Mathematics inevitably involves a lot of self-teaching; if you're just planning to sit there and wait for the lecturer to introduce you to important ideas, you probably need to find yourself another career. So, like a lot people here, I try to educate myself on important concepts that aren't covered in the standard curriculum. Of course, sometimes this involves going back to revise material that you already half know, to understand it properly this time. My question is really how to do this successfully.
Question. How do you revise material that you already half-know, without getting bored and demotivated?
Honestly, I haven't worked out how to do this yet.
Take group theory, for example.
If I pick up an advanced book, it'll usually assume a lot of background knowledge and I'm immediately lost.
But if I pick up an introductory book, it'll usually go painstakingly through some really elementary stuff, for example a book on group theory will go on for awhile about sets, functions, permutations etc, then there'll be a philosophical interlude about sets with further structure, eventually we'll get the definition of a group, then there's a chapter about, you know, subgroups, quotient groups, Cartesian product of groups, homomorphism of groups, Cayley's representation theorem, blah blah. At some point while reading the basics that you already know, you just get super bored and decide to skip forward. But in doing so, you've missed a few definitions/notations/ideas that were hidden in the stuff you skipped somewhere, and when you skip forward you end up kind of lost and just not really on the same page as the author.
This kind of thing happens to me with lots of subjects; not just group theory, but ring theory, real analysis, probability theory, general topology, I could go on. I usually end up feeling really demotivated pretty quickly and I eventually forget my plans to revise the subject. My question is basically how to avoid this.