I have a question about how to take notes. Should I copy down every theorem, or can I only copy down important ones and refer to the book for the rest?
Same goes for proofs. Should I try to memorize as many as many theorems and proofs as possible, should copy them down, or what?
On
Math is a delightfully introspective subject. I know more about my innermost thoughts and the deeper workings of my mind through hours staring at a whiteboard than a psychologist could gather in a hundred years.
From this, I know that I cannot take notes. Ever. If I have any form of paper in front of me during a lecture, then I have a blank slate for my thoughts and math, usually unrelated to whatever topic I'm trying to learn, will soon have pervaded the paper. Instead, I have to prepare extensively before class and simply listen to lectures I attend.
I include this as a helpful guide toward developing your study/lecture/learning habits. You, and only you, can figure out what works best. My suggestion is to experiment. Once you find a method that works well for you, stick with it.
There are, however, five rules that should point you in the right direction:
Probably contrary to most of your academic experiences thus far, you cannot rote memorize math past this point and expect to do well. We cannot "just know things." For a related, anecdotal reference on the habit of just learning machinery, see this post.
As far as I know, there are only two ways to be successful in math:
You will only understand the theorems if you use them enough to know them by heart. Play with them! If you intend to become a mathematician, these will be your toys for the rest of your life. If you do not wholeheartedly want to work/play with these for the rest of your life, then you might want to consider changing your major.
Of course, if you don't want to go into mathematics, none of this applies and you should, by all means, rote-memorize everything.
My opinion on notes has steadily changed over time. Here is what I think now.
Warning up front: if you go into class tired or distracted, my suggestion is to write everything down that is written on the board, or even more if your presenter has a more oral style. The reason is that you (a) probably aren't very good at discerning what is important in your state, and (b) are very likely to fall asleep if you do nothing but listen.
Assuming that you are focused, the single most important thing happening in a math class is your inner monologue. You can't write it all down, of course; it goes too fast. But whenever I find myself saying something different than the presenter or asking a question, I write it immediately; even if I am already in the middle of writing something else.
In theory, I think that this is all one ever really needs for notes. In practice, some other things are useful. The gist of what follows is: your notes should contain approximately the difference between the entire class and what is in the book.
Examples: Many. Some will be standard enough that they are in the book, but oftentimes the examples are chosen specifically because they are really insightful, even if they don't appear so on the surface. I don't want to say "all" because sometimes a presenter will go overboard, but you want lots of them.
Pictures: All. Again, your book is less likely to have them, and they are often very useful.
Definitions: Only nonobvious ones. There are lots of technical, obvious definitions. Don't bother with these. Also don't bother writing "for all natural numbers $n$", or somesuch, unless it is radically important that it is not, say, an integer, but this fact isn't obvious from the context.
Theorems: Really important ones, and anything starting with the phrase "The Following Are Equivalent". Usually a presenter will make a special effort to distinguish the groundbreaking proofs; I mean on the scale of Mean Value in analysis, Rank-Nullity in linear, or Lagrange's Theorem in abstract. You don't technically need these because they're in the book, but since these are the ones you should remember long after the class ends, some extra effort would be good.
Proofs: I take a pretty radical stand here. If your textbook is not Rudin-scale terse, I don't think it is necessary to write any proof given in a lecture*. You will be writing proofs in homework, so it's not about practice. Any proof given in class is either simple enough that you can do it yourself or significant enough that it will be given by the textbook; and even if not, you can find it online. Your attention is much better spent by concentrating very hard during the lecture and gaining an intuition about the argument. Try to keep an example running as you go through it**. Your goal is to get a very big picture of the proof.
Do not memorize proofs unless you have a very specific reason to do so (i.e. you will be asked to quote the proof of the first Sylow theorem on the midterm). Copying down proofs from a book may have some benefit for you, it doesn't for me. Be rigorous on the homework, and if your professor does not assign all the problems in the section, think about how you might solve the others.
[* One notable exception: a proof that takes an entire day or multiple days is worth at least sketching. But I still would advise against being too detailed here.]
[** actually, this is something I am experimenting with: I'm auditing an Algebraic Geometry course and I'm taking notes by phrasing all of my theorems as examples (e.g. instead of "A ring without zero divisors has the cancellation condition" I would write "$\Bbb Z[i]$ has the cancellation condition" and phrase the proof through that lens). I'm not sure if it is actually useful yet.]