It’s not uncommon for me to talk with other students in my math department about their advisors’ mentoring styles, and what type of advice they give. This certainly includes discussion of what courses we’re all taking, and why. At least at the University of Tennessee, it seems a lot of the other students are being advised to take as few courses as possible, at least after completing the written preliminary examinations. I think the idea of this is simple: the less time spent on classes, the more time is left for research.
Interestingly enough, my advisors have challenged this common advice.
At this point I’ve not only completed the written examinations, but also the oral speciality examination, which bestows upon me the title of “PhD candidate.” However, my advisors are encouraging me to push beyond the minimal course load each semester, and welcome me to take courses that might not be immediately useful for our research. I’ve been working with this team for close to a year and a half at this point, but we just this week had the discussion about why they think it’s a good idea to take more courses.
They said that taking more courses will help me develop independence as a mathematics researcher. Supposedly, if I explore a wider range of math concepts at some depth, it will help me “depart from their lines of reasoning;” that is, it will become easier for me to develop original arguments in research, and possibly solve problems that other cannot. And even if it’s not a new problem solved, but an old problem solved in a new way, that in of itself often has a lot of merit (at least in mathematics). As a corollary of this, I expect it will also become easier for me to identify what types of concepts, ideas, and proof techniques excite me most in mathematics, and begin carving my own niche as a researcher. My friend told me that his advisor used the analogy “the advisor is like the coach, and the student is like the athlete.” I think it’s a perfect fit to describe this phenomenon.
Ultimately, I’m not saying that one approach is right or wrong, just that there is more than one approach. I also think it largely depends on the PhD student’s intended career path, as taking more courses has a varied level of benefit depending on the next step. Sometimes the extra time isn’t entirely best off used for research, but also for building other skills, such as programming, data analysis, and the like.