After reading Daniel Apon's question, I started thinking that it might be useful (especially to junior researchers and graduate students like me) to ask a broader and more general question so we can learn from the experience of more senior researchers.
So here is the question:
What practices have you found most useful in your research?
I don't want to restrict it to any particular type of advice, so any advice on research practice is welcome.
One thing I found useful is to allocate time and designate a space for doing specific research activities.
When I was at Princeton U, I loved sitting at the Engineering library that is well lit, bright and spacious, to read and to think of new ideas. When I verified my 139 pages paper, I used to do it in a room in the biology library at Weizmann that had no computers and no other people, only a desk, chairs and a window to an inner garden. When I go over introductions or notes, I like doing it in coffee shops.
There are several reasons why I found this to be a good practice for me:
(1) Just pondering about a good environment for me for an activity fills me with anticipation for this activity, or at least somewhat prepares me for it.
(2) The fact that I decide to do something specific at this time, and I have the space I need for doing that, induces simplicity, clarity and good order.
(3) Knowing what I like, what I care about, and also what distracts me and what is not good for me, I create environments that make it is easier for me to do what I need to do.
For every question that you can't solve there's an easier variant that you can solve; for every question that you've just solved, there's a harder variant that you still can't solve. Going back and forth across the "boundary of solvability" is extremely useful as it (1) allows you to progress in baby steps (2) gives you a clearer picture of the landscape.
Manuel Blum has this extraordinary page on advice to a beginning Ph.D student. Read it slowly though, for there is much to absorb.
Update: Let me add this piece of advice by Dijkstra, his Third Golden Rule for successful scientific research:
"Never tackle a problem of which you can be pretty sure that (now or in the near future) it will be tackled by others who are, in relation to that problem, at least as competent and well-equipped as you."
He presents this interesting zen-ish inference: A corollary of the third rule is that one should never compete with one's colleagues.
This inference had an huge influence on me; but it took me some time to dig out this reference.
There is plenty of things that can be given as advice, but I tell 3 things to all my students (in no specific order):
I'm sure many have already seen these, but so they are collected in one place:
Nielsen's Principles of Effective Research
Richard Hamming: You and Your Research
Matthew Might: The illustrated guide to a Pd.D
Tao: Ask Yourself Dumb Questions
The last one has multiple links that click through to other blog entries about research advice.
One thing I think I can offer, from myself, is this: Don't take rejections, or acceptances, personally. A sense of self-worth should come from far more important things than the temporary disposition of a mathematical result.
Write everything, all the time. In TeX, preferably. Whether you are considering a question or proving a lemma, put it in a digital format as soon as possible. Write the necessary background. Try to keep the thoughts organized in a narrative.
Having all of these things in a digital form makes paper-writing much easier, but still a lot of work. In my experience, it helps a lot to start again from scratch. This allows for a clean start to find an improved organization. Also, proofs are always easier the second or third time around.
This may sound a very trivial thing, but I would recommend trying to take a break before you go to bed. Try to take a break from research and clear your head. It's very tempting to decide to take a few minutes to work on some pet problem in the comfort of your bed before you go to sleep, but in my experience it absolutely guarantees insomnia and leaves you totally unfit for proper thinking the next day.
This is more of a do as I say not do as I do thing, because I find myself drawn to working late at night far too often, and I can say from experience that it is a practice best avoided.
I'll start with some advice my advisor Amit Chakrabarti gave that I found quite useful:
(1) Keep a research journal and write in it every day.
Its easy to forget about the details of what I was thinking about 2 weeks ago. Revisiting previous thoughts/problems/miniresults enables me to avoid reinventing the wheel. Also, I found it especially useful to write down places where I was getting stuck and not making progress. Revisiting later often let me break through whatever I was trying to solve. In a different way, writing in a journal forced me to spend those extra ten minutes at the end of the day to follow through and finish up whatever task I had left that day. Otherwise, I probably would've just put it off until the next day.
(2) Set aside time to read research papers, even if they're not directly related to your current research.
I get tunnel-visioned, so if I don't do this I tend not to keep up with current research.
Finally, it's been very useful to me to think about my research right before going to sleep/in the shower/while exercising/on the subway/etc. Some of the other commenters seem to disagree here, especially with the right-before-bed part, but I've found it to be very useful. I tend to look at the problem from different directions in these kinds of situations, and I might spot some insight I wouldn't have at work.
Sylvain's already mentioned as much, but I think it's worth emphasizing how important hard work and persistence are. Also, I can add:
A mathematician is a machine for turning coffee into theorems.Addendum: Since considering the comments of others about coffee, I have discovered the wonders of a fresh cup of hot tea. :) Allow me amend this comment by saying:
Get up and talk to colleagues and fellow grad students, take breaks and lunches together and talk about that interesting problem you're exploring. Most research benefits from multiple points of view, and most researchers are happy to discuss intriguing and knotty thought experiments.
Some people dislike (or fear) to talk about their projects, perhaps out of concerns for poaching, but most people when I was in grad school were pretty open. Unless the door was closed, in which case either I or they were hard at work without wanting interruptions.
There is envisioning: describing to yourself in a detailed way what you are going to do, before you start doing it. It works extremely well when you have a complicated task ahead of you, like writing a paper, preparing a presentation, planning a class, etc.
I found that when I have a very clear vision in my head of what I'm going to do, actually doing it becomes easier and faster and the outcome is better.
It decreases dramatically the chances to miss something.
It makes you think about the right order to do things.
It helps simplifying: we can't hold in our heads too many things at once, so it makes you concentrate on what's important, and modularize.
So things just go more smoothly when you plan.
I would like to second the advice given by Justin. There are a lot of great suggestions in the replies, but many of them are just about effective work habits and don't specifically have anything to do with research. I was thinking about this recently after viewing a talk by Steven Johnson on the subject of his latest book, "Where Good Ideas Come From". (One version of the talk can be found at TED.) His basic premise is that there are certain environments which act as natural incubators for new ideas. He gives the example of the coffee house in Renaissance England, which aside from the effects of the coffee itself, brought people of varied backgrounds together and encouraged the exchange of ideas. It is this cross-pollination of ideas which gives rise to new, novel ideas.
Now quite possibly this hasn't been mentioned in most of the other responses because it is just assumed that everyone is already going to be doing this, but I thought it would be worth emphasizing the point.
Richard Feynman's must-read book "Surely You're Joking, Mr. Feynman!" is a mine of good research practices. My favorite Feynman's advice is on how teaching helps to do good research, see these excerpts. If nothing more, this is at least very motivating when you have TA/teaching duties that seem to "obstacle" your research.
