How do you revise material that you already half-know, without getting bored and demotivated?

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.

Solution 1:

A few tips that you might find useful:

1. 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.

2. 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.

3. 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.

Solution 2:

1. Write about it.

Try to write about the topic you know on your notebook or post it on a personal blog. I personally think that the latter is better because it will benefit anyone on the Internet who got stuck on some difficult concepts or need other's perspective to understand it.

2. Do the exercises.

Solving problems is fun. Especially some difficult one because they force you to think. The best feeling for a problem solver that I believe is... after spending some time on the difficult one and you are finally able to see through it.

Solution 3:

When one half-knows something, in order to not be bored re-reading elementary facts about it, the best thing to do, in my opinion, is to use it trying to solve problems involving it. This approach will automatically tell him what is the concept to be relearned and arouse a curiosity: "Why am I not able to use this concept, although I think to know it?" Jumping directly to this concept, the reviewing, dictated by such a curiosity and not simply felt as being one's own duty, is performed critically and with more interest.

The kind of problems I mentioned can be disparate: practicing to relate what he half-knows with what he very well knows from other fields, or playing with what he half-knows through experiments (computer?), or subscribing to related arguments on SE seeing what visions people have on them.

So the secret is recognizing what is not known among those thing one thinks to know.

Solution 4:

I think I'll give my opinion on this matter, precisely because I have found myself having the same experience – and to a certain extent, overcoming it. A small amount of background: I am by no means an experienced mathematician, I am entering a master's program in the fall and intend to continue with a Ph.D. program afterwards. As such, I am taking the Mathematic GRE subject test in the fall with the intention of boosting my score to the $n^{th}$ percentile for $\lvert 99-n\rvert<\epsilon$ for all $\epsilon>0$. Whether this comes to fruition or not, the review process is annoying. I think I have found some effective techniques for reviewing otherwise dull material – Calculus, Elementary Linear Algebra, Basic Algebra, etc. Here are some thoughts.

(1) First and foremost, when reviewing a more basic theory, I try to see how it ties in to more advanced subjects I have been learning about. For instance, much of the theory of Calculus of Several Real Variables lends itself to generalization to $C^\infty-$ manifolds and Riemannian Manifolds. I have been trying to make sure all of these more basic results are integrated into my understanding of Manifold theory. I have found quite a few gaps in my understanding of the latter in so doing.

(2) If you're reading something that you really do know quite well, see if you can recreate it from first principles, and see how slick you can make your proofs. Not only will this help you organize and remember the information, it will help you as an expositor in the future if you record these notes. Alternately, you can create some sort of lecture series on a blog someplace, as mentioned by another user in the answers.

(3) If you are in a university setting, sometimes tutoring material you want to review is quite beneficial for both parties. If you aren't that comfortable with the material, then maybe don't charge money, but find a friend who is struggling, and explain the material to them. If they are particularly inquisitive, you may finding them asking you things that you can't immediately answer.

(4) You can always just jump to the more advanced subjects that do interest you, and refer back to the basics as necessary. Sometimes, seeing the utility of the elementary material can help motivate you to study it. Indeed, for me, a lot of basic manifold theory used tricks from multivariable calculus that I was not as familiar with as I should have been. This certainly made me review the basics.

(5) The last piece idea I will propose is to seek out hard problems. And I do mean rather difficult problems. Sometimes, racking your brain for ideas when you are stuck forces you to review the material at hand without even realizing it. Moreover, these sorts of problems will improve your general problem solving skills, whilst allowing you to review basic materials, thereby killing two birds with one stone.

I hope this helps somewhat.