Notices of the American Mathematical Society
Welcome to the Notices of the American Mathematical Society.
With support from AMS membership, we are pleased to share the journal with the global mathematical community.
PDFLINK |
Do Mathematics Every Day
A fact of life in our profession is that we often spend extended periods of time facing two opposite challenges. The first challenge many of us face is multimonth periods of time spent with extremely full schedules with teaching commitments, committees, seminars, and the whirlwind of activity of the academic year. This poses obvious challenges to our research productivity and schedule. The second challenge, which may receive less attention, is long stretches of time when our schedules are completely open, and here the challenge is for us to find ways to organize our time efficiently and be productive. While the second situation may be considered “optimal” for research and getting a lot done, it certainly comes with its own set of potential pitfalls. In this article, I want to share some thoughts on how to keep our mathematical fires burning consistently, and also how to balance an ambitious research schedule against the need to recharge and to be mindful of our well-being.
Semester-pocalyse. We often face stretches of time when we have far more on our plates than we can reasonably do. This might be for a stretch of weeks, a couple of months, or even a whole semester. If you are reading this as an early career person, I’m sure you can relate, but unfortunately this only gets worse with more seniority. So what can you do to keep your mathematics moving forward when overloaded with myriad other duties? My advice is deceptively simple.
Do mathematics every day. By this, I mean to consciously set aside time for mathematics every day, and to do this no matter what else is going on (barring a major crisis or something really exceptional going on). I used to do this intuitively—I’ve always enjoyed working in coffee shops, and even when I’ve been really busy I would often relocate to a coffee shop for an hour and work on mathematics—the change of scenery would free me from feeling the need to reply to emails and remove the chance of interruptions. My caffeine dependency thus led to me doing consistent daily research even during the busiest of times!
In recent years, I’ve carved out research time more intentionally and systematically. This came from a breakthrough in my bass-playing. I’ve been a dedicated bassist since I was a teenager, but my playing had been languishing due to a lack of practice for several years (correlating with starting a tenure-track position and having kids). What used to be possible for me in terms of finding practice time was no longer possible. I rebooted my playing during the pandemic, taking an online course on the Discover Double Bass platform with jazz bass maestro John Goldsby, who had this to say about practice: “Practicing 15 minutes a day, every day, is better for your development than two hours of hectic bass playing every weekend.” This was a lightbulb moment for me. For years, I’d felt that I was too busy to practice. The idea that I could make progress by doing 15 minutes of serious practice each day was a revelation. What kind of crisis would have to be going on that I couldn’t find 15 minutes in my day? It would have to be quite a crisis if I couldn’t find just 15 minutes! Another aspect of this is that 15 minutes is the floor, not the ceiling—if the going is good, you just keep on going. Over the course of a year or two, this approach worked wonders for my playing, and now I’m swinging my way through jazz standards and burning my way through the bluegrass songbook. I got to thinking about how to apply this to mathematics research.
A common mistake that we make in research is the same one I was making with my music practice. It’s a common trap to feel like we need a substantial block of uninterrupted time to get in the zone and make progress. For research, maybe we are waiting for a free day or even a free week so we can really get in the zone. Those are certainly optimal conditions, and those big blocks of time are great to have when they’re available. However, it’s important to let go of the idea that you need to wait for optimal conditions to do research. Being able to make progress in nonoptimal conditions is the way to produce research consistently over the long-term.
My advice to you during busy periods is to decide how much research time each day can be a realistic floor for your research time. Is it one hour? Is it half an hour? Maybe even carving out two hours each day would be realistic depending on your schedule. The point is to figure out what can realistically be done EVERY workday based on your schedule for the semester, and to stick to it as religiously as possible. This will keep your research moving along, and when a block of uninterrupted time does open up, you are ready to make the most of it. Otherwise, when that uninterrupted block of time comes up, you risk needing most of the time just to get back up to speed on your projects, and there may be little time remaining to make further progress.
Let your subconscious do some of the work. A common misconception is to think that we’re only working when we’re sitting in front of the computer, or reading a paper, and actively concentrating. In fact, the subconscious often does a lot of the heavy-lifting. The “active work” of concentrating on reading a paper, or working through a calculation, is sometimes most useful as a process of getting the ideas into our head. We may not know how to solve a problem when we’re consciously trying to do it. When we step away from a problem we’ve concentrated hard on, and come back the next day, often our subconscious has been working away in the meantime, and suddenly we know what to do next. This is another benefit of the ‘Do math every day’ approach. Each day that you learn some mathematics, even for a relatively modest amount of time, your subconscious will continue to work away on it while you are teaching your classes or busy with other duties.
When not to do mathematics every day. Burnout happens in our profession. We’re running a marathon, not a sprint, and it’s important to have the tools to keep on going indefinitely. My previous advice to do math every day aims to stimulate your subconscious to keep on working on math even while you’re doing other tasks. Sometimes you just need to rest and to let your mind be quiet. Without rest, stress becomes cumulative and can lead to chronic anxiety issues. The classic recommendation by psychologists is to give yourself the following periods of downtime
Taking a full day away from research each week is certainly important. It’s probably difficult for most people to take off a full week every quarter; I haven’t been doing that. However, that is the classic evidence-based advice from psychologists, and maybe we would all benefit from it when possible. Taking at least a couple of week-long breaks in the year seems realistic and in the spirit of this advice. During these times, you want to rest your brain as completely as possible so you can come back fresh and strong when you return to work.
Balancing semester duties. When things are really busy, it is an unfortunate fact of university life that other tasks squeeze our research time. Our teaching and service work have hard deadlines that push these duties to the top of our priority list. To counteract this, it’s good to create some firm deadlines in our research too. Regular meetings with collaborators are a great way to provide this, since it’s natural for us to want to have some progress to report at the meeting. Giving a talk on your research, or an expository talk on an adjacent subject can be another great way to generate momentum. Deadlines like these can guide how we use our research time and help us use it efficiently. While adding research deadlines is useful for direction and motivation, it can be freeing to balance this by occasionally going “rogue” and thinking about some interesting mathematics even if it is not directly useful for a current project. For example, learning what a colleague is working on or spending a little time with an interesting paper that jumped out at you from the daily arXiv listing is a great way to stay productive when you need a break from your main research projects and deadline-oriented work. If you’re lucky, you might even find some inspiration for a great new project.
If service has taken over your schedule to the point where research feels almost impossible, then I would advise looking at what you can let go of in the future. Personally, I always try to complete the service assignments I committed to as best I can, even if I find out the hard way that I took on more than I should. However, if I have taken on too much, I try to make a change for the following year. Most things we sign up for, committee assignments for example, are for a fixed amount of time. If you find yourself doing an unreasonable amount of service during the semester, then in most cases it is absolutely reasonable to let some things go the following year to preserve more of your time for research. Alternatively, if you anticipate having an unusually demanding service load, and you’re unwilling or unable to let anything go, you can request to teach courses you’ve taught before to reduce time spent on teaching preparation.
Wide open spaces—summer and sabbatticals. Being productive on research when we are NOT busy with teaching, committees, etc., is an underrated challenge. Many of us oscillate between the extremes of our schedules being completely full and mostly empty as semesters begin and end, and it can be a challenge to adapt. We often begin our summers, or a semester without teaching duties, with high hopes for extraordinary research productivity. Indeed, these are the times when one can really make a lot of research progress, but these periods can also be overwhelming, particularly when paired with lofty self-expectations. The “do math every day” mantra still applies, but now the challenge is to use your research time well and also to be mindful about staying healthy.
I recommend creating a research “master plan” for your summer or semester. Not only should you create a list of the broad projects that you would like to work on, but you should write down a couple of pages worth of specific research tasks, e.g., “understand paper X; polish the proof of Lemma 3.7; work on the introduction of paper Y; work on the main technical result needed for project Z,” and research-related tasks, e.g., “schedule regular meetings with collaborator A; email expert B asking them a question.”
If your to-do list solely consists of “do great math research today,” that will likely be overwhelming, and some days it will be unclear where to start. We often lose some perspective and mental clarity if we’ve just spent a couple of days working on a technical lemma, or doing a MathSciNet deep dive into the literature on a topic we’re learning. When we hit a wall on writing up a technical lemma, or trying to absorb all the information in a literature search, it can be valuable to change gears. That is where your to-do list of specific tasks becomes useful to fall back on. If you hit a wall with one task, or you’re unsure how to proceed, just consult your list and pick whatever task looks good to you to work on that day. Some people go further and schedule their days into blocks of time for specific tasks. If that style works for you, then great. Personally, I prefer to leave my time unstructured, but to use my list of specific tasks to guide me and keep me on track and productive.
It’s also important to keep up with non-math activities to help you stay mentally sharp and healthy. This could be exercise, music, social activities, or just getting outside. This is a good idea for overall well-being, but it’s also good for the mathematics itself. Judiciously chosen time spent on non-math activities helps our subconscious organize and process all the hard work we do.
Let me finish with a quote from legendary record producer Rick Rubin
Take the example of an album. If you’re a musician struggling with ten songs, narrow your focus to two. When we make the task more manageable and focused, a change occurs…. Going from two to three is easier than going from zero to two. And if you happen to get stuck on three, then skip it and get four and five done. Complete as many elements of the project as you can without getting hung up…Often the knowledge we gain from finishing the other pieces becomes a key to overcoming earlier obstacles.
References
[ 1] - Edmund J. Bourne. The Anxiety and Phobia Workbook, Seventh Edition. New Harbinger Publications, 2020.
[ 2] - Rick Rubin. The Creative Act: A Way of Being. Penguin, 2023.
Daniel J. Thompson is a professor of mathematics at The Ohio State University. His email address is thompson.2455@osu.edu.
Article DOI: 10.1090/noti3008
Credits
Photo of Daniel J. Thompson is courtesy of Neal Havener.