Undergraduate mathematics programs have traditionally produced excellent students. The number of undergraduate mathematics majors increased during the 1980s, but most of these students either ended their education with a bachelor's degree or chose graduate or professional study in other fields. Fewer students continue their studies in the mathematical sciences than in other sciences and engineering. From 1977 to 1986, the attainment rate (the percentage of bachelor's degree holders who receive PhDs) in mathematics was 4%, somewhat lower than the 5% rate in all natural sciences and engineering (NRC, 1990b, p. 48). Mathematical sciences B.S. graduates who earn doctorates often earn them in other sciences or in engineering.

The fact that many excellent students who complete bachelor's degrees in the mathematical sciences either do not continue their studies or go on to graduate studies in other fields is an indication that the pool of candidates for doctoral study in the mathematical sciences is not being fully tapped. The programs visited by the committee show that the pool of highly qualified domestic doctoral students can be increased by active recruiting and that the completion rate (percentage of students who complete the PhD) can also be increased. In this chapter, human resource issues—ranging from recruitment of first-year graduate students to placement during the postdoctoral period—are discussed. Both the successful and the detrimental features described here are based on observations made in the committee's site visits.

The conventional approach to recruiting consists of sending out announcements and posters to other institutions, listing teaching assistantships and fellowships in professional society publications and elsewhere, responding to inquiries, reading applications, and selecting students with the best credentials. Since students commonly apply to a department based on its research reputation alone, and not on other characteristics of the doctoral program, this type of passive recruiting works for programs with the best reputations. However, passive recruiting attracts only those undergraduates who are already committed to graduate study in mathematics. Departments typically compete for applicants only from the pool of students who have already decided to attend graduate school in the mathematical sciences, a group that includes too few domestic students, even fewer women, and almost no students from minorities. The committee believes that the custom of enrolling only those students who decide on their own to attend graduate school and to complete a doctoral program sustains the imbalance in the nationality, race, and

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Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States
4
HUMAN RESOURCE ISSUES
Undergraduate mathematics programs have traditionally produced excellent students. The number of undergraduate mathematics majors increased during the 1980s, but most of these students either ended their education with a bachelor's degree or chose graduate or professional study in other fields. Fewer students continue their studies in the mathematical sciences than in other sciences and engineering. From 1977 to 1986, the attainment rate (the percentage of bachelor's degree holders who receive PhDs) in mathematics was 4%, somewhat lower than the 5% rate in all natural sciences and engineering (NRC, 1990b, p. 48). Mathematical sciences B.S. graduates who earn doctorates often earn them in other sciences or in engineering.
The fact that many excellent students who complete bachelor's degrees in the mathematical sciences either do not continue their studies or go on to graduate studies in other fields is an indication that the pool of candidates for doctoral study in the mathematical sciences is not being fully tapped. The programs visited by the committee show that the pool of highly qualified domestic doctoral students can be increased by active recruiting and that the completion rate (percentage of students who complete the PhD) can also be increased. In this chapter, human resource issues—ranging from recruitment of first-year graduate students to placement during the postdoctoral period—are discussed. Both the successful and the detrimental features described here are based on observations made in the committee's site visits.
RECRUITMENT
The conventional approach to recruiting consists of sending out announcements and posters to other institutions, listing teaching assistantships and fellowships in professional society publications and elsewhere, responding to inquiries, reading applications, and selecting students with the best credentials. Since students commonly apply to a department based on its research reputation alone, and not on other characteristics of the doctoral program, this type of passive recruiting works for programs with the best reputations. However, passive recruiting attracts only those undergraduates who are already committed to graduate study in mathematics. Departments typically compete for applicants only from the pool of students who have already decided to attend graduate school in the mathematical sciences, a group that includes too few domestic students, even fewer women, and almost no students from minorities. The committee believes that the custom of enrolling only those students who decide on their own to attend graduate school and to complete a doctoral program sustains the imbalance in the nationality, race, and

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gender distributions among doctoral students and postdoctoral associates in the mathematical sciences.
Active recruiting can result in the enrollment of a high percentage of domestic graduate students and a large number of women and minorities.
The program is able to attract so many American students because it recruits so actively. It recruits primarily from small liberal arts colleges and has over the years developed contacts with faculty members in these schools. It works at maintaining friendly relationships with these contacts, accepting students who are highly recommended, in order to keep the pipeline open. The process is as follows. In each November and December the entire faculty sends letters to friends at four-year colleges and state universities requesting names of qualified students. Application materials are sent to all those recommended and are followed up with telephone calls.
Committee Site Visit Report
The department has a history of successfully recruiting female students. Word of mouth, the department's general reputation, and post-application visits and telephone calls by the graduate chairman and other faculty help attract female students.
Committee Site Visit Report
Networks with departments at the colleges and smaller universities in a region can be set up. The faculty of a doctoral program and the faculty of the other departments may trade visits to get acquainted with both programs and people.
Recruiting is very active and comparatively aggressive. The vice-chairman for graduate students routinely makes three or four telephone calls to prospective students, and students often expressed appreciation for this aggressiveness. Their graduates exist throughout the educational system and they send their students back. Alumni contact is important.
Committee Site Visit Report
Faculty members should be encouraged (by travel support and other means) to speak with juniors and seniors at the colleges and universities they visit professionally. Former doctoral students and other faculty contacts in undergraduate programs, in particular, programs in local teaching departments, can be used effectively.
The recruiting process has involved sending faculty members out to initiate contacts with many of the liberal arts colleges of the region. In many cases, pipelines with some of the best mathematics programs are now established, and [former] graduate students are asked to return to their undergraduate

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schools to talk about the program. They often get partial support from the department for this. Promising applicants are often invited for one-day interviews/visits at the department's expense and are given the red-carpet treatment.
Committee Site Visit Report
Prospective students can be invited to visit a department. Getting to know the faculty can cause students not already committed to graduate study to decide to choose it.
The 20 to 30 top applicants for the PhD are brought to a workshop in early March. They arrive Saturday and spend Sunday hiking with faculty. On Monday and Tuesday, faculty give lectures presenting their research interests, and the nature of the graduate program is explained. By all accounts, this is an extraordinarily effective recruiting mechanism. It builds enthusiasm for the program and an esprit de corps among prospective students. The workshop is followed up by telephone calls. One student told us that a major factor influencing his decision to come was that the department's persistence in trying to reach him convinced him that he was really wanted.
Committee Site Visit Report
Information that shows the success of the new PhDs produced by the department in obtaining good jobs can be very convincing to prospective students.
During the winter, the chair and associate chair visit approximately 10 schools each to give talks and to speak with interested junior and senior students. A typical talk is about one hour, with the first half hour devoted to some mathematical topic accessible to undergraduates. The second half hour is split roughly between a general discussion of graduate education in mathematics and discussion of the special features of their program. Because they put such a high priority on tracking their students' first employment following graduation, they can share this information with students during and after the talks as well as in application materials.
Committee Site Visit Report
Paradoxically, one source of doctoral students that seems to be underutilized is the pool of mathematical sciences undergraduates at the research universities themselves. While some of the best undergraduates at a particular research university are encouraged to pursue doctoral study, there is usually no systematic recruiting of the remaining mathematical sciences undergraduates at a research university by anyone, the university's own mathematical sciences doctoral program or that of any other university.
In spite of the current tight employment market, recruiting is important. The students recruited today will quite possibly enter a much more favorable job market when they

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graduate. Even if the overall number of doctoral students is reduced to fit a smaller market that is not as favorable as some predictions indicate, it is important to increase the number and percentage of domestic doctoral students by active recruitment.
ADMISSION
Successful programs generally adopt one of two strategies for admission. The first strategy is a highly selective policy that results in a student body with high ability, preparation, and motivation. The doctoral programs with the best reputations for research can be highly selective about the students they admit, since prospective students often equate its reputation for research with the overall quality of a doctoral program. For other departments, a selective policy can be pursued as long as the department actively recruits students and ensures that the pool of candidates is sufficiently large. A second strategy is a less selective, more open admission policy that provides opportunities for talented but less well prepared students to study in the mathematical sciences. In successful programs, this strategy is coupled to a commitment to provide the additional support and opportunities for appropriate course work necessary to give the students a chance to succeed.
Many programs, under the pressure of decreasing interest in graduate study in the mathematical sciences among domestic students and the need for teaching assistants to staff undergraduate courses, have a somewhat open admission policy. Underprepared students admitted in this manner are often not provided with opportunities for appropriate course work, and many of them discontinue their studies. If recruiting is active, the pool is likely to be large enough so that students need not be admitted for the purpose of filling positions as teaching assistants.
DOMESTIC STUDENTS
There has been a sharp decline in the number of American doctoral students in the mathematical sciences during the past two decades. In the 1970–1971 academic year, 1009 (82%) of the 1238 mathematical sciences PhDs granted in the United States went to U.S. citizens, whereas in the 1990–1991 academic year, only 461 (43%) of the 1061 new PhDs went to U.S. citizens.
The drop-out rate from the mathematics career pipeline (beginning at the undergraduate level and terminating at the doctoral level) averages 50% per year, which is markedly higher than the corresponding rates in the other sciences and in engineering. Many American undergraduates take the minimal number of mathematics courses necessary to complete their major, which is a large factor in fewer of them going on to graduate school: limited exposure to mathematics in undergraduate school leaves American students at a

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disadvantage in the competition with foreign students for admission and support as well as in their studies. Yet it is clear that many American students are talented and would make excellent graduate students with additional study and seasoning.
… the Department believes that American students are often not very well prepared for graduate study in comparison to foreign students and that they need “minority nurturing.” Students who come from small schools are accustomed to individual attention and therefore are given special attention during the first semester, when nearly all of the drop outs in the program occur. Classes are small and faculty doors are open.
Committee Site Visit Report
Although the major portion of funding for educating mathematicians and statisticians comes from universities, federal programs also affect the community, often beyond the intended primary purpose. Many federal research programs in the mathematical sciences have had the effect of attracting students to the mathematical sciences but have also had an unintended effect of decreasing the attention paid to teaching, because more time spent competing for grants often means that less time is spent on teaching. While this seems to be less of a problem in sciences with undergraduate laboratories, it may be a contributing cause for the decline in the number of domestic graduate students in the mathematical sciences.
One of the earliest federal programs for domestic doctoral students was the Predoctoral Fellowship Program at the NSF. The mathematical sciences participate, as do all the disciplines supported by the NSF. The Predoctoral Fellowship Program attracts some of the best undergraduates, and, for this reason, it is unlikely that the program increases the pool of students interested in graduate study. Any mathematics student who is competitive for this program would also be assured of full support in an elite program. Nevertheless, this program continues to demonstrate the importance attached to doctoral education by the federal agencies.
The recently introduced Graduate Assistance in Areas of National Need (GAANN) Program at the Department of Education is structured differently. Graduate programs compete for a block of fellowships that a department can award to domestic students who enroll in its program. Having the fellowships to award has caused many departments to start actively recruiting graduate students, with special emphasis on American undergraduates. Although the short-term effect of the GAANN Program might be to shift American students to graduate programs with GAANN support, the long-term effect should be to increase the domestic pool by attracting some of the bright mathematics majors who would otherwise enroll in graduate and professional programs in other sciences, engineering, medicine, or even law.

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The committee visited a number of programs in which the proportion of domestic students was between 50% and 75%. All of them were actively recruited.
The large number of enthusiastic domestic students was one of the striking aspects of the program. We believe that the factors bringing this about include (1) the esprit de corps initially established during the recruiting workshop and then strengthened during the first year, (2) active faculty interest and accessibility, (3) a supportive atmosphere, (4) a system of qualifying exams that is perceived as well defined, reasonable, and fair, and (5) the attraction of interdisciplinary studies.
Committee Site Visit Report
WOMEN AND UNDERREPRESENTED MINORITIES
Although female U.S. citizens constitute almost one-half of undergraduate majors in the mathematical sciences, they still accounted for only 24% (112 out of 461) of the U.S. citizen recipients of PhDs in the 1990–1991 academic year. The PhD completion rate for underrepresented minorities is lower than that for women. Of the 461 U.S. citizen recipients of PhDs in 1990–1991, only 10 were African-American, 6 were Hispanic, and 2 were Native American. The job market projections for the next decade point to a need for greater representation by women and underrepresented minorities in order to meet U.S. requirements for people with expertise in the mathematical sciences (NRC, 1990b).
The losses of mathematical talent are not evenly distributed among racial and ethnic groups or the sexes. At each critical juncture in the pipeline more women and minority students drop out than do white males. (NRC, 1990b, p. 35)
Few research universities have more than one or two women in their mathematics departments, and even fewer have minority faculty. Thus women and minority role models are scarce in the mathematical sciences. In several of the committee's site visits, graduate students, both male and female, pointed out the paucity of women on the faculty. But in some other departments, committee members' questions on this topic were met with indifference.
The concerns of women interviewed by the committee were similar to but more pronounced than those of domestic students in general.
Women students report neither more nor less support from mentors than they believe is received by male students. However, more female than male students appear to drop out from frustration or other personal reasons after they have completed the first exam.
Committee Site Visit Report

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My impression is that the PhD completion rate is low. A woman student attributed this mainly to lack of encouragement. She was interested to learn that a group of [domestic] male students had expressed the same concerns about lack of encouragement.
Committee Site Visit Report
The committee observed in its site visits that departments that provide a positive atmosphere and expect women and underrepresented minorities to succeed often find that they do indeed succeed.There is an unusual amount of individual attention given [in the program], and the students appreciate it. Most of them came because of the interest expressed in them, an interest that does not diminish once they arrive. For women and minorities, the completion rate is 100% and [the rate] appears to be unusually high for all students. Every student seems to know where she/he is going at all times. Faculty are perceived to be extremely approachable, and we heard the term “open door policy” used frequently by students to describe the department.
Committee Site Visit Report
The same type of active recruiting that is effective for domestic students in general is effective for women and underrepresented minorities. Nevertheless, some universities choose to “jump start” a program.
There is a substantial and fairly effective program to help draw more women and minority students. This program is accorded 10% of the admission and support resources. After the admissions committee drafts its list of offers, 10% of the (total) offers are selected from the remaining applicants, with special attention given to women, minorities, students from weaker institutions, and so on. A similar procedure is followed in the award of support.
Committee Site Visit Report
At present several fellowship programs are directed toward women and underrepresented minority students. The NSF Minority Predoctoral Fellowship Program awards a small stipend to the undergraduate school from which the fellow has graduated. The Department of Education's Ronald E. McNair Post-Baccalaureate Achievement Program provides support to juniors and seniors from first-generation college families and encourages them to pursue graduate study.

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FOREIGN STUDENTS
Foreign students, now a majority in many doctoral programs, often have a higher level of mathematical experience than do American students. In many foreign countries, undergraduate students take many more mathematics courses than do most American undergraduates. In addition, some foreign students have completed a master's degree before coming to an American graduate program. The result is that, on entrance to a doctoral program, foreign students commonly have a one-to two-year advantage over American graduate students. Most domestic students and even many faculty members are not aware of this situation. When placed in the same introductory courses, foreign students thus often perform better than the American students. The committee believes that such a disparity can contribute to an increase in the drop-out rate among domestic students, especially among women and underrepresented minorities.
There was some tension in the first two years between American and foreign students, the latter typically arriving with a more advanced level of training. Some of the American students confessed to being somewhat intimidated by this difference in the beginning.
Committee Site Visit Report
Those foreign students who have already had first-year graduate courses should not be allowed to repeat these courses for reasons related to language and acculturation but should be placed in higher-level courses immediately.
Colleges and universities have of necessity had to rely increasingly on foreign nationals to teach undergraduate mathematics. The long-term ramifications and impact of this practice need careful study.
PLACEMENT
Every department should take an active role in the placement of new PhDs, an effort that currently is often viewed as the responsibility of the student and the thesis advisor alone. The networks used for recruiting students can be used for placement in academia. Faculty and alumni contacts can be used to facilitate placement in academia and industry. Relationships established in industry-related research and apprenticeship programs can be used to help place new PhDs in industry.
A large fraction of new PhDs go to faculty positions at institutions that view teaching as their primary mission. In 1990–1991, 80% of the new doctorates employed in the United States took positions in academia, and 50% of those took positions at institutions that did not offer a doctorate (McClure, 1991). Placement would be facilitated if students were prepared by their educational experience to become effective teachers. Graduates should

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be prepared by their programs of study to make the transition from highly active research programs to institutions with a smaller degree of research stimulation.
In addition to many tenure-track positions, non-tenure-track instructorships and assistant professorships are available at research universities. Temporary appointments with large teaching obligations are a growing phenomenon. It is not uncommon for new PhDs to hold two or more of these temporary positions before obtaining a tenure-track position. In such positions, time is spent mainly on teaching and seeking a job for the following year. Universities could treat temporary faculty as postdoctoral associates with teaching duties. Hiring in areas of strength and having senior faculty members act as research mentors would be a “win-win” strategy for both the department and the temporary faculty.
A new PhD with a broad applied background has opportunities in government, business, and industry and is less dependent on temporary positions in academia. In 1990–1991, 140 (20%) of the new doctorates taking jobs in the United States took non-academic positions in government, business, and industry (McClure, 1991). This figure represents a large increase over the 107 (16%) new doctorates in 1989–1990 who took jobs in government, business, and industry (McClure, 1991). However, most of the increase came from those holding degrees in statistics, operations research, and scientific computing. Very few new PhDs from pure or applied mathematics departments included in Group I or II (Jones et al., 1982; see also AMS, 1988, Notices, 35 (April), 532–533) found industrial employment.
The shortage of opportunities for postdoctoral positions in the mathematical sciences makes postdoctoral study an unlikely next step for many new PhDs. Federally supported postdoctoral positions in the mathematical sciences totaled 188 in 1988 (at which time there were 1280 federally supported postdoctoral positions in physics and 2587 in chemistry), and only one-half to one-third of these are available each year, enough to serve less than 10% of the new PhDs. Since many colleges and universities seek to make their tenure-track appointments to individuals who have completed a postdoctoral fellowship, the paucity of such fellowships reduces the opportunity for the many other excellent candidates.