How do academics manage to do research when they are lecturing and running courses, helping students, writing up assignments and tests, marking assignments and tests, supervising graduate students, going to meetings, doing admin, having personal lives, etc. ?
How does anyone "find the time" to do anything? The answer is that mathematicians don't "find the time", the time is already there and they make use of it. Your premise that those other activities you mentioned already fill up 100% of the time is simply incorrect. In particular, summers, breaks and other periods with no teaching offer a convenient and relatively distraction-free environment for doing research. But even teaching is not mutually exclusive with doing some research.
I am sure you, me and many others are doing research when they eat,before they sleep,when they are watching a football game.
I think you do not "find time". You use the time you have.
The important thing is that you do not have to do research for three hours a day.
I usually say to myself "Ok today you have time to think about problem X for 30 minutes. Use 20 minutes of them.Others will not". And indeed,sometimes in these 20 minutes an idea may come.
(P.S.)When you teach you become better at mathematics.
(P.S. 2) I miss those times when I was undergraduate and I could spent 5 hours for research every day.
Really it depends on how you put your priorities. The list you have made is a perfect summary of legitimate excuses one can make to abstain from thinking of mathematical problems when you are in no mood for them. In reality, the activities you listed do not need to take too much time unless you schedule them all over the week and all over the day. Teaching usually takes 3 days a week (2 if you are lucky), leaving you 4 other days. Within those 3 days, you normally teach about 3 hours per day, leaving you another 13 (assuming you are awake 16 hours per day). To make a decent assignment or test takes about half an hour if you have a clear idea what skills you want to check. Grading is a nightmare if you give 20 problems on each test and try to distinguish between 3 and 4 on a 10 point scale but it gets much easier and faster if you give 6 problems, use a 4 point scale, and have faith in the law of large numbers that will level out minor grading deviations over the semester.
This leaves just students, committees, and personal life. In my experience, those activities take exactly as much time as you are willing to spend on them. You have all options between opening your door at the beginning of your office hours and locking it at the end and running after each individual student in each of your classes urging him to come to your office to discuss his performance, etc. As to the committees, normally you are not obliged to be on more than two and those usually meet for at most 3 hours (very few people can endure longer meetings) once every two weeks or so. As to the personal life, you are the only one to choose its style and intensity though some choices have long range implications.
In other words, you can control your time to a great extent, so, when you are really into something and are making progress on a problem, you can easily downsize everything else to the bare minimum and free up a lot of time. The problem is that such moments are extremely rare and most of the time you feel like a helpless idiot who cannot figure out the simplest thing and should be kicked out for being a useless wreck. Then, to justify your existence, you start stretching other activities and can easily fill all available time with them, creating an illusion of a busy and productive day and thus keeping your self-esteem afloat.
You need to make a schedule that takes into account your needs to be able to the research work you're supposed to do. You can then minimize the time spent on activities that you don't find all that useful (like grading homework, making homework assignments, etc.), while making sure you get enough quality time for research. What you want to avoid is having your research time broken up into small pieces during which you can get hardly anything useful done.
Many people would ideally want to have large blocks of several hours of uninterrupted study time. E.g. you can put a 3 hour block from 9 am till 12 pm, and another one from 2 pm till 5 pm. Then all your other work that you don't value as much as research work can be crammed into the two hours from 12 pm and 2 pm. This means that you're assigning less quality time for things such as grading homework, e.g. you can do that while eating lunch. On some days you may have to spend a few hours doing other things like teaching, on those days you can perhaps only fit one 3 hour research block into your schedule.
Assigning quality time for research is very important. If you try to get to a more efficient schedule where you work in smaller blocks, you may benefit on the short run, but you'll tend to not go the extra mile with your research work, which will then suffer on the long run. Suppose e.g. that you have just one hour to work on a problem, then you'll tend to not start doing something that would be half finished when investigate a marginal issue. If an approach to attack a problem is not working well and to get to the bottom of that you could write a program to do some simulations, then just the idea of having a half finished program will put you off from even embarking on that, and because it's just to investigate why the method isn't working, you'll tend to just ditch that approach and move on. However, by systematically taking such decisions, you may miss valuable insights on the long run.
You can't predict when you would gain a valuable insight. You won't notice what you have missed, but after several years you may notice that your colleagues have had more success with their research work than you.
The other answers are spot on - you do not need to find time, but use it.
On the other hand, there are very busy times, e.g. terms where I teach too many courses or courses that take of lot of time to prepare. In these times I usually try to reserve at least half a day per week where I do not put any appointments and do research. I make very few exceptions and most of the time these slots are quite productive.
Other simple tricks to work quietly is to simply close the office door for your research periods or work at other places…
Great answers up here saying to use the time you have. But what if there is really no time? You have to sacrifice something to work on your research project. In my experience I see people in academia sacrificing the following, from the most frequent to least:
- Sleep time
- Time spent with friends
- Time spent socializing
- Leisure (there is usually not much free time left)
- Nothing. They just hack up a few low quality publications for check marks.
- Work time
My favorite is actually #7, but I'm surprised how few people sacrifice other work for more productive research work. Drop a class or few, re-negotiate work load, talk your way out of bureaucracy, paperwork, admin meetings, useless conferences, etc. Do research during office hours. Let students do simpler parts of your research work. Finally, change a university to one that values research more and willing to pay you for it.