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I am a former computer engineering student that switched to mathematics this semester. The only math I have is basic calculus and linear algebra, both thought in a "engineering way". Now I really want to learn mathematics, and I work with it most of the day. I have thought out two general strategy's for learning mathematics. (my idea about what doing mathematics means)

i) Spend most time on problems in the book. ii) Do the more easy drilling exercises and try to generalize them as much as possible.

I am afraid that I wont get through what the courses expect me to if I only do step i). I also thought about reading very abstract stuff and force my mind to try to understand it, like a kind of puzzle activity. Is this setup good, any other advice?

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Doing drilling problems in the book is definitely a good way to consolidate your knowledge of the material, but you should not make it a top priority. Indeed, mathematics is more than just a collection of techniques that one should master. It is an exploration of unfamiliar ideas, which opens up new vistas of research.

It is perfectly alright to work on a few drilling exercises, but if you feel that you are not being mentally challenged enough, study those problems that force you to think directly about concepts and not just techniques. Abstraction is something that every good mathematician must deal with, so you have to train your ability to generalize ideas. Of course, no one can jump straight into abstraction like the great Alexandre Grothendieck, so start from the specific and work your way to the general. It takes time, but the rewards will be great.

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Thanks, what about proving things and deriving things? If person A is able to derive a result B, then is this process a good indicator that A understands B? –  user29163 Feb 1 '13 at 10:46
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If you can derive results on your own, then it shows that you have acquired a certain level of understanding. However, don’t just stop there. Try several approaches. Ask yourself: Is there another way of proving the result? Can a proof be found that allows me to generalize the result? –  Haskell Curry Feb 1 '13 at 17:49

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