TITLE: How do you revise material that you already half-know, without getting bored and demotivated?
QUESTION [180 upvotes]: 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.

REPLY [2 votes]: you've missed a few definitions/notations/ideas that were hidden in the stuff you skipped somewhere

Half-known material has gaps that you don't know about - unknown unknowns.
So when you have trouble with a more advanced problem because of a gap, you won't know what the gap is, where the gap is, how many gaps there are, or even if there really is a gap after all (and you're having trouble for some other reason).
People advocate doing advanced material, then filling in the gaps, but this hasn't worked for me. Perhaps if you have a really solid background in the basics, it can work. Or, if you have great intuition.
Therefore, I think it's a mistake to half-learn material, and it's important to go through the material in the conventional order. Then, even if you are not fully on top of the material, you'll at least know what the material is, and have a good chance of recognizing what the gaps are - known unknowns.
One might think that skimming it, to get an overview of the material would help, but it doesn't help you recognize when it's needed - for that, you need the details.
Doing advanced material can make you feel advanced, without actually being advanced...
It does have one great advantage: when you stumble, it gives you motivation for learning the underlying material properly - and to stick throuhh the boring stuff.
Sometimes, it turns out that the boring and obvious stuff isn't boring or obvious after all - like the choice of definitions, the introductory context, or why something works.
The "boring and obvious" stuff would have seemed boring and obvious to you, even if you read it first (i.e. you hadn't read advanced material, to half-know it). This is a problem with mathematical exposition, and I also don't know what the answer is either.
But realizing that maybe there's more to it than you realize might make it more interesting. Like seeing a world in a grain of sand.