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Hello, recently I've been trying various attempts regarding how to approach a math book to learn in the best way. Should one memorize the theorems and proofs so that one can recite them? I tend to sometimes forget proofs and such after some time, so my question is the following.

When you're trying to learn from a book and it's a new subject, how do you usually do? Do you learn all theorems and proofs? Do you write it down? Or do you just read? This is a general question, I'm aware of that, I do however hope that this question will fit the community.

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Your profile says that you are an undergraduate, and this question suggests that you are just starting to learn math on your own without a teacher (an important step!) . Because of this, I think you might get more useful responses if you posted this question to math.stackexchange.com instead of here. –  Andy Putman Aug 13 '10 at 16:15
    
In general, the way of doing this varies a lot from one person to another. You can ask others how they do it, but in the end you will have to find the way that works best for you. –  Gerald Edgar Aug 13 '10 at 16:21
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Personally I often find that I think I completely understand a proof, only to discover that this is not the case if I try to replicate it without notes. For that reason, I find it is often a good idea to sometimes test my understanding by writing down what I've learnt in my own words. Note this is different from just memorizing and reciting theorems/proofs though! –  Adam Aug 13 '10 at 16:31
    
The approach will vary from person to person, and the level at which you're working. However, I think there is one close-to-universal rule: if you're serious about what you're reading, you keep a pencil and notepad handy. –  Thierry Zell Aug 13 '10 at 16:34
    
The automatically-generated "Related" items at the side now include these... mathoverflow.net/questions/27221/… mathoverflow.net/questions/12734/learning-speed-in-math –  Gerald Edgar Aug 13 '10 at 16:34
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closed as off topic by Andrew Stacey, Steve Huntsman, Noah Snyder, Andy Putman, Harry Gindi Aug 13 '10 at 17:33

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2 Answers

I am a big believer in learning by exploration: It builds independence and confidence... something you need if you plan to pursue further your math journey. That means when you find an interesting result or theorem in a book, explore it's consequences, see the details of the proof, etc. It helps your understanding.

Most important when learning math: do a lot of exercises. If possible, do a lot of difficult ones. Really, do them, there is no learning if don't get your hands dirty! Don't expect to solve every single difficult problem you try, but seriously, trying them will help you tremendously. As a rule of thumb, I never spend more that half an hour on a problem, unless it's really something very motivating.

And last: Life is sequence of choices, and exploration takes time, a lot of it! Which means: you learn much faster by reading (and paying attention to what you read, they call it focus), but you don't master something unless you get your hands dirty, as I said.

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A more precise formulation is: "what is the role of rote learning in mathematical study?" A purist such as Pólya would say "almost none". You have learned a theorem well when you know it back-to-front, and being able to recite it is extremely shallow in comparison. Actually one outstanding mathematician spoke of the first month of studying a new topic like being rote learning: but that was J. E. Littlewood explaining how he approached getting into a new research area. It of course depends what you're trying to achieve, but I doubt you'd find many good mathematicians really suggesting learning faster than you can expand your understanding. (There may be interesting "cultural differences" in ways of explaining "understanding", though, and these could form the basis of a more reasonable question.)

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