This is good. I think two principal components should be emphasized when studying mathematics. 1. Proper preparation. There are textbooks at even the graduate

This is good. I think two principal components should be emphasized when studying mathematics. 1. Proper preparation. There are textbooks at even the graduate level which have no formal prerequisites and which are largely self-contained. Technically speaking, someone with no prior background but a strong mathematical maturity could tackle these, but it might take them an inordinate amount of time to really grasp the material. For example, if you understand things like mathematical induction and proof by contradiction, you can learn analysis before you've been exposed to calculus, or category theory without abstract algebra. But it's far from ideal because you'll probably need to go over the same material several times and struggle with it. Furthermore, even the same subject within "advanced" mathematics can have wildly different depths of coverage depending on the author. Pinter's A Book of Abstract Algebra is probably approachable for anyone currently reading this comment, or even high school students. Dummit-Foote is a step beyond that, and appropriate for undergraduates who are already immersed in a math degree. But Lang or MacLane-Birkhoff would be significantly more challenging without first building up to them. Sometimes this is not just a question of depth, but also of pedagogical style. You can get a lot of satisfaction by learning analysis from Rudin for the first time, but it's really a rough go of it if you're not prepared for the terse definition->theorem->proof->remark->definition->theorem->proof->remark style of writing. On the other hand, Tao's Analysis I and Analysis II are much more approachable (similarly, some writers, like Halmos or Munkres, are praised for their exposition in introducing otherwise complex material). Ideally someone looking to study a subject should introspect about whether or not they are prepared for that subject overall. Once they've confirmed they are, they should read the first 10 pages of five or so well-recommended textbooks on the subject at their level, then choose to stick with the one that has the most approachable exposition style for them. 2. Proper study. When studying any given textbook (or videos, lectures, etc) it's really important to understand that mathematics is an active discipline. You cannot learn it by reading it. The process that has worked for me is the following: first, read through a chapter without taking any notes. Do so quickly, but not quite so quickly as skimming. When you come across things you don't know, compartmentalize them a bit and keep moving forward to the end of the chapter. The idea is to let the chapter's new material percolate a little before you begin actively tackling it. Next, start over at the beginning of the chapter and write down every single definition and theorem as you read. Before reading the author's proof of any given theorem, try to prove it yourself for at least 10 minutes. Then compare your work to the author's, and copy their proof meticulously in order to learn the method. Continue on to the end of the chapter. Finally, there will probably be anywhere between 5 - 20 exercises at the end of the chapter. Solve a meaningful fraction of these exercises, and don't look up the solutions to any of them until you've struggled with them for a good half hour or so (each). When you do look up the solutions, make sure you check multiple proofs for the same exercise so you can understand how the chapter's material can be applied in different, flexible ways. Mathematics has always exemplified a central belief of mine, which is that humans learn under conditions of optimal struggle. Even though it feels like being mired in hopeless complexity while you're struggling to complete a particularly difficult problem, you're actively learning the subject by doing it. But it's a question of efficiency. You want to aim for a subject and a presentation of that subject which is difficult enough to be just out of your current capabilities, but not so difficult that you can't follow its exposition.