The Fall semester, as many of us know, is a popular semester for writing letters of recommendation and advising seniors on their future. As a math professor in a small liberal arts school, I am aware that many (most?) of our majors are not going to go on to graduate school. Certainly, it is not true that math majors can only have careers in academia. There are many options for them and this fact has led to higher enrollments in the mathematical sciences (I think). But this doesn’t mean that we should discourage students from pursuing academic careers, and in this post I will share some of my thoughts on when and how to advise students to pursue higher degrees in mathematics.
First and foremost, you want to encourage students who are ready to succeed in graduate school. It’s not enough to have good grades. To me it comes down to a combination of mathematical maturity and love of mathematics. You can’t have just one of those things. I do, however, encourage students that I think are not so sure about graduate school and who could use a bit more preparation to do a postbac program (such as the one offered by Smith College ) or even to sit in on some advanced courses in a university with a graduate program. It’s also a good idea to send them to REUs early on or something like the Budapest semester in mathematics , since that will give them a better idea of what graduate school will be like. This is especially important for students in small schools like mine that don’t have a graduate program or very many advanced courses. At a larger research university, undergraduates may have a better exposure to graduate school and an idea of whether they want to go.
I recently read a very good Slate article that I have now shared with a few of my students. The main point I take from this article, and the advice that I have given students in the past, is that if you really enjoy mathematics, thinking about mathematics, doing research in mathematics, then getting a Ph.D. is not a waste of time. In fact, you can only be better off afterwards. The article was in response to other recent articles mentioning how academic jobs are very tough to get and that maybe Ph.D.s in science were not worth the effort. But there are still employment options, essentially the same that you had when you finished college but now you are smarter and more mature to boot. Usually, and the Slate article mentions this, too, you get paid enough to get by, so even though you may not be getting rich you will not be more in debt by the end, either. One thing I make sure to tell my students (and this is something I told myself before accepting to go to Texas for a Ph.D.) is that it is also completely OK to change your mind. Deciding to pursue a Ph.D. is not a prison sentence, and if they realize that they do not enjoy math as much as they thought, or that they don’t really want to spend so much time learning more mathematics, then there is no reason to stay.
Of course, other things besides not enjoying learning math can happen, like not passing quals or prelims or whatever they’re called these days. It is my opinion that people who truly want to stay in graduate school find a way to do so. I have also had friends who have left for a few years, come back, and are very successful. Other friends have switched from one institution to another that was a better fit. This reminds me of a talk Kathryn Leonard gave at MathFest this past August, “I failed, and no one died”. In it, she explains how we need to teach our students (and learn for ourselves) to distinguish between failure and Failure. A Failure is when an airplane pilot fails to land his airplane or a surgeon botches a procedure. These are bad places to fail at what you’re doing. In mathematics, we are constantly faced with little-f-failure. We are working on a research problem that we can’t prove, or we don’t pass our qual, or we get a bad grade on our Real Analysis midterm. These are not huge problems, and no one is going to die. They are also indicative that you need to change something that you’re doing. Either try a new approach to your research problem, figure out how to study for your qual, find a study group for your next midterm. These failures could also indicate other issues, like maybe your problem is much more difficult than you thought (maybe you need to assume GRH!) , and in the other two situations, maybe you realize that you don’t care enough to struggle. This is important, because math is difficult, and if you don’t enjoy it then it is very hard to get through some of these obstacles. This goes for many difficult occupations, by the way.
Just like with “failure”, “success” is also measured in different ways. I think of myself as being successful, because I got the job I wanted to get, which was teaching math and doing research at a good, small liberal arts school, but some people would say that unless you are at a top research institution you really have not made it as a mathematician. I have a friend who recently got his Ph.D. and went on to teach high school, because that is what he wanted to do. Some people were surprised because he was really good and they felt he should be pursuing a more “prestigious” path. But let me ask you who is more successful: the person who loves their job even though some people don’t think too highly of it, or the person who got the job everyone covets but doesn’t enjoy one moment of it? And this gets me again to the point that even though getting an academic job after getting a Ph.D. is becoming more difficult, this is not the only respectable or desirable job option. There are so many things one can do with a degree in mathematics! I always point unsure students (especially the ones that want to explain to their parents why they want to major in math and maybe pursue a higher degree) to this wonderful website by the BYU mathematics department. It is more geared towards undergraduates, but many of these career paths (probably all, actually) are still reasonable pursuits post-Ph.D.
My advice is probably biased because I enjoyed my own graduate school experience so much. This is why I do try to remind students of the possible obstacles and remind them to give themselves permission not only to fail, but to change their minds. I know of people that did not find the experience as rewarding as I did, and they tend to advise students not to go to graduate school (unless they are extremely gifted and confident). But I ask them to try to think, just like I’m doing, about the possibility of a different experience for their students. At the end of the day, there is no way of knowing what will happen. I saw my fair share of superstars fizzle out during their first few years of graduate school, and some very quiet kids from not well-known schools blow everyone away and become superstars later on.
I am aware that not everyone agrees with me on this topic, and in fact I have had many heated (but friendly) debates with colleagues about this issue. My point of view is perhaps overly optimistic, but hey, it’s my point of view. That said, I do want to encourage everyone to post their own thoughts, experiences, and advice in the comments below. Maybe some of you with more experience (and who have advised graduate students, for example) have a different perspective. We would love to hear what you think.