23
votes
3answers
1k views

Nearly all math classes are lecture+problem set based; this seems particularly true at the graduate level. What are some concrete examples of techniques other than the “standard math class” used at the *Graduate* level?

In the fall, I am teaching one undergraduate and one graduate course, and in planning these courses I have been thinking about alternatives to the "standard math class". I have found it much easier ...
36
votes
19answers
4k 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 ...
19
votes
10answers
1k views

Learning through guided discovery

I have been working through Kenneth P. Bogart's "Combinatorics Through Guided Discovery". You can download it from this page: http://www.math.dartmouth.edu/news-resources/electronic/kpbogart/ I've ...
6
votes
12answers
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 ...
5
votes
3answers
1k views

Battle of the brains; cultural mathematics

Firstly, I apologize if my question is long. Three years ago, I watched a video with the name Battle of the Brains. It was a wonderful video about challenging some famous peoples to solve some ...
15
votes
7answers
1k views

Unexpected applications of the fact that nth degree polynomials are determined by n+1 points

I had a funny idea for proving an identity in Euclidean geometry. While it didn't end up being a very nice proof strategy in my case, I would still like to collect nice examples of where the proof ...
12
votes
17answers
2k views

Short Course Suggestions For High School Students

I am planning to teach a course for talented high school students at a summer camp and I need suggestions for possible topics. The students usually have different backgrounds but most of them are ...
7
votes
2answers
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 ...
3
votes
2answers
888 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 ...
40
votes
44answers
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 ...
22
votes
18answers
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 ...
7
votes
6answers
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 ...
55
votes
73answers
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 ...
8
votes
4answers
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 ...
10
votes
6answers
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 ...
174
votes
71answers
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 ...
11
votes
17answers
2k 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 ...
17
votes
16answers
4k 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 ...
5
votes
3answers
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 ...
6
votes
4answers
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 ...
61
votes
15answers
5k 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 ...
0
votes
1answer
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 ...
6
votes
3answers
4k 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 ...
32
votes
12answers
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 ...
364
votes
176answers
96k 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 ...
36
votes
11answers
4k 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 ...
86
votes
53answers
18k 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 ...
4
votes
6answers
3k 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: ...
56
votes
16answers
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 ...
102
votes
30answers
30k 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, ...