For someone self-studying mathematics: Do the exercises. Especially the hard ones. Struggle for hours. Don’t worry about solution manuals. It should be painful.
That is how people learn how to pass tests instead of learning the material. Passing tests is important in college, but if you want to learn on your own it doesn't matter.
Yes, but I'd argue that the path you took was a very suboptimal path to reach the same understanding since you almost surely didn't reach it. So now you'd have to redo almost everything in order to reach the same goal making it a very huge waste of time. Of course I cannot be sure of that, but I have seen so many people who learned the way you did and they had really poor understanding. In my experience only the people who try to challenge themselves actually gets there.
Passing math classes requires almost no understanding at all btw, even getting good grades in them doesn't.
Along the same lines: in a math textbook or math-heavy paper, try to prove the Lemmas or theorems yourself, when it’s sensible to attempt it. Probably more suitable in a textbook where they build up to a larger theorem starting from smaller lemmas — try to prove them yourself. Even if you don’t prove them the first time, on a re-read you can try to prove it yourself, as a way to test your understanding, and of course this also deepens your understanding.
This is good advice. But for me, the hours’ struggle is more useful if not contiguous.
I get the best results from struggling with a given problem for no more than 1 hour continuously, and then going away and coming back sometime after my next sleep.
By struggling I mean “not making visible progress” - if progress is happening then just keep rolling.
This sounds like how I approach leetcode. Leetcode feels a bit like math to me anyways. Coding isn't the hard part, understanding the algorithm in conjunction with the data structures is.
Coding is the hard part if you are bad at coding but good at maths and algorithms. Writing accurate code for solving a hard problem in under 30 minutes requires you to be really good at coding.
