Toward Student-Centric Graduate Training

By Yara Skaf and Reinhard Laubenbacher

Career opportunities have never been better for people with Ph.D.s in the mathematical sciences. In addition to traditional employment in mathematics departments and non-academic engineering-based positions, a wide array of new careers in academia and beyond are now available due to the rapidly increasing use of mathematical and computational techniques in many fields. Medicine, education, and finance are prime examples of such fields. The great variety of requisite skills and backgrounds for these careers demands greater customization of student training programs. We intend to address possible ways in which academic departments can achieve student-centric program structures.

A 2018 report from the National Academies of Sciences, Engineering, and Medicine [2] offered several recommendations for reform, including more student-centric programs that account for students’ career goals and interests and provide extensive advising. This advice is timely for the mathematical sciences community. Mathematics is unique; it is a research field in its own right that simultaneously provides a universal language and technology that now touch virtually every other scientific, technological, and social aspect of society. This pertinency leads to unprecedented career opportunities for highly trained mathematicians and mathematics educators, even as academic job prospects expand only moderately [3, 4]. At the same time, it challenges the mathematics community to adjust its training paradigms to keep up with these developments. Here we present some thoughts as to how such an adjustment might look. An expanded, more nuanced version of this article is available online [5].

The traditional model of doctoral education in mathematics effectively prepares trainees for tenure-track academic careers with a dual research/teaching mission (with various degrees of emphasis on research). But many Ph.D. students will ultimately pursue career paths that diverge from this trajectory and thus require different preparation. Programs that acknowledge this reality and adapt their training accordingly will better serve their students and allow them to take full advantage of the myriad exciting career opportunities inside and outside of academia. They will also attract a broader range of students who are interested in advanced mathematics training, thereby growing Ph.D. programs. Discussions about the effective training of doctoral students for modern careers are by no means new, and a robust body of literature exists on the topic [5].

Program Structure

A survey of the literature supports the following observations. First, most Ph.D. programs in the mathematical sciences take a “one-size-fits-all” approach, meaning that the program elements and requirements are essentially the same for every student. The vast majority of the top 50 Ph.D. programs in mathematics—as ranked by U.S. News & World Report [7]—still follow the traditional “algebra, analysis, topology” preliminary examination schedule that occupies the first two years of graduate school, with only minor variations. Second, the interests and career trajectories of graduate trainees extend far beyond tenured positions in academic institutions to a wide range of additional opportunities for which current training does not provide optimal preparation [1, 6]. To add perspective, we briefly contrast the key features of mathematics programs with two other fields that might provide useful ideas.

Mathematics: In addition to dissertation research, standard requirements consist of preliminary examinations in several core subjects, qualifying examinations in a chosen focus area, and substantial coursework. Because stipends are predominantly funded through teaching endeavors, graduate students must tutor, grade, and otherwise assist with multiple undergraduate classes every semester — or teach an entire class themselves.

Biological sciences: In contrast to mathematics programs, most programs in the life sciences require minimal coursework and no preliminary exams. Instead, students must put together a thesis proposal. Mandatory classes are usually more like seminars or journal clubs in which students read and discuss research papers; these courses contain little to no new theory beyond what is taught in undergraduate curricula. Students immediately begin rotating through different faculty laboratories to select an advisor for their Ph.D. research project, which is well underway by the end of the first year. They are also expected to assist with grant proposals, prepare manuscripts, and present original research.

M.D./Ph.D. programs: M.D./Ph.D. programs train students to conduct translational research that generates or applies basic scientific results for improved medical care. As such, the Ph.D. portion of these programs tends to emphasize research activities almost exclusively. A typical M.D./Ph.D. program includes four years of biomedical science Ph.D. training, which is interspersed with four years of medical school. To enable this accelerated completion of a Ph.D. in four years rather than five or six, most programs have very few general requirements.

Many factors contribute to the significant differences between these three types of programs, including incoming students’ preparedness for research projects. For example, most first-year graduate students in biology are capable of driving and executing significant research projects on their own simply because the theoretical foundation for biology is constructed long before graduate school. Another contributing factor is the difference in professional expectations. Graduates in the life sciences or M.D./Ph.D. programs are expected to have several peer-reviewed publications by graduation; be well prepared for the demands of professional positions; and possess the skills to design research projects, write papers and grant applications to support their research, and deliver presentations at conferences. In contrast, most mathematics programs do not strongly emphasize these skills.

Figure 1. Two separate models for doctorate education in applied mathematics. 1a. Conventional approach to graduate education. Students access a predetermined set of training resources in a standard manner that is independent of any specific needs or interests. 1b. Training activities are organized into four tracks around students’ personal career objectives. Clockwise from top left: academic research, private sector, education, and engineering-based industry. Each track weights available resources differently. Figure courtesy of the authors and SIAM.

A Proposal

Mathematics Ph.D. programs should strive for completely personalized training schedules that are built around the unique interests and career aspirations of each student. Such personalization should extend to coursework (or other means of acquiring expertise); professional training like teaching or internships in business, industry, or government; and the research component. Students’ individualized needs should guide the relative emphasis of major program components so that they can spend most of their time on activities that will best encourage their professional development. As an intermediate step, programs could therefore develop several distinct “tracks,” each with a structure that is tailored toward a particular career trajectory (see Figure 1). The forthcoming breakdown offers examples of each track.

Academia: This track is identical to the current paradigm that trains students for traditional careers in academia; few changes are required.

Teaching and Education: This track is intended for students who are interested in careers that emphasize education, such as faculty positions at liberal arts and community colleges, educational research, or the development of novel teaching methodologies in the private sector. Institutions should adjust the research component in this track to account for a reduced emphasis on original mathematical research in students’ future careers. This path should also include appropriate internship and training opportunities in professional environments, e.g., teaching opportunities at local community colleges.

Non-academic Careers: This track prioritizes hands-on experience in the type of interdisciplinary work that is integral to non-academic careers. A portion of the courses in this track fall outside of the mathematics department to ensure that students gain adequate background in the application of mathematical methods. Students should also participate in semester- or year-long internships; online sites such as SIAM's Career Resources page and the BIG Math Network can provide guidance.

Academic Research: Mathematics Ph.D.s can now pursue research careers in academic settings outside of mathematics departments. These settings include a variety of departments and institutes that heavily rely on grants and contracts. For instance, biomedical and computational biology research provides many opportunities in data science and mathematical/computational modeling within medical schools and biomedical and biological research institutes. Though M.D./Ph.D. students seldom pursue graduate training in mathematics, more might choose to do so if programs were better able to adapt to the unique requirements of a four-year Ph.D. timeline.

Challenges

Students who take part in personalized programs must immediately start designing curricula that are right for them, making periodic adjustments as necessary. Intensive mentoring is crucial and should extend far beyond the efforts of current mathematics Ph.D. programs. Such programs also need to inform students of different career opportunities early in their training, and faculty mentors should know about the broader universe of available careers and resources. Student passage from one track to another must be flexible, and mentoring efforts should prepare students to transition to the next phase in their careers.

If readers have thoughts, questions, or suggestions about the aforementioned proposal, we encourage them to comment on the online version of this article or contact the authors directly at [email protected] and [email protected].

Acknowledgments: We are most grateful to the many members of the mathematics community who provided extensive feedback on an earlier version of this manuscript to greatly improve the final text.

References
[1] Langin, K. (2019, March 12). In a first, U.S. private sector employs nearly as many Ph.D.s as schools do. Science Careers. Retrieved from https://www.sciencemag.org/careers/2019/03/first-us-private-sector-employs-nearly-many-phds-schools-do.
[2] National Academies of Sciences, Engineering, and Medicine. (2018). Graduate STEM education for the 21st century. Washington, DC: The National Academies Press.
[3] National Center for Science and Engineering Statistics. (2019). Survey of doctorate recipients. National Science Foundation. Retrieved from https://nsf.gov/statistics/srvydoctoratework/#tools&tabs-3&infdsts&rSR&rWP&sd.
[4] Rieley, M. (June 2018). Big data adds up to opportunities in math careers. Beyond the Numbers: Employ. & Unemploy., 7(8). 
[5] Skaf, Y., & Laubenbacher, R. (2021). Student-centric graduate training in mathematics: A commentary. Preprint, arXiv:2109.07661.
[6] Terry, M. (2019, March 20). Report: When it comes to hiring PhDs, private sector and academia are about even. BioSpace. Retrieved from https://www.biospace.com/article/report-when-it-comes-to-hiring-phds-private-sector-and-academia-are-about-even. 
[7] U.S. News & World Report. (2018). Best mathematics programs. Retrieved from https://www.usnews.com/best-graduate-schools/top-science-schools/mathematics-rankings.

Yara Skaf is an M.D./Ph.D. student in the Department of Mathematics and the Laboratory for Systems Medicine in the Department of Medicine’s Division of Pulmonary, Critical Care, and Sleep Medicine at the University of Florida. She is developing tools from topological data analysis and applying them to the analysis of electronic health records. Reinhard Laubenbacher is director of the Laboratory for Systems Medicine and a professor in the Department of Medicine’s Division of Pulmonary, Critical Care, and Sleep Medicine at the University of Florida.