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I'm currently self-teaching analysis by reading From Real to Complex Analysis by R. H. Dyer and D. E. Edmunds. My strategy at the moment is to go through the book slowly, coming up with examples of objects that satisfy each property mentioned, and proving every lemma and theorem (and doing every exercise) myself, before looking at the proof provided in the textbook.

As I go, I'm creating a LaTeX document containing questions that I can look back at to remind myself of all of what I need to remember (e.g I have "state theorem X" and "prove theorem X" as questions for every theorem, so I can regularly remind myself how to prove things) and the answers to those questions, including proofs for every theorem, and my solutions to the exercises, in the document.

It's working so far, and I feel like I'm gaining a really good understanding of the material, but it's incredibly slow. In fact, my solutions document is longer than the section of the textbook I have gone through so far. I have time, as in I'm not in university, so I can take 6 months to finish the first chapter if I want, but I also don't have time, in that I'm busy studying for other things as well. I also would like to get to the more exciting content that is later in the book, but I don't want to feel out of my depth when I get there.

Is this a good strategy for learning maths from a textbook?

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    $\begingroup$ My impression is that if you're enjoying it then this kind of approach is great for grokking fundamental material. But be aware that there is another mode of learning which you'll have to also become comfortable with eventually, which involves learning in a "big picture first", coarse-to-fine manner and learning by talking to people who know a lot about something and absorbing their knowledge. When you encounter people who seem to have an incredibly vast amount of knowledge, and who seem to learn fast, I think it's largely because they're comfortable with this second top-down mode of learning. $\endgroup$
    – littleO
    Mar 15, 2020 at 12:14
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    $\begingroup$ @littleO's response is excellent. I just want to add that what you're doing is similar to "inquiry based learning" or IBL, and you may want to check out textbooks that are specifically designed for IBL classes. I learned from "Closer and Closer: Introducing Real Analysis." As you get to the more advanced material in the book, the proofs may get near impossible for you to come up with on your own. (E.g., there exists a function that is continuous everywhere and differentiable nowhere - hard to prove if you've never seen the construction!). An IBL textbook can help walk you through. $\endgroup$
    – kccu
    Mar 15, 2020 at 12:22
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    $\begingroup$ It is always good to first try to do the proof without any help. If you do not make any progress, you can look for hints. And only if this neither helps, look at the complete solution. $\endgroup$
    – Peter
    Mar 15, 2020 at 13:10
  • $\begingroup$ @littleO thank you, this is very helpful. I've heard other people talking about learning like that. I assume I wouldn't be able to make the "fine" part of the learning as thorough as my current strategy, but it probably strikes a good balance between mastering what you know and knowing a lot. $\endgroup$
    – esechanota
    Mar 15, 2020 at 13:14
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    $\begingroup$ Ravi Vakil made some interesting comments about learning math in grad school that I posted here: math.stackexchange.com/a/3532246/40119 $\endgroup$
    – littleO
    Mar 15, 2020 at 13:18

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This is certainly a good approach to master the content of your book, but you could probably improve speed as follows:

  1. Unless you are really a $\LaTeX$ champion, use pencil and notebook for most of your work (you can still use $\LaTeX$ for the really important things).
  2. Keep the examples carefully. This is worth a special notebook (and even a $\LaTeX$ file Examples.tex if you wish).
  3. Don't duplicate what is already in the book, in particular proofs. You may try to prove the results by yourself, but once you get the idea, your notes should only sketch the proof by giving hints like: "use compacity" or "apply the intermediate value theorem". You should arrive to the point where these hints allow you to recover the entire proof instantaneously.
  4. Do the exercises, but not in $\LaTeX$. First to go faster, but also because you may discover later on that your proof can be improved, or that it can be greatly simplified by some result you haven't studied yet.
  5. If you have the opportunity to do so, from time to time ask a mathematician to check your proofs - or ask on this site ! Sometimes, you may discover that some correct but pedestrian proof can be improved by a very simple observation. And once you know the trick, you will save a lot of time.
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    $\begingroup$ (+1) especially for #3, even more especially for the last sentence of #3, although I'd probably replace "allow you to recover the entire proof instantaneously" with "are sufficient for you to know how to recover the entire proof without undo difficulty". $\endgroup$
    – Dave L. Renfro
    Mar 15, 2020 at 18:46

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