How to fill gaps in my math knowledge?

Solution 1:

You want to look at one or both of the following, excellent, resources:

• Thinking Mathematically: By Mason, Burton, and Stacey. This will help tremendously with being able to approach problem statements "mathematically", for example, how to deal with, think about, or read, mathematically, "word problems", and problem statements, break them down, approach them, and solve them.
• How to Prove it: A Structured Approach: by Daniel Velleman. Excellent survey of logic, proof strategies, and proof-writing, with many hands-on examples and practice problems to develop these skills.

Both will be excellent preparation for and throughout college.

ADDED:

• Another "classic" is Polya's book How to Solve it. It's a gem for better understanding and developing the skills and strategies needed when in problem-solving.

Also, for practical purposes: you might want to take a look at Paul's Online Notes, for some pre-college and "undergraduate level" lecture notes and coursework, compiled and maintained by Paul Dawkins, from years of college teaching at Lamar University.

Solution 2:

I recommend to try Coursera course "Calculus - Single Variable" by Robert Ghrist. I have just finished this course. He is starting it again in May.

Solution 3:

I think the lecture notes by William Chen are good introductory material. The Trillia Group has decent textbooks, free for personal use.

For "understanding" highschool math better, look around for the materials used in preparing for maths olympiads (hard problems in that general area).

If you want some historical perspective, the books by William Dunham ("The Mathematical Universe" and several others) are a delight to read. Paul Nahin's "An imaginary tale -- The story of $\sqrt{-1}$" is a nice read too.

But as in anything, I've learned the hard way that to study something "because I might need it later" is a waste of time: You might never get to use it, the use could be so far in the future that by when you need it you already forgot, or just things have changed so much (new techniques, outlook, your own knowledge) that what you learnt isn't right/useful/the best way anymore. And studying something without a clear, nearby use is rather uninspiring.