Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States (1992)

Successful programs have a focused, realistic mission. A program that has clearly defined goals, usually of a limited nature, is said to have a focused mission. A program that (1) educates doctoral students and/or postdoctoral associates for the jobs available to them after leaving the program and (2) can actually be carried out given the available human and financial resources is said to be realistic. Agreement on the mission by faculty, administrators, postdoctoral associates, and students is crucial. The mission must be consistent with the strengths of the faculty, and there must be adequate resources for the program, including a technical library, computer facilities, financial support by the university administration, a strong faculty, and, most importantly, doctoral students and postdoctoral associates capable of meeting the expectations of the program.

In this chapter and throughout the report, “model” means a “program with a focused mission.” Site visits were carried out at programs that fit into the first four of the five models—standard, subdisciplinary, interdisciplinary, problem-based, and college-teachers—discussed in this chapter. The college-teachers model, which was known to committee members from other sources, is included for completeness and because of the growing emphasis on undergraduate mathematics education.

The “standard” doctoral/postdoctoral program in the mathematical sciences at American universities is modeled on the doctoral/postdoctoral programs at traditional, pre-World War II, European universities. It supports research in a broad range of areas, with depth in each one. The goal is to prepare talented, well-motivated doctoral students and postdoctoral associates for careers as mathematical scientists at research universities.

Many standard doctoral/postdoctoral programs assume that they will instruct the best students in advanced mathematics, that natural selection will ensure that those sufficiently able will write theses, and that the best of these will take positions at research universities and do research. And the cycle will repeat itself. This approach gives too little formal recognition to the fact that there are undergraduates to teach, that the graduates of the programs teach in colleges or secondary schools, and that positions in government laboratories, business, and industry require applications of the mathematical sciences. In other words, the current multifaceted responsibilities of the profession are all but ignored. This description is, of course, an exaggeration of the views that many have toward their graduate programs but, in many cases, not by much.

The principal goal of the graduate program is to produce academic mathematicians. When pressed, some faculty acknowledged that employment out of academia did not absolutely indicate failure, but the bias was apparent. Unfortunately, not many recent graduates have obtained high-quality positions in academe, and that fact is bemoaned by faculty and students alike.

Committee Site Visit Report

Most standard programs do well in preparing their best students and postdoctoral associates for the academic research job market, but very few prepare any of their students well for jobs in teaching, government, business, or industry. In the current tight academic employment market, a program that attends to the larger needs of its students may find that its graduates succeed in obtaining employment, whereas programs with a narrower focus may have graduates who experience difficulty.

The faculty come from prestigious research institutions and feel that their mission is to train graduate students to do research. This leads to some frustration on the part of the faculty, who lament a rather weak applicant pool and struggle to maintain the standards they consider vital to the credibility of their research enterprise.

Committee Site Visit Report

In a standard program, research and study are viewed as solitary activities. Unless the program is very large, it is likely that there are only one or two professors in each area. Students may carry out their research without the benefit of interaction with a group of other students and professors working in their area of interest.

Most doctoral programs at American universities began by adopting the standard-model. A number of these programs struggle because they cannot attract the graduate students necessary to function as a standard-model program. The committee believes strongly that a successful program based on an alternate, specialized model, four of which are mentioned below, is highly preferable to a struggling, unsuccessful standard program. The health of the profession will always depend partly on well-established standard programs at a small number of centers. The committee believes, however, that efforts to broaden experience and provide a more supportive learning environment would improve these programs also.

In the subdisciplinary model, the department concentrates much of its faculty and resources in a few subdisciplines of the mathematical sciences. This can be done in both pure and applied areas. Recruiting strong, well-prepared students for subdisciplinary programs requires considerable effort to ensure a proper fit. The main advantage of the subdisciplinary model is that clustering of students and faculty working on related topics enables them to assist each other in their common goals.

The most successful circumstances seemed to be those in which there was a coherent group of faculty with closely related research interests and active seminars. In such cases their students became part of a cohort of students who could give some critical attention to each other's work. As in many other situations, the creation of some kind of social bonding seemed to greatly enhance the quality and effectiveness of the work environment.

Committee Site Visit Report

Some of the doctoral programs with the best reputations for research are subdisciplinary programs. A smaller department is more likely to be successful if it adopts the subdisciplinary model or one of the other three models outlined below.

While a subdisciplinary program often consists of a whole department, an interdisciplinary program is usually only one among several programs in a department of mathematics, statistics, or operations research. It utilizes department faculty with interdisciplinary interests and mathematically oriented faculty in cognate disciplines. The curriculum, which often involves course work in one or more other departments in science or engineering, trades depth in the mathematical sciences for greater breadth overall. Students can choose thesis advisors from the mathematical sciences department or the other department. Faculty in both departments often adopt a cooperative approach to directing PhD research.

The strong esprit de corps that is usually part of such a program can be used effectively to recruit higher percentages of domestic students and to increase the PhD completion rate. Women and minorities are often successful in interdisciplinary programs because of this positive atmosphere.

Graduates of interdisciplinary programs sometimes move into other disciplines or take positions in industry. These programs succeed in bringing mathematically well-trained

students into fields in which they can effectively use their talents and at the same time promote the transfer of mathematical knowledge to these fields.

In a problem-based model, a specific application or set of applications is used as a unifying theme for courses and research. Unlike an interdisciplinary program, a problem-based program usually concerns itself with the strictly mathematical aspects of an applied problem. Mathematical modeling is a common focus in a problem-based program. The committee observed programs in discrete modeling for economics and social sciences and in continuum modeling for engineering. All of the problem-based programs visited by and known to the committee concentrate on applied areas of mathematics, statistics, and operations research, but a problem-based program in pure mathematics could be set up. The problem-based programs that the committee visited concentrate on the training of professionals knowledgeable in applications of value in academia and in industry.

An attraction of the problem-based model is that the students are immersed in research-related activities from the beginning. Student internships in regional industries are often an integral part of this kind of program. Industrial researchers often visit the department. Post-PhD employment opportunities in industry are common, but graduates also obtain positions in academia.

The college-teachers model is designed to prepare teachers at two-and four-year colleges. A college-teachers program is to be distinguished from a program that confers doctor of arts or doctor of education degrees. Breadth of course work and an emphasis on professional development in pedagogy are, in addition to a research apprenticeship, parts of such a program. Most new PhDs from standard programs currently take jobs in college teaching but are often ill prepared for their teaching duties. New PhDs from a college-teachers program are attractive candidates for employment at four-year colleges because they are prepared to be teachers. The need for college-teachers programs could increase as the need for college faculty increases toward the end of the 1990s and into the next century.

Two human resource problems, recruitment and placement, are alleviated by having a specialized program. The specialized programs that the committee observed were able to place their graduates in appropriate jobs more easily than standard programs. In

recruiting domestic doctoral students, postdoctoral associates, and junior faculty, specialized programs must be careful to pick the “best” candidate only if he/she fits into the areas of specialization of the program. Nevertheless, specialized programs are typically able to recruit domestic students, including women and underrepresented minorities, more effectively than standard programs, partly because of their ability to articulate their mission and place their graduates effectively. Placement is easier because, as the committee saw in its site visits, many specialized programs maintain close contact with regional colleges and industry as well as with former graduates. Industrial internships are a prominent feature of many of these programs, especially those in statistics and operations research.

Although the education in interdisciplinary, problem-based, and college-teachers programs is typically broader than that provided in standard programs, the depth can be somewhat less. This seems not to cause problems for students taking positions in industry or four-year colleges, but graduates seeking permanent positions at research universities sometimes need to further their education in a postdoctoral position.

More than simply alleviating human resource problems, the focused mission of a specialized program also promotes clustering of faculty, postdoctoral associates, and students, which helps create a positive learning environment and promote relevant professional development.

In the doctoral and postdoctoral system of mathematical sciences education in the United States, both standard and specialized programs are needed in theoretical as well as applied areas, and all of these kinds of programs can be successful. However, programs that do not have the human or financial resources to run a successful standard program should consider whether a specialized model might better fit their needs.