TITLE: How Do You Actually Do Your Mathematics?
QUESTION [111 upvotes]: Better yet, what I'm asking is how do you actually write your mathematics?  
I think I need to give brief background: Through most of my childhood, I'd considered myself pretty good at math, up through the high school level. I easily followed mathematical concepts introduced in my classes and even did a few competitions; I definitely wouldn't say I was a star of the caliber one meets when one ventures out into the bigger ponds, but I thought I was decent and convinced myself I would major in math when I entered college.
That changed after a couple of years when I hit my first Real Analysis class that used Rudin's book; that was the first class, I think, I took that really required more than "expand-a-definition" type proofs and my struggle to find intuition and understanding there impacted my mathematical self-confidence.   I eventually switched majors, with a bit of regret.
One thing that got me, I think, was the veritable explosion of superscripts and subscripts that one encounters for the first time in Real Analysis.  I'd often find myself struggling to set up the machinery of what I was trying to prove, lost in the notation.  How do good mathematicians format their work on paper so as not to get lost in the $i$s, $j$s, and $k$s and keep track of what they're investigating?  I remember dealing with subsequences of sequences to show that limits did or did not exist got especially hairy in this way...writing things like $s_{n_{k_{\epsilon}}}$ and remembering what my goal at each "level" was difficult.  I'd be interested in knowing if aspiring mathematicians and/or professional mathematicians scribble marginalia or have a system to overcome such problems.
Another thing that got me were what I personally called "consider..." statements.  Many times, on this site, the most talented commenters will say "Consider $f(n)$" or "Consider transformation $T:$ $U \rightarrow V$" that in the first case gives a summation that wonderfully telescopes/has an obvious bound, or in the second case transforms the problem into a trivial application of the rank-nullity theorem, or something like that.  Mathematics is a subject replete with geniuses , I understand that, but how do mere mortals investigate such functions and "massage" them into doing what they want?  When good mathematicians get intuitionistic ideas, what (explicit) steps do they take to formalize them, especially when it is likely that first idea is murky or wrong? (Aside: I've been given "use numerical examples" as advice before, but sometimes I think to myself, "I've been dealing with $\mathbb{Z}$ since I was 6 years old, and not so much with Dedekind's definition of the real numbers...") 
There's lots more I could ask, but I want to keep this question tractable, so I guess I might summarize by asking: How do you [professional and aspiring mathematicians] organize your math "notebook", and what perhaps idiosyncratic methods do you employ to be original and clever within it? I know there will be no strict formulas anyone can give; mathematicians are scientists of the abstract; I understand that the subject is acclaimed partly because it's so intellectually and individually demanding.   But I think even acclaimed scientists draw on Springer's Protocols and Nature Methods...There seems to me a bit of a jump between the dryly algorithmic way one is taught to do math in high school and the more abstruse methods at the undergraduate level.  I'd be interested if anyone here could help me bridge that gap, if only for my personal fulfillment.
(Apologies in advance if the question is ill-posed or too subjective in its current form to meet the requirements of the FAQ; I'd certainly appreciate any suggestions for its modification if need be.)

REPLY [21 votes]: How do good mathematicians format their work on paper so as not to get lost in the is, js, and ks and keep track of what they're investigating?

To be honest, part of this is just getting used to juggling several things in your head at once. Fortunately, as with many other skills, this is trainable: try, for example, doing Sudoku puzzles with a pen.
There are, of course, other ways to keep your work organized. The first thing to do when solving a problem is to write down all of the data you're given and write down the goal of the problem. The second thing is to unravel all of the definitions, working through all the quantifiers. This will get you surprisingly far, at least in real analysis. And then the third thing is to actually think about it. If you can't hold all the quantifiers in your head at once, practice reading the statements aloud slowly until you can (and see the first paragraph).

When good mathematicians get intuitionistic ideas, what (explicit) steps do they take to formalize them, especially when it is likely that first idea is murky or wrong?

Write down an argument that follows the lines of the intuition, at least for a simpler case or version of the problem. If it doesn't work on the simple version, it probably won't work on the hard version. If it does, you can try to figure out whether it extends to the original question and, if not, what the barrier is. 
This is easier said than done. The bottom line is you need to practice the skill of turning your intuitions into proofs. This comes in a few steps: first you need to learn how to prove things, then you need to learn how to train your intuition to help you prove things more easily. Like anything else, this takes hard work and practice and there isn't a magical shortcut. 
Terence Tao has written some very clear stuff on this and related subjects.

How do you [professional and aspiring mathematicians] organize your math "notebook", and what perhaps idiosyncratic methods do you employ to be original and clever within it?

I'm not completely sure what this means.

There seems to me a bit of a jump between the dryly algorithmic way one is taught to do math in high school and the more abstruse methods at the undergraduate level. I'd be interested if anyone here could help me bridge that gap, if only for my personal fulfillment.

Many people I know bridged the gap through competitions such as the AMC. Training for competitions is not everybody's style, but it is a chance to be exposed to interesting topics not covered in the high school curriculum and also a chance to hone problem-solving and proof-writing skills (at the Olympiad level). A book I benefited from enormously while doing this is Engel's Problem-Solving Strategies, which is geared fairly specifically to Olympiad preparation but is a great source of problems and elementary techniques for solving them. For more general advice, Polya's How to Solve It comes highly recommended (although I have not read it myself).