notes-life-learning

to learn mathy book stuff well, you gotta do all of:

• understand it, e.g. have an intuition about the meaning of everything and about why you do each step, which comes from reading about it slowly and thinking deeply about what you are reading, and re-explaining it to yourself and others. In order to remember this understanding, once you have the understanding you need to re-read the material a few times.
• learning how to redevelop it yourself from first principals, and to advance it, which comes from (a) being posed applications of the material where you don't immediately know how to use the material to solve the problem, and figuring this out for yourself rather than being told or reading the answer, and (b) writing down new things and exploring their deductive consequences, and (c) doing projects applying it to new things and actually finishing them
• be fluent in it, which comes from memorizing the key formulas, and doing lots of practice problems

Unfortunately there are two parts of that process in which learning can be counterproductive:

• if the first time you learn, you read the material quickly without stopping to think deeply about it, you'll eventually learn the 'received wisdom' intuition but it'll be harder to develop your own idiosyncratic intutions because you'll be stuck in the blinders of thinking about the material the same way everyone else does; whereas if you think deeply about the material when you first encounter it, when it still seems strange and confusing, you can more easily find unconventional perspectives on it, and then you can still learn the conventional perspective later.
• if you read or watch how to apply the material to various applications, then you can't ever figure out those applications for yourself, so it is harder to find material for that kind of practice

Mathy stuff takes a ton of time, and if you want to spend extra time doing these things upon your first encounter with each bit of new material, it takes even more. So it is not always possible to do so.

Mathy stuff, like anything but especially, is learned automatically by your mind 'in the background' after your exposure to it; by this i mean that if you encounter some new material and it seems really confusing and then you sleep on it and come back to it, it often seems magically much easier to understand. For this reason i highly recommend trying to give mathy material a pre-read long before you are actually supposed to get to it. It's hard to find time to do this but i think it's actually more efficient, because then more time will have passed between your first and second read-thrus, allowing more background processing to occur in the meantime.