I wonder how good you can get at maths just by casually reading books. You need to work on problems for hours and hours to get a grasp on the theories.
Maybe I'm unique in this regard, but I always took it as sort of implied that "reading a math book" entails "reading the book and working (at least some of) the exercises".
The tricky part is once you get to math where you can't trivially check your answer by "substituting back in" or "using a calculator" or whatever. Doing proofs, for example. Without a teacher, how do you know if your proof is correct? So far the only thing I've really found to do for that is to post on MathOverflow or one of the "learn math" related sub-reddits. I've often wondered if learning to use an automated theorem prover / proof assistant of some sort would be helpful, but that's such a huge undertaking in its own right...
Maybe I'm unique in this regard, but I always took it as sort of implied that "reading a math book" entails "reading the book and working (at least some of) the exercises".
The tricky part is once you get to math where you can't trivially check your answer by "substituting back in" or "using a calculator" or whatever. Doing proofs, for example. Without a teacher, how do you know if your proof is correct? So far the only thing I've really found to do for that is to post on MathOverflow or one of the "learn math" related sub-reddits. I've often wondered if learning to use an automated theorem prover / proof assistant of some sort would be helpful, but that's such a huge undertaking in its own right...