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~j The U.S. Labor Force and Higher Education More new jobs will require more postsecor~dary mathematics education. The rate of growth in mathematically based occupations is about twice that for all occupations. Overall college enrollments are expected to decline until 1995. Minorities and women, now less likely to choose mathematically based occupations, will constitute larger shares of new workers. Shifting interests of college students and high attrition have reduced the number of students in the natural sciences and engineering pipeline. Introduction The need for a mathematically educated citizenry has grown steadily over the past century and has accelerated in the past two decades. This rapid growth is projected to continue into the next century. There are increased needs for mathematical knowledge arid skills in all current areas of use-practical, civic, professional, and cultural- but of the labor force that has attended college nearly doubled from 22% in 1965 to 40% in 1984, and more than 50% of the new jobs created between 1985 and the year 2000 will require some college education (Figure 2. 11. At the same time, the share of the current labor force with less than four years of high school has dropped from 43% to 20%. Mathematical sciences education is needed by all indi- viduals in the labor force, although to varying degrees. the needs for future workers have received the most atten- Problem solving and numerical reasoning are becoming lion. Recurring predictions of the two chief requirements essential in increasing, numbers of jobs, and higher levels for future workers higherlevels of skills and adaptabil- of mathematical competency are required of those in- ity-imply that more, and possibly different, postsecon- valved intechnologicaldevelopment end implementation. dary education and especially mathematics education will As developments within the mathematical sciences occur be needed. These projections are noteworthy in view of the signifi- cant rise over the past two decades in the educational attainment of the civilian labor force. The average worker in the labor force 20 years ago had a high school diploma. Today the average worker's level of education has in- creased to include almost a year of college. The percentage and their applications to many areas expand, more special- ized mathematical knowledge becomes a prerequisite for mathematics-related professions. Much of the added responsibility for meetin, this larger need for mathematically educated people for the work force is borne by colleges and universities. The extent to which this need can be met is central to this report, and 9

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A Challenge of Numbers 80 60 40 ~ . - ~ _ 20- . _ a_ To 1965 1984 2000 (new jobs) High School (or less) College (one or more years) FIGURE2.1 The educational requirements ofthe work force are increasing. SOURCES: Bureau of Labor Statistics (BLS, 1985) and Hudson Institute (BLS, 1987~; see Appendix Table A2.1. some of the problems to be overcome are new and forrni- dable. By most assessments, the current flow of talent produced by U.S. college and university mathematical sciences programs is insufficient. Improving that flow will be complicated by a decreasing pool of U.S. students and by significant changes in the ethnic and racial composition of that pool, given a continuation of the current low inter- est in and attractiveness of the study of mathematics by groups that will have the most significant population growth. Many mathematical sciences programs are over- extended and preoccupied with the large increases in remedial and precalculus enrollments of the past IS years, and the teaching system is outdated, both in curricular content and in the methods and technology used in instruc- tion. More Skills and Greater Adaptability The U.S. economy is projected to add 21 million new jobs between 1986 and 2000, after having added 3 1 million newjobs from 1972 to 1986, as reported in WorIfforce2000 (BLS, 19871. The report goes on to state: "For the first time in history, a majority of all new jobs will require postsecon- dary education" (BLS, 1987, p. xxvii). Adding the number of 1985 workers overa~eSOwhohavemorethanfouryears of college work to the projected number of new jobs that will require more than four years of college yields more 10 than 12 million jobs for college graduates by the year2000. Assuming that current trends continue, this number is close to the total number of new graduates expected between 1985 and 2000. Thus the overall need for college graduates in the work force will be met if most of these college graduates enter the labor market and if their degrees are in the correct areas. However, these conditions are not likely to be satisfied, and significant shortfalls have been pre- dicted in science and engineering. Recent National Sci- ence Foundation predictions point to a shortage of about half a million scientists and engineers by the year 2000, with shortages of 400,000 scientists and 275,000 engineers predicted for 2006 (AAAS, 1989~. WorIfforce 2000 (BLS, 1987) addressed the issue of requirements for increased skills by stating: The new jobs in service industries, all those avail- able, will demand much higher skill levels than the jobs of today. Very few new jobs will be created for those who cannot read, follow directions, and use mathematics. ... The fastest growing jobs will be in professional, technical, and sales fields requiring the highest education and skills level. Of the fastest- growing job categories, all but one, service occupa- tions, require more than the median level of educa- tion for all jobs. Of those growing more slowly than average, not one requires more than the median education. Ranking jobs according to skills, rather than edu- cation, illustrates the rising requirements even more dramatically. When jobs are given numerical ratings according to the math, language, and reasoning skills they require, only twenty-seven percent of all new jobs fall into the lowest two skill categories, while 40 percent of current jobs require those limited skills. By contrast, 41 percent of new jobs are in the three highest skill groups, compared to only 24 percent of current jobs. In meeting the projected needs for more highly educated workers, two problem areas have been noted. First, there is a growing mismatch between the emerging jobs, which

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The U.S. Labor Force and Higher Education will call for increasingly higher levels of skill, and the people available to fill them. Second, the labor market will be a place of churning dislocation, with companies coming and going and jobs changing and being redefined as the United States copes with rapid technological change and an increasingly competitive global economy. The ability of workers to adapt will be critical for success (BLS, 1988b; Richman, 1988~. Adaptability and education are virtually synonymous for workers. Acquired job-specific skills become secon- dary; knowledge, writing, problem solving, and numerical reasoning are critical. Better-educated workers already experience significantly shorter periods of unemployment after losing jobs. Unemployment rates of college-educated workers are approximately one-third the overall rate (NAS, 1987b). Even within the same organization, adaptation to new work environments has become commonplace. To com- pete successfully, U.S. companies must be able to rely on workers to develop, learn, and adapt to new technologies. These abilities depend on the education of the workers, and since many new technologies are mathematically based, mathematics education is critical. Growth in Science-Based Occupations According to projections of the Bureau of Labor Statis- t~cs, eight of the ten fastest growing jobs will be in science- based occupations by 1995. Before 2000, industries will need many more computer programmers and operators, systems analysts, scientists, and engineers. The increase in the number of jobs requiring scientific or technical skills- many mathematics-based is estimated to be significant and is predicted to occur at a rate much higher than that for all jobs (CPST, 1988~. This projection is based on various circumstances pertinent to specific fields, including ad- vances in technology and new applications, the increased importance of quantitative analysis in decision making, shortages of and replacements for doctoral degree holders, and replacements for people transfemng to other occupa . . . tlons or retlnng. The projected increase in the demand for all scientists, BOX 2.1 Degree Programs In Mathematics Of the more than 3,300 higher education institu- t~ons in the United States, more than 2,500 have programs in mathematics. Approximately 1,000 of these are two-year institutions, and most of the remain- ing 1,500 offer programs leading to a bachelor's de- gree with a major in mathematics. About 425 institu- tions offer master's degrees in mathematics, and 155 offer programs for the doctoral degree. Degree programs in mathematics education are fre- quently different from Dose in mathematics and may be located in a different a~rn~nistrative unit such as a college of education. Some institutions may offer separate degrees in applied mathematics or mathe- matical sciences, andjointhachelor's degrees inmathe- matics and computer science are becoming more common. engineers, and technicians between 1986 and 2000 is 36%, compared to a 19% increase in overall employment de- mands. For scientists the expected increase is 45% and for mathematical scientists 29%, compared to 7657c for com- puter specialists, 32~o for engineers, and 36% for techni- cians (NSB, 1987~. These increases are projected from a 1985 base that was generally higher than the average base for the past 25 years. The High Technology Recruitment Index, maintained by Deutsch, Shea & Evans, Inc., monitors demand for technical expertise based on the number of recruitment advertisements directed to scientists and engineers. Data have been collected since 1961, the year used as the base of 100. The index has averaged 106 and has ranged from alowof44iI1 1971 toahighoflS8in 1966. Dunng 1987- 1988, the index hovered at a moderately high level between 115 and 125, but it fell to about 100 in 1989. In addition to there being a reasonably favorable out- look for mathematics-related jobs, such jobs are some of the most desirable, accordin;, to an article (Shogren, 1988) that reported on a study in the The Jobs Rated Almanac. 11

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A Challenge of Numbers Several factors, only one of which is job outlook, are used to measure job desirability. Other factors are salary, stress, work environment, security, and physical demands. When these other factors are considered, the best 5 of 250 jobs rated-(1) actuary, (2) computer programmer, (3) com- puter systems analysts, (4) mathematician, and (5) statisti- cian are mathematics-based Each of these requires an intensive mathematics background at the undergraduate level equivalent to a bachelor's degree in computer sci- ence, mathematics, statistics, or actuarial science (Shogren, 1988). Much of the responsibility for meeting these challenges to provide workers with more and possibly different kinds of education rests with the U.S. higher education system. BOX 2.2 Degree Programs in Statistics The American Statistical Association (ASA) pub lishes lists of U.S. degree programs in statistics and in other areas with an emphasis in statistics (e.g., mathe- matics, business administration, and public health). The information below is taken from the 1987 list. The degree programs are housed in 252 departments in 197 institutions as follows: 174 of the departments have names that include the designations statistics, mathematics, mathematical sciences, or combinations of these; 35 of the departments are biological units with de =,ree programs titled biostatistics or biometry; 30 are departments of business administration; and 13 are scattered among agriculture, psychology, education, and engineering. The list of programs includes: 131 bachelor's degree programs at 123 institutions; 217 master's degree programs at 172 institutions; and 164 doctoral degree programs at 122 institutions. 12 Higher Education in the United States Higher education in the United States is extensive, diverse, and increasingly expensive. More than 64 million (about 1 of 4) people in the United States are involved in giving or receiving formal education at all levels. Of these, 14.4 million are involved in higher education at 3,340 institutions. These institutions spent an estimated $112 billionin 1986-1987. Such expenditures, end thus the costs of higher education, have increased significantly in 1985, the cost to students of attending college was 3.5 times the cost in 1966 and twice the cost in 1975 (NCES, 1987a). Ofthe 3,340U.S. institutions of highereducation, 1,309 offer less than four years of work, typically two years. Of the others, some offer as the highest degree the bachelor's degree (707), a first professional degree (93), a master's degree (566), some degree between a master's and a doctorate (153), and the doctorate (473~. Some 37 institu- tions do not grant degrees (NCES, 1987a). Most of these institutions offer degrees in the mathematical sciences (see Boxes 2.1 and 2.2~. On average, of 100 people involved in higher education, 86 are students, 9 are administrative or support staff, and 5 are faculty members. In 1985 women students outnum- bered men students by 6.4 million to 5.8 million (NCES, 1987a). Some 7.1 million were classified as full-time, while 5.1 million were classified as part-time. There were 10.6 million undergraduates and 1.3 million graduate stu- dents. Minorities made up 18YG of the students in 1986 compared to 15% in 1976 (NCES, 1988a). College enrollments peaked in 1983 at 12.5 million after increasing by 40% from 1970 to 1980 and increasing slightly in the early 1980s. During, the period from 1970 to 1985, the percentage of adults with at least four years of college increased from 11% to 19%. Undergraduate enrollments have declined since 1983, but graduate enroll- ments have been steady since the 1970s, with small in- creases in the middle 1980s. Total undergraduate enroll- ment in colleges and universities increased 13.4% in the decade ending in 1986, while durin, this same period the total number of 18- to 24-year-olds decreased. In 1984-1985, higher education institutions awarded

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an 40 X '.,. ii..: I',,,: ,'. ' it ,. ..:., '; ............ , . i.; . ...... 20 _ A_ O ~ ....... - The U.S Labor Force and Higher Education FIGURE 2.2 Percent distribution of undergraduate enroll- ments by race and ethnic group. SOURCE: National Center for Education Statistics (NCES, 1987a). 979,000 bachelor's degrees, 286,000 master's degrees, and 32,700 doctoral degrees. The most popular areas for the bachelor's degrees were business and management, engineering and engineering technology, social sciences, education, and the health professions (see Appendix Table A2.61. The leading areas for master's degrees were educa- tion and business and management, and for doctoral de- grees were education, the social and behavioral sciences, the life sciences, the physical sciences, and engineering (NCES, 1987a). The Pool of Potential Students and Workers Between now and the year 2000, the U.S. population will grow more slowly than at any time since the 1 930s, and the average age will increase to 36, six years older than the averge at any time in the history of the nation. More women will enter the work force, minorities will be a larger share of new workers, and immigrants will represent the largest share of the increase in the work force since World War I. In fact, native white men, constituting 47% of the 1985 labor force, will constitute only 15% of the new workers between 1985 and 2000 (BLS, 1 987~. If current trends con- tinue these projections indicate reduced numbers of both college students and persons choosing mathematically based occupations. The traditional pool of college students, persons be Wh~te - ~ Other Hispanic ~ Black 0] ~ . , 1976 1978 1980 1982 1984 1986 1985 1990 1995 2000 2010 FIGURE 2.3 The pool of college students is changing, 18- to 24-year-old population. SOURCE: Bureau of the Census (BOC, 1986~; see Appendix Table A2.2. tween the ages of 18 and 24, is shrinking, and the fraction of minorities in this shrinking pool is increasing. Minori- ties have been less likely than majority whites to enroll in college in the traditional age range of 18 to 24, but they are slightly more likely to enroll as older students. College enrollments are expected to decline through the late 1980s and early 1990s for two reasons. First, the 18- to 24-year-old group in the U.S. population peaked at 30 million in 1981, is now declining, will reach a low of 24 million in 1995, end then will climb beck to about the 1970 level by the year 2000 (BOC, 1986~. The decline will not be uniform geographically but, in general, will take place north and east of a line extending from northern Florida to northern Idaho. South and west of that line, there will be increases. Geographic mobility will thus complicate the effects of the decline. Second, demographic and socioeco- nomic projections predict a population that will have a lower college attendance rate if current patterns persist (Figure 2.21. The fraction of the total 18- to 24-year-old population represented by blacks and Hispanics will in- crease from 22% in 1985 to 27% in 2000 and to 30% in 2010 (Figure 2.3), and these two groups have had a lower college-attending rate than has the general population. The possible decline in enrollments is expected to be mitigated by two factors. Enrollments from the 25- to 34- year-old group are expected to stay strong. The fraction of this age group enrolled in education has approximately 13

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A Challenge of plumbers doubled from 1.6 million in 1970 to 3.2 million ire 1985. This age group constituted one-quarter of all enrollments in 1985. The second factor is the growing enrollments of foreign nationals. The number of these students attending U.S. colleges and universities has increased by about 50% in the past ten years (NCES, 1987a). In 1986 about one-third of the U.S. 18- to 24-year-olds who were high school graduates were enrolled in college, and this fraction represented about one of every four in the total cohort of that age. The relationship between this segment of the population and those enrolled in colleges and universities is not direct, for, in recent years, while the 18- to 24-year-old population was decreasing, enrollments in colleges and universities were increasing. However, the group aged 18 to 24 still enrolls in college at a rate more than three times the rate for the group aged 25 to 34 and remains the traditional and principal pool of college enrol- lees. Of those who do not attend college, some will drop out of high school and others will graduate from high school but not enroll in college. In 1985 high school dropouts cor~- stituted 14% of 18- and 19-year-olds. This fraction was higher for Hispanics and blacks but lower for whites (Figure 2 41. Not only were blacks and Hispanics less likely than whites to graduate from high school, but also 40 35 30 % 25 20 10 1975 1980 1985 those who did graduate were less likely to enroll in college (Figure 2.5~. College participation rates for 25- to 34-year-olds who completed high school also showed a variation among racial and ethnic groups. In this age group 8-lO~o of high school graduates were enrolled in college. Blacks and Hispanics were slightly more likely than whites to be enrolled as older students. Persistence in College Enrollment Approximately ~ of 6 high school seniors persists through four years of college in the traditional pattem. This ratio is much lower for blacks (1 of 10) and for Hispanics (1 of 15~. Data describing persistence indicate which students stay in the general education pipeline and, for those who do not stay in the pipeline, when they exit. Although persistence has been linked to degree attainment, the results should be interpreted with some caution. Persis- tence and traditional pattern here refer only to those who enter a four-year college following high school graduation and remain enrolled full-time. Several studies have shown that the majority of students who earn a bachelor's degree do deviate from this traditional pattern of enrollment. Delaying entrance, switching to part-time status, dropping 1' 40 ~ Hispanic Black 35 Total ~ Mite 1975 1980 198: White Total Hispanic O Black FIGURE 2.4 Percent of 18- and 19-year-olds who are high FIGURE 2.5 Enrollment in institutions of higher education school dropouts, by ethnic group. SOURCE: National es a percept ofbigh school graduates. SOURCE: National Center for Education Statistics (NCES, 1987a). 14 Center for Education Statistics (NCES, 1987a).

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The U.S. Labor Force and Higher Education _ _ . Out, taking a leave of absence, or transferring to a two-year college are some of the many ways students might diverge. A group of 1980 high school graduates was surveyed by the U.S. Department of Education for six years following high school graduation (NCES, 19891. Ofthose surveyed, about one-third had never enrolled in postsecondary edu- cation. Blacks and Hispanics were less likely to have attended college; this tendency is also reflected in the different enrollment rates of 18- to 24-year-olds. Another one-third did start college, but not in the traditional way. Of the remaining one-third those who did enroll full- time in college about one-half persisted through four years of college, and three-fourths of these eventually attained degrees. Whites (56%) and Asians (61%) were more likely to persist after enrolling in college than were blacks (44%) and Hispanics (42%) (see Appendix Table A2.8~. At no one point in the four years of college were students more likely to get off track. The lower persistence rates for blacks and Hispanics reflected the cumulative effect of fewer students continuing at each point (academic year and summers) rather than high attrition at any one identifiable stage (NCES, 1989~. Shifting Interests of College Students In the last 20 years student interest has shifted drarnati- cally among various college academic majors. This shift has been monitored in three different ways: (1) surveys of entering freshman on probable major and career, (2) enroll- ments in courses by field, and (3) degree production by field. Each of these measures points to similar trends. Students are more interested in fields of study that are job related, and of these job-related fields, the higher paying ones are more popular. The proportion of entering freshman intending to major in business, computer science, and engineering has in- creased in the last 20 years, while at the same time fresh- man interest in the sciences, the humanities, education, and mathematics has decreased, in some cases dramatically. The declining interest in education is an exception to the general trend of the growing popularity of professional fields; factors such as low salaries, poor working condi tions, low esteem, and broader career options for women have more than offset any increased interest in education. According to the Cooperative Institutional Research Pro- gram (CIRP), which conducts annual surveys of entering college freshmen, career-related fields have gained in popularity at the expense of virtually every field tradition- ally associated with a liberal arts education. A comparison of anticipated majors of college freshman with the distribu- tion of baccalaureate degrees conferred confirms this shift- ing of interests (Figure 2.61. There is a strong correlation between the anticipated major and the general distribution of degrees conferred, even though there is considerable shifting of majors after students enter college. The increased interest in marketable skills has raised serious issues and basic questions about the purpose of higher education that are beyond the scope of this report. However, these issues do concern the mathematical sci- ences because one aspect of the lack of interest in the mathematical sciences is the more general lack of interest in a traditional liberal arts education. The cutting issue raised by Astin et al. in The American Freshman: Twenty Year Trends (CIRP, 1 987b, pp. 26-27) and a host of others conceded with higher education and its future is "whether the higher educational community should adapt passively to these 'market' trends in student expectations, or whether the inherent dangers in such trends should be recognized and curricula revised accordingly. Should colleges simply phase out their programs in the humanities, cut back on their social science and education programs, and expand their offerings in business arid technology?" Data on enrollments in courses by major field of study display the same general trends of shifting interest as do data on the intended majors of freshmen, in spite of the numerous changes in major made during college. Enroll- ments and degree production by field generally have mir- rored what students say they are interested in as entering freshman. In 1982 the most popular areas in terms of enrollments by field were business and commerce and combined engineering and computer science. These three fields accounted for one-third of all enrollments. The number and the distribution of baccalaureate degrees conferred reflect the general enrollment patterns. 15

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A Challenge of Numbers From 1971 to 1985, the total number of bachelor's degrees conferred increased 17%, from about 840,000 to 979,000. Business and management, computer and infor- mation sciences, engineering, and the health sciences all showed remarkable and consistent increases in the number of degrees conferred and also accounted for a larger share of all degrees conferred. The number of education (-50%), English (-47%), mathematics (-39%), and social sciences (-32%) degrees each declined precipitously. And fine arts, the life sciences, physical sciences, and agriculture showed either a mixed or a fairly constant production of bachelor's degrees (Figure 2.61. With the growing popularity of certain majors and the loss of appeal of others, the distribution of bachelor's degrees conferred has changed dramatically since the early 1970s. In 1985 the distribution of degrees conferred was roughly 25% in business and management and 10% each in the social sciences, engineering, education, and the hu- manities, including English; the physical sciences, mathe- matics, computer science, and the life sciences each ac- counted for less than 5 percent of the degrees conferred. Natural Sciences and Engineering The total production of natural sciences and engineer- ing bachelor's degrees has grown steadily in the last 25 years, but there have been wide differences in the changes in the various fields. The category natural sciences and engineering includes the fields of physical, mathematical, life, and computer sciences and en~ineenng but does not include the social sciences. Since the early 1970s, the number of degrees awarded has grown considerably in engineering and computer science, has remained relatively constant in the physical sciences, and has plunged in mathematics. Aggregate data on science and engineering degrees masks these important differences between the subfields. A comparison of the total number of people in the natural sciences and engineering pipeline with degree production at different levels in various fields highlights some of these differences. In 1985 natural sciences and engineering degrees ac- counted for 212,300 of the 979,000 bachelor's degrees 16 conferred, or about 22%. Trends in the production of natural sciences and engineering degrees have followed the same general pattern described above- job-related degrees have increased in the last 10 to 15 years, and those in the arts and sciences and not specifically job related have either remained constant or have decreased. For students continuing on to doctoral degrees, confer- ral of a bachelor's degree can be viewed as a midpoint in the educational process. From this viewpoint, the pipeline begins seven to eight years earlier in high school. At this critical stage students either take the requisite courses to continue in the pipeline or they drop out of the science and engineering track. At venous points in the pipeline, losses occur, and students are not likely to return once they have left. After the conferral of the bachelor's degree, seven to eight years are required to complete work for the doctoral degree for those who do continue. The time required to educate a scientist or an engineer can extend to at least 15 years from the time a student first has some choice in the selection of courses in high school to the awarding of the doctoral degree. As an illustration of this lengthy and leaky process, consider the fact that of 1,000 students who were high school sophomores in 1977, only 2 will have continued in the pipeline to receive a doctoral degree in science or engineering by 1992. Of these 1,000 high school sophomores, 180 were interested in science or engineering as sophomores, but by the time they were seniors only 150 were still interested. Only slightly more than half of these, 85, continued on to college . . . . . . . . . . Wit n t ne intention or maJonng in a science or engineering field. Approximately 51 of these 85 received a bachelor's degree in a natural science or engineering field. Of those who received bachelor's degrees, about 15 continued on for graduate work, and 11 of the 15 received a master's degree. Of these 11 who remained in the system, 2 will continue their studies to successfully complete a doctoral degree (NSF, 1987b). Analysis of degree production for select science and engineering fields shows different patterns for the various fields. The time elapsed from receiving the baccalaureate to earning the doctorate has recently lengthened slightly and shows a variance by field, rangin;, from a low of 5.6

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The U.S. Labor Force and Higher Education L 12% 10% 8c7. 6 .~ 4 ~ \ )' ad ) ~ ~, 4% ~ 1 1 2'Xc ~ _ ,. l l l 970 1975 1980 1985 10% 8% 2050 1 5c/o 1 Oslo 5% . : '1~ _ 1 ~ . 1970 1975 1980 1985 % ~, , . 1970 1975 1980 1985 FIGURE 2.6 Shifting interest in selected majors. Left: Anticipated college major of entering freshmen. Right: Bachelor's degrees of exiting seniors. SOURCES: Cooperative Institutional Research Program (CIRP, 1987a) and National Center for Education Statistics (NCES, 1987b). 20% 15% Education Social sciences En;,lish Mathematics 10% _ _ x.. . ._~ ..... ~ 0% Fine Arts Life sciences Agriculture O Physical sciences ~ Business Engineering Computer science 1971 1975 1980 1985 2% 1% 0% 1971 19751980 1985 25% 20%' / 15% V . 1971 1 975 1980 1985 Social sciences Education English/Letters Mathematics Life sciences Fine arts Agriculture & Home Economics 0 Physical sciences Business Engineering Computer science 17

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A Challenge of Numbers years in 1970 for chemistry to a high of 11.9 years in 1986 for the health sciences. For most fields, the time to com- plete a doctorate after receipt of a baccalaureate has been between 6 and 8.5 years. For puIposes of simplifying analysis between fields, 7 years from receipt of the bacca- laureate to receipt of the doctorate, 5 years from the master's to the doctorate, and 2 years from the baccalaure- ate to the master's are used below as averages for all fields. The numbers of degrees conferred from 1971 to 1985 in selected fields for the three different levels~octorate, master's, and baccalaureate were analyzed to compare attainment percentages from one degree level to a higher level. Allowing for the lags of 2, 5, and 7 years, the total number of degrees at a higher level was divided by the total number at a lower level to give an attainment percentage. Those percentages are given in Table 2.1 (see Appendix Tables A2.10 and A4.11 for more details). There is no adjustment for entry into degree programs by students from outside the United States, as there is none for U.S students changing fields between degrees. Since about half of the doctorates in engineering and in the mathematical sciences are awarded to non-U.S. students, adjustments recognizing this would significantly lower the analogous attainment rates. Taken for one field alone, these rates are not very meaningful, but comparisons between fields are of interest. In the mathematical sciences, since the numbers of degrees awarded at all three levels have increased and decreased together that is, the lags have not been mean- ingful the rates are less significant. However, compari- sons between fields reveal more similarity between the mathematical sciences and engineering than between the mathematical sciences and the other sciences. And attain- mer~t rates for advanced degrees are lower for the mathe- matical sciences than are those for all of the natural . . sciences and engineenng. The Challenges and the Responsibility The needs of the nation's labor force, the shrinking and changing pool of workers, the shifting interests of students, and the projected shortages of scientists and engineers provide a matrix of challenging numbers for U.S. higher education. The major responsibility for meeting the chal- lenges rests with the mathematical sciences component of higher education, the subject of Chapters 3 and 4. Mathematics has always been a major part of higher education, but its fundamental role in society has expanded significantly in recent years. The complex circumstances in higher education and in the work force described above have combined with equally complex circumstances within the mathematical sciences to produce layers of formidable and interconnected problems that must be solved to meet the nation's needs for mathematically educated workers. TABLE 2.1 Attainment rates of advanced degrees for selected fields, 1971 to 1985 Master's/Bachelor's (2-year lag) Doctoral/Master's (5-year lag) Doctoral/Bachelors (7-year lag) All natural science end engineenn;, 22% 21% 5% Engineering 33% 17% 6~c Life sciences 14% 54% Sac Physical sciences 25% 56% 15% Mathematical sciences 21% 18% 4% SOURCES: National Center for Education Statistics (NCES, 1987) and National Science Foundation (NSF, 1987b). 18