A few months ago an acquaintance of mine (Cornell / Harvard Law School, so a smart guy) emailed me and asked if I had any advice for how to explore mathematics further:

Wanted to see if you’d recommend any online math courses (available free) lots of MIT stuff available and the like but wasn’t sure which one was good.

My response (edited for clarity):

I wrote the advice below and then remembered that you’d done math competitions in high school, so I’m not sure how obvious some of it may seem to you:

1) What you should spend your time on really depends on what your goals are. For example, when I was studying math I really was only interested in how I might use math to help me in money-making pursuits, and as a result I was interested in getting a good high-level understanding of how the field worked, what mathematicians were up to, what the different advanced fields were about, and why I’d seen so many mathematicians who were good at learning other things. My thinking was that once I understood the field at a high-level, I would know where to dive deep (if necessary). On the other hand, if you’re just studying math for the fun of it, then you might want to just jump into a type of math that sounds interesting and start learning about it.

2) I have a section on my website where I’ve gathered some of the stuff I’ve run across: NW’s Thoughts – Mathematics

3) For a high-level understanding of what’s going on in the field of math:

It’s not free, but Timothy Gowers is a great writer and I remember finding this (short) book very helpful:

Mathematics: A Very Short IntroductionI also found this (short) book very helpful:

Letters to a Young Mathematician4) For learning the general ability of problem-solving that makes mathematicians tend to be pretty good at tackling other subjects:

How to Solve It

I was at one point trying to create a summary of the book here: NW’s Thoughts – How to Solve ItThe movie “Hard Problems” was at one point on YouTube. I found it helpful in at least three ways: 1) I found it fun to watch, and I think it’s good to have fun while spending time thinking about mathematics, 2) it made the ‘genius’ math kids seem less mysterious, and 3) there was a very memorable scene where I saw one of these ‘genius’ math kids going through a tough problem and he is speaking out-loud about his approach to solve it, and the big trick is to just solve a simpler version of the problem first and only then start to solve more-complicated versions of the problem that are closer to what your final goal is. Hard Problems

5) As for learning actual math, the conclusion I came to after a lot of research was that the most important fields to be familiar with were 1) probability, 2) combinatorics, and 3) linear algebra (with the order of importance depending on your situation).

KhanAcademy has videos on all of those.

This video series on Linear Algebra was highly recommended by someone, but I haven’t watched it myself yet: YouTube – The Essence of Linear Algebra

If you’re not already aware of it, I also highly recommend checking out the ‘Art of Problem Solving’ website, as far as I know it’s the premier hangout spot for school-aged math competitors, and there are a lot of friendly people there of all ages.