In the period since World War II, research in the mathematical sciences has flourished in the United States. Large numbers of graduate students and researchers from around the world have come to this country to study and to work in pure and applied mathematics, in well-established and new fields. The American system of doctoral and postdoctoral study and research in the mathematical sciences is considered by many to be an unqualified success, in contrast to the system of pre-college and undergraduate education.

The current scarcity of highly qualified domestic graduate students, often attributed to mediocre pre-college education and to related problems in undergraduate education, is seen by many as an unfortunate circumstance but not as a major problem. In the 1990–1991 academic year, only 43% (461 out of the adjusted total of 1061 reported in McClure, 1991, p. 1093) of the recipients of PhDs in the mathematical sciences from institutions in the United States were U.S. citizens (McClure, 1991), whereas during the 1960s, 82% of the recipients of such PhDs from U.S. institutions were U.S. citizens (NSF, 1988). Among the U.S. citizen recipients of PhDs in the 1990–1991 academic year, less than a quarter were women and less than a twentieth were from underrepresented minorities. American mathematical sciences departments, research laboratories, and industry are relying increasingly on students, faculty, and professional researchers from abroad because fewer and fewer American students are being attracted to study in the mathematical sciences and because the education that many of those students receive leaves them ill equipped to compete with their foreign counterparts.

Noting the scarcity of highly qualified domestic students and the current tight employment market, some maintain that the chief problem in the doctoral and postdoctoral system is overproduction of PhDs, a problem that should be solved by encouraging students to choose other disciplines and by reducing the number of doctoral students. Our over-reliance on academia for jobs for new PhDs is often not considered to be a problem, nor is the matching of doctoral education with the positions that graduates take considered to be a priority. Increasing production of PhDs since 1987, international events that have increased immigration of students and professional mathematicians to the United States, and a recession in the economy have indeed combined to produce what is now the most difficult employment market for PhD mathematicians since the 1970s. Further complicating the current picture are the indications that the demand for mathematical scientists will rise as the many mathematical scientists hired in the 1960s start to retire over the next decade (NRC, 1990b). The long-term growth in demand for mathematical scientists in academia, government, business, and industry and the expectation that the wave of immigration of mathematical talent to the United States will eventually taper off suggest that the country will be best served by a positive outlook that emphasizes

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Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States
1
INTRODUCTION
In the period since World War II, research in the mathematical sciences has flourished in the United States. Large numbers of graduate students and researchers from around the world have come to this country to study and to work in pure and applied mathematics, in well-established and new fields. The American system of doctoral and postdoctoral study and research in the mathematical sciences is considered by many to be an unqualified success, in contrast to the system of pre-college and undergraduate education.
The current scarcity of highly qualified domestic graduate students, often attributed to mediocre pre-college education and to related problems in undergraduate education, is seen by many as an unfortunate circumstance but not as a major problem. In the 1990–1991 academic year, only 43% (461 out of the adjusted total of 1061 reported in McClure, 1991, p. 1093) of the recipients of PhDs in the mathematical sciences from institutions in the United States were U.S. citizens (McClure, 1991), whereas during the 1960s, 82% of the recipients of such PhDs from U.S. institutions were U.S. citizens (NSF, 1988). Among the U.S. citizen recipients of PhDs in the 1990–1991 academic year, less than a quarter were women and less than a twentieth were from underrepresented minorities. American mathematical sciences departments, research laboratories, and industry are relying increasingly on students, faculty, and professional researchers from abroad because fewer and fewer American students are being attracted to study in the mathematical sciences and because the education that many of those students receive leaves them ill equipped to compete with their foreign counterparts.
Noting the scarcity of highly qualified domestic students and the current tight employment market, some maintain that the chief problem in the doctoral and postdoctoral system is overproduction of PhDs, a problem that should be solved by encouraging students to choose other disciplines and by reducing the number of doctoral students. Our over-reliance on academia for jobs for new PhDs is often not considered to be a problem, nor is the matching of doctoral education with the positions that graduates take considered to be a priority. Increasing production of PhDs since 1987, international events that have increased immigration of students and professional mathematicians to the United States, and a recession in the economy have indeed combined to produce what is now the most difficult employment market for PhD mathematicians since the 1970s. Further complicating the current picture are the indications that the demand for mathematical scientists will rise as the many mathematical scientists hired in the 1960s start to retire over the next decade (NRC, 1990b). The long-term growth in demand for mathematical scientists in academia, government, business, and industry and the expectation that the wave of immigration of mathematical talent to the United States will eventually taper off suggest that the country will be best served by a positive outlook that emphasizes

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Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States
attracting more domestic students into the mathematical sciences and giving those students proper foundations for their future careers.
A positive outlook that serves the interests of the profession and the country can be translated into actions intended to achieve the following two broad objectives:
All students, including the majority who will spend their careers in teaching, government laboratories, business, and industry rather than in academic research, should be well prepared by their doctoral and postdoctoral experience for their careers.
Larger percentages of domestic students, and, in particular, women and underrepresented minorities, should be attracted to the study of and careers in the mathematical sciences.
In this report, a “successful” program is understood to be one that accomplishes these two objectives. The needed renewal of the profession, as pointed out in the “David I” report (NRC, 1984), A Challenge of Numbers (NRC, 1990b), and the “David II” report (NRC, 1990c), requires larger percentages of domestic students. Although statistics invariably oversimplify the situation, the following two “completion rate” statistics concerning percentages of domestic students are useful in judging a program's success: (1) the percentage of students who entered the program five years earlier and who have received their doctorates, and (2) the percentage of students who completed their second year of graduate study four years earlier and who have received their doctorates. The first of these two types of completion rate is an appropriate measure of the success of highly selective programs, while the second is appropriate for less selective programs. The committee observed a number of programs for which both rates were well above 50 percent.
PURPOSE AND SCOPE OF THIS REPORT
The charge to the committee was to determine what makes certain doctoral and postdoctoral programs in the mathematical sciences successful in producing large numbers of domestic PhDs, including women and underrepresented minorities, with sufficient professional experience and versatility to meet the research, teaching, business, and industrial needs of our technology-based society. The mathematical sciences are considered to be pure mathematics, applied mathematics, statistics and probability, operations research, and scientific computing. Computer science, a separate discipline, is not included among the mathematical sciences.
The doctoral period considered in this report extends from the first year of graduate study through completion of the thesis, regardless of whether or not the student obtains a

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Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States
master's degree. The postdoctoral period is the first five years after receipt of the PhD. A postdoctoral associate, postdoctoral fellow, or, in common parlance, “postdoc,” is a recent PhD who has a fully funded position to do research. Since such a small number of new PhDs in the mathematical sciences enjoy postdoctoral fellowships, this report concerns not only postdoctoral fellows but also junior faculty working during the postdoctoral period.
The purpose of this report is to present and disseminate information about types of mathematical sciences doctoral and postdoctoral programs that succeed in attracting large numbers of domestic students, including women and underrepresented minorities, and succeed in giving their students academic and professional experience that is relevant to their future careers. There are U.S. programs that provide high-quality doctoral education to student bodies that are 80% American, have nearly 50% women, or have 30% underrepresented minorities. This report, based on the committee's insights gained in site visits to 10 universities, describes characteristics of these programs. What these programs do differently and what they and others consider to be their successes and their frustrations is information that this report seeks to make available to the community so as to encourage doctoral/postdoctoral program models that are relevant to the needs not only of academic research but also of teaching, government, business, and industry, and to increase the quality and number of domestic PhDs, especially women and underrepresented minorities.
This report follows on and is complementary to a number of studies by the National Research Council that examine the health of U.S. mathematical sciences research and education, including Renewing U.S. Mathematics: Critical Resource for the Future (NRC, 1984), Everybody Counts (NRC, 1989), A Challenge of Numbers (NRC, 1990b), Moving Beyond Myths (NRC, 1991b), Renewing U.S. Mathematics: A Plan for the 1990s (NRC, 1990c), and Actions for Renewing U.S. Mathematical Sciences Departments (NRC, 1990a). One includes the following pertinent summary.
Graduate and postdoctoral training programs offered by mathematical sciences departments are key to the successful renewal of the profession and reform of mathematics education. Successful programs can attract individuals to a career in the mathematical sciences and can develop highly qualified teachers and researchers to stimulate, nurture, and train future generations. Is our present graduate and postdoctoral educational system in mathematics working well? The answer seems clearly to be that it could be much better. The community could attract more students to the study of the mathematical sciences, and more students entering graduate programs could succeed in obtaining doctorates. With nurturing and continued attention through good postdoctoral programs, more of these young people could develop into good mathematicians—some as teachers, some as researchers, and many as both. (NRC, 1990a, p. 13)

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Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States
CONTENTS OF THIS REPORT
Chapter 2 gives a brief historical perspective of the mathematical sciences in America, with emphasis on doctoral and postdoctoral training.
Chapter 3 describes how some programs in the present system achieve success. Three characteristics of successful programs—a focused and realistic mission, a positive learning environment, and relevant professional development—are introduced in this chapter. The issue of having a high-quality faculty—a sine qua non of a successful program—is acknowledged but not discussed in detail in this report.
The heart of the report is Chapters 4–7, which treat the three characteristics of successful programs and human resource issues that must be taken into account. Chapter 4 discusses human resource issues; in particular, those related to domestic students, women, and underrepresented minorities are examined. A number of specialized missions for doctoral/postdoctoral programs are described in Chapter 5. Chapter 6 discusses a positive learning environment. Relevant professional development is described in Chapter 7.
Chapter 8 describes how faculty, departments, professional societies, and federal agencies can work together to create more successful programs.
A guide for self-evaluation by departments forms Appendix A. Appendix B includes advice to prospective doctoral students on how they can best choose a doctoral program. Appendix C is a brief discussion of master's degree programs in the mathematical sciences, a feature that may form a part of a well-rounded graduate program in the future as doctoral programs become more oriented toward wider job markets, including business and industry.