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.

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    Use the next level book to identify the gaps in your knowledge. Find the first gap, fill it, then return to the advanced book. Repeat the process. – quasi Aug 6 '17 at 8:38
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    If you already know a little bit about what you're reading, go directly to the exercises section. Use the text only as a reference – leibnewtz Aug 6 '17 at 8:44
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    Skip the boring sections, but do the exercises to be sure you didn't miss anything important in the "boring" sections. – rtybase Aug 6 '17 at 9:51
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    I'm pretty sure you mean review and not revise. Those two words are definitely NOT synonymous. – Wildcard Aug 8 '17 at 0:35
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    @goblin, aha. Today I learned a new definition; I suppose you are in the U.K. "(Brit.) reread work done previously to improve one's knowledge of a subject, typically to prepare for an examination." It definitely does not mean that in the U.S. But you are using the word perfectly correctly; I stand corrected. :) – Wildcard Aug 8 '17 at 4:25

14 Answers 14

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.

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    +1 for teaching. Rabbi Chanina said, "I have learned much from my teachers, more from my colleagues, and the most from my students". – Meni Rosenfeld Aug 7 '17 at 0:13
  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.

  1. 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.

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    "2. Do the exercises." What, we're actually supposed to do those? – PyRulez Aug 6 '17 at 17:58
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    I wish Coxeter's works had exercises. – Anton Sherwood Aug 7 '17 at 6:02
  • It depends. The 2. is completely optional. I believe for a passionate problem solver they will do whatever problems they come across. Plus, being able to do the problems will be a better sign of understanding. That aside, I was expecting someone to write 'make mathematics videos on Youtube' in the answer. Lol. It will take a long long time. – Crazy Aug 7 '17 at 15:52

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.

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.

Go to math.SE. Select the corresponding tag. Pull up unanswered questions. Answer them. By the end, you'll know the material very well!

If you sort my answers chronologically, you'll see the times that I was reteaching myself Galois Theory, elementary Group Theory and Calculus on Manifolds.

I very often struggle with this exact problem. I have found two solid ways to help tackle the issue of having to properly learn a subject I have already visited once.

First Method: Absorb the information in as many different ways as possible. Reading and rereading text books can get dull - if you manage to find video tutorials (watching), podcasts (listening) and exercises (doing). Depending on the subject this can be easier said than done, but after a bit of searching I am still yet to find a subject without material in different mediums.

Second Method Reapply the knowledge learn in different ways: Most people find the best way to learn is by doing, but nobody likes doing the same exercises/project over and over again. Have a think of some new way to apply the knowledge, you might fail at completing a project like this but the important thing is to learn whilst failing. You might also succeed in solving new problems with your knowledge - the best feeling in the world!

Most Important No matter how bored you get always remind yourself of the reasons why you are doing it. To graduate, to solve that new problem, to get that job etc... It's always easy to get bored on the journey when you forget the destination.

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    +1 for last paragraph – Math_QED Aug 7 '17 at 8:03

Try reading the book backwards, in the sure knowledge that:

  1. you are getting ahead of the game by knowing the answers first
  2. the ideas and theories you don't know yet will come up as you read back, and
  3. you will know what use the theory will have when you get there.
  4. The index can be used for cheating in the backward read.

It worked well for me with Physics text books. It's probably the same as reading a detective novel for academic study (studying English Literature anyone?) versus reading it for pleasure (given that it was written for pleasure;-)

  • Nice idea! $\;\!$ – goblin Aug 8 '17 at 14:01
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    Also, read the introduction. I am often surprised at how much solid structure is given in the introductions to books that really does set out a framework for the rest of the book. It often contains the 'Why is this important' part. – Philip Oakley Aug 8 '17 at 14:03

I find it works for me to stick with the harder bit of material..it may be slow at first but if I spend enough time with material that feels like "the next step", it ends up improving my understanding of what I already (should) have learned.

Just remember, you are more capable than you think.

Some of the other answers may have mentioned it, but let me single out one method that will help you: backtrack

Go the chapters with material that you don't know, and get engaged with new ideas.

  1. If you can follow, excellent! you are learning.
  2. If you can't, figure out why; what notions are you missing? Go back and study that.
  3. If it is kind of new, but familiar, it means that you are getting a new perspective on an old concept. This is a golden moment! Focus on understanding how the two points of view complement each other. These are the times when you do your mathematical growth.

I had a professor, Max Shiffman, who was a brilliant lecturer. To him equations had personalities, anatomies, and histories. He was a bit excentric though. For example, I often saw him waiting for a local bus and he would be waving his pipe at the bus sign, talking away.

I am working very hard right now on a group theory problem, and, (at my age?), I find it hard to keep all of the details in mind. One thing I do that helps is imagine that I am standing in front of a classroom lecturing. Walking home from work one day, I stopped in front of a garbage can and explained to it some of the intricacies of the Chinese Remainder Theorem. I looked over and saw a woman driving by, slowly, in her car and shaking her head back and forth. Oh Max, I am so sorry.

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    > For example, I often saw him waiting for a local bus and he would be waving his pipe at the bus sign, talking away. Could have been rehearsal for lecturing. If an actor were doing such a thing, we'd assume so. – Kaz Aug 9 '17 at 5:34
  • @Kaz - For the sake of brevity, I left a lot of details out. You should read the link I provided. Dr. Shiffman was my favorite professor. I took every undergraduate class he taught that my schedule allowed. Not because they were easy, but because he made them easy. He was also very odd from a 19 year old's point of view. – steven gregory Aug 9 '17 at 6:32
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    There's a similar thing in programming called rubber duck debugging. – hyperpallium May 29 at 1:47

Try to teach it to yourself, like your dream teacher would. Also try to draw stuff if this is applicable.

This is challenging and will keep you motivated.

I had this exact problem when revising for my A-levels in Maths and Physics.

My boredom in the easy stuff that I knew would stop me getting to the more difficult stuff and revising for that.

In the end I started to jot notes and pictures and equations about all the bits and found that I could condense it down more and more until it was pretty much all on one page.

It gave me a new perspective on the subject and I actually learned things while revising purely from that method.

My advice would be to start with a new page and to doodle and draw little bits and bobs about your subject rather than trying to write it all in words. Picturing and doodling it that way gives your brain a new way to work with that information.

Do as many exercises as you can, type them down, and reread them from time to time, thinking whether there is still room of improvement, and whether there are easier and shorter ways to do it. If you think your answer are polished to close to perfect, do more exercises. If the easy ones are too easy, do the more difficult ones.

The textbook exercises that I wrote as homework when I took courses in the undergraduate years are very helpful. For one thing, typing them down in the first time serves as a critical examination of what you think learned. I often think something works, but when I type them down, I see I was mistaken. Moreover, when I reread, I find my previous understanding is poor, or simply wrong. In revising them, the nature of electronic files makes it easier to revise and reorganize them. Your work may also help others who is learning the same material, or show of your level of knowledge when you need to.

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.

  • This is a reasonable point-of-view, but can you give some specific advice? Your answer is a bit too open-ended to be really useful... – goblin May 29 at 7:21
  • @goblin I agree, but I think that's how it is. The advice boils down to (1) (retrospectively) don't have half-learned it; instead, to have done it in the proper order (2) to overcome boredom, be aware there's more to it than you're seeing - be open, look closer; specifically, at the significance of definitions and intro context (which can otherwise be boring) in the mathematical exposition. I've resisted review, thinking "it's obvious, I won't learn anything, a waste of time, am I so stupid I have to review this?".... and then gained several things, new or forgotten (some boring parts too). – hyperpallium May 29 at 8:14

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