I believe that a lot more people would be interested in math if they saw what it really was. These links are places where you can see what it "really is."

My suggestion is to seek out one of the in-person programs that offer classes, especially Mathcamp or ESP. Having a real teacher presenting material makes it much more exciting and accessible. Mathcamp, in particular, creates an environment of cool math unlike any other.

If you have suggestions for what else to include on this list, let me know!

Programs During the Year

• The MIT Educational Studies Program offers a number of programs in the Boston area (and beyond!). These programs aren't limited to just math --- they cover all kinds of topics from physics to literature, dancing to video game design, and much more. In particular, MIT ESP offers Splash each November, twenty hours of amazing courses taught by MIT students, and the High School Studies Program (HSSP), a ten-week long program that meets once per week. Their webpage has more information.
• The Stanford Educational Studies Program is similar to that at MIT, although their programs have been smaller.
• cSplash was a program put on at the Courant Institute at New York University, modeled on MIT's Splash but offering only math courses. Look for it to return and perhaps expand.
• Math Circles are local programs that typically meet about once per week. Some, like the Boston Math Circle, offer classes where the goal is to introduce students to new branches of mathematics and other beautiful ideas in math. Others focus on a problem-solving approach, introducing nice shorter problems for students to sharpen their skills. There are many math circles out there and I'm not going to try to make a complete list --- I suggest Googling "math circle" along with your particular location.

Summer Camps

• Canada/USA Mathcamp is the largest mathematics camp out there right now. I've been attending every summer since 1999 --- first as a student, then as a junior counselor, and now as a mentor/instructor. Mathcamp emphasizes the beauty and variety of mathematics; it presents many different topics at all kinds of difficulty levels that students can choose from. Classes are taught by graduate students and research mathematicians.
• Other summer camps are all over the place. Some of the best known are HCSSiM (Hampshire), which has a wonderful free spirit and does lots of great stuff; PROMYS (Boston University) and Ross (Ohio State), which emphasize number theory developed through dedicated student work, and SUMaC (Stanford) which guides students carefully through one or two topics.

Web Sites

• Artofproblemsolving.com is the largest online community of high school students in math. It was founded by one of the authors of the well-known problem solving book "The Art of Problem Solving," and manages to create a wonderful community for high school students to join in and discuss mathematics. The focus here is on problem solving.
• Math Mudd Fun Facts, hosted at Harvey Mudd College. A collection of fun and amazing results and mathematics. Just clicking "random" repeatedly can lead to hours of fun!
• John Baez's website is a huge resource of advanced mathematics made approachable. Baez is a mathematical physicist who's made large contributions to a field of math called "category theory," a kind of meta-mathematics that studies how mathematical theories work (and then provides new insights into those theories). In particular, check out Baez's This Week's Finds in Mathematical Physics.
• Frequently Asked Questions in Mathematics, a collection of very interesting mathematics. This can answer questions you never even knew you had. The biggest limitation is that the selection of topics is pretty limited, and it hasn't been updated in over a decade.
• Cut-the-Knot, "interactive math miscellany and puzzles." Lots of good interactive applets and interesting discoveries to make.

Books

• Excursions in Mathematics by C. Stanley Ogilvy. A light and fun book.
• Proofs from the Book by Martin Aigner and Gunter Ziegler. A wide selection of mathematics chosen because it's absolutely gorgeous; widely varying in difficulty level.
• What is Mathematics? by Richard Courant and Herbert Robbins. More serious, this provides an excellent introduction to many branches of mathematics.
• Numbers, by Ebbinghaus et. al. A collection of all the places that numbers of all sorts show up in mathematics. Very interesting. Some of the material is very easy to understand, some of it very hard.
• Geometry: Plane and Fancy by David Singer. A relatively axiomatic, pretty gentle introduction to non-Euclidean geometry.
• Geometry of Surfaces by John Stillwell. Lots of interesting geometry here at a basic level, great at demonstrating how much more there is to geometry than what's done in high school.
• The Joy of Sets by Keith Devlin. A good introduction to the flavor of real set theory.
• Outside In, a video by Silvio Levy and William Thurston. Gorgeous. If you can get your hands on it, see it.

Other Notes

I'm always happy to answer math questions. Feel free to contact me: dzaharo2_removeme@math.andme.uiuc.edu.