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al Introduction and Historical Perspective Mathematics has become essential and pervasive in the U.S. workplace, arid pro- jections indicate that its use will expand, as will the need for more workers with a knowledge of college-level mathematics. However, socioeconomic and demographic projections as well as circumstances within the college and university mathematical sciences system suggest that an adequate supply of appropriately educated workers is not forthcoming. Development of mathematical talent will be impeded by the low general interest in mathematics as a college major; the relatively small numbers of minorities and women studying and practicing mathematics; a shortage of qualified faculty to deal with huge enrollments in low-level courses and students with widely varying levels of preparation; and the difficulty of maintaining the vitality of the mathe- matical sciences faculty. The MS 2000 Project and the Scope of This Report Because a healthy flow of mathematical talent is important for the nation's welfare, the National Research Council initiated in 1986 the project Mathematical Sci- ences in the Year 2000 (MS 2000) to assess the status of college and university mathematical scier~ces and to design a plan for revitalization and renewal. This report describes the circumstances and issues surrounding the people in- volved in the mathematical sciences, principally students and teachers. The description is not complete because comprehensive data are not available, but most data that are relevant and available are included and are adequate to describe the circumstances in the mathematical sciences. Two additional descriptive reports~ne on curriculum and the other on resources-are forthcoming,. Together these three reports will form the basis for the the MS 2000 Committee's final report, which will contain recommen- dations for actions to achieve revitalization and renewal of the college and university mathematical sciences enter- pr~se. This report is concerned with all students of collegiate mathematics. However, mathematics majors have a spe- cial role to play because they are the source of the new faculty members necessary to renew and sustain the sys- tem. And increases in the need for mathematics in the workplace in turn fuel a need for more academically skilled workers. A dramatic demonstration of this need is the 'For the purposes of this report the discipline referred to as the "mathematical sciences'' includes mathematics, applied mathematics. and statistics. A broader definition is generally used in the taxonomy of scientific disciplines. For a discussion of the mathematical sciences research community see Reneu ing U.S. Mathematics s: Critical Resou' ~ e fo'- the Future (National Academy Press. Washington. D.C.. 1984). pp. 77-85. Computer science is not a branch of the mathematical sciences. but its close ties with mathematics. both intellectually and administratively, have significantly affected college and university mathematical sciences over the past two decades. This report does not attempt to describe circumstances in computer science, but references tO computer science are necessary because of these ties and their effects.

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A Challenge of Numbers doubling of the number of scientists and engineers in a single decade (Figure 1.1~. Understanding students and teachers in the mathe- matical sciences who they are, what they learn and teach, and how they use what they learn-requires understanding the vast and diverse system in which they work. The mathematical sciences programs in U.S. colleges and universities account for nearly logo of all collegiate teach- ing in the United States and nearly 3097c of all collegiate teaching in the natural sciences and engineering. Eac term, approximately 3 million students are taught by more than 40,000 full-time and part-time faculty members and 8,000 graduate teaching assistants in 2,500 institutions. To better understand this system and how current circum- stances evolved, a review of events of the past 30 years is helpful. Three Roller Coaster Decades For centuries, mathematics has been recognized as interesting,, challenging,, and essential for the support of science and engineering. Within this century, mathematics has become much more broadly applicable and important. Giant strides toward reco;,nition of its significance were made during World War II. After World War II, U.S. mathematicians branched out, studyin;, and developing (in thousands) 4000 · . 3 000 r 2000 1000 1 976 - 11~ Engineers · Scientists 1986 FIGURE 1.1 Total number of scientists and engineers. SOURCE: National Science Board (NSB, 19871. 2 new areas in many directions very successfully. This period of innovation and the concurrent expansion of college and university mathematics programs positioned mathematics as a key participant in the nation's emphasis on science spurred on by the 1957 launch of Sputnik. Thus began three decades of extraordinary change-a decade of expansion, followed by a decade of adjustment and depres- sion, followed by a decade of partial recovery. The decade following Sputnik's launch was one of expansion for U.S. mathematics. Statistics became more widely recognized as a distinct discipline and began to flourish. Then as now, to a slightly lesser extent most of the research in mathematics and statistics was per- formed in universities. College enrollments increased, faculties expanded, and positions were plentiful. The number of bachelor's degrees in mathematics awarded annually tripled, and the number of graduate degrees increased fivefold in this decade. Support for specialized research programs, which was available from federal agencies for individuals, was ideally suited to the mathe- matical research mode. In the late 1 960s, immediately following the dramatic expansion of science and mathematics programs, the na- tion's interest and attention shifted to social issues. Al- though more students continued to enter colle ,e as access to higher education expanded significantly, many came without adequate preparation for college mathematics and with questions about the relevance of learning any. Over the 20-year period from 1965 to 1985, college enrollments doubled, and mathematical sciences enrollments more than kept pace. However, most of the increase in mathe- matics enrollments was at the lower levels, with remedial enrollments in high school mathematics taught in college leading the way. The surge in the numbers of decrees awarded in the mathematical sciences in the late 1960s and early 1970s and the lack of establi shed employment markets for mathe- maticians outside of academe created more degree holders than there were jobs, especially at the doctoral level; in addition, part of the response to increased enrollments in mathematics courses was to let student-faculty ratios in- crease. A depressed employment market resulted that

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Introduction and Historical Perspective fasted nearly a decade, into the early 1980s. To some extent this depression was spread across all science and engineer ing fields. Statistics was an exception, with some modest White Males increases in degrees granted and a better nonacademic 74% employment market. College and university mathematical sciences facul ties were changing. Increasing responsibilities for teach ing precalculus and high-school-level courses, the predic tion that college enrollments would soon decline, and the perception that mathematics Ph.D.s were plentiful changed employment practices on college faculties. The changes included the creation of positions with heavy teaching loads for full-time faculty and the use of more part-time and temporary teachers. Many faculty members had little time and motivation for personal scholarship; some lapsed into inactivity. Teaching introductory algebra and calculus to students majoring, in other areas became more widespread and restricted the independent growth of mathematics and mathematicians. Some faculty members did not teach what they thought about their research- and also had little enthusiasm or latitude to think about what they taught. These forces reduced the attention to curriculum develop ment and redo. In response to nationally articulated goals in the mid-1960s, the fraction of Ph.D.s on mathe matical sciences faculties had increased significantly to nearly 80% in four-year institutions, but a seeming mis match between training and duties prompted a reversal of this effort. In particular, new doctoral degree holders, educated for research, were mismatched with the teaching positions available. Consequently, both teaching and research suffered. In research universities, graduate students, plentiful in the 1960s, assumed a large share of the teaching responsi bilities. Inflation on a weak mathematics employment market and better opportunities in other areas such as computer science spread quickly among U.S. students, and the numbers choosing mathematics as a major area of study began to decline. This decline was partially offset at the graduate level by increases in the number of non-U.S. students that, combined with the significant decline in the number of U.S. students enrolled in mathematics, changed / _ ~- _ ~ - _ ~ White Females 19% Non-White Males 5% Non-White Females 1% FIGURE 1.2 Ph.D. degrees in mathematics, 1986-1987. SOURCE: American Mathematical Society (AMS, 1987~. to nearly one of two in 1988. This trend, coupled with the heavy teaching burden carried by graduate students, cre- ated teaching problems across the country. Factors other than the poor employment market also reduced the number of mathematical sciences majors. One factor was the predominance of white middle-class males in the study and practice of mathematics. Relatively few women and minorities were choosing mathematically based careers and curricula, although more women, more blacks, and more Hispanics were entering college. The fraction of bachelor's degrees earned by women did increase-from about one-third of the total in the mid- 1 960s to almost one- half by 1 98~but the increase was smaller at the master's level and smaller still at the doctoral level (Figure 1.21. That comparatively few blacks and Hispanics choose mathematically based careers has continued to be the case. The number of Native Americans choosing such careers is small but does reflect approximately this aroup's share of the total U.S. population, while Asian-Americans continue to show a preference for these careers. In the 1960s and 1970s, little attention was given to creating employment opportunities for mathematical sci- entists in the nonacademic workplace. Academic employ- ment in a research environment was the dominant destina- tion for degree holders in mathematics, and these opportu- nities had diminished. Thus, not only were there rough transitions to the workplace for those with bachelor's and master's degrees, but Ph.D.s, too, were not suitably matched the non-U.S. representation from about one of five in 1970 with the teaching jobs that were available in colleges. 3

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A Challenge of Numbers Mathematical sciences enrollments in introductory and remedial courses continued to increase in the 1970s, fueled by added mathematics requirements in the curricula of fields such as business and by shifts to majors that required more mathematics. This development reflected an increasing need for mathematics in the workplace, both for professionals in other areas and for mathematical scientists. Mathematics was emerging as more important in professional education, achieving a new prominence that complemented its centuries-long role in human intel- lectual development. Problem-solvina ability and adapta- bility dominated the requirements of new jobs. Said another way, liberal arts education especially mathemat- ics education was becoming closer to professional edu- cation. However, the nonacademic employment market for mathematical scientists continued to be poorly under- stood and was invisible to many. Departments across the country met the increased enrollments of the 1 970s with a variety of types of faculty members and the same traditional courses, mostly because they were busy and lacked resources (Figure 1.31. Many temporary and part-time teachers were hired on an ad hoc basis tenn after term. Thus began a dismantling of the buildup to a high fraction of faculty with Ph.D.s that had just been achieved. The responsibilities of departments became more diverse and more difficult to carry out (thousands) 2.000 1.500 1 .000 Advanced _., ~:~ I, I: ,. ~ , ,: :' ''lo .... :'::': ::' O:. ~ ~ ~ . ~ . ~ .: ~ :: :. I- . ~ . ~ ..~ .: ... , ~, : ~. ::. I.: :, ~ I. ~ : ~.. ~: ..~. ., ' 'a 1 1 111 111 :1 965 1970 1975 1980 1985 FIGURE 1.3 Left: Total undergraduate enrollments in mathematical sciences departments. Right: Mathematical sciences faculty at colleges and universities. SOURCE: Conference Board of the Mathematical Sciences (CBMS, 1987). because of heavier involvement in coordinating activities, fewer experienced and involved teachers, large remedial and placement problems' and fewer mathematics majors. No other collegiate discipline teaches as many students with such widely differing levels of preparation as does mathematics, and most of the students are expected to use the mathematics in subsequent courses. An overwhelming combination of problems of collegiate teaching-unmoti- vated and underprepared students, unenthusiastic teachers, language problems in the classroom, outdated and irrele- vant curricula and courses, large classes, heavy teaching loads, too few resources, and little use of modern technol- ogy-came together in the 1 970s and resonated in mathe- matics classrooms across the United States. By the early 1980s, the number of degrees awarded annually in the mathematical sciences had fallen by nearly 50% at all three levels (Figure 1.41. Occurring simultane- ously with the decline in the numbers of degrees awarded were increases in enrollments and in reliance on part-time faculty. Belatedly, institutions decided that the high mathe- matical sciences enrollments would persist, and they began to employ regular faculty members. By this time there were too few U.S. citizens among the new doctoral degree holders to meet the demand; indeed, there were overall shortages of candidates. There were (and are) no surplus ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ . . Hi. ~ . pUU1b U1 Illa~l~rnaucal sciences tn.~.s no large number 40,000 30.000 20.000 1 0~000 o 1970 1980 1985

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Introduction and Historical Perspective BOX 1.1 Computer Science Computer science has developed since World War I} from roots in mathematics and electrical engineering. It has become a separate academic discipline within the past two decades and has developed its own sources of students and federal fundin;, for research. (See the 1984 David Report (NRC, 1984) for a fuller analysis.) Computer science is not considered to be a part of the mathematical sciences, and that is the position taken throu Shout this report. However, older reports on the mathematical sciences may include computer science data, and computer science and mathematics continue to be administered by the same unit in most colleges and universities (see Box 3.3~. Because of these historical, administrative, and intellectual ties, the emergence and rapid growth of the discipline of computer science have had a significant effect on the mathematical sciences. At points in this report some of these effects are conjectured, but only the effects of computer science on the mathematical sciences are considered. No attempt is made to describe the conditions in computer science. Nevertheless, the grouping of computer science with the mathematical sciences in many college and university departments and the dual teaching roles of many faculty members are facts. The 1985 CBMS Survey (Box 3.3) concluded that in four-year colleges and universities, half (49%) of all computer science course sections were taught in departments with mathematics and the other half (51 Tic) in com- puter science departments. (This breakdown did not include courses in computing, taught by many units in busi- ness and engineering, for example.) Thus approximately 270,000 students enrolled in computer science courses were taught in mathematics departments in fall 1985. This compares to estimated enrollments of 1,827,000 in mathematics and statistics courses in these departments in fall ~ 985. Thus approximately 13% of the teaching, in these departments was in computer science (the 1985 Annual AMS Survey results give an estimate of 10% rather than Who, but the 1984 Annual AMS Survey yielded 12% see Box 3.2~. In two-year colleges in 1985, approximately 10% of the 1 million students enrolled per term were enrolled in computing and data processing. The 1985 CBMS Survey listed 27,500 bachelor's degrees awarded by departments of 'mathematics" and 400 awarded by departments of statistics in the period July 1984 to June 1985. Of these 27,500 degrees, 40% were awarded in computer science (8,700) or jointly with computer science (2,5003. The 1985 CBMS Survey reported that of the 3,750 Ph.D.s on the nation's full-time computer science faculty, 41 % had their doctorates in mathematics. Of the 2,200 Ph.D.s on the part-time computer science faculty, 61 % had their doctorates in mathematics. Of the total full-time computer science faculty, 35% had their highest decrees in mathematics, and of the total part-time computer science faculty, 42% had their highest degree in mathematics. Of the 5,650 members of the full-time computer science faculty, 2,O50 were employed by 'mathematics'- departments. Of the 5,350 members of the part-time computer science faculty, 3350 were employed by 'mathematics" departments. This translated to 3,150 full-time equivalents (FTE) offaculty members teaching one- half of the computer science sections in "mathematics" departments and 4,250 FTE of faculty members teaching the other half in computer science departments. It is noted that computer science departments are concentrated in the universities where teaching loads are lower. s

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A Challenge of Numbers of postdoctoral positions and no candidates from other disciplines who fit the faculty needs. Ad hoc hiring practices continued, partly because of a lack of suitable candidates for regular faculty positions. From another perspective, by 1970 a large infrastruc- ture of mathematical sciences graduate study and research had been established across the country and was spread through more than lSO universities. Success in research was clearly the principal criterion for respect within this community, and the research environment was clearly the best in the world. However, federal support formathemati- cal sciences research became less available, as did other governmental and institutional support (NRC, 1984), and the employment market was very depressed. There were many discouraged faculty members and persons seeking faculty positions. Many defected to other areas. By 1980, the mathematical sciences infrastructure was clearly weak- ening. During the 1970s and continuing until the present, many departments' programs, especially at four-year col- leges, contained a mixture of mathematics, statistics, and computer science. Planning, was confounded further by conflicting trends within these three disciplines. Computer science was booming, statistics was growing steadily, and mathematics was struggling to adjust to a depressed em 00000 0000 000 ~ he.. 1~^ 00 1950 1956 1962 1968 1974 1980 1986 o Coo ~ Too 00 ° 04C.°0° c, ~ .e ,~ ,G, ~ Bachelor's Master's FIGURE 1.4 Mathematical sciences degrees awarded SOURCE: National Center for Education Statistics (NCES, 198Sa). ployment market, fewer majors, and huge enrollments in introductory courses. Computer science was emerging as a separate aca- demic discipline. Many computer science programs had been formed within mathematical sciences departments, and the number of majors and the course enrollments were rising rapidly. Since there were far too few people with academic degrees in computer science to fill the available faculty positions and because of computer science's close connections to mathematics, many mathematics faculty members were able to cross over to computer science. And students who once might have been mathematics majors began to choose computer science as a major. By 1988 separate computer science departments had been estab- lished in most large universities, but in smaller institutions the hybrid department was still the rule. Approximately half of all computer science enrollments continue to be in these combined departments, thus competing for faculty time and energy and the interest of the students (see BOX 1.1~. While much of the ferment over the past two decades also affected other academic disciplines, especially the sciences and engineering, the impact on the mathematical sciences was more extreme. The declines in the numbers of mathematics degrees awarded were relatively larger, and the declines in the numbers of majors in other science and engineering disciplines turned around much more quickly in the early 1980s. Mathematics has been the slowest field to recover although there has been some recovery-one reason being the close ties between mathe- matics and academe. No other science or engineering discipline depends as heavily on academic employment for its graduates, especially those with doctorates, as does Doctorate mathematics. Consequently the health of the mathemati- cal sciences enterprise is very closely tied to the health of education, especially higher education. By 1982 a number of serious problems posing a risk to the general health of mathematics had become apparent. The numbers of degrees awarded were near the lowest; course enrollments were the highest, with the heaviest concentration at the lower levels; teaching loads had in- creased dramatically; federal support for research was at a

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Introduction and Historical Perspective BOX 1.2 Sources of Data Several sources of data were used to compile this report. The main sources include: · American Mathematical Society (AMS); · Conference Board of the Mathematical Sciences (CBMS); · National Center for Education Statistics (NCES) of the Department of Education; · National Research Council (NRC); and · National Science Foundation (NSF). In general, the mathematical professional societies' (AMS and CI3MS) data relate only to the field of mathe- matical sciences and do not allow any comparisons across fields. When field comparisons are made, the sources of data are usually the NRC, NSF, or NCES. Inconsistencies do arise, partly because of different survey populations. For instance, some data on mathematical sciences include data on computer science. Where possible in this report, mathematical sciences data have been separated from computer science data. It is not feasible to reconcile or explain all the differences Analysis of data in detail reveals differences that cannot be reconciled, but the implications of these differences appear to be minor. Nevertheless, the different sources have beers found to be consistent enough to depict the general circumstances in the mathematical sciences enterprise. The tables in the text are numbered consecutively within each chapter, as are the figures (mostly graphs). Tables giving the data used to construct the figures presented in this report are included in the report's appendix and are numbered to correspond to the relevant text chapter rather than to a particular text figure or table. For example, Table A4.3 is the third table in Appendix Tables that contains data for Chapter4, while Table 4.3 is simply the third table in the text of Chapter 4. The sources of the information shown in the tables and figures are given according, to the standard referencing system used throughout the report. very low point, especially in core areas of mathematics; and faculty morale was frequently low. Responding to the expansion of the previous two decades and the associated problems had talcen most of the faculty time and energy. During that period, whole new areas of mathematical sciences had developed, including operations research, discrete mathematics, mathematical biology, statistical design and analysis, and nonlinear dynamical systems. In fact, the term "mathematical sciences" itself had become a part of the taxonomy of science. In spite of these new developments in the mathematical sciences, new applica- tions, and new opportunities for using technological ad- vancements, the teaching of mathematics had essentially not changed. Both the curricular content and its delivery had remained static. Mathematical sciences departments were not able to simultaneously cope with enoImous instructional loads, maintain excellence in faculty scholar- ship, and allocate resources to innovations or even known improvements. The forces at work were too diverse and too disparate. National Efforts Toward Renewal In 1982 the mathematical sciences community began to address these problems on a national level. In 1984 the National Research Council (NRC) published Renewing U.S. Mathematics: Critical Resource for the Future (re- ferred to as the David Report; NRC, 1984), documenting 7

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A Challenge of Numbers the weakening of federal support for research in the mathe- matical sciences. That report was the first of a series of efforts within the NRC and in professional societies to assess the health of the mathematical sciences and to design a plan for renewal. The NRC project Mathematical Sciences in the Year 2000 (MS 2000), of which this report BOX 1.3 Statistics The discipline of statistics is included in this report as a part of the mathematical sciences, pnnci- pally because statistics has an intellectual base in mathematics, mathematics students are the principal source of statistics graduate students, and significant federal funding for academic research that develops fundamental statistical concepts and methods comes from the "mathematical sciences" units of federal agencies (NRC, 1984~. Degree programs in statistics are mostly gradu- ate degree programs. The number of students en- rolled in statistics and the number of undergraduate statistics majors are much smaller than the analo- gous numbers for mathematics. In major universi- ties, statistics usually constitutes a separate aca- demic department, but in other institutions statistics is likely to be taught in the same unit as mathematics (see Box 3.3~. In addition, statistics courses are taught in a variety of administrative units, including business, engineering, medical sciences, and social sciences. Partly because of the close administrative and intellectual ties between statistics and mathematics, much of the data on statistics in colleges and univer- sities in this report is a;,gre~,ated with analogous data on mathematics. Some disag~regation is possible and has been done when possible in this report. However, in general, the data are dominated by those for mathematics, and caution must be used in draw- ing conclusions about statistics from the aggregated data. 8 is a part, was initiated in 1986. MS 2000 is an effort to assess the state of college and university mathematical sciences and to design a national agenda for revitalization and renewal. The events of the past three decades, detailed here, lend urgency to this effort. A major step in broaden- ing the audience for this message and including all of mathematics education was taken by the NRC in publishing Everybody Counts early in 1989 (NRC, 1989~. The issues and implications identified in this report and the two additional descriptive reports on curriculum and resources will assist the MS 2000 Committee in presentin;, an a ,enda that will ensure a healthy flow of mathematical talent into the next century. Contents of This Report Fundamentally this report concerns students and teach- ers. The events and forces described above indicate the complexity of this simple-sounding enterprise and how the current predicaments have developed. Box 1.2 describes the sources of the data used to compile this report and explains the relationship between the text tables and fi ,- ures and the additional data presented in the report's Appendix Tables. Box 1.3 details characteristics of the statistics component of college and university mathemati- cal sciences and describes the the context in which infor- mation in that area is provided. Chapter 2 describes in broad strokes the larger communities of the U.S. labor force and higher education, which both encompass the mathematical sciences enterprise. Chapter 3 describes the major components of, trends in, and utilization of college and university mathematical sciences. Chapter 4 focuses on mathematical sciences majors, both undergraduate and graduate. Chapter 5 describes mathematical scientists in the workplace; colleges and universities are a principal topic, since academe is still the dominant employer of mathematical scientists. That situation, however, is chang- in=. The increased use in various professions of the mathe- matical sciences adds to their traditionally important uses in everyday life, civic activities, and our rich intellectual culture.