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|3 Majors in Mathematics and Statistics Attrition from the mathematical sciences pipeline is approximately half per year after the ninth grade. The number of degrees awarded annually at each of the three degree levels increased sharply after 1960, peaked in 1970, and then declined sharply to about the mid-1960s level and about the average levels of the past 40 years. Many students decide to major in mathematics after entering college, and the circumstances In graduate and undergraduate programs are closely connected, being simultaneously subject to the same forces. Relatively few women, blacks, and Hispanics receive degrees in the mathematical sciences, especially at the graduate level. Graduate students, nearly one-half of whom are not U.S. citizens, are mostly supported by teaching assistantships. Introduction The prebaccalaureate "pipeline" population in mathe- matics and statistics is elusive as students move in and out of degree programs. The attrition rate for this population is certainly not uniform from year to year, but, for recent years, assuming an attrition rate of 50% per year from the ninth grade onward gives surprisingly close estimates of the actual number of bachelor's, master's, and doctoral degrees awarded in mathematics and statistics (Figure 4.1). As indicated in Figure 4.1, in 1972 there were an~roxi ", , Halving twice more on the way to the master's degree yields 3,500 degrees (there were 2,700 reported in 1982), and three more halv~ngs yield 437 doctoral degrees in 1985, which closely approximates the 400 U.S. citizens who received doctorates in 1986. (It is noted that the assumption of five years from the baccalaureate to the doctorate in the model is short of the average of six years of registration, and there are no adjustments for non-U.S. students below the doctoral level.) The losses of mathematical talent are not evenly distrib- uted 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. In the eighth grade the mix of students is roughly equal to their representation in the population. Fifteen years later this representation is highly skewed in favor of non-HisDanic white males. with mately 3.6 million U.S. students in the ninth grade. An attrition rate of 50% per year yields about 225,000 college freshmen in 1976. An attrition of 50% in each of the next four years of college yields about 14,000, approximately the number of bachelor's degrees awarded in mathematics 78% of doctoral degrees being earned by 42~o of the and statistics in 1980. (The actual number reported for population (Figure 4.2). The highest losses in the pipeline 1980 was 11,000 (NCES, 1988a), but this figure was near occur among blacks and Hispanics, two segments of the the minimum reported since 1960 end has increased since.) population that are increasing. 35

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A Challenge of Numbers 10,000,000 1,000,000 100,000 1 ~ : = 10,000 1.000 100 Ninth Graders . (3.6 i~IrO~--~ Frestmen \~(294~000) -~l,< B.S.I \_(11 ~ C=; ..... .. ~ : r degrees 1 000) _ _ M.S. pegrees ~ . ~_. ~r~s 1972 1976 1980 1982 1986 FIGURE 4.1 Students in the mathematical sciences pipe- line-about half are lost each year. SOURCES: National Center for Education Statistics (NCES, 1987a), Cooperative Institutional Research Program (CIRP, 1987b), and Ameri- can Mathematical Society (AMS, 1986 to 1988~. The numbers of mathematical sciences degrees awarded annually to U.S. citizens and others at the bachelor's, master's, and doctoral levels have shown similar patterns since 1950 (Figure 4.3~. Steep increases occurred from the mid- 1 950s to peak production in the early 1 970s. Equally steep decreases then occurred until the early 1980s, and moderate increases have occurred since. Although similar patterns exist in other areas of science and engineering, the numbers of mathematical sciences degrees have been the slowest to rebound from the declines of the 1 970s and early 1980s, and degree production is still very low even when compared to that for the other sciences and for engineering. Current numbers of degrees awarded annually are approxi- mately the same as those for the early 1960s and very near the averages for the past 40 years: 15,000 bachelor's degrees, 3,000 master's degrees, and 800 doctoral degrees. These numbers are each 40-45% lower than the analogous numbers for peak production in the early 1970s. This supply appears to be slightly short of current demand, and significantly higher demands at all three levels are pro- jected. The composition of the population of degree recipients has changed over the past 15 years. The most dramatic change has been in the new-doctoral-de=,ree population, in which the fraction of U.S. citizens has tumbled from four- f~fths to less than one-half. The other significant change 36 has been in the representation of women among degree holders, particularly at the baccalaureate level. The per- centage of women among the annual recipients of bache- lor's degrees has increased steadily so that current levels almost represent parity. This is not the case, however, at the master's degree and doctoral degree levels. Recent increases in the number of bachelor's degrees appear to be fueled by students changing to a major in mathematics after entering college, since the number of en- tenng freshmen expressing an intent to major in mathemat- ics remains very low, at less than 1%. Consequently, pro- jections of future numbers of degrees based on the current number of planned majors are tentative. That current num 40 MY White Females (non-Hispanic) 8 17 8th 12th B.S. M.S. Ph.D. Grade Grade a Bachelor's b Master's b Doctorate b High school record indicates possible choice of mathematics as a college major b Degrees in mathematics FIGURE 4.2 A representation of U.S. students in the mathe- matics pipeline. SOURCES: Bureau of the Census (BOC, 1982 and 1986), National Center for Education Statistics (NCES, 1987a), and American Mathematical Society (AMS, 1986 to 1988~.

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Majors in Mathematics and Statistics 30000 25 ~O 2n nno 10.000 onn n 6.000 1 2t)0 1950 1956 1962 1968 1974 1980 1986 1950 1956 1962 1968 1974 1980 1986 1950 1956 1962 1968 1974 1980 1986 FIGURE 4.3 Number of mathematical sciences degrees awarded by U.S. institutions, 1950 to 1986. Left: Bachelor's degrees. Middle: Master's degrees. Right: Doctoral degrees. SOURCE: National Center for Education Statistics (NCES, 1988a). her does not predict significant increases in degree produc- tion in the near future (Figure 4.4~. There is also a striking similarity in the trends in the numbers of degrees awarded at the three levels in that no noticeable phase lag exists from bachelor's to master's to doctoral degrees. This suggests that the attitudes about ma- joring in mathematics originate in the colleges and univer- sities and that the undergraduate and graduate programs are simultaneously subject to the same forces. Undergraduate Majors The number of baccalaureate degrees awarded in mathe- matics rose from 11,000 in 1960 to a high of 27,000 in 1970, declined steadily throughout the 1970s and early 1980s, and by 1986 had rebounded slightly to the current level of 16,000 (Figure 4.53. The fraction of freshmen anticipating a mathematical sciences major has dropped sharply, from 3.3% in 1970 to 0.6% in 1987, and the share of degrees awarded to women has increased steadily, from about one-th~rd in the mid- 1 960s to almost one-half (46%) in 1986. Over the past 20 years total mathematical sciences enrollments and the number of mathematical sciences majors have reflected disparate trends. For example, the number of bachelor's degrees awarded per 1,000 mathe- matical sciences course enrollments in four-year irlstitu- tiorls has varied from 7 to 23, and from 4 to 15 if two-year college enrollments are included (Table 4.11. At the undergraduate level there have been close ties between the mathematical sciences and computer science (see Box 1.1~. The decline in the number of degrees in mathematical sciences since 1970 has occurred at the same time as a comparable increase ire the number of degrees in computer science (Figure 4.6~. Joint majors in mathemat- ics and computer science and overlapping employment op- portunities are additional indications of the interconnec- tions. When or why students decide to major in mathematics is not clear, although evidence suggests that many make TABLE 4.1 Mathematical sciences bachelor's degrees per 1,000 mathematical sciences enrollments, 1965 to 1985 1965 1970 1975 1980 1985 Four-year enrollments only 20 23 15 7 9 All enrollments 15 15 9 4 6 SOURCES: Adapted from Conference Board of the Mathematical Sciences (CBMS, 1987) and National Center for Education Statistics (NCES, 1987b). 37

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A Challenge of Numbers these decisions after entering college. Comparing the number of full-time freshmen who anticipated a mathe- matics major with the actual number of mathematics de- grees conferred four years later shows clearly that recently many students have decided to major in mathematics after entering college. The expected number of bachelor's degrees in the mathematical sciences is correlated with freshmen interest, and decreases at the same rate as that level of interest, since freshmen enrollments have re- mained relatively stable. Several studies have shown, however, that broad deci- sions about science versus nonscience careers are made by the time a student graduates from high school, and high achievement in mathematics at the secondary school level is predictive of a mathematics-related career (NAS, 1 987a). Knowledge of mathematics has been said to be central in separating scientists from nonscientists, and its signifi- cance is increasing in areas of business and management. Major losses, as high as 50%, take place during the college years in the general science and engineering talent pipe- line, but recently it appears that college students have been slowly migrating toward, rather than away from, mathe- matics as a field of study. Prior to 1975 many more entering students planned a mathematics major than actually received a mathematics degree. But in each year since the early 1980s the number 5 4 3 % \~ : 2 1 1966 1973 1980 1987 FIGURE 4.4 Percentage of entering college freshmen expect- ing to major in mathematics. SOURCE: Cooperative Insti- tutional Research Program (CIRP, 1987b). 38 30,000 - 25.000 - 2O,OOO 5,000 1O,OOO 5,000 1 1970 1972 1974 1976 1978 1980 1982 1984 1986 FIGURE 4.5 Bachelor's degrees awarded in the mathemati- cal sciences, 1970 to 1986. SOURCE: National Center for Education Statistics (NCES, 1988a). Of students who have entered college intending to major in mathematics has actually been lower than the number receiving a bachelor's degree in mathematics four years later. The interest of entering freshmen in mathematics as a probable major has been very low in the 1 980s, but degree production has increased (Figure 4.7~. Thus the antici- pated major of college freshmen as measured by the Coop- erative Institutional Research Program's survey (CIRP, 1987b) is not a very good forecast of the number of mathe- matics degrees awarded four years later. The discrepancy between expected and actual major and the recent chang- ing migration of talent into rather than out of the mathemat- ics pipeline make any discussion of persistence in the choice of major somewhat speculative. High achievers, as would be expected, are more likely to persist in their freshman choice of a mathematics major. A 1985 study of students who were college freshmen in 1981 showed that most freshmen who had planned to major in mathematics and had achieved an "A" grade point average after four years of college did in fact major in mathematics. Students with lower grade point averages were less likely to continue with their freshman choice of a mathematics major and to switch instead to other fields, most often to the social sciences or business (Table 4.21.

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Majors in Mathematics and Statistics One notable trend, mentioned above, is the increased representation of women among recipients of baccalaure- ate degrees in mathematics. In 1950 less than one-fourth of the 6,000 mathematics degrees were awarded to women, and in 1986 almost half of 16,000 were. This progress in the representation of women at the baccalaureate level, however, has not translated to corresponding progress at the graduate level. Even though the fraction of women among baccalaureate degree recipients has been at more than 40% since 1975, only 27% of the current full-time graduate students enrolled in mathematical sciences doc- torate-granting institutions are women. This is a priority issue of the Association of Women in Mathematics: "The widespread and successful participation of women in undergraduate mathematics must be better understood in order to counteract the attrition that occurs at other stages of the educational process, most notably in adolescence and in the graduate years" (AWM, 1988, p. 9~. Blacks and Hispanics receive few of the bachelor de- grees awarded in the mathematical sciences. The ethnic and racial composition of U.S. citizens receiving bache- lor's degrees in 1985 in the mathematical sciences (and in all fields, for comparison) is given in Table 4.3. Non-U.S. citizens received only a small fraction (SYo) of these degrees. In 1982, according to an opinion survey, most senior college and university academic administrators (not de- partmental chairs) did not think that the quality of under 40 000 30,000 20,000 10,000 Computer Science ~ I_ Mathematical Sciences . _. , . i . . . . . . . . . . . 1970 1974 1978 1982 1986 FIGURE 4.6 Number of bachelor's degrees awarded in mathematicaland computer sciences, 1970 to 1986. SOURCE: National Center for Education Statistics (NCES, 1988a). TABLE 4.2 Changes in mathematical sciences majors by undergraduate grade point averages,1981 freshman cohort Undergraduate grade point averages A A- or By B or less Percent persisting in freshman choice Percent defecting to Biology Physical sciences Social sciences Engmeenag Business Education Technical fields Other 87 N 13 N N N N N N 22 5 N N 28 N 19 8 19 N 3 14 20 14 ~2 15 17 a N means no significant number. SOURCE: Holmstrom, E. I., unpublished data. graduate science and engineering students had declined during the previous five years. The majority felt that there had been no change in quality, and one-four thought there had been improvement. These administrators also thought that the most able students who were shifting fields were shifting toward rather than away from science and en gineering (ACE, 1984~. However, a more recent survey of mathematical sciences departments regarding the quality of undergraduate majors in mathematics indicates that lack No of quality is a major concern. The 1985-1986 CBMS Survey asked mathematical sciences departments to rate the lack of quality and the lack of quantity of undergraduate majors as problems in their programs (CBMS, 1987~. About half of the responding departments rated the lack of quantity as a major problem, and a slightly higher percentage rated lack of quality as a major problem (Table 4.4~. Undergraduate mathematics majors are the primary source of students for graduate programs in the mathemati cal sciences. As is true for doctoral degree holders in the physical sciences and engineering, almost three-quarters of those who receive doctorates in mathematics also have 39

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A Challenge of Numbers ~_ _ a bachelor's degree in the same field. Nearly 20% of recent mathematics graduates have enrolled as full-time graduate students, but often in areas other than mathematics or statistics (see section headed "Doctoral Degree Recipi- ents"~. Those not continuing their studies have found favorable employment opportunities. In 1986 a low general unemployment rate (2%) and a high science and en~ineenng employment rate (74%) characterized the labor market for recent mathematics bachelor's degree recipients. This compared with rates for bachelor's degree holders in all fields of 4% and 64% for general unemployment and for science and engineering employment, respectively. Industry employed more than half of recent mathematics graduates, one-fourth found teaching positions, and the remainder worked for govem- ment. On average, of every five mathematics graduates employed in a science or engineering field, two found work in a mathematics or statistics field, two in computer sci- ence, and one in engineering, economics, or some other field. The 1986 median annual salary of $24,100 for mathematics bachelor's degree holders was just below the TABLE 43 1985 bachelor's degrees awarded in mathematical sciences average of $25,000 for all science and engineering fields, trailed the average for engineering (530,000) and for computer science ($28,000), but topped that for all other science fields. Salaries were higher by about $7,000 in industry and government than they were in educational in- stitutions. Degrees for Secondary School Mathematics Teachers Two different but related trends are primarily respon- sible for the current low number of degrees being awarded to prospective secondary school mathematics teachers. Interest in the study of both mathematics and education declined sharply in the 1970s and early 1980s. This drop in popularity has translated to fewer degrees in both fields (Figure 4.83. The following is ~ discussion of the numbers arid some characteristics of persons receiving degrees as preparation for secondary mathematics teaching. A related discussion of school mathematics teachers in the workplace is in Chapter 5. The loss of interest in teaching degrees dipped to its WhiteBlack HispanicAsianIndian Total U.S. Non-U.S. Total Total, math. sci. 12,162766 25788059 Percent 14,124 761 14,885 By race in the U.S. 86. 1% 5.4% 1.8% 6.2 So 0.4% 100% By citizenship 95% 5%100% Men 54% Women 46% Total, all fields 979~477 Percent By race in the U.S. 88.0~o 6.1% 2.8% 2.7% 0.4% 100% By citizenship 97% 3% Men Women SOURCE: ACES, unpublished data. 40 100~o 49% 51%

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Majors in Mathematics and Statistics TABLE 4.4 Summary of responses on quality and quantity of undergraduate majors Mathematics UniversitiesPublic Four Year Private Four Year l -Statistics - Universities Lack of quality Lack of quantity 38% major 15% no/minor 39% major 18% no/minor 62% major 6% no/minor 54% major 20% no/minor 39% major 7% no/minor 42% major 9% no/minor 31~o major 9% no/minor 22% major 21% no/minor NOTE: The possible responses were 0, 1, 2, 3, 4, or 5 with 0 meaning "no problem" and 5 meaning "major problem," and the others indicating gradations between these. The percentages given above for "major" represent the 4 and 5 responses, while "no/minor" represents the 0 and 1 responses. By subtracting the sum of these percentages from 100, one can get the percentage of 2 and 3 responses. SOURCE: Conference Board of the Mathematical Sciences (CBMS, 1987~. lowest level in 1982 arid has rebounded slightly since. The reasons for the loss of appeal of teaching are varied and complex, but they include low salaries, broader career options for women, and lack of prestige associated with teaching (OTA, 1988b). Additionally, market conditions for teachers throughout the 1970s were not favorable because of a perceived general oversupply that actually was present in some disciplines but not necessarily in mathematics. The recent turnaround is supported by in- creased public awareness, higher salaries, and a more favorable market. However, increases are not expected to be sufficient to meet anticipated demand. The number of baccalaureates conferred In education and the number in mathematics have declined as dramati- cally as the interestlevels of entering, freshmen. From 1971 to 1985, education degrees awarded annually decreased by 50%, and mathematics degrees decreased by 39%. Fifteen years ago, one in five baccalaureate recipients majored in education; today, the number is less than one in ten. Most new teachers were education majors in college, but some were single-subject (such as mathematics) ma- jors who did supplementary work in education. The single- subject majors appear to be increasing,. This switch from education to subject majors reflects the current questioning of the usefulness of an education decrees "The utility of the education major is under serious consideration at the moment and several groups have proposed a wide-ranging overhaul of teacher edllcatiorl'' (OTA, 1988b, p. 54~. Only a small portion of those majoring in education have specialized in mathematics education and hence are likely to become high school mathematics teachers. About 1 To of the bachelor's degree holders in education, 0.5~o of the master's degree holders, and 0.4~c of the doctoral 60,000 50~000 40.000 30,000 20,000 1 0,000 - _ OI I I I I 1 1 ~1 ~ ~ 1971 1973 1975 1977 1979 1981 1983 1985 Number of freshmen entering four years earlier and intending ~\ Exnected Nllmber of Decrees* \ \' _ ; Actual Number of Degrees to major in mathematics. FIGURE 4.7 Expected versus actual number of bachelor's degrees in mathematical sciences. SOURCES: Cooperative Institutional Research Program (CIRP, 1987b) and National Center for Education Statistics (NCES, 1987a). 41

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A Challenge of Numbers degree holders have specialized in mathematics. Since the early 1980s, education departments have produced be- tween 1,000 and 1,500 candidates for high school mathe- matics teaching each year. Recently, there has been an upswing, in the number of bachelor's degrees awarded in mathematics education (see Appendix Table A4.4~; how- ever, these figures are small when compared to the 16,000 school districts that need mathematics teachers. In addition to education majors specializing in mathe- matics, the other main source of high school mathematics teachers is undergraduate mathematics majors. A survey of recent mathematics graduates conducted by the NSF in 1986 found that about one-quarter of bachelor's degree recipients taught in educational institutions, presumably high schools, and one-third of master's degree holders also taught, some in high schools and others in college (NSF, 1987a). Standardized test scores for students in the field of education are lower, on the average, than are those for students in other general areas of study, but data on the relative qualifications and abilities of students attracted to education are limited. The SAT mathematics and verbal scores for those planning education majors are lower than the average scores for all SAT takers. But no breakdown of these scores is available by intended level or field of teaching, and so it is not possible to determine how those planning to teach secondary school mathematics compare TABLE 4.5 1987 SAT scores by intended college major Mathematics Verbal Business and commerce Educationa ~ . ngmeerlng Language and literature Mathematics Physical sciences TOTAL 459 437 554 518 602 576 476 - a Includes elementary and secondary education majors. SOURCE: College Entrance Examination Board (CEEB, 1987~. 42 TABLE 4.6 1986 GRE scores by undergraduate and intended graduate major ~- Undergraduate major Educationa En~glneer~ng Math. sciences Physical sciences Other humanities Total Quant.Verbal 451441 674464 657484 637505 517343 540489 Intended graduate mayor Quant. Verbal 464 457 671 461 657 481 639 504 523 542 540 489 a Includes elementary and secondary education majors. SOURCE: Educational Testing Service (ETS,1987~. with others, both in education and mathematics. The 1987 SAT scores by intended major for selected fields are given in Table 4.5. Graduate Record Examination (ORE) scores for 1986 for students planning graduate work show that students who majored in education scored below the national aver- ages, almost 90 points on the quantitative portion and 47 points on the verbal portion (Table 4.61. Those who planned graduate work in education also had below-aver- age scores in both the quantitative and verbal components of the ORE. The above discussion reflects a serious lack of informa- tion about students preparing for careers in teaching. There is little information that distinguishes entrants into the various teaching fields, and there is a wide variety of degree programs indicative of the differing certification requirements across the country. 408 408 456 537 475 507 Both the quantity and quality of graduate students are 430 considered problems by mostmathematical sciences gradu ate departments. In 1986 most of the approximately 18,000 mathematical sciences graduate students were en rolled in doctorate-:,ranting institutions. Over 40% of those enrolled full-time were norl-U.S. citizens (Figures Graduate Students

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Majors in Mathematics and Statistics - 4.9 and 4.10), and that fraction was greater in the top graduate (Group I) institutions (Figure4.1 1~. Fewer women, blacks, and Hispanics enroll in graduate programs than their share among bachelor's degree holders would pre- dict. The dominant mode of support is graduate teaching assistantships, with the result that nearly one-half million students~ne of twelve each year are taught solely by graduate assistants. Three-fourths of the graduate students have undergraduate degrees in mathematics. Of the 18,000 graduate students enrolled in mathemati- cal sciences programs in 1986, almost 12,000 were en- rolled full-time in doctorate-granting institutions, and the remainder were either part-time students or were in mas- ter's-grantin;, institutions. From 1975 to 1986 the total number of students enrolled full-time increased by about 1~700. or 17%. This total gain of 1,700 students actually represents a loss of U.S. students ( 1,400) and a gain of non- U.S. students (3,100~. The number of U.S. students en- rolled reached a low in 1981 and has increased since to the current level. The number of non-U.S. students has almost tripled. The change in the composition of graduate stu- dents has been significant in the mathematical sciences, even when compared to other science and en~ineenn~ fields experiencing the same general trends (Figure 4.91. In the mathematical sciences the number of U.S. stu- dents enrolled was actually less in 1986 than in 1975 (Figure 4.10~. This drop in U.S. students combined with the increased numbers of non-U.S. students has had an impact on both teaching and learning. Since most oradu- ate students in the mathematical sciences are teaching assistants, this increasing reliance on non-U.S. students has affected undergraduate teaching and is believed to be affecting the choice of mathematics or mathematics-re- lated majors. Because undergraduate education in many countries is much more specialized than it is in the United States, non- U.S. students have often been exposed to much more mathematics arid thus come to graduate school with a more sophisticated understanding of the field than do their U.S. counter ts. For instance, in China a baccalaureate de- gree in mathematical sciences is comparable to the tradi- tional master's degree in the United States (MAA, 19831. Such countries have opted for depth of specialization over the breadth of a liberal arts education. Besides the decreasing number and proportion of Americans, the other swilling characteristic of mathemati (thousands of decrees) 10 ~ 8 6 O Education Mathematics Am_ 1 1 ' I 1966 1970 1975 1980 1985 1987 150 100 50 o 1971 \Education Mathematics 1975 1980 1986 FIGURE 4.8 Left: Interest in mathematics and education among entering college freshmen. Right: Degrees in mathematics and education among exiting college seniors. SOURCES: Cooperative Institutional Research Program (CIRP, 1987b) and National Center for Education Statistics (NCES, 1987a). 43

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A Challenge of Numbers cat sciences graduate students in doctorate-grantin5, insti- tutions is that almost three-quarters are men. However, there are about 1,000 more women currently enrolled full- time in doctorate-granting institutions than there were in 1975, and the fraction of women has increased from 21% in 1975 to 27% in 1986 (see Appendix Table A4.6~. Small numbers of non-Asian minority students are enrolled in graduate programs; however, in terms of percentages, more such students are enrolled in master' s-grantina insti- tutions than are enrolled in doctorate-granting institutions (see Appendix Table A4.9~. Mathematical sciences graduate students depend heav- ily on institutional teaching assistantships for support dur- ing their studies. In 1986, 70% listed institutional support as theirmajor source of income, 17% were self-supportina, 8% reported having federal SUppOIt, and 5% listed other outside support (Figure 4.121. There were gender differ- er~ces in the sources of major support for full-time graduate students in doctorate-grantin~ institutions. Women were more likely to have institutional and self-support than were men, and they were less likely to receive federal and other outside support than were the male graduate students. The various types of support for graduate study include fellowships, traineeships, research assistantships, teaching, assistantships, and other types of support. Of the graduate students in the mathematical sciences in 1986, 59% sup 1 9.000 o~ooo 8.000 6~000 4 000 2.000 o 1 975 1 977 1 979 50 - ~ 40 30 20 10 O _. ~ ~ 1975 _-- ~ Mathematical Physical Engineering Total sciences sciences FIGURE 4.9 Percent offull-time graduate students in doctor- ate-grantin~, institutions who are non-U.S. citizens, 1975 and 1986. SOURCE: National Science Foundation (NSF, 1988a). ported themselves with teaching assistantships, To had research assistantships, 7% hadfellowships, 1% had~ainee- ships, and24% reported other types of support. Nearly half the teaching assistants taught their own classes, while the others conducted recitations, tutored, and graded papers. Compared to physical sciences and engineering, graduate students, mathematical sciences graduate students are much more likely to teach and much less likely to have research assistantships. One-quaIter of all science and engineering graduate students have research assistantships, and despite increases in recent years, Drily 9% of those in mathematics do. Although about one-quarter of all science and engi- neenng students have teaching assistantships, fully three so Total _ - ~- % 40 20 10 ) ~ 1 98 1 1 983 1 985 1 97, ~ [Doctoral r ~ ~ ~ _ it_ I I I I . . ~ . ~. . . . 1 979 1 98 1 1 983 1 983 O Group I Masterts FIGURE 4.10 (Left) Mathematical sciences graduate students enrolled full-time in doctorate-granting institutions, 1975 to 1986. (See Appendix Table A4.5.) FIGURE 4.11 (Right) Percent of non-U.S. citizens as mathematical sciences graduate students by type of institution, 1977 to 1986. (See Box 3.2 or note on Appendix Table A5.13 for explanation of Group I.) SOURCES: National Science Foundation (NSF, 1988a) and Conference Board of the Mathematical Sciences (CBMS, 1987~. 44

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Majors in Mathematics and Statistics . . TABLE 4.7 Summary of responses on quality and quantity of graduate students Mathematics Public four-year colleges Universities - Statistics - Universities Lack of quality Lack of quantity 50~o major 2% no/minor 52% major 14% no/minor 44% major 21% no/minor 53% major 24% no/minor 56% major 14% no/minor 55% major 20% no/minor NOTE: The possible responses were 0, 1, 2, 3, 4, or 5 with 0 meaning "no problem" and 5 meaning "major problem," and the others indicating gradations between these. The percentages given above for "major" represent the 4 and 5 responses, while "no/minor" represents the 0 and 1 responses. By subtracting the sum of these percentages from 100, one can get the percentage of 2 and 3 responses. SOURCE: Conference Board of the Mathematical Sciences (CBMS, 1987~. fifths of those in mathematics do. Some of the differences in support among, fields are due to the nature of the field, and particularly to whether laboratory work is significant or not it has not been significant in the mathematical sciences. "Other types of support," a category that includes self-support, was highest among engineers at 38%, lowest among physical scientists at 9%, and in between among mathematical sciences graduate students at 24% (Figure 4.13~. The percent of graduate students with fellowships was about the same in the mathematical sciences, the 100% ~ 80~c 60% 40~G .20C/o 0% Male Female Self support Other outside support - 111115 Federal Institutional FIGURE 4.12 Source of major support for mathematical sci- ences graduate students in doctorate-granting institutions, 1986. (See Appendix Table A4.7.) SOURCE: National Science Foundation (NSF, 1988a). physical sciences, and engineerin ,. Almost three-quarters (74%) of prospective graduate students who take the GRE and state an intention to do doctoral work in mathematical sciences have an under- graduate major in mathematics (ETS, 1987~. Another 22% have majored in another science or engineering field, and 4% are conscience majors. Among those intending to do graduate work below the doctoral level, the large majority (68%) have a baccalaureate degree in mathematics, 26% have majored in another science or engineering field, and 6% have a baccalaureate degree in a conscience field. Even though most students who pursue graduate-level work in the mathematical sciences have backgrounds that include degrees in the same field, the quality of incoming students is often not what graduate programs expect. The 1985-1986 CBMS Survey asked responding mathematical sciences departments to rate the lack of quality and the lack of quantity of graduate students as problems in the pro- ~rams (CBMS, 1987~. More than half of the responding graduate departments rated both the lack of quantity and the lack of quality as major problems (Table 4.71. Master's Degree Recipients The production of mathematical sciences master's degrees has followed the same pattern as that for both 45

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A Challenge of Numbers bachelor's and doctoral degrees, peaking in 1970 after a steady climb and dropping since then. The number of degrees steadily increased from about 1,000 in 1950, peaked at S,600 in 1970, and has decreased since with a slight rebound in the last few years. In 1986 approximately 3,200 master's de;,rees were awarded. The current level of production is the same as that in the mid- 1 960s, repre- sents a drop of 44% from the peak production of 1970, and is about equal to the average for the past 40 years (Figure 4.14). Perhaps what is most notable about recipients of mas- ter's degrees is just how little is written or known about them. There are several major surveys that track doctoral degree holders, but these include few details on master's degree recipients. Little is known about how master's degree holders fit into the broader mathematical commu- nity. Since most master's degree recipients about two- thirds-work in nonacademic settings, the utilization of the master's degree in the workplace is very different from the utilization of the doctorate. Total, Sciences and Engineering Total. Sciences Engineering Physical Sciences Mathematical Sciences % 20 40 60 80 100 Other types of support ~ Traineeships Fellowships ~ Research assistantships ~ Teaching assistantships FIGURE 4.13 Types of major support for graduate students in doctorate-granting institutions, 1986. (See Appendix Table A4.8.) SOURCE: NationalScienceFoundation(NSF, 1988a). 46 6.000 Coon 4,000 3,000 2 30 ........ ........ ; ....... . .... .................. ............. ............... ... . . ..... .. ... .................. .... ... . . . .. . ..................... ... ..................... ............................. _ 970 FIGURE 4.14 Master's degrees awarded, mathematical sciences. SOURCE: National Center for Education Statistics (NCES, 1988a). Most institutions that offer doctoral degree programs in mathematics or statistics also offer programs for master's degrees. In mathematics, in addition to the 155 institutions offering doctoral degrees, another 273 institutions grant a master's degree as the highest decree (see Box 3.21. In sta- tistics, there are approximately 217 master's degree pro- grams at 172 institutions. In 1986 three-fourths of the 2,700 mathematical sci- ences graduate students in master' s-granting (not doctor- ate-granting) institutions were enrolled part-time. Mas- ter's-granting institutions included a higher proportion of women and U.S. citizen graduate students, than did doc- torate-granting institutions. However, the proportion of non-U.S. students enrolled full-time in master' s-granting institutions increased from 13% in 1977 to 34% in 1986, or by about 700 students. The gender composition of the master's degree recipients changed from 20% women in the mid-1960s to 35% women in 1986. The majorincrease in the representation of women occurred dune=, the period from 1965 to 1975; since then the number and proportion of degrees awarded to women have been relatively steady at about 1,000 and 35%, respectively (see Appendix Table A4.11.

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Majors in Mathematics and Statistics 100 80 60 % 40 20 o . , _ _ _ _ ........... ~_ 11' Total of 3,200 Master's Degrees in 1986 Other Pure Applied HI Statistics and Actuarial General FIGURE 4.15 I aster's degrees in mathematical sciences, distribution by subfield. SOURCE: National Center for Education Statistics (NCES, 1988a). By subfield, two-thirds (65%) of the 3,200 mathemati- cal sciences master's degrees awarded in 1986 were in gen- eral mathematics, about 16% were in statistics or actuarial science, and 12% were in applied mathematics (Figure 4.159. This distribution has not changed much in recent years except for increases in applied mathematics since the early 1980s. The ethnic and racial composition of U.S. citizens receiving master's degrees in 1985 was almost identical to the composition of graduate students enrolled full-time in doctorate-granting institutions in 1986. Although non- Asian minorities were more highly represented in part- time enrollments and in master' s-granting institutions, their shares of degrees awarded were similar to their frac- tions of full-time graduate students in doctorate-granting institutions. Non-U.S. students received one-quarter of al master's degrees awarded in 1985 (Table 4.81; just four years earlier in 1981, their share of the master's degrees was 18%. Mathematical scientists, like engineers attain doctoral degrees after master's degrees at a relatively low rate compared to that for other scientists (see Chapter 2~. But a significant proportion of master's degree recipients in the mathematical sciences do continue on to study for a doctor- ate. Based only on the numbers of decrees awarded at the three levels, about one in five bachelor's recipients con- tinue on for a master's degree, and also one in five master's degree recipients continue on for a doctoral degree. Less than one in ten women continue on for a doctorate from a master's degree, but nearly one in four men do (Table 4.91. Among the already-reduced attainment rates for mathe- matical science students compared to students of the other sciences, the graduate degree attainment rate for women is exceptionally low. The attrition of women aloe=, the path from the bachelor's to the doctoral degree is significantly higher in the mathematical sciences than in other science fields (NBC, 1983~. Demand for mathematical scientists at the master's degree level has increased since 1976. According to a TABLE 4.S 1985 master's degrees awarded in mathematical sciences programs U.S. Citizens White Black Hispanic Asian Indian Total U.S. Foreign Total Total 1,873 53 49 164 7 2,146 685 2831 Mer1 1,170 34 28 108 4 1,344 499 1843 Women 703 19 21 56 3 802 186 988 Percent By race in the U.S. 87.3 By citizenship SOURCE: NCES, unpublished data. 2.5 7.6 0.3 100 76 24 100 47

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A Challenge of Numbers TABLE 4.9 Attainment rates of master's and doctoral degrees Percent Master's to Bachelor's (2-year lag) All natural sciences and engineering, 22 Mathematical sciences Men Women Percent Doctorate to Master's (5-year la;,) 21 21 24 17 19 24 9 SOURCE: Appendix Table A4.11 and National Science Foundation (NSF, 1987b). survey by the NSF, recent mathematics master's degree lOOO recipients had a very low general unemployment rate (1.5%) and a high employment rate in science and engi neering occupations (90%) (NSF, 1987a). This compared with a general unemployment rate of 2. loo for all recent 800 science and engineering master's graduates and an em ployment rate of 84% in science and engineering. Median annual salaries for 1984 and 1985 mathematics master's graduates showed men receiving $4,000 more than women 600 in 1986, and industries on the average paid graduates $7,000 more than did educational institutions. Close to half of master's graduates worked in the business and industry sector, two-fifths worked in educational institu tions, and the remainder worked in government or other sectors. Doctoral Degree Recipients The number of doctorates awarded by U.S. universities in the mathematical sciences rose from about 200 in 1950 to a high of about 1,000 in 1973 and has fallen since to about 750 (Fi Sure 4.161. In 1973 nearly four of five of these doctorates were earned by U.S. citizens. In 1988 fewer than half were. In the period from 1977 to 1987, more than one-third of the approximately 9,000 doctoral decrees awarded in the mathematical sciences went to non-U.S. cltlzens. 48 Percent Doctorate to Bachelorts (7-year la;,) 4 5 1.5 ....... . ~ . ~\ . ~ 200 ... -Mu S. ~-j 738 . I.. Pda;Ies. 1~ 1 . . ~ ~ ~ ~ _ ~ _ 11 l l ~ 1 ~ 1 ~ 11 ~ 11 ~ 1111@11 1 i l 1B 111 1 ~ ! l ~ 15 . l ~ ~ . . , en ~ 73 77 82 87 ,76 a@: i73 FIGURE 4.16 Ph.D. degrees in mathematics. (See Appendix Table A4.16.) SOURCE: American Mathematical Society (AMS, 1976 to 1987).

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Majors in Mathematics and Statistics TABLE 4.10 Ratio of new doctorates in mathematics to new doctorates in selected other fields, 1970 to 1985 Chemistry Physics/Astronomy Biology Engmeenng 1970 1975 1980 1985 0.55 0.65 0.48 0.37 0.74 0.88 0.76 0.64 0.36 0.33 0.20 0.18 0.36 0.38 0.30 0.22 SOURCE: Edward A. Connors. Original data were taken from National Science Foundation (NSE, 1988c). Certain of the trends in the number of mathematical sciences doctorates awarded are not unique; some of these Other comparisons point out that the decline in mathe- same patterns have been observed in the total number of matical sciences doctorates since the mid-1970s is more exceptional. Edward A. Connors notes that each of the ratios of the number of new doctorates in mathematics to the numbers in chemistry, physics and astronomy, biology, and eng~neenng has declined when computed for the years 1970, 1975, 1980, and 1985 (Connors, 1988~. For ex- ample, in 1970 there were approximately half as many degrees in the mathematical sciences as in chemistry, but this had dropped to one-third as many in 1985 (Table 4.101. A possible reason for this decline ~ ratios is the decision of those holding baccalaureates in mathematics to switch to doctoral study in other areas, and data are given to support the contention. The shifts increased dune=, the late 1960s and through the 1970s but appear to have leveled off in the 1980s (Table 4.11). The number of women U.S. citizens earning doctorates subfield of mathematics. research doctorates awarded by U.S. universities. This total was about 10~000 in 1960, rose to a maximum of nearly 34,000 in 1973, and then fell slightly until it stabi- lized at about 31,000 ire 1978. The percent of U.S. citizens among the recipients was nearly 85% in the early 1970s and had fallen to 72% in 1986. The number of doctorates awarded in the physical sciences, including the mathemati- cal sciences, declined dunog the 1970s but increased dunug the 1980s. Statistics was an exception to the decline as degree production in that field was approximately stable, but in mathematics the decline was extreme. Mathematics is further exceptional in that it has been the slowest to stem the decline of the 1970s. The percent of U.S. citizens among, new doctorates awarded in the physical sciences stood at 63~o in 1986, about 12 points higher than in the TABLE 4.11 Mathematics majors going on to doctoral study in other areas of science and engineering, 1960 to 1985 Mathematics majors earning ~ = ~ doctorates in science/engineennga doctorates in mathematics Ratio 1960 290 207 0.71 1965 639 484 0.76 1970 1362 924 0.68 1975 1310 883 0.67 1980 1139 609 0.53 1985 959 506 0.53 a Includes mathematics and the social sciences. SOURCE: Edward A. Connors. Original data were taken from National Science Foundation (NSF, 1988c). 49

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A Challenge of Numbers TABLE 4.12 Ethnic representations among new mathematical sciences doctorates, U.S. citizens, 1977 to 1986 Number of new U.S. citizen Percentage of new Approximate doctorates in mathematical U. S. Citizen percentage of sciences Doctorates U.S. population Asian-Americans 224 4.3 2 Blacks 80 1.5 12 Hispanics 34 0.6 7 Native Americans 26 0.5 0.5 Total, U.S. citizens 5,249 NOTE: For changes since 1974, see Appendix Table A4.12. SOURCE: American Mathematical Society (AMS, 1976 to 1988~. each year in the mathematical sciences has been essentially constant since 1973, usually in the range 80 to 90. The percent of U.S. citizens receiving doctorates that is ac- counted for by this group has risen from 10~o to 20% because the total number of U.S. citizens receiving mathe- matical sciences degrees each year has fallen from 774 to 362. Overall, among both U.S. and non-U.S. citizens, women constituted 10% of the new doctoral degree hold- ers in 1973 arid 17% in 1987. Among all doctorate recipients in 1986, women were better represented, havin, received 35% of the degrees; but in the physical sciences women were awarded only 16% of the new doctoral degrees and in engineering only 7~G. Blacks and Hispanics receive inordinately few of the doctorates awarded in the mathematical sciences. AMS 100% Annual Survey results showed that during the ten-year period from 1977 to 1986, these two groups combined received about 237c of the doctorates awarded to U.S. citizens (Table 4. 121. Analogous data (NRC, 1987) for all research doctor- ates awarded by U.S. universities in the ten-year period from 1977 to 1986 are given in Table 4.13. Additionally, since the NRC report includes information on the visa status of non-U.S. citizens, a second comparison is al- lowed. Including those non-U.S. citizens with permanent visa status changes the representations very little except when Asian-Americans are included. Then the change is dramatic. Asian-Americans constituted 46% of the non- U.S. citizens with permanent visas who received a doctor- ate from a U.S. university between 1977 and 1986. TABLE 4.13 Ethnic representation among all new research doctorates, U.S. citizens and permanent residents, 1977 to 1986 Number of Percentage of doctorates to U.S. citizen U. S. citizens doctorates Number of doctorates to U. S. citizens or permanent residents Percentage of U.S. citizen or permanent resider doctorates Asian-Americans 4,579 1.910,404 4.2 Blacks 9,903 4.210,821 4.3 Hispanics 4,970 2.15,697 2.3 Native Americans 790 0.3792 0.3 SOURCE: National Research Council (NRC, 1987~. 50

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Majors in Mathematics and Statistics 100 80 60 % 40 20 o Male Female O Applied mathematics Topology - Analysis Probability/ Statistics Algebra FIGURE 4.17 Doctoral degrees in mathematical sciences, distribution by subfield and sex. SOURCE: National Science Foundation (NSF, 1983); see Appendix Table A4.15. The five major subfields in the mathematical sciences are algebra, probability and statistics, analysis, topology, and applied mathematics. These five subtields accounted for 65% of all doctorates awarded to both men and women in the period from 1960 to 1982. The most popular subfields in the mathematical sciences? as measured by the number of degrees awarded, have not been the same for men and women (Figure 4.17~. In the two decades begin- ning in 1960, the three most popular subfields for women were algebra, probability and statistics, and analysis. For men the most common choices were analysis, probability and statistics, and applied mathematics. Women were more than twice as likely as men to opt for either algebra, probability and statistics, or analysis rather than for applied mathematics. A broad interpretation of the mathematical sciences includes programs in other departments, such as operations research, computer science, statistics, and mathematics education. For example, operations research might be taught in departments of mathematics, engineering, or business. In the last decade, the total number of doctoral decrees awarded annually in the broadly interpreted area of mathematical sciences has been relatively constant at about 1 ,400 (Figure 4. 18~. The trend has involved student shifts away from mathematics and education and toward com- puter science, with the numbers choosing statistics and operations research remaining steady. The median time from entry into graduate school to receipt of the doctoral degree was 7.3 years for the 1986 mathematical sciences doctoral degree recipients and has been essentially the same since 1958. The median for all fields was 10.4 years in 1986. Generally, this median is in the 7- to 9-year range for the sciences and engineering and in the 9- to 13-year range for the social sciences, the humanities, and education. The median registered time, which is the time spent from entry until completion of enrollment in graduate school, was six years for those who received mathematics doctorates in 1986. Three-quarters of the 1986 mathematics doctoral de- gree recipients had their bachelor's degrees ire mathemat- ics, and the same fraction (73%) had master's degrees. In all the physical sciences, 73% of those with doctorates had bachelor's degrees in the same field, but only 52% had master's degrees. For all fields these percents were 55% and 79%, respectively. About 200 postdoctoral appointments in mathematical sciences were made at doctorate-graIlting institutions in 1986. In recent years the number of appointments has ranged from a low of 110 in 1981 to a high of 225 in 1985 (NSF, 1988a). An NRC survey showed that the number of doctoral degree holders with postdoctoral study plans increased from about 10% to 23~c of the total in the past tern 1,600 1,400 1,200 1,000 800 600 400 200 O r ' ~ ~ ~ ~ . . . . _ 1976 1978 1980 1982 1984 1986 C1 Mathematics Education 1~ Operations Research Statistics Computing Mathematics FIGURE 4.18 Number of doctorate recipients in broadly interpreted mathematical sciences. (See Appendix Table A4.14.) SOURCES: National Research Council (NRC, 1987) and American Mathematical Society (AMS, 1976 to 1987). 51

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A Challenge of Numbers years (NRC, 1987~. This is stillfar belong the 40~o in all the physical sciences who planned postdoctoral study. The NRC survey asked new doctoral degree recipients to check one of five reasons as the most important reason for either taking or deciding against a postdoctoral appointment. In this survey, two dominant reasons were given for deciding against postdoctoral study in mathematics: "No postdoc- toral available" was given by 38%, and "attractive employ- ment" was given by 42%. The other three choices were "little or no benefit," "inadequate stipend," and "other." The "no postdoctoral available" response was given most often in mathematics (38%) and the next most often in the humanities (31%~; the response for all fields was 20%. The dominant reason given in favor of postdoctorate by those with mathematics doctorates was gaining additional expe- rience, the reason given by 69% of those planning postdoc- toral study. This reason too, led in all fields, with the overall percentage at 56%. Excluding those who have postdoctoral positions, about three-quarters of doctorate mathematical scientists are currently employed in academe. The fractions taking 52 nonacademic employment are higher for applied mathe- matics and for statistics and lower for mathematics. There have not been major changes in the employment patterns of doctoral degree holders over the years. Approximately 20% find employment in business and industry arid the rest are employed in government. An increased demand in aca- demia for mathematical scientists is likely because of the current low supply and an increase in retirements. Patterns and Prospects At all degree levels, several patterns are apparent. The numbers of degrees awarded annually have decreased significantly since peaking in the early 1970s but have shown some increases recently. The attrition rates in degree programs are high. Relatively few women, blacks, and Hispanics receive degrees, especially graduate de- grees, and non-U.S. citizens are close to achieving a majority among graduate students. There are increased demands for mathematical sciences degrees in the workplace, and these increases are projected to continue.