On Challenges and Opportunities of Graduate Advising

1. Introduction

One of the most fulfilling aspects of my career in mathematics has been the opportunity to serve as the thesis advisor for a wonderful group of people over the past 30 years. This includes undergraduate Honors theses, Masters theses, and PhD dissertations, though in this article I am going to focus on doctoral advising as it poses a series of challenges that are coming into an even sharper focus in recent years. One of the reasons for this is increasing efforts by the mathematical community to address the issues of gender and cultural diversity in their ranks. Another is the effort to keep the PhD degree relevant in the age of uncertain job prospects in academia and increasing opportunities in the finance, high tech, and big data sectors of industry. In this essay, I shall describe my approach to graduate advising—the process that goes far beyond the supervision of the PhD dissertations. I shall also endeavor to describe some of the challenges and opportunities that have repeatedly arisen since I started advising graduate students 23 years ago.

I did not enter the world of graduate advising with a solid idea of what the process is supposed to be like. As a graduate student, I was exposed to a wide variety of advising strategies. My own PhD advisor, Chris Sogge, who shaped my views on research mathematics and many other things in profound ways, always emphasized the importance of independence from the early stages of the educational process. This approach played into my strengths as I always viewed mathematics as a creative process that is much closer to my lifelong interest in reading and writing poetry than to a purely technical pursuit. As a result, I developed an independent research program very early and credit this fact to my survival in this turbulent and uncertain profession. To this day, I consider developing an independent perspective an essential part of the development of a graduate student. In addition, once I encountered my own PhD students with a wide variety of backgrounds, interests, and goals, I came to the conclusion that every student requires an individual approach in order to bring out the best in them. This raises the question about the balance of responsibilities between the advisor and the advisee and all the complexities that it entails.

2. My Approach to Advising

2.1. The guiding paradigm

When I was a small child, my grandmother described to me how shoemakers were trained in the small Jewish shtetl she came from. A fairly young person would start out by sweeping the floors and observing the master shoemaker at work. They would then be given simple tasks to do, like putting the shoes into boxes or measuring the shoe size of potential clients. Slowly, over a period of time, the apprentice would be taught how to cut the materials, create basic designs, and, eventually, create their first pair of shoes, first in collaboration with the master, and, ultimately, on their own. Several decades later I recalled these stories when I started advising graduate students and quickly realized that the process need not be all that different. While every student is different and requires a different approach, the apprenticeship scheme is the most compelling model I have ever found to inform my advising process.

In practice, the apprenticeship scheme works as follows. A typical student starts working with me during their second or third year of graduate school, after they have passed most or all of their preliminary examinations. The first thing I do is sit down with a student several times over a period of several days or even weeks to figure out what types of problems they may be passionate about. Once the general class of problems is identified, I typically run a reading course on the background material, at the end of which the student is given a few more papers to read and a concrete problem to solve. This first problem is typically something I have an idea of how to solve, and it forms a foundation of a basic creditable PhD dissertation. If the student is able to complete the first stage in a reasonable amount of time, they are given a second, harder problem, where they are expected to exhibit a greater degree of independence. Strong students who pass the second stage are set loose on open problems with much more uncertain outcomes.

Every graduate student I work with is encouraged to develop practical skills like programming, applied statistics, and basic data analytics techniques. On one hand, this puts them in a position to make a good living if this is where their preferences or the vicissitudes of the job market take them. Moreover, technology has been playing an increasing role in my own research, and that of my graduate students, both due to forays into data science and an increasing computer experimentation to shed more light on my theoretical projects.

2.2. The pitfalls of the “sink or swim” approach

One of many age-old dilemmas in mathematics is whether the “sink or swim” approach works. Also known as the “survival of fittest,” or, in the immortal words of Dolph Lundgren, “if he dies, he dies!” This paradigm occasionally pops up in both elite and not-so-elite institutions alike. Sometimes it works, and sometimes it leads to serious problems for all involved, with literally everything in between, but the idea never seems to go away and is probably destined to be debated forever. For obvious reasons, the draconian approach is less likely to fail in top graduate programs, but even there it has been known to lead to unwarranted complications. Among the many problems I have with the “sink or swim” approach to advising is that it does not, in my opinion, lead to the discovery of the most productive or talented mathematicians. The background, psychology, and previous experiences of the individual play a tremendous role in determining how they respond to a particular advising style, independent of their mathematical ability. In many cases, a student with a considerable amount of talent and drive simply crumbles in the face of excessive early expectations and insensitive treatment (see, e.g., 1, 3 and the references contained therein). Students who have faced hidden or open discrimination due to their gender, race, ethnicity, or LGBTQ+ status are statistically particularly vulnerable. Some of my top students, who have developed into strong and productive research mathematicians, may not have survived in the “sink or swim” world that still too often pervades academia.

2.3. Generating ideas

One of the greatest challenges I have faced as a PhD advisor is getting the students to generate their own ideas. Some were naturally predisposed toward creative reasoning, but the point I make to all my students is that one can significantly improve one’s ability to come up with new directions through frequent practice. I ask my students to come to my office with at least two or three new thoughts, and I stress that it is absolutely fine if those ideas turn out to be wrong or well-known. I describe the graduate, or even upper-division undergraduate, experience to my students as the process of moving from being a consumer to a creator of mathematical ideas. The degree of resistance I face in these situations varies from student to student, but persistence typically pays off and often results in graduate students developing a well-developed independent perspective by the time they receive their PhDs. This serves them very well whether they end up going into academia or industry.

The key question implicit in the discussion of generating ideas is, how does one begin this process? A typical approach is for a student to examine the research papers they are reading in the area where they are conducting their investigations and look for alternate directions and approaches. This is certainly an important part of the process, but the point I continually emphasize to students of all levels is that creative questions can already be asked about elementary aspects of mathematics going back as far as high school algebra or even before! In an ideal world, in which we do not live, mathematics graduate students would have had years of generating ideas and anticipating deeper concepts behind them, but the current state of mathematics education in the US and the lack of support structures for mathematical activities in elementary school, high school, and sometimes even college means that many rudimentary skills need to be learned later. The good news is, this can be done successfully, and an important part of this process is to continually review elementary mathematics, derive all the basic formulas from scratch, and learn to be creative with simple concepts.

2.4. Undergraduate research and outreach

Another way graduate students can sharpen their skills and improve their creative processes is by getting involved in supervising undergraduate research. Universities all over the world are increasingly recognizing the importance of undergraduate research for all students, not just the most talented ones, as a way of preparing them in a more realistic fashion for the world that lies beyond the college walls. While such activities should be organized and run by experienced faculty, graduate students can contribute tremendously by assisting the students with the projects, and by helping to design the various research directions. This process plugs directly into the creativity theme we have been discussing. Mathematics-related community outreach can also help by forcing graduate students to conceptualize their mathematical understanding in a way that can be communicated to people from less-privileged backgrounds who were never exposed to mathematical ideas in a coherent and systematic way. The vertical integration concept implicit in these recommendations should, in my view, increasingly become the guiding paradigm behind university curricula (see, e.g., 4).

2.5. Recruiting graduate students

From the very beginning of my advising experiences, I have been deeply convinced that recruiting makes a tremendous difference in terms of bringing in quality graduate students. If an undergraduate student has the credentials to get into a PhD program in mathematics at Princeton, Harvard, MIT, or another school of that level, no amount of recruiting is likely to persuade them to go to a school of a lower level, but this encompasses only a small sliver of potential graduate students. A large number of very talented students remain after top schools have had their pick, and waiting for these students to apply to your institution on their own initiative is not the strategy I recommend. The experience of the University of Missouri math department, which in the 2000s recruited two PhD students, Roman Vershynin and Svitlana Mayboroda, who have gone on to speak at the International Congress of Mathematics, as well as Atanas Stefanov, Milena Stanislavova, Xiaochun Li, Dimitry Bilyk, Peter Honzig, Roman Shvidkoy, Vlad Yaskin, Doowon Koh, Krystal Taylor, and a number of others who are now well-known tenured faculty at strong PhD-granting institutions, shows that recruitment can fundamentally transform the level and quality of a mathematics graduate program.

The obvious question is, how does one go about recruiting graduate students? Sending out posters, creating an easy-to-navigate departmental web page, and bringing in prospective students for on-site visits are certainly essential, but one should not stop there. An effective recruitment strategy is not limited to the graduate director and encompasses as many people in the department as possible. Virtually every active research mathematician has numerous friends and acquaintances throughout the world of mathematics who have regular contact with strong undergraduate students at their institutions. Contacting these people and asking them to spread the word about your PhD program can go a long way, but one can even go further. Speaking to prospective students directly and getting across your level of enthusiasm for your graduate program and your own research program is a very effective recruiting tool in my experience. This can be accomplished in several different ways. For example, when a research mathematician gives a seminar or a colloquium talk, they can request to meet with upper-class mathematics majors to tell them about graduate studies in general and their home institution in particular. Another way is to work with the graduate director to get in touch with the students who have been accepted to the graduate program but have not yet made the final decision on whether to accept.

I cannot emphasize enough that graduate recruitment is not a zero-sum game. In the process of recruitment, excellent students who may not have received the encouragement and support that is commensurate with their talent and accomplishments may be persuaded to pursue their research dreams. This is especially important with regard to prospective students who are women or members of underrepresented groups. One of the most tired, feeble, and disingenuous excuses often heard in the mathematics world is that it is difficult or impossible to recruit women and members of other underrepresented groups at various levels because there are allegedly so few candidates and everybody else is trying to recruit them as well. In addition to lamenting the relative absence of diversity in mathematics graduate programs, an activity the mathematics world got exceedingly good at over the decades, we should put in a much greater effort to do something about it. This effort must involve every single one of us, and should not just be delegated to the Diversity and Inclusion administrators at our home institutions.

3. Conclusion

I believe that effective PhD advising is a multifaceted process that is not limited to overseeing the content and structure of the dissertation. Initial recruitment before, job placement after, and facilitation of intellectual development and psychological support during the PhD years are all essential components of advising in the modern university. In an ideal academic world, in which we do not live, much better support structures would exist that would make this process much easier. In the meantime, it is up to the advisor to guide the student toward filling the existing gaps in their mathematical background, and facilitating the creation of a work environment that would allow the student to succeed. No matter what support structures and safeguards are put in place, the PhD advisor is destined to remain the central figure in the development of their graduate students.