> cheap alternative Here from my experience is a still cheaper alternative: My experience? From 9th grade on, concentrated on math. In college also took a l

> cheap alternative Here from my experience is a still cheaper alternative: My experience? From 9th grade on, concentrated on math. In college also took a lot of physics. Got a Ph.D. in applied math. Taught math in two big universities. Applied math to US national security (mostly Navy work), FedEx, and computing. From that some advice: (1) Get a good outline of math, that is, a list, with short descriptions, of the main subjects, and for each subject get 2-3 of the best books. One source: Course descriptions at the Web sites of some of the best universities. In this, draw ONLY from the very best, e.g., Princeton, etc. Fact of life: Mostly the teaching and books are, in a word, junk. The worthwhile teaching and books are under 10% of the total. It's simple: A big fraction of the profs are much less good than the best but still teach and write books. Sorry 'bout that. Soooo, did I mention, draw ONLY from the very best. For cheap, pay no tuition. Get the books used, cheap. (2) Study the books, and work a lot of exercises. Sometimes go through quickly and get only some high points; other times go slowly and try to understand nearly everything there. Sorry, at least in math, it's "not a spectator sport". Maybe a course with lectures can help, but really to do well with the material still have to do that sitting alone with the books. Sorry 'bout that. (3) Work to understand the psychological, social, national, economic, power, political, competitive, publicity, and business aspects -- these have important effects, and the real world of the subjects is not 100% pure knowledge. E.g., get some understanding about how US/EU academics works. Use this understanding to help make decisions about career directions. (4) As you proceed, think at least a little about how might do some original research building on what you have learned. Eventually you will find some results to publish, and that will open some special doors for you.