For years I have been mentoring undergraduate students (and others) in math research projects. For example, see the new REU program I recently posted about. I therefore spend time thinking about the issues that beginning researchers have to overcome. One issue stands out as the most common and most problematic problem that beginning researchers need to overcome: learning to think hard about the research for long stretches of time without being distracted. For lack of a better name, I will refer to this as the problem of willpower.
In this post I will write some of my own thoughts about the willpower problem. I am still struggling with it myself (who doesn’t?) and will be happy to hear your own opinion about handling it.
Even without getting into academic research, everyone is familiar with the issue of procrastination from school and work. While studying for an exam, the idea of peeling 200 grapes might become unusually tempting. One suddenly has to check online what happened with that friend of a friend they briefly met ten years ago (or is that just me?). Or perhaps there is a blog post about procrastination that must be written right now…
When one gets to working on academic research, they most likely already figured out how to overcome the above issues when doing homework or studying for an exam. But then they find out that the same problem pops up several orders of magnitude larger. There are several obvious reasons for that:
- Unlike learning material or doing an assignment, it is not clear whether what you are trying to do is possible. It might be that the math problem you are trying to solve is unsolvable. Or perhaps the problem is solvable but the tools for handling it would only be discovered in 200 years. Or perhaps it is solvable now but after several months of making slow progress, some renowned mathematician will publish a stronger result that makes your work obsolete. These scenarios are not that rare in mathematics and related theoretical fields. Are you still going to spend months of hard work on a problem with these possibilities in mind?
- Unlike exams and most jobs, there are no clear deadlines. It is likely that nothing horrible will happen if you will not work on research today, or this week, or this month. There might not be any short term consequences when spending a whole month watching the 769 episodes of “Antique Roadshow”.
- It’s hard! Working on an unsolved problem tends to require more focus and deeper thinking than learning a new topic. Also, part of the work involves trying to prove some claim for weeks/months/years and not giving up. It is surprising to discover that reading a textbook or doing homework becomes a way of procrastinating – it is easier than thinking hard on your research.
(This might give the impression that theoretical research is a horrible career choice. It is much more stressful than one might expect, and requires a lot of mental energy. However, most people who have been doing this for a while seem to agree that it is one of the more satisfying, challenging, and fulfilling jobs that they can think of. I think that I am more excited about and happy with my job than most of my non-academic friends. But I digress…)
So how can one overcome the issue of willpower? While there are many good resources for similar academic issues (writing guides, career advice, etc.), I am not familiar with any good sources on this topic. I am also not an expert on this issue. All I will do here is write some of my current observations and personal opinions. I assume that some of these are naïve and will change over the years.
- Brainstorming is not a solution. For most people it is much easier to discuss a problem with others than to focus on it on their own. Sessions of working with someone obviously have many advantages, but they are not a solution for the willpower problem. One needs to spend time and frustration thinking hard on the problem on their own. Otherwise, they are unlikely to get a good understanding of the topic and get to the deeper issues. Brainstorming sessions become much more effective after first spending time alone and obtaining some deeper understanding and intuition.
- Collaborations do help. Unlike a brainstorming session, long-term collaborations do seem to help with the willpower problem. Not wanting to disappoint a collaborator that I respect, I will have extra motivation to work hard. Having someone else that is interested in the problems also helps keep the motivation high.
- Reserve long stretches of time for research work. Like most people, I constantly have a large amount of non-research tasks, from preparing lectures to babyproofing the house. It is tempting to focus on the non-research tasks since these require less focus and are easier to scratch of the to-do list. When this happens I try to place in my schedule long stretches of time dedicated to research. I try to find times when I am unlikely to be tired or distracted. Sometimes I turn off the wireless and phone during these times. To quote Terence Tao:
“Working with high-intensity requires a rather different “mode” of thought than with low-intensity tasks. (For instance, I find it can take a good half-hour or so of uninterrupted thinking before I am fully focused on a maths problem, with all the relevant background at my fingertips.) To reduce the mental fatigue of transitioning from one “mode” to another, I find it useful to batch similar low-intensity tasks together, and to separate them in time (or space) from the high-intensity ones.”
- Procrastination with writing tasks is a separate issue. While beginners often have a hard time sitting to write and revise their work, this seems to be a simpler problem. The magic solution seems to be writing a lot (not necessarily research work). After a lot of practice, writing becomes a task that does not require a lot of mental energy or deep concentration, is easy to do, and is mostly fun.
- Find the research environment that works best for you. This is an obvious observation, but I would still like to state it. Different people have different environments that work better for them: Some need a quiet environment while others focus better in a crowded coffee shop, some focus better in the morning while others prefer the middle of the night, and so on.
- Find ways to keep yourself highly motivated. Everyone seems to be at least somewhat motivated by being successful and by their ego. Everyone seem to be at least somewhat motivated by an urge to discover the mathematical truth. However, most people seem to need additional motivation when things are not going well. Some people get extra motivation by being surrounded with hard working people. Others become more motivated by reading biographies of successful mathematician and scientists. And so on.
So what are your thoughts? Do you have any tips? Any sources worth reading?
books and 
subscribers. Each subscriber read at least one book from the library. Prove that there must exist two disjoint sets of subscribers who read exactly the same books (that is, the union of the books read by the subscribers in each set is the same).
