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How is it that you read a mathematics books? Do you keep a notebook of definitions? What about theorems? Do you do all the exercises? Focus on or ignore the proofs?

I have been reading Munkres, Artin, Halmos, etc. but I get a bit lost usually around the middle. Also, about how fast should you be reading it? Any advice is wanted, I just reached the upper division level.

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Usually I read sections in math books in multiple passes. On the first pass I just read until I am confused or finished. Then I go back over it slower, sometimes taking notes on important ideas. I think the first pass is really helpful because it will help you see the big ideas that your are leading up to. – Danikar Jan 15 at 5:20
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Do read and understand all proofs; and do at least many of the exercises. Only when you manage to do the exercises as well, you get the book; and doing them will often make you read chapters again as you finally understand what they really mean. Personally, I do every last exercise in books I self-study (as you, say, Munkres chapters 1-5, 9, 11; and currently reading Artin); but that is a bit obsssive. None, though, you just cheat yourself: you read that book; but you know little. – gnometorule Jan 15 at 5:20
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Too bad you didn't mention you were reading Axler's linear algebra book. He says in the preface: "You cannot expect to read mathematics the way you read a novel. If you zip through a page in less than an hour, you are probably going too fast." – Tyler Jan 15 at 5:21
I typically skim a chapter and then go through it (maybe the next day after things have had a chance to set in) with pencil in hand until everything makes sense. In my opinion, problem sets are a must. Many authors leave interesting results that aren't quite theorem-caliber as exercises. As for speed, if you find yourself constantly consulting previous chapters, then you're moving too quickly. That's my take on it, at least. – Scott Kirila Jan 15 at 5:52
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@TylerBailey on the other side of the coin, if you are spending an hour on a single page you have probably lost track of the big picture. There is a time to have such intense focus, but it shouldn't be on your first few reads. – orlandpm Jan 15 at 8:06
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This method has worked well for me (but what works well for one person won't necessarily work well for everyone). I take it in several passes:

Read 0: Don't read the book, read the Wikipedia article or ask a friend what the subject is about. Learn about the big questions asked in the subject, and the basics of the theorems that answer them. Often the most important ideas are those that can be stated concisely, so you should be able to remember them once you are engaging the book.

Read 1: Let your eyes jump from definition to lemma to theorem without reading the proofs in between unless something grabs your attention or bothers you. If the book has exercises, see if you can do the first one of each chapter or section as you go.

Read 2: Read the book but this time read the proofs. But don't worry if you don't get all the details. If some logical jump doesn't make complete sense, feel free to ignore it at your discretion as long as you understand the overall flow of reasoning.

Read 3: Read through the lens of a skeptic. Work through all of the proofs with a fine toothed comb, and ask yourself every question you think of. You should never have to ask yourself "why" you are proving what you are proving at this point, but you have a chance to get the details down.

This approach is well suited to many math textbooks, which seem to be written to read well to people who already understand the subject. Most of the "classic" textbooks are labeled as such because they are comprehensive or well organized, not because they present challenging abstract ideas well to the uninitiated.

(Steps 1-3 are based on a three step heuristic method for writing proofs: convince yourself, convince a friend, convince a skeptic)

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I am currently using this approach on Rotman's Algebraic Topology text. It has been extremely helpful because instead of explanations he gives explicit constructions, from which the reader is left to extract an explanation. An equation may be a concise answer to a question, but it is rarely a great explanation. – orlandpm Jan 15 at 7:32
Ok. I am reading Baby Rudin atm. In the first chapter I went through all the proofs in great detail. Was this a mistake? I think Chapter 1 is different to the others because the proofs are very minimalist and axiomatic (see, for example, the proof on page 10). However, the proof is enlightening because there are axioms you would expect to need but the proof avoids. I think the rest of the book takes slightly more of an intuitive approach by the looks of it, but there are still concrete proofs everywhere. Would you recommend going through it page by page, or to use your method? – Adam Rubinson Jan 15 at 12:40
Now that I think about it, your method does sound ideal for Baby Rudin. – Adam Rubinson Jan 15 at 12:42

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