**573**

votes

**218**answers

145k 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 ...

**261**

votes

**72**answers

97k 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 ...

**175**

votes

**35**answers

42k 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 ...

**143**

votes

**30**answers

55k views

### Real-world applications of mathematics, by arxiv subject area?

What are the most important applications outside of mathematics of each of the major fields of mathematics? For concreteness, let's divide up mathematics according to arxiv mathematics categories, e.g....

**122**

votes

**58**answers

25k 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 ...

**122**

votes

**37**answers

115k views

### Too old for advanced mathematics? [closed]

Kind of an odd question, perhaps, so I apologize in advance if it is inappropriate for this forum. I've never taken a mathematics course since high school, and didn't complete college. However, ...

**116**

votes

**7**answers

12k 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 ...

**113**

votes

**28**answers

33k 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 ...

**104**

votes

**27**answers

15k 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 ...

**89**

votes

**17**answers

8k views

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

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 ...

**72**

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 ...

**70**

votes

**18**answers

17k 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 ...

**69**

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 ...

**67**

votes

**7**answers

10k 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 ...

**67**

votes

**8**answers

11k views

### How do you not forget old math?

I am trying to not forget my old math. I finished my PhD in real algebraic geometry a few years ago and then switched to the industry for financial reasons. Now I get the feeling that I want to do a ...

**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 ...

**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 ...

**61**

votes

**18**answers

6k 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 ...

**59**

votes

**12**answers

11k 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 ...

**58**

votes

**15**answers

6k views

### Sophisticated treatments of topics in school mathematics

Sophisticated mathematical concepts typically shed light on sophisticated mathematics. But in a few cases they also apply to elementary mathematics in an interesting way. I find such examples ...

**58**

votes

**10**answers

13k views

### Is Euclid dead? [closed]

Apparently Euclid died about 2,300 years ago (actually 2,288 to be more precise), but the title of the question refers to the rallying cry of Dieudonné, "A bas Euclide! Mort aux triangles!" (...

**58**

votes

**15**answers

9k 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

**10**answers

7k 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 ...

**56**

votes

**8**answers

8k 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 ...

**55**

votes

**20**answers

9k 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 ...

**54**

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 ...

**52**

votes

**9**answers

13k views

### Relating Category Theory to Programming Language Theory

I'm wondering what the relation of category theory to programming language theory is.
I've been reading some books on category theory and topos theory, but if someone happens to know what the ...

**51**

votes

**47**answers

18k 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 ...

**50**

votes

**13**answers

6k 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 ...

**49**

votes

**17**answers

11k views

### Parodies of abstruse mathematical writing

Perhaps under the influence of a recent question
on perverse sheaves,
in conjunction with the impending $\pi$-day (3/14/15 at 9:26:53),
I recalled a long-ago parody of abstruse mathematical language
...

**46**

votes

**18**answers

15k 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 ...

**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 ...

**45**

votes

**34**answers

7k 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 ...

**45**

votes

**19**answers

7k views

### Are there proofs that you feel you did not “understand” for a long time?

Perhaps the "proofs" of ABC conjecture or newly released weak version of twin prime conjecture or alike readily come to your mind. These are not the proofs I am looking for. Indeed my question was ...

**44**

votes

**11**answers

6k 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 ...

**44**

votes

**5**answers

3k views

### How do you mentor undergraduate research?

Lets say you had an undergraduate who wanted to do some advanced work and some research, possibly for a thesis, or things like that.
There are two slightly more specific groups of questions I have ...

**43**

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 ...

**43**

votes

**1**answer

3k views

### Probability that a stick randomly broken in five places can form a tetrahedron

Edit (June 2015): Addressing this problem is a brief project report from the Illinois Geometry Lab (University of Illinois at Urbana-Champaign), dated May 2015, that appears here along with a foot-...

**42**

votes

**42**answers

12k views

### What should be offered in undergraduate mathematics that's currently not (or isn't usually)? [closed]

What's one class that mathematics that should be offered to undergraduates that isn't usually? One answer per post.
Ex: Just to throw some ideas out there
Mathematical Physics (for math students, not ...

**42**

votes

**18**answers

6k views

### How can an extremely mathematically talented young person be helped to fulfill his/her potential?

Obviously, this question is not a research level mathematics question at all. But, I've just met an extremely mathematically talented 11 years old student and I don't know how I can help him. For ...

**36**

votes

**12**answers

8k 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 ...

**36**

votes

**15**answers

7k 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 ...

**36**

votes

**4**answers

4k 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 ...

**35**

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 ...

**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 ...

**34**

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 ...

**34**

votes

**14**answers

3k 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.
...

**34**

votes

**1**answer

2k views

### Hilbert's Hotel

Hilbert's Hotel is a famous story about infinity attributed to David Hilbert (1862-1943).
Is it documented that Hilbert's Hotel is in fact due to Hilbert, and if yes, where?

**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 ...

**32**

votes

**12**answers

13k 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 ...