**36**

votes

**31**answers

5k 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 ...

**5**

votes

**1**answer

603 views

### Can one live without actual infinity? [closed]

The title of this question is the exact title of one of the sections of a book written by Alexandre Borovik: Mathematics under the Microscope. Under the title, we read:
How should we approach the ...

**62**

votes

**20**answers

7k 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 ...

**1**

vote

**0**answers

190 views

### Is Independent University of Moscow recognized? [closed]

What graduate schools recognize the degree from Independent University of Moscow? It is not a university strictly speaking and their degree doesn't have any official status in Russia, but they claim ...

**39**

votes

**18**answers

5k 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 ...

**27**

votes

**11**answers

2k views

### Are there elementary-school curricula that capture the joy of mathematics?

UPDATE: Wow, thank you everyone for the great insights!
A couple of months ago I stumbled across Paul Lockhart's essay A Mathematician's Lament and it made perfect sense to me. I'm not meaning to ...

**94**

votes

**37**answers

64k 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, ...

**15**

votes

**4**answers

2k 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 ...

**396**

votes

**178**answers

103k 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 ...

**192**

votes

**72**answers

77k 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 ...

**10**

votes

**7**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 ...

**12**

votes

**10**answers

2k 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 ...

**4**

votes

**5**answers

570 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 ...

**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 ...

**56**

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 ...

**87**

votes

**53**answers

19k 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 ...

**2**

votes

**3**answers

634 views

### Assessing effectiveness of (epsilon, delta) definitions [closed]

There is much discussion both in the education community and the mathematics community concerning the challenge of (epsilon, delta) type definitions in calculus and the student reception of them. The ...

**46**

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 ...

**17**

votes

**5**answers

3k views

### Pacing for learning new material

I'm beginning to run into work where I have to do a significant amount of learning of math by myself, with a book rather than with a teacher. Now, I do know that doing problems tends to be the best ...

**107**

votes

**29**answers

32k 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, ...

**6**

votes

**4**answers

2k 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 ...

**28**

votes

**2**answers

900 views

### Schubert calculus, as lowbrow as possible

Starting in a week I'm going to be an instructor at a summer program for exceptionally mathematically talented high school students, and I'm going to be teaching a class on Schubert calculus. The ...

**13**

votes

**19**answers

11k 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 ...

**28**

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: ...

**33**

votes

**1**answer

2k views

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

The following problem was brought to my attention by a doctoral dissertation on Mathematics Education, but - as far as I know - the solution remains unknown.
I have already asked this question on ...

**22**

votes

**6**answers

8k views

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

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

**6**

votes

**12**answers

4k 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 ...

**4**

votes

**3**answers

502 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 ...

**-6**

votes

**1**answer

526 views

### V.I. Arnold's high school problem [closed]

According to his interview to the Notices of the AMS, when Vladimir I. Arnold was 12 years old (in 1949) his teacher I.V. Morozkin, gave to his classroom (apparently 6th grade of a soviet primary ...

**50**

votes

**15**answers

6k 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 ...

**11**

votes

**5**answers

578 views

### Recreational mathematics: where to search?

I am not sure I can strictly define recreational mathematics. But we all feel what it is about: puzzles, problems you can ask your mathematical friends, problems that will bother them for a couple of ...

**54**

votes

**10**answers

10k 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!" ...

**1**

vote

**3**answers

175 views

### Application for functions of the shape $r = f(\theta)$

A fairly ubiquitous object in elementary calculus is a function of the shape $r = f(\theta)$, where $r$ is the radius and $\theta$ the argument. Common examples include the cardiod and limacon, and of ...

**13**

votes

**2**answers

581 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 ...

**44**

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 ...

**23**

votes

**3**answers

1k views

### Is “problem solving” a subject to be taught?

I am witnessing a new curriculum change in my country (Iran). It includes the change of all the mathematics textbooks at all grades. The peoples involved has sent me the textbook for seven graders (13 ...

**38**

votes

**7**answers

4k 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 ...

**12**

votes

**4**answers

868 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 ...

**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 ...

**4**

votes

**2**answers

4k 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 ...

**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 ...

**8**

votes

**7**answers

1k views

### Review papers in mathematics

Are there review papers, literature reviews in mathematics that describe the recent developments in various fields for a newcomer? Or is the prerequisite knowledge always provided in research ...

**57**

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 ...

**16**

votes

**12**answers

1k views

### Insightful books about elementary mathematics

What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic non-abstract algebra, plane & solid geometry, trigonometry, etc, in an insightful ...

**18**

votes

**2**answers

1k views

### Papers better than books?

Not so long ago I took a class called "Discrete analysis". I remember that I couldn't find any "novice" level material on Mobius functions in combinatorics. So then I went to the roots and read Rota's ...

**7**

votes

**2**answers

705 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 ...

**37**

votes

**19**answers

5k 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 ...

**1**

vote

**2**answers

326 views

### Can one branch of mathematics be completely learned from the perspective of another branch of mathematics? [closed]

This arose from a discussion with a friend (people involved are two engineers) who argued that every result in mathematics should be transformable into another branch. For example, he argued that ...

**82**

votes

**7**answers

8k 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 ...

**2**

votes

**3**answers

295 views

### Good Books on the history of Zero

I am looking for books that discuss the origins of the zero, specifically the differences in the use and concept of the zero number among different civilizations (considering also the Mesoamerican ...