to the grad ...

**40**

votes

**11**answers

5k views

### Interesting results in algebraic geometry accessible to 3rd year undergraduates

On another thread I asked how I could encourage my final year undergraduate colleagues to take an algebraic geometry or complex analysis courses during their graduate studies.
Willie Wong proposed me ...

**5**

votes

**1**answer

3k views

### How would You encourage graduate students to learn algebraic geometry and/or complex analysis?

Hello,
I am the 3rd year undegraduate student of mathematics.
After I obtain a bachelor degree I want to study maths at graduate level, especially algebraic geometry and complex analysis.
This fields ...

**46**

votes

**13**answers

5k views

### How do you approach your child's math education? [closed]

My son is one year old, so it is perhaps a bit too early to worry about his mathematical education, but I do. I would like to hear from mathematicians that have older children: What do you wish you'd ...

**13**

votes

**5**answers

2k views

### Why is a topology made up of 'open' sets? Part II [closed]

Because the display was getting quite cluttered, I thought I'd post a second part to this question separately. I hope the Gods of Math Overflow don't take too much offense. I'll go now into some ...

**3**

votes

**2**answers

1k views

### How should I find a tutor for math-overflow level mathematics? [closed]

Searching for maths tutors online finds people willing to teach up to A-level. I'm looking for help at a more advanced level.
At the moment I'm trying to teach myself category theory from downloaded ...

**34**

votes

**7**answers

2k views

### Informal online seminars or reading groups via videoconferencing?

Does the following exist, and if not, does anyone besides me wish it did? A web site where a mathematician (say) could find other mathematicians who want to study the same book or paper, and arrange ...

**146**

votes

**35**answers

33k views

### Why is a topology made up of 'open' sets? [closed]

I'm ashamed to admit it, but I don't think I've ever been able to genuinely motivate the definition of a topological space in an undergraduate course. Clearly, the definition distills the essence of ...

**19**

votes

**9**answers

1k views

### How do you motivate a precise definition to a student without much proof experience?

When introducing students to highly technical definitions for seemingly intuitive concepts (e.g., homotopy, continuity), how do you motivate the necessity of the definition? On the one hand, you ...

**46**

votes

**4**answers

2k views

### What algorithm in algebraic geometry should I work on implementing?

This summer my wife and one of my friends (who are both programmers and undergraduate math majors, but have not learned any algebraic geometry) want to learn some algebraic geometry from me, and I ...

**2**

votes

**1**answer

729 views

### Text/structure for an analysis course for students with pre-existing understanding of some applied aspects of analysis

Greetings,
I'm teaching a one-off course (perhaps never to be repeated) in a curriculum that's in transition, and I'm looking for advice on a textbook, or stories from people who have taught similar ...

**8**

votes

**4**answers

2k views

### How to teach introductory statistic course to students with little math background?

Next semester I will teach an elementary statistic course for the first time (which I am actually quite excited about). A brief description can be found here. I am told to expect very little math ...

**6**

votes

**8**answers

4k views

### Best way to teach concept of real numbers using a hands-on activity?

I know a middle school math teacher looking for some suggestions for hands-on activities to teach the concept of real numbers. I'm new to this site, so this may be a little off topic.

**12**

votes

**1**answer

579 views

### Teaching Methods and Evaluating them

Hey,
As a lowly graduate student, I'm on a committee (I'm not sure how important my role really is) trying to evaluate how effective different approaches teaching undergraduates. We are looking at ...

**102**

votes

**55**answers

22k views

### What are your favorite instructional counterexamples?

Related: question #879, Most interesting mathematics mistake. But the intent of this question is more pedagogical.
In many branches of mathematics, it seems to me that a good counterexample can be ...

**7**

votes

**3**answers

2k views

### Specializing early

Topic: this is a mathematics education question (but applies to other sciences too).
Assumptions: my first assumption is that most mathematical concepts used in research are not intrinsically more ...

**4**

votes

**6**answers

4k views

### Interesting applications of max-flow and linear programming

Max-flow and linear programming are two big hammers in algorithm design: each are expressive enough to represent many poly-time solvable problems. Some problems are obvious applications of max-flow: ...

**18**

votes

**13**answers

3k views

### Do you find your students are less competent in basic algebra and arithmetic, and, if so, do you believe that this is due to overuse of calculators at an early level? [closed]

So first I gave my class the quiz problem: Compute $$\lim_{h\rightarrow 0} \frac{\frac{1}{3+h} - \frac{1}{3}}{h}.$$ Upon finding that they could not do that (no real surprize) I asked them to compute ...

**4**

votes

**0**answers

443 views

### Probability in Math Education

Why is probability an under-emphasized subject in most math programs? Why does it seem that the hot topics these days are category theory and algebra? What do you think about the following: A student ...

**18**

votes

**10**answers

3k views

### Research Experience for Undergraduates: Summer Programs

Some time ago, I found this list of REU programs held in 2009.
The main aspects that characterize such programs are: (a) a great deal of lectures on specific topics; and, admittedly more ...

**103**

votes

**28**answers

27k views

### Cool problems to impress students with group theory [closed]

Since this forum is densely populated with algebraists, I think I'll ask it here.
I'm teaching intermediate level algebra this semester and I'd like to entertain my students with some clever ...

**42**

votes

**34**answers

6k views

### Are there any books that take a 'theorems as problems' approach?

Are there any books that present theorems as problems? To be more specific, a book on elementary group theory might have written: "Theorem: Each group has exactly one identity" and then show a proof ...

**6**

votes

**4**answers

3k views

### Applications of Euler-Cauchy ODEs

The Euler-Cauchy ODE (2nd order, homogeneous version) is:
$$
x^2 y'' + a x y' + b y = 0
$$
Looking in various books on ODEs and a random walk on a web search (i.e. I didn't click on every link, but ...

**13**

votes

**2**answers

2k views

### Teaching and students

Sometimes I get stumped by students' questions in my classes I teach. I am an algebraist by training and have just started teaching. Sometimes I have to teach analysis courses. My question is: Is it ...

**34**

votes

**4**answers

3k views

### Motivation for concepts in Algebraic Geometry

I know there was a question about good algebraic geometry books on here before, but it doesn't seem to address my specific concerns.
**
Question
**
Are there any well-motivated introductions to ...

**14**

votes

**5**answers

6k views

### How seriously do professors take teaching evaluations?

Do they ever know who writes them? How seriously do departments take teaching evaluations? If a professor knows which student wrote a particular evaluation....would they be biased (e.g. be nicer, ...

**24**

votes

**6**answers

14k views

### What are the advantages and disadvantages of the Moore method?

Describe your experiences with the Moore method. What are its advantages and disadvantages?

**35**

votes

**15**answers

6k views

### Strong induction without a base case

Strong induction proves a sequence of statements $P(0)$, $P(1)$, $\ldots$ by proving the implication
"If $P(m)$ is true for all nonnegative integers $m$ less than $n$, then $P(n)$ is true."
for ...

**5**

votes

**1**answer

1k views

### In studying maths, do you tend to go through books one at a time? [closed]

Or do you read a lot of books on the same subject (say Topology)? Consider these two cases:
Case 1: Suppose you want to study point set topology. You pick up a good book and work thoroughly on it.
...

**33**

votes

**21**answers

7k views

### Nontrivial question about Fibonacci numbers?

I'm looking for a nontrivial, but not super difficult question concerning Fibonacci numbers. It should be at a level suitable for an undergraduate course.
Here is a (not so good) example of the sort ...

**9**

votes

**2**answers

5k views

### Mathematics for machine learning

I would like to know what mathematics topics are the most important to learn before actually studying the theory on neural networks.
I ask that because I will start to learn about neural networks and ...

**4**

votes

**7**answers

808 views

### Reference for elementary and “cool” statistics or financial math

I signed up for a Math Mentorship Program (for high school students) this term, but one of the students assigned to me is more interested in Statistics and Finance - something that would help him to ...

**18**

votes

**12**answers

6k 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 ...

**66**

votes

**18**answers

15k views

### Depressed graduate student. [closed]

How does a depressed graduate student go about recovering his enthusiasm for the subject and the question at hand?
Edit: I am not that grad student; it is a very talented friend of mine.
Moderator's ...

**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, ...

**18**

votes

**13**answers

4k views

### Pedagogical question about linear algebra

Last semester I taught a linear algebra class that is intended to introduce young students (at a sophmore-junior level) to "abstract mathematics". It seems that a major conceptual hurdle for many of ...

**21**

votes

**4**answers

3k views

### Curriculum reform success stories at an “average” research university

Greetings all,
There's a never-ending story that many of us have sunk our teeth into. How do we go about teaching subjects like calculus and analysis "well?" Most universities that I'm familiar ...

**54**

votes

**15**answers

7k views

### What's a nice argument that shows the volume of the unit ball in $\mathbb R^n$ approaches 0?

Before you close for "homework problem", please note the tags.
Last week, I gave my calculus 1 class the assignment to calculate the $n$-volume of the $n$-ball. They had finished up talking about ...

**58**

votes

**16**answers

5k views

### One-step problems in geometry

I'm collecting advanced exercises in geometry. Ideally, each exercise should be solved by one trick and this trick should be useful elsewhere (say it gives an essential idea in some theory).
If you ...

**5**

votes

**4**answers

1k views

### Choice of adviser

Not sure how to tag this one so feel free to edit and add tags.
When I initially started graduate school my choice for an area of study was quite nebulous. I had only figured out enough to know that ...

**8**

votes

**10**answers

8k views

### Math journal for high school students?

I recently discovered The College Mathematics Journal and enjoyed reading through some of the articles on fun applications of mathematics. I'd like to send some of the articles to my younger sister, a ...