My own personal routine:

I start reading, whenever I feel there is a problem presented in the text (mostly theorems and examples but sometimes questions arise in the text itself), I’ll try solving it. I invest time in this endeavor proportionally to my belief that it can be solved in a reasonable amount of time. If this time runs out, I’ll start reading a little of the solution and trying again. I usually postpone the end-of-chapter exercises to “later” because of spaced repetition and lack of time (I have to have studied a certain amount of chapters for exams so I have to prioritize the quantity.). Sometimes interesting ideas/quasi-problems/ways of looking at things come to mind during the study, and I think on them as well.

When I’m solving a problem, I’ll usually walk around and try to solve it mentally, because I have this uncertain belief that if my math needs paper and pencil, then I can hardly use it “at will.” I’m the kind of person who forgets stuff and needs to repeatedly derive them from first principles when needed, and I can’t quite do that if those derivations rely on writing stuff out. (I also hate memorizing theorems. It feels stupid. :D )

By the way, if the solution is not in the text and I’m stumped, I websearch around. I hate the professors who give grades for the text’s problems and thus disincentiviz solution manuals economically and socially. These take-home exercises correlate more with people’s social skills and network than their effort or knowledge. Quizzing people instead is a much better choice.

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I read math for personal enjoyment, so note that I don't have many checks on my understanding, besides my ability to read more math and feel like I understand.

As I read the book I mentally keep track of how difficult it feels and how much things make sense. If things feel more difficult than I'll make notes as I go through. The notes are more for the purpose of moving slowly through each statement --- otherwise I might skip sentences. I'll draw pictures and label everything in the picture and maybe do a really simple example.

A lot of my focus is on creating mental visualizations and creating a mapping from the definition to the visualization, or from the written proof to the visualization. I'm not sure how to describe it, but different aspects of the visual might seem 'looser' and ambiguous or 'tighter' and well defined, and this guides my thinking. I'll go back and forth trying to work out an example on paper and figuring out how the picture changes until it seems well defined. For example, I've been doing some topology lately, and have been constructing different pictures of lines or shapes and imaginary manifolds and removing different size pieces of them to get an idea of what is considered connected and what is not connected.

Previously I used to do lots of exercises, but I've found the above thing seems to help me learn better and make the exercises easier when I do them, so I've been doing fewer exercises. With exercises where I can look through the book and find a proof that use a similar method and modify it to solve the problem, and then I feel like I've just copied it instead of learning anything, so I often skip these types of problems. For something I'm learning to use for something else (I'm learning math to understand physics better) I do the simple check-your-understanding exercises and move on.

At the end of a chapter of few, I make a 'note sheet' where I add definitions and examples to a single (or several) sheet of paper, as though I was in college and preparing for an exam.

What I haven't figured out is that sometimes I come across small things that I'm not able to understand. Sometimes I ask questions on stack exchange, but find this really breaks up my routine, so if they seem like confusions with minimal impact I move on.

2 comments, sorted by Click to highlight new comments since: Today at 6:41 PM

1. I am not familiar with any empirical work on the subject.

2. Some textbooks give advice in the beginning, esp. about how ideas are organized. (Which can be helpful if you don't want to read the whole thing - it can explain dependencies between chapters, like 5 requires 1-3, but not 4, etc.)

3. Speculative:

  • Every subject has basic 'building blocks'. If a subject builds on an earlier subject, those remain important. Example: For polynomials there's: 1, x, x^2, x^3, etc. (All the other polynomials can be assembled out of those.) In calculus, there are functions that basically map between them (give or take multiplying/dividing by a constant factor), that are important.
  • you know times/addition tables? That's a great format for two dimensional functions that take positive integers as input. (Though f(1-10, 1-10) might not all be necessary if there's symmetry, and depending on the function, other constants might be more useful to know, like 1, 0, -1, e, pi, the primes, etc. For 3d, you might want to involve color or variables for associations or distinctions.)
  • These answers are all about how you study a math textbook. Answers based around other people might capture low hanging fruit, for one reason or another.

Don't feel like I have enough to contribute to post my own answer but I'm disappointed and slightly surprised this question isn't getting more engagement.