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4 The Problem of Renewal The key problem facing the mathematical sciences today remains what it was in 1984: renewal. The pressing concerns of renewal are, Where will the mathematical talent come from? How can young talent be attracted to and retained in the career path? How can researchers be helped to remain active and be encouraged to serve as mentors for the next generation? The problem of renewal is crucial because of the increasing demand for mathematical scientists as educators and re- searchers. DEMAND FOR MATHEMATICAL SCIENTISTS Mathematics and familiarity with mathematical modes of thought are the foundations on which are built education in other scientific disci- plines, and increasingly education in various areas of business, eco- nomics, and social science. Mathematical scientists are needed as educators to satisfy this growing demand, as well as to provide the increasingly sophisticated training of new mathematical scientists needed in increasing numbers by many quantitative areas of our complex society. United States Ph.D. production (supplemented by a large influx of foreigners) is at present barely sufficient to meet the current needs of our educational institutions, and demographers warn that faculties will need to grow after the year 2000 as the children of the baby-boom generation reach college age. Mathematical scientists are necessary also as researchers, because the expanding use of mathematics in all quantitative fields, the height- ened mathematical sophistication of users, and the explosive growth 55

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56 RENEWING U.S. MATHEMATICS in computer modeling are all fueling the demand for mathematics research.2 Since World War II the trend toward quantification has affected not only traditionally quantitative areas but also such fields as biology, business, and economics. This trend seems to be continu- ing and even to be increasing, and it may be regarded as a natural phase of development that follows after observation, classification, and other qualitative methods alone become inadequate. Mathematics is vital to this progression because it is the language in which funda- mental concepts and relationships can be precisely specified, manipu- lated, and extended for greater understanding. The spread of com- puter modeling has also generated a commensurate demand for mathe- matical expertise: mathematical scientists often provide critical steps in the process of developing computer models and algorithms, and they also address issues such as convergence criteria, error bounds, and expected asymptotic behavior, which are important for purposes of validation and control. Mathematicians need broad training in order to be responsive in this research environment. Thus, to avoid serious declines in scientific and technological educa- tion, as well as shortages of urgently needed mathematical scientists at all levels, mathematical sciences Ph.D. production will likely have to increase in the near future. Assuring that the profession can attract bright young people is a goal to be addressed now, before large incre- mental demands for additional faculty and new mathematics strike. SHORTFALL IN SUPPLY The 1989 book Prospects for Faculty in the Arts ~ Sciences,3 by W. G. Bowen and J. A. Sosa, warns of near-term faculty shortfalls in U.S. colleges and universities. For instance, the authors project 9300 fac- ulty openings in mathematics and the physical sciences in the period 1997 to 2002, but fewer than 7500 candidates, with the result that a maximum of only 80% of available faculty slots will be filled, assum- ing current student to faculty ratios. Bowen and Sosa project a very flat supply of mathematics Ph.D. degree holders seeking U.S. aca- demic employment over the next 15 years, averaging just 356 annu- ally. Doctoral degree production in the mathematical sciences declined steadily over many years, falling from a peak of 1281 in 1972 to a low of 688 in 1985. During recent years the percentage of U.S. citizens receiving a Ph.l). in the mathematical sciences has clipped below 50%. Some evidence, albeit inconclusive, suggests that the increase in sup-

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THE PROBLEM OF RENEWAL port given to graduate students and postdoctoral researchers since the early 1980s is beginning to have an effect. After three years of essen- tially flat Ph.D. production, data from the spring and summer of 1989 show that the total number of Ph.D.s awarded had increased by 12% over the previous year, to equal approximately the level of production in 1978. In acldition, women constituted a record! 24% of the U.S.- citizen doctoral degree recipients. Whether or not these changes mark the beginning of a bona fide turnaround, the rate of influx of talent into the field will remain a high-priority concern for a number of years. REASONS FOR THE SHORTFALL Changing Demographics The problem of renewal is made more difficult by the shifting demo- graphics of the United States. The report Workforce 2000 (Hudson Institute, Indianapolis, Ind., 1987) has brought to public attention the dramatic changes occurring in the U.S. population and in the work force on which the economy depends. Its message that only 15% of net entrants to the work force between 1985 and the year 2000 will be native-born white males has surprise`] many people, driving home the point that-in the future groups other than white males will provide much of the new talent for the nation. This is a major issue in all the sciences, which are now so heavily dominated by white males. Science cannot continue to depend on the brain power of white males; their participation rate in the sciences would have to increase greatly to offset their declining numbers among work force entrants. Therefore, all branches of science must greatly increase efforts to attract and cultivate women and minorities. In mathematics, where women hold less than one Ph.D. in five and the numbers of blacks and Hispanics are almost vanishingly small, such efforts will need to be very intensive. The issues were vividly por- trayed in the human resources chapter of the recent NRC report Every- body Counts.4 Cultural and Educational Problems The problem of renewal in the mathematical sciences, exacerbated by changing demographics, must be seen and attacked in a broader con- text than that of graduate student recruitment and support. Attention 57

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RENEWING U.S. MATHEMATICS must be paid to the entire mathematical pipeline, a requirement emphasized in Everybody Counts: The underrepresentation of minorities and women in scientific careers is well documented and widely known. Less widely known is the general under- representation of American students in all mathematically based graduate programs. Evidence of disinterest in mathematics permeates all racial, socio- economic, and educational categories, although the level of disinterest varies greatly among different groups. Young Americans' avoidance of mathematics courses and careers arises from immersion in a culture that provides more alternatives than stimulants to the study of mathematics. Without motivation and effective opportunity to learn, few students of any background are likely to persevere in the study of mathematics.... Developing more mathematical talent for the nation will require fundamental change in education. Our national problem is not only how to nurture talent once it surfaces, but also how to make more talent rise to the surface. A1- though more must be done, the United States is reasonably successful in tam ping and channeling the highly visible talent springs which develop without special support from formal schooling. But these sources are inadequate to our national need. We must, in addition, raise the entire water table.5 The forthcoming final report of the MS 2000 project will detail many crucial recruitment and educational reforms needed into the twenty- first century. All will require substantial input from the mathematics profession for Planning and implementation. v What Discourages Talent How do young people choose a career in the mathematical sciences? A very few young people with mathematical talent come into contact with interesting aspects of the subject and become committed to mathe- matics at an early age, but this is very much the exception. Most young people who decide to study mathematics make the commitment much later, balancing their aptitudes against the possible disciplines to pursue while weighing the quality of life offered in each profession. Obtaining information about mathematical careers is often difficult for the prospective mathematics major, because most teachers and other students are poorly informed about the possibilities. An unusu- ally enthusiastic high school teacher, a professor in the early years of college, or some family friend or relative in the profession is the usual adviser. The picture they convey to the young student necessarily contrasts the joy of doing mathematics with the difficulty of obtaining support for graduate and postdoctoral studies, the heavy teaching loads even in the predoctoral years, and, after becoming established in 58

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THE PROBLEM OF RENEWAL research, the sudden decrease in the midyears in the ability to obtain research support. All this is in sharp contrast to career opportunities in the other sciences and in engineering. The Leaky Pipeline The mathematical sciences career path includes education from secon- dary school through the completion of the Ph.D., and professional development beyond that. In assessing the career path, this commit- tee considered the quality of undergraduate and graduate eclucation, the efficacy and efficiency of the process by which students become researchers and teachers, and the opportunities for continued profes- sional growth throughout a mathematician's career. Renewal efforts critically depend on what takes place in doctorate-granting depart- ments, and that is the milieu this committee addresses. Although problems exist throughout the career path, this discussion focuses on those that directly affect the production of research and researchers. On a national level, evidence such as that shown in Figure 4.1 docu- ments the leaky educational pipeline in mathematics. About half the students in the mathematics pipeline are lost each year. (Only U.S.- citizen Ph.D.s are shown in Figure 4.1 because the chart represents the flow of U.S. students through the mathematics pipeline.) In high school and college, mathematics acts as a filter rather than as a pump; students are deterred, and mathematical talent is not identified and encouraged. As for graduate studies, the ratio of doctoral degree recipients to bachelor's degree recipients is lower in the mathematical sciences than in many other fields: over the period from 1971 to 1985, this ratio averaged 4% for the mathematical sciences, whereas it was 6% for engineering, S% for the life sciences, and 15% for the physical sciences.6 Clearly, talent and productivity are being lost throughout the mathematics pipeline. Many career path shortcomings affect the production of Ph.D. mathe- maticians. In graduate school, only 18% of mathematical sciences students receive research funds to support themselves' compared to 28% in the social sciences, 45% in engineering, and 50% in the physical sciences (see Table 2.4~. Upon receipt of a doctorate, the neophyte mathematical scientist does not generally have the benefit of postdoc- toral research training, but moves directly into an assistant professor- ship. Only 21% of mathematical sciences junior faculty (assistant professors and instructors) active in R&D received federal support in 59

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10,000,000 1,000,000 100,000 Number of Students 10,000 1,000 100 Nine Graders ~(3.6 million) ~1 ~ ~ ~ 1 __ . RENEWING U.S. MATHEMATICS , ~1 Freshmen ~ (294,000) 'AREA_ . __ 1 === = ~ __ ~ B.S. Degrees (113~) M.S Degrees (2,700) Ph.De Degrees (400) 1972 1976 1980 Yet FIGURE 4.1 U.S. students in the mathematics pipeline. 1982 1986 SOURCE: From Mathematical Sciences in the Year 2000 project, reprinted from Na- tional Research Council, A Challenge of Numbers: People in the Mathematical Sciences (Na- tional Academy Press, Washington, D.C., 1990), p. 36. 1987, compared with 53% in chemistry and 67% in physics and astron- omy.7 Further along the pipeline, only 18% of Ph.D. mathematical scientists in four-year colleges and universities surveyed in 1985 could call research their primary activity, as compared to 33% of chemists and 42% of physicists and astronomers.8 The correlation between research funding for junior researchers and research activity in later years is striking. In addition, with the current reward structure' it appears that the 82% of mathematical scientists who consider some- thing other than research their primary activity are undervaluecl. Mathematical sciences departments have adapted to these conditions but are unable to overcome them. While preparing this report, the committee asked four department chairs to write essays giving anec- dotal accounts of problems and solutions. The problems described often stemmed from the fact that departments in the mathematical sciences have the broadest mission in the university, comprising re- search and graduate education, undergraduate education, upper- and lower-division service, and community outreach and education. Par- tial solutions often came from cooperation between departments and their administrations in choosing priorities among these missions and setting mutually satisfying goals. 60

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THE PROBLEM OF RENEWAL How an increase in funding ~ in this case, from the university- can lead to marked improvements in overall departmental quality is re- flected in one department chair's statement: Impending shortages of mathematicians and resulting increased competition between universities have made it more vital than ever to establish a first-class senior faculty.... [This] enabled w to argue for and achieve a general increase in salary levels.... In several cases research activity has improved as a result of the new climate in the department. Indeed, in some instances this has resulted in federal funding for those who had been off the rolls for some time. Faculty improvement has begun to have an effect on our resources for the future, especially with regard to the quality of our graduate students.... we now find a small group worthy of any institution. The Reward Structure Another chair's essay pointed out a problem with the academic re- ward structure: "A number of faculty develop instructional material, textbooks, and software [yet they] receive little recognition for these efforts, and a portion of the faculty attach negative weight to these activities. With our many missions, we have a responsibility to re- ward excellence in a broad range of activities." Another stated that "promotion or tenure without grant support is extremely difficult." Later, the same writer noted, "The implementation of [better courses for elementary and secondary school teachers] will require the partici- pation of active mathematicians, although this is not always easily achievable because of the possible detrimental effects on one's ca- reer." The current reward structure may be inferred from the results (Table 4.1) of a 1985 Conference Board for the Mathematical Sciences (CBMS) survey, which asked university department chairs to rate the impor- tance of various professional activities to promotion or salary deci- sions. ADDRESSING THE SHORTFALL Recruitment Recruitment requires an active effort on three fronts: improving the quality of the career path within mathematics, improving the external appeal of the profession, and performing recruiting drives. The first two lay the groundwork to maximize the effectiveness of the third, which is not discussed here. The quality of the mathematical sciences 61

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RENEWING U.S. MATHEMATICS TABLE 4.1 Department Chairs' Valuation of Professional Activities, 1985 Valuation by Department (Percent) Mathematics Statistics Professional Activity High. Low. High. Low. Published research 96 0 100 0 Talks at professional meetings 42 5 25 11 Supervision of graduate students 34 7 81 0 Classroom teaching performance 70 3 71 6 Undergraduate/graduate advising 9 22 21 21 Service to dept., coil., or univ. 31 5 31 11 Activities in professional 22 8 31 6 societies or public service Expository or popular articles 22 13 14 19 Textbook writing 9 35 12 50 ~ "High" means 4 or 5 on a scale of importance running from 0 to 5; "Low" means 0 or 1. SOURCE: Adapted from National Research Council, A Challenge of Numbers: People in the Mathematical Sciences (National Academy Press, Washington, D.C., 1990). career path must be improved in order for recruitment efforts to suc- ceed. Bowen and Sosa state: While many variables affect decisions to pursue graduate study, students are surely more likely to seek Ph.D.'s, and to think seriously about teaching and research vocations, when employment opportunities in academia are attrac- tive. The historical record offers strong support for this simple line of reason- ing.... [T]he number of newly awarded doctorates in almost every field increased dramatically in the 1960s. It is no coincidence that those were also the years when the number of academic appointments was growing rapidly, faculty salaries were rising, and financial aid for graduate study was widely available. Subsequently, the grim academic labor markets of the 1970s were accompanied by a sharp decline in the number of new doctorates earned, especially by U.S. residents.9 It is reasonable that, in the absence of strong counter-influences, these same correlations hold for individual fields. A career in the mathe- matical sciences suffers by comparison with those in other fields, due to long-term effects of funding imbalances, so that recruiting efforts in mathematics are hindered. The 1984 National Plan and this committee's updated recommendations address precisely these problems. 62

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THE PROBLEM OF RENEWAL Equal in importance to the effort to improve the career path is the need to improve the external appeal of the profession. The mathemat- ics profession must reach out to students and the general public to show the value and accessibility of mathematics; the image of rigid, unquestionable theorems should be complemented by that of excited, creative, and inspired people developing new mathematics. This is part of the role of the Board on Mathematical Sciences' the loins Policy Board for Mathematics, and the professional societies, but is also a challenge for individuals throughout the mathematical sciences com- munity. The beauty, history, and excitement of mathematics are sel- dom conveyed to students; in fact, too often they receive the impres- sion that mathematics is all 150 years old and stagnant, and that individuals cannot contribute except in limited and long-term ways. Thus the field appears uninteresting and intimidating. Mathematics educators at all levels can reverse these negative images. Students who see computer-oriented work as glamorous need to learn that, without mathematics, the power of computers could not be ap- plied to many real-world problems. Mathematical work crucial to global warming studies, aircraft design, or medical imaging devices should capture the attention of students who think that mathematics is irrelevant to modern developments. Two high school student win- ners of the 1988 Westinghouse Science Talent Search carried out new mathematics work, exemplifying the fact that newcomers can contrib- ute. Advances such as waveless and Karmarkar's algorithm show that it need not take decades for research to bear fruit. Finally, students should know that some 525,000 persons have received some mathe- matical sciences degree in the United States in the last 40 years. Most are still in the work force, yet three-fourths of them are working in areas other than the mathematical sciences. Clearly, a mathematical sciences degree provides a flexible foundation. Many parts of the 1984 National Plan would aid recruitment by im- proving the career path to bring it more in line with those of other sciences, by increasing the attractiveness of a life in mathematical research, and by increasing the cadre of active, enthusiastic research- ers' who serve as recruiters as well as mentors and positive role models. Replenishment Recruitment is just the start of renewal. If renewal is to be achieved, young people who choose to specialize in the mathematical sciences 63

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RENEWING U.S. MATH~ATICS must first be well trained and then must be encouraged to remain active throughout their careers. The 1984 National Plan addresses both of these goals. The former can be accomplished by providing more research time for graduate students and postdoctorals, and by ensuring support for established researchers who will act as mentors. The latter can be attained by supplying a sufficient number of grants to encourage continued active research and professional development. The laboratory sciences generally provide longer periods of direct interaction with faculty mentors than do the mathematical sciences: close contact in a research context begins early in a graduate student's career and extends beyond the doctorate for additional training. Beginning graduate students in the laboratory sciences may learn as much from advanced graduate students and postdoctorals as they do from the principal investigator- the group provides a mutually sup- portive and nurturing learning environment for all. The challenge for the mathematical sciences is to create an analogous environment for their own graduate students and postdoctorals. The 1984 National Plan stipulates financial support for graduate stu- dent and postdoctoral research training, and, by recommending an increase in the number of established, funded investigators, provides for an environment that fosters mentor-apprentice interaction. Profes- sors with good abilities and track records as mentors should be en- couraged by, for example, being provided with their own postdoctoral funds, either individually or in groups. An apprenticeship period is particularly important now because breadth is becoming vital to research in the mathematical sciences. Research- ers have become more problem oriented, and so each investigator must be familiar with a wide variety of mathematical tools: unifica- tion of the field relies on, and demands, broader experience. Many mathematical scientists must also be adept in areas of engineering, biology, management science, or other disciplines. This breadth is apparent in much of the work profiled in Appendix B. The mathe- matical sophistication of researchers in many fields is increasing, with the result that mathematical sciences research problems are appearing more frequently in a nonmathematical context. The graduate student and postdoctoral research time stipulated in the 1984 National Plan would provide the quality training needed to renew the intellectual base of mathematics. 64

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THE PROBLEM OF RENEWAL Unless its members possess diverse skills and interests, the field will not be able to respond to the demand for new mathematics arising in novel areas. The scientific and technological competitiveness of the United States is ever more dependent on our national ability to re- spond quickly to new developments, with minimal time to intellectu- ally '`retool." Unifying the mathematical sciences and linking them more strongly with other quantitative fields are important goals in todays globally competitive environment. Replenishment of the field also demands that the potential of well- trained Ph.D. degree holders be realized. This can be fostered in part by supporting a larger cadre of individual investigators, including young investigators, as recommended in the 1984 National Plan. Research funding can also enable travel and workshop attendance, which expose researchers to new ideas, quicken their response to new research directions, and provide intellectual invigoration that can improve their ability to act as mentors. The availability of summer support would encourage a larger cadre of university mathematical sciences faculty to remain active in research. Although not all of these faculty members will carry on research throughout their careers, far too many currently cease such efforts within the first few years after receipt of the Ph.D. Broadening the Reward Structure The multiple roles played by mathematical sciences departments- providing general and specialized education for a large fraction of the college and university population, influencing society's mathematical knowledge through elementary and secondary school teacher training and expository writing, and developing new mathematics and future mathematicians are all essential. Therefore, a corresponding reward structure is needed, so that people are encouraged to devote their energies to whichever of these valuable tasks best suit them. This change would make more efficient use of the human resources in the field. A revamped structure should include rewards for the following activities: Teachingparticularly conscientious and effective teaching at all levels, including the lower-division service courses and courses for prospective teachers. Tailoring courses for nonmajors, directing and improving the teaching abilities of graduate student instructors, and 65

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RENEWING U.S. MATHEMATICS designing computer-oriented classes and their software are activities that must be encouraged. . Mentoring guiding and enhancing the education of undergradu- ate majors, graduate students, and postdoctorals. Good mentors devote time and energy to this process, honing their methods and maintain- ing broader interests than those necessary to do research alone, so as to give their apprentices the breadth and depth required of today's mathematicians. . Outreach collaborating with people in other fields, recruiting good students, particularly from among women and minorities, and communicating with local education professionals and with industry. Since the long-term health of mathematics depends on maintaining strong ties with the other sciences, recruiting top-quality people, and satisfying the mathematical needs of society and industry, depart- ments and universities should encourageand reserve funds for all of these outreach efforts. In short, there should be a broader spectrum of respectable careers available to people educated in the mathematical sciences. NOTES 'Current mathematics faculties cannot accommodate this growth. The number of full-time mathematics faculty at research universities actually decreased by 14% over the period 1970 to 1985; the simultaneous 60% increase in course enrollment was handled by tripling the number of part-time faculty. 2The demand for nonresearch mathematical scientists is also growing. The number of secondary school mathematics students will rise before the college population does, requiring more mathematics teachers. And the projected demand for mathematical scientists at all levels is expected to increase by 29% between 1986 and 2000, compared to a 19% growth in overall employment. Many of these people will be employed in science and engineering. Considering all degree levels, the employment of mathemati- cal scientists has already tripled over the period from 1976 to 1986, showing a 10% annual growth rate second only to that for computer specialists among the science and engineering fields. 3Bowen, W.G., and Sosa, J.A., Prospects for Faculty in the Arts ~ Sciences (Princeton University Press, Princeton, N.J., 1989). National Research Council, Everybody Counts: A Report to the Nation on the Future of Mathematics Education (National Academy Press, Washington, D.C., 1989). sNational Research Council, Everybody Counts, 1989, pp. 17-18. 66

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THE PROBLEM OFRENEWAL 6Data from Table 2.1, National Research Council, A Challenge of Numbers: People in the Mathematical Scicnecs (National Academy Press, Washington, D.C., 1990). 'Data from the Survey of Doctoral Recipients project office, National Research Council (personal communication). 'Doctoral Scientists and Engineers: A Decade of Change, NSF 88-302 (National Science Foundation, Washington, D.C., 1988). 9Bowen, W.G., and Sosa, J.A., Prospects for Faculty in the Arts & Sciences, p. 162. * 67

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