Up-to-date advice on the best way to take notes (maths) I have read some old discussions about this topic and would like to get some up-to-date advice if possible. I'm going to start university next year (maths), and I know how important is to have a set of well-organized, well-taken notes, since you have an extremely larger bounch of material to study (compared to high school). Then my questions are:
--what are the best techniques to take good notes "in real-time" (if you know what I mean) and without getting distracted (I mean, without getting everything stright from the blackboard to my notebook without passing through my head)?
--should I use a tablet/computer to take notes? If so, which app/program do you suggest? Is it advisable to write mathematics with a stylus on a tablet, or it is better to use something like mathamatica? Is it possible to use such devices in real-time?
Any piece of advice on this topic (from professors or students) is really welcome. Thank you in advance!
 A: I don't think there can be any answer to that that's independent on the kind of lecturing you're going to encounter.
If you're in a good course where all the technical facts you need to learn are already in the textbook (or in handouts), and the point of the lectures is to provide perspective and intuition about the material in the textbook, then the best way to take notes is not to take any! Dedicate your entire mental capacity to following the presentation and getting an internal idea of what is happening. Then after the lecture, read the corresponding sections in the textbook critically to determine whether the lecture gave you any insights that are not already there.
On the other hand, if you're in a course where you need to learn things that was only ever said during lectures (or written on the blackboard), then you need to take notes.
But there's no one-size-fits-all.
A: This is highly tangential, but hopefully still useful.
If you want to do well in university mathematics, you'll need to keep pencil-and-paper with you at all times; both to do homework assignments in your spare time, and to scribble your own stuff and pursue the extra-curricular questions that most interest you.
To this end, I recommend the following.


*

*A workbook filled with lined paper, such that the pages are easy to tear out. I'll explain why in a moment. 128 pages is a good size; in which case, you'll probably need a new workbook every couple of weeks.

*A mechanical pencil (you know, the one's that don't require sharpening).

*An eraser. Smaller erasers are best because they flex less, so you can erase faster.

*A small stapler.

*A 30cm ruler.

*An A4 display book (you know, the one's full of plastic slips).
How it all works together. Suppose you need to prove a theorem $\varphi$. You state the theorem $\varphi$ at the top of the page, and start proving it. If you make a small mistake, use the eraser. If you make a big mistake, tear it out and start again. Every time you complete a page, you tear out and put it on the left of the page you're about to start writing on. That way, you can see what you've written so far. When you've finally completed the proof, staple all the pages of the proof together; this stapled collecion constitutes the complete proof of $\varphi$. Then slip the stapled pages into your A4 display folder. 
Also, if you need to open a subproof, use the ruler to form an "indent," basically just a vertical line spanning the length of the subproof. You can also form "sub-subproofs" etc.
Like I said, this is tangential to the question you asked, but hopefully still useful.
A: It's really an art, not a science. I can tell you what has worked for me:


*

*Use a legal pad with three-ring punches where the sheets rip off clean at the top. I prefer a white one. After class, tear all your notes out from that day and put a paper clip on them. Store them in a manila folder dedicated to that class at your house. Also in this folder should be graded homeworks and other class materials.

*When you write notes, put the date of the class at the top of the
first page of notes, so later when you've got a bunch of various days of notes, you can tell at a glance the chronology. 

*Number each page of notes from that day in the
    lower left-hand corner (so if they get mixed up, you can quickly get
    them back in the right order). Start back over at page 1 with each new class. If you end up having to insert a page in the middle,
call that page "6b" or something, and change the previous one to
"6a", so you can tell there should be a page between 6a and 7.

*If you're on page 3, say, and you want to refer to something on page
1, mark it with an equation number or a star or something and save
yourself time by abbreviating it: for instance, writing "then using
(1.5), we get that (1.1) becomes (1.3)" is a lot quicker than
writing out all those equations again.

*Directly after each class, do the following:
--Go over the notes quickly and write on the top of the front page, by
the date, keywords representing the topics covered that day. For
instance, "Poisson's formula", or "Proof that $e^{i\pi}=-1$". This
way you'll be able to tell in which set of notes a topic is covered
when you're looking for it later.
-- Go over the notes and isolate the things that require follow-up work
for you, and put those in your to-do list (you should have one!) For
instance, "Understand second fundamental form". Then later, use your
resources to take care of these. Do not just stow the notes away
and promise yourself that you'll "go over them" later. Unless you
isolate specific things that you need to do, they will just pile up
and turn into lumps of stuff you haven't taken care of.
Remember that the art of organization is the art of being honest with yourself: what are you really going to go back and do? What parts of the notes are you really going to look over and use later? Are you writing notes with the goal of advancing your understanding, or just because you want to feel like you're doing something?
For general organization tips (and getting the most out of notes is largely about organization), I recommend reading Getting Things Done by Allen.
EDIT: After a number of years, I would now recommend something like Notability for capture, and Anki for retention. Above all, don't rely on your intuition for when good learning is happening. Read the evidence (e.g. the book "Make it Stick" or www.learningscientists.org). Notes are generally low-utility. (Though of course you do need notes to capture information.)
A: I am one of those people who just transcribes the professor's board using a standard black pen, white-out, and fast writing. I do happen to be blessed with fairly good handwriting, and tremendously blessed to have good memory. I have given my notes to friends, who sometimes felt awkward about taking them, until they learned that I never look at my notes again (unless I need to figure out what the homework assignment was, and that's really rare), so giving them away isn't a problem. By the time I finish writing the notes down, I have them pretty well memorized, so I don't bother to organize them.
It drove my Mechanics I professor (confessions time, I'm an engineering student) nuts that I would take notes for 5 classes (including Multivariable Calculus and Linear Algebra) in one single-subject college-ruled notebook. It works for me, though. However, I don't recommend the practice to anyone who doesn't basically memorize their notes. In fact, the reason I do the single-notebook rule, is because I don't have space in my backpack for five notebooks.
A: If the teacher is thorough on the board, you can take pictures using your smartphone.  There are apps that will then automatically transfer these to your computer in real time.  That way you can pay more attention to the material and less to transcribing it.
If you have your computer with you, you can then use it to review notes the teacher has since erased in case you need to in order understand something later in the lecture.  Also, if the teacher says anything crucial without writing it on the board, then you can type that up and use it to annotate your pictures.  
Make sure whatever app you use to take pictures is noiseless, as you are likely to be prohibited from taking pictures if doing so proves distracting to the rest of the class.  In my experience, most teachers don't complain about noiseless picture taking.  
Note of caution:  Use this method as a means to pay better attention to the material in class; not as an excuse/means to procrastinate.  
I realize some people learn as much or more by taking notes than they do from reading them.  In my view, since we can think and read more quickly than we can write, that method is inefficient - but if that learning style suits you then my method may not be for you.  
A: Apart from taking notes with pen and paper there is of course the possibility to use LaTeX. For me that was a great way to learn LaTeX since sooner or later I had to do it anyway. This method of taking notes also has the advantage of being very readable and readily distributable to your fellow students.
On the other hand I also have to give a warning. In the beginning this will use a lot of your attention and it will be hard to keep up with the lecturer. Only after some training of your muscle memory will this become easier. Also if there are charts and diagrams you still need to have a piece of paper to draw them and insert them later.
To succeed in live transcription with LaTeX there are some tools that where really helpful for me:


*

*an intelligent keyboard layout, that allows me to directly input unicode characters (e.g. NEO)

*a suitable editor, which provides handy short-cuts for tex constructs (e.g. vim with the latex-suite plugin)


These help you to handle the overhead of tex-commands.
