Although the United States is considered a world leader in mathematical sciences research and in doctoral and postdoctoral education, concern is growing about whether the needs of the profession and of an increasingly technological society are being met. Many doctoral students are not prepared to meet undergraduate teaching needs, establish productive research careers, or apply what they have learned in business and industry. This inadequate preparation, continuing high attrition, and the declining interest of domestic students, the inadequate interest of women students, and the near-absent interest of students from underrepresented minorities in doctoral study are problems that transcend the current difficult job market.

One of the principal strengths of the American educational system is its diversity. No single paradigm for education at any level—doctoral/postdoctoral, undergraduate, secondary, or elementary—is imposed. Different programs can all be successful in accomplishing the same goals. This system encourages innovation and the development of local solutions that satisfy the needs of the profession and the country. Such local solutions can then spread, improving education everywhere.

A consequence of this system is that there are at present doctoral and postdoctoral programs that succeed in preparing their students for careers in teaching and industry as well as in academic research and succeed also in attracting large numbers of domestic students, including women and students from underrepresented minorities. But these programs have remained largely unnoticed by both the community and students. In an educational system that encourages innovation, the spread of successful innovative methods now seems to be stymied.

This observation was the starting point of this study. A study of the American doctoral and postdoctoral system of education in the mathematical sciences was proposed at a meeting of the Board on Mathematical Sciences in April 1990. With the support of the Alfred P. Sloan Foundation, the Committee on Doctoral and Postdoctoral Study in the United States was established in July 1990. Its charge was to determine what makes certain programs successful in producing large numbers of domestic PhDs with sufficient professional experience and versatility to meet the research, teaching, and industrial needs of our technology-based society and then to make this information available to the mathematical sciences community. The committee members were chosen to be broadly representative of the mathematical sciences.

Since the committee had neither the charge nor the resources to do a complete quantitative and qualitative study of all of the mathematical sciences doctoral/postdoctoral programs in the United States, it proceeded differently. In a two-day meeting in

September 1990, the committee reviewed data on these programs, clarified the definition of a successful program, and selected a diverse set of programs in 10 universities for two-day site visits, which were carried out in October-November 1990 and January-February 1991. These programs were in both small and large, and in both public and private, universities. They were also geographically diverse. They were all in the “top 100” and included four departments in the “top 20.” Achieving unanimity on the selections was no small task since the committee members had widely divergent opinions—some based on philosophical differences between the disciplines of the mathematical sciences—about what was important in doctoral and postdoctoral programs.

To minimize bias in the site visits, a uniform plan was prepared in advance for two days of observation and discussion by teams consisting of two (three for larger departments) members of the committee. Before the site visits began, a field test of this plan was conducted at a university not among the 10 universities chosen for the study, and, as a result, adjustments to the plan were made. Information to be collected before the site visit included the information packet sent to prospective graduate students, orientation material sent to new graduate students, orientation material sent to new faculty, completion rates for the different programs (with separate breakdowns for women and minorities), information on first-year courses, sample qualifying and candidacy examinations, a description of the duties and responsibilities of teaching assistants, a breakdown of funding sources for graduate students, and list for the previous five years of doctoral students, postdoctoral associates, and junior faculty appointments with information on their advisors, career tracks, and so forth.

The allocation of time during the visits was 50% for students, 25% for faculty, and 25% for administrators. At the beginning and end of the visit, the team held discussions with the chair(s) of the mathematical sciences department(s). The team also spoke separately with the following people and groups: director of the graduate programs, members of the graduate committee, graduate students (in small groups and in informal settings), women and underrepresented minority students (if enrolled in sufficient numbers), thesis advisors, junior faculty, postdoctoral associates, graduate secretary, supervisors of teaching assistants and of the teaching assistant orientation program, and the person(s) responsible for admissions and recruitment. The team visited the departmental and institutional libraries, the computing facilities for students, and students' office space and their space for socializing. It observed the opportunities for social interaction among students and between students and faculty.

Since the committee was not in any sense evaluating the programs at the universities it visited and because it wished to receive the full and open cooperation of the departments, no program identification is made in this report, nor are the site visit reports included. However, quotes from site visit reports are incorporated to illustrate the committee's findings. The statistical data presented during the site visits are not presented in this

report, since an extensive statistical portrait of a restricted number of departments would not serve the committee's purpose.

After the site visits were completed, the committee met in March 1991 to analyze the results. At the end of two days of discussion, the committee was unanimous in concluding that the widely differing programs that had been visited by different teams did indeed have common characteristics that produced success. Moreover, the committee decided that one report was appropriate for all of the mathematical sciences.

The committee recognizes the limitations of this methodology. The result is a report that is often impressionistic in nature, but one that represents unanimous impressions of committee members with widely varying disciplinary and intellectual backgrounds. Within these limitations, the committee believes that it has a strong message to deliver in favor of adjusting the American doctoral and postdoctoral system of education in the mathematical sciences so that it responds better to the needs of the profession, students, and the society. The audience to which the report speaks is all U.S. doctoral and postdoctoral programs in the mathematical sciences, and, in particular, those programs that have limited human and financial resources. The report suggests that even with limited resources success can be achieved if, among other things, a program focuses its energies rather than trying to implement a “standard” or traditional program that covers too many areas of the mathematical sciences. It also notes that departments with the best faculty do not necessarily have the most successful doctoral and postdoctoral programs. A quality faculty is necessary for a good program, but of equal importance are students and researchers that can benefit from the program.

The committee was assisted in preparing this report by many people at the National Research Council and in the mathematical sciences community. The cooperation and assistance of the faculty, students, postdoctoral associates, and administrators at the 10 institutions visited by the committee were invaluable. While the committee wishes that these people and universities could be identified, this study was structured to keep their identities anonymous, since the intention was not to certify or advertise these programs but rather to use them as examples of what is possible.

The Board on Mathematical Sciences gratefully acknowledges support from the Alfred P. Sloan Foundation for this report.