How do I get good at Math?

Question

How do I get good at math?

I'm a freshman in college, and I've always done OK in math. I never had any good teachers in High school, and I always have done the bare minimum. Over the course of this year, I really came to understand how important math is and how much potential I have to enjoy it. I'm taking a discrete 2 course and I love it. (I'm a Computer Science major)

Even though I passed Calculus 3, I don't feel I understand Calculus. I have an excellent memory and was able to memorize formulas and apply them when I was supposed to. I don't understand anything though.

I came to the realization that I should put some effort into getting better at math. Over spring break, I went through almost all of Khan Academy's exercises just to make sure that I have a good grasp on the very basics. I'm going to finish them up in a couple of days.

what should I do next?

My plans are to start at the beginning of my Calculus book (Thomson's Early Transcendental) and work my way through the book (one part a day), reading everything and doing every problem. I don't know how good of a book this is though. I've done some googling and a lot of people recommend Spivak's and Apostle's Calculus books. Is it worth purchasing one of these (or something else) if I really want to comprehend the material? Are they good for self-teaching?

I usually have some sort of project like this going on (except never math based), and I'm completely willing to put time into this goal. Some direction would be nice though.

Answer 1

If you're interested in the subject, you should avoid getting trapped into thinking that mathematics consists only of those things you find in the curriculum of a university. That might mean you should look at some popularizations before deciding which things to try to learn. I read David Bergamini's Mathematics as a kid. It gives a different and far more truthful impression of the subject from what you'd get by doing the "bare minimum" in school. Not up to date but I still think it's worthwhile.

I like C. Stanley Ogilvy's Excursions in Geometry. It's amazing how much he can do with so little needed in prerequisite knowledge. Some of it is extraordinarily beautiful. (Follow the link and you'll see that one of the five-star reviews on amazon.com is mine.)

Some of the books published by the Mathematical Association of America are probably worth looking at.

At a more advanced level---say upper-division-undergraduate level--- (Part of the point of some of the comments above, is that you shouldn't only work at a more advanced level. But it's also necessary to do stuff at a more advanced level.) if you like discrete math, maybe Brualdi's textbook on combinatorics and Wilf's Generatingfunctionology. If you like stuff with lots of engineering and scientific applications, maybe Strang's linear algebra book. Very applied. (Here's the difference between "pure" and "applied" mathematicians: the former know about spectral decompositions of real symmetric matrices; the latter know about singular-value decompositions.) Regardless of whether you want "pure" or "applied" stuff, Dym & McKean's book on Fourier series and integrals can teach you something. (It's not very good for learing the analysis background; it's superb for reading about lots of examples of uses of Fourier theory.)

To be continued, possibly in a separate answer later, maybe........

Answer 2

If you are looking into building more pure math based tools, it may be worth your time to check out Munkres's book on Topology. One of the more popular examples of a famous topological proof is turning a sphere inside out. A visual representation is found here http://youtu.be/R_w4HYXuo9M

The book itself starts out with the foundations for set-theory and methods of proof which you will need in higher math. After completing the first 80 or so pages, you will have covered enough material to begin understanding what the notion of a space is. Be prepared for testing your patience, because this material does not come quickly to many. Also, after completing the first section of the book, you are given an option to stop reading it, or move on to algebraic topology. It seems like the goal of algebraic topology is to categorize topological spaces even further based upon an algebraic group. I believe he covers the algebra you need for it, but if you want to divert off to another book, then I highly recommend Algebra: Chapter 0 by Paulo Aluffi.

This monstrous 728 page book provides a comprehensive overview of many fields in abstract algebra, many of which is useful for cryptography. Also it covers quite a bit of category theory, which is the basis for Haskell programming.

If you wish to read start topology later on, you may want to purchase munkres anyway and read the preliminary pages that I stated above. I was able to buy the international edition for only $15, which is extremely affordable as far as textbooks go.