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.
A few tips that you might find useful:
Study a text book that covers more or less the same material but via a different approach. For example, if you studied group and ring theory from Dummit and Foote, you might enjoy revising the material using Aluffi’s book “Algebra: Chapter 0”. It covers pretty much the same material but emphasizes from the beginning a more modern and categorical approach. By relearning the old material from such a book, you’ll not only relearn the material but learn a lot of new material (category theory) and a different way of looking at the old results. For complex analysis, I can recommend “Complex Analysis: The Geometric Viewpoint” which puts familiar results in complex analysis in the context of differential geometry and curvature.
Teach it. I found the best way to improve your knowledge in areas that you learned once and haven’t used much since is to teach them. This can mean teaching or TAing a class, giving private lessons, writing a blog or answering questions on math.stackexchange. Teaching gives you “external” motivation to look at old results, clarify them as much as possible and extract their essence so that you can explain everything to others as clearly as possible. This way, when you do it, you don’t feel like you’re doing it only for yourself.
Study more advanced material which uses the material you want to revise. For example, if you want to revise measure theory, you can learn some functional analysis. Since many examples in functional analysis come from and require knowledge of measure theory, you’ll naturally find yourself returning all the time to those areas of measure theory which you don’t feel comfortable with (if there are any) and filling the gaps. If you want to revise the implicit function theorem, study some differential geometry. This way, the revision won’t feel artificial or forced because you’re actually studying new things and, in the process, revising the things which come up naturally.