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I'm a first year graduate student of mathematics and I have an important question. I like studying math and when I attend, a course I try to study in the best way possible, with different textbooks and moreover I try to understand the concepts rather than worry about the exams. Despite this, months after such an intense study, I forget inexorably most things that I have learned. For example if I study algebraic geometry, commutative algebra or differential geometry, In my minds remain only the main ideas at the end. Viceversa when I deal with arguments such as linear algebra, real analysis, abstract algebra or topology, so more simple subjects that I studied at first or at the second year I'm comfortable. So my question is: what should remain in the mind of a student after a one semester course? What is to learn and understand many demostrations if then one forgets them all?

I'm sorry for my poor english.

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    That's quite normal. The most important thing is that you acquire the capacity to find what you need when you need it. Therefore it is perfectly OK if in the long haul only the general ideas survive in your mind. The rest you can always look up or study again if you so need.2012-05-14
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    I regularly forget the details of things I've learned if I don't use them very often. If you at least remember the general ideas then you'll recognize when you need it again, and the advantage of having taken a course is that you'll have a set of notes you can relearn it from, and (at least I find) it will be *much* easier to learn for the second time.2012-05-14
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    Do lots of problems too! It's much easier to recall things you have actively used. Another technique which can be helpful is to try to prove statements independently of the book. If you succeed, then you understand it a lot better, and if you don't, once you read the "official" proof, you will appreciate it more.2012-05-14
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    Education is what remains after one has forgotten what one has learned in school. Albert Einstein2012-06-15
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    If there is a song on the piano that I haven't heard or played in a while. I find that, even if I can't play it immediately, I have not actually forgotten it. If I just play up to a moment I don't remember, and guess at what comes next over and over, it comes back. Memory is fickle, but it does its job well.2013-06-19

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