[jasonmorton]

Here is a proven approach for at least the first part, building foundations and being ready for graduate work. Many Berkeley Ph.D. students passed through this route. Get the book "Berkeley Problems in Mathematics." It contains historical problems from the Berkeley math prelim exam, and solutions. Now don't look at any solutions yet.

This is the exam all Berkeley math Ph.D. students must pass within three semesters of arriving to stay in the program, and the fail rate is about 50%.

You will also need reference books, advanced undergraduate and beginning graduate textbooks. Buy, download, or borrow as appropriate.

Pick a problem (start with the older ones, they are easier). Set aside 30-60 mins and try to solve it. No devices, no references at all, go to a library or a coffee shop without your devices. Dont' give up till time is over. If you cannot (usually the case), still don't look at the answer. Hit the reference books (don't look up the problem online either, it will go right to the answer and you won't learn much). Read and try to understand enough so that you can solve the problem. It is ok if you solve it this way (in the course of reading about it).

For bonus points, students studying for the exam will typically take entire old exams (available from the Berkeley website), take that to the library and just sit down for three to six hours and try to solve all the problems correctly. Then self-grade harshly. When you can do that for a recent exam (and get a good score), you will have more or less mastered undergrad math to the point that you could teach it.

Most important: you have to struggle to solve problems. Reading a solution is about as useful as watching someone else lift weights: you get minor tips on form but not any stronger.

[axiom92]

Thanks for the excellent analogies!