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I am learning a lot of material on my own and I enjoying it but I have a constant problem: I really don't know how much time to spend actually reading theorems, corollaries and stuff and how much time to spend solving the exercises. I like both but at times I feel I spend too much time reading and postponing solving exercises on the excuse that "I must first build foundation". A friend of mine who is working on similar topics only reads definitions and theorems once and jumps into exercises and at times I feel he understands the stuff better than me. Help!

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What I usually do is this: I read theorems and proofs but with a pen and some paper. It is extremely hard to follow a proof (unless it is elementary) without writing something down. I usually try to simplify proofs, definitions, etc. as much as possible because I am dyslexic but apparently it is a good strategy for everyone. Try to focus on the key works. Richard Feynman used almost the same method. This is why almost every professor of physics in the world has a copy of Feynman lectures. I also use a lot of symbols when convenient e.g. $\exists$, $\forall$, $\Rightarrow$, $\Leftarrow$, $\Leftrightarrow$, etc. But doing a few exercises also helps because you understand the concepts better. For example, when you first see the definition of the floor of $x$ i.e. $\left\lfloor x \right\rfloor$, you might think it is obvious but you still need try a few values of $x$ to convince yourself. This way you will actually remember. Your friend will most definitely forget everything by the end of the semester, and you do not need to understand the proof of a theorem to do an exercise that uses that theorem.

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The OP’s friend may in fact be learning very effectively and with excellent retention: people have a very wide range of learning styles. There is even a teaching method, the Moore method, that forces students to work in a fashion very similar to the OP’s friend. And sometimes you very much do need to understand the proof of a theorem in order to do an exercise that uses the theorem, especially in advanced courses. – Brian M. Scott Oct 27 '12 at 23:05
Yes. My discrete math instructor used this method. We submitted summaries for each section online, presented proofs in class, etc. I spoke with a few people who took the class with me and no one even remembers what a truth table is. In general if you are forcing yourself to remember theorems and concepts from a particular area of math, then you are probably not interested in that area. If that happens all the time -- you probably do not like math at all. – glebovg Oct 27 '12 at 23:11
As I said, it varies enormously from person to person. My first encounter with topology, back in the spring of 1966, was largely taught Moore method and I loved it. Essentially writing my own textbook taught me an enormous amount, and that class is very likely one of the reasons that I became a topologist rather than some other kind of mathematician. No, it doesn’t work well for everyone, but neither does any other method, and for some people it works very well. The last two sentences of your comment may well be true, but they don’t seem to me to be relevant to the OP’s question. – Brian M. Scott Oct 28 '12 at 0:41

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