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I am an undergraduate CS major with strong interests in applied math and theoretical computer science. In the past, I've done reasonably well grade-wise in all math-related (that is, pure math, applied or theoretical CS) classes, but I feel that I still haven't taken away as much as i could have from most.

As people who have often taught math courses and had to deal with the inevitable fact that no lecture will be universally effective, what are your suggestions for how I (as a student) can best learn in these classes.

A few problems I've experienced regularly:

When professors try to present long and difficult proofs on the blackboard. I always find it ridiculously hard to understand proofs in real time or to understand verbal and visual explication of the proof simultaneously. I have to look the proof up in a textbook, and the comprehensibility of textbook proofs varies widely.

More generally, accessing the "kernel" of the proof that really makes it comprehensible is sometimes difficult, especially when it's presented more formally. I tend to think of proofs in terms of algorithms, and proofs that don't fit this well tend often evade me.

Definitions, even, (especially in pure math) tend to blend together and become obscure. I've re-learned the basic definitions of probability waaaay too many times.

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closed as off topic by Mark Sapir, Andrés Caicedo, coudy, Andy Putman, Harry Gindi Jan 21 '11 at 22:00

Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here.If this question can be reworded to fit the rules in the help center, please edit the question.

Why would you expect that "long and difficult proofs" be easy to understand in real time, and why do you think you should not have to look the proofs up in textbooks? In any case, I am not quite sure your question is a good match for this site. – Mariano Suárez-Alvarez Jan 21 '11 at 19:36
You might consider asking this question on (although a version of it may have already been asked there). – Qiaochu Yuan Jan 21 '11 at 19:41
I would disagree that it's off-topic. Many topics here are professors asking how they should teach; why not a student asking how he should learn? – DoubleJay Jan 21 '11 at 20:53
@DoubleJay: this site is geared towards research mathematicians and graduate students and their interests. Research mathematicians have to teach, but they do not have to learn (in classrooms, anyway). So I can understand the argument that this is off-topic. – Qiaochu Yuan Jan 21 '11 at 21:26
I agree with Qioachu, but +1 to DoubleJay for making a symmetry argument. This tends to work out better in mathematics than in rhetoric. :) – Pete L. Clark Jan 22 '11 at 10:31
up vote 5 down vote accepted

This is not a universal recipee for anything, rather a few random points.

1) Make sure your background matches the course expectations. If not, work on it before even thinking of taking an advanced class.

2) Read ahead, not behind. Most teachers will tell you what's coming next and if you come to the class knowing half the story already, you can concentrate on the other half and gain double time for absorbing it.

3) Ask questions, ask questions, and ask questions. Don't sit and try to digest everything on your own.

4) Learn each proof to the level that your professor can wake you up at midnight and you'd be ready to present it right away. Keep in mind that there are millions of theorems but only thousands of proofs, hundreds of proof blocks, and dozens of ideas. Unfortunately, no one has figured out how to transfer the ideas directly yet, so you have to extract them from complicated arguments by yourself.

5) Solve problems, solve problems, and solve problems (not the ones that ask you to do something according to the ready scheme, of course, but the ones that ask you to prove something that is not clear from the beginning). You need to learn how to create simple proofs before you can understand the complex ones.

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Regarding 2), 3) and 4), and some statements by the OP: I was unsatisfied with a certain proof given in a logic course I was taken. It's not exactly that I didn't understand it, but that it looked an awful lot like the proof that every ideal is contained in a maximal ideal, and it bugged me that I couldn't find a more direct connection than the use of Zorn's lemma. I did some reading and realized that it not only resembled that proof, it was a special case: all the professor was doing was proving that every ideal in a Boolean ring was contained in a maximal ideal, and I worked out the... – Qiaochu Yuan Jan 21 '11 at 22:02
...precise connection at… . At this point I could indeed give the proof to a professor at midnight because the idea is the same as one I had already seen. ("Every ideal" above should be "every ideal of a commutative ring.") – Qiaochu Yuan Jan 21 '11 at 22:05
You don't need commutativity ;) – darij grinberg Jan 21 '11 at 22:17
Even though this topic was closed, this answer is exactly what I was looking for. Thanks! – DoubleJay Jan 22 '11 at 2:11

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