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~ College and University Mathematical Sciences College and university mathematical sciences constitute a vast and diverse system that accounts for approximately loo of all higher education. The system is strong at the top but is weakening at all levels. Precollege indicators predict mild improvements after a long decline. The transition from high school to college mathematics is one of the most troublesome in education. Enrollments in mathematical sciences courses have doubled in the last 20 years, but the increases have all been at the lower levels, with remedial enrollments leading the way. Introduction The academic mathematical sciences consist princi- pally of mathematics and statistics. Also included are programs labeled applied mathematics, many areas of applied statistics, and the more mathematical parts of operations research, mathematical biology, engineering, and economics. The boundaries are by no means distinct. For example, it would be very difficult to determine any reasonably precise boundary between applied mathemat- ics and theoretical physics or between mathematics and computer science. As recently as ten years ago, computer science was frequently included in the mathematical sci- ences, but that is no longer the case. Academically, the boundaries are not important and in fact are better disre- garded. However, for description, administration, and policy, general boundaries need to be understood. The diversity of the profession is illustrated by the large number of professional organizations that have an interest in college and university mathematical sciences (see Box 3.1), and much of the information about the people in the mathematical sciences comes from the professional or- garlizations, in particular, the annual surveys conducted by the American Mathematical Society (AMS) and the sur- veys of the Conference Board of the Mathematical Sci- ences (CBMS). Boxes 3.2 and 3.3 describe the nature of these surveys. College and university mathematical sciences in the United States constitute a vast enterprise with serious and diverse responsibilities that are critical to the welfare of the nation and to the maintenance of at least the disciplines of mathematics and statistics. There are mathematical sci- ence programs in at least 2,500 institutions of higher education, and these provide nearly 10% of all the teaching in U.S. higher education and approximately 30% of the teaching in the natural sciences and engineering. Each term, approximately 3 million students are taught by ap- proximately 50,000 teachers, about 27,500 of whom are full-time, 14,500 are part-time, and 8,000 are graduate assistants. 19

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A Challenge of Numbers . Of the 2,500 institutions that have mathematical sci- ences programs, about 1,000 are two-year institutions, another 1,000 offer a bachelor's degree as their highest mathematical sciences degree, nearly 300 offer master's degrees as their highest degree, and nearly 200 offer a doctoral degree in the mathematical sciences. (Most of the institutions with master's and doctorate programs also have a baccalaureate pro~rarn.) Many of these graduate institutions have departments and degree programs in each of mathematics and statistics. Some also have separate programs in applied mathematics or operations research. In addition to these programs, many institutions have programs closely related to the mathematical sciences in other academic units, for example, biostatistics or biom- etry in health science areas; operations research in engi- neering; business statistics, management science, orecono- metrics in business; statistics in social sciences; and mathe- matics education in education (see Boxes 2.1 and 2.2 for a breakdown of institutions by decree program for mathe- matics and statistics). Most of the institutions granting bachelor's degrees, some Granting master's degrees, and a few grantin;, doc- toral degrees have only one department in the mathemati- cal sciences, and that department frequently houses a program in computer science. In the two-year institutions, the unit that contains the mathematical sciences may con- tain other areas of science or technology. The responsibilities of college and university mathe- matical sciences are broader tears those of any other aca- demic area. These diverse responsibilities include provid- in~ courses for general education, service courses for other disciplines, programs for middle and secondary school mathematics teachers, and courses for elementary school teachers; educating, college and university mathematical sciences faculty members, mathematical science research- ers, and applied mathematical scientists; and nurturing the continued development of the disciplines of mathematics and statistics. Strong at the Top Mathematical sciences education and research at the 20 highest levels in the United States are generally considered to be the strongest in the world. This strength comes from both the education of U.S. students as researchers and the immigration of mathematical scientists into the United States. The U.S. environment for mathematical research is clearly one of the best in the world. But a major concern of the mathematical sciences community, a concern that has far-reaching consequences, is how to preserve this strength. Edward E. David, Jr., has summarized the current situation as follows: "American mathematics is strong- way out of proportion to its numbers, way out of proportion to its level of support today. But will it be able to sustain and renew itself in the future? Unfortunately that problem has not gone away. It is in fact more Dressing than ever" (AAAS, 1988b). rid = Winners of the Fields Medal, the world's most presti=- ious award for research in mathematics, have often been mathematicians from the United States. Awarded to out- standin, research mathematicians every four years since it was established in 1936 by Professor J. C. Fields to recognize existing work and the promise of future achieve- ment, this monetary prize and medal usually go to mathe- maticians who are less than40 years old. Ofthe 30 winners of the Fields Medal, 9 were born in the United States and an additional 7 were affiliated with U.S. institutions at the time they won the award. Despite such positive indicators of the position of U.S. mathematical sciences research in the world, several other indicators have implications that are mixed or inconclu- sive. Prestigious awards, increased collaboration, and development of new fields are signs of the vitality and strength of this enterprise, but the share of publications and citations attributed to U.S. mathematicians has been stead- ily dropping, in fact, dropping faster than that for any recorded U.S. research field. Scholarly productivity is probably more narrowly and rigidly defined in mathematics than in any other science and engineering field. The productivity of research scien- fists has been assessed by monitoring (1) the number of articles published and (2) the number oftimes these articles are cited. In 1984, U.S. researchers produced 37% of the world's research articles in mathematics. This compares to

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College and University Mathematical Sciences .. . . . _ BOX3.1 ProfessionalOrganizations The seven general professional organizations whose primary interests are college and university mathematical sciences are the following: American Mathematical Association of Two-Year Colleges (AMATYC). Established in 1975, AMATYC's interests are, as the name implies, the mathematics and professional issues in two-year colleges. AMATYC cur- rently has approximately 1,900 members. American Mathematical Society (AMS). Established in 1888, AMS's interests have centered on research and graduate study in mathematics. AMS currently has approximately 23,000 members. American Statistical Association (ASA). Established in 1839, ASA's interest is general statistics, including mathematical statistics and applications in various disciplines. ASA currently has approximately 15,000 mem- bers. Institute of Mathematical Statistics (IMS). Established in 1930, IMS's interestis mathematical statistics. IMS currently has approximately 3,000 members. Mathematical Association of America (MAA). Establishedin 1915, MAA's interests have centered on issues in undergraduate mathematics. MAA currently has approximately 27,000 members. National Council of Teachers of Mathematics (NCTM). Established in 1920, NCTM interests have centered on the teaching of mathematics, both at the college and precollege levels. NCTM has approximately 76,000 members, of whom about 38,000 are secondary school teachers and 3,000 are college faculty members. Society for Industrial and Applied Mathematics (SIAM). Established in 1952, SIAM's interests have centered on research and applications of mathematics. SIAM currently has approximately 7,000 members. There is overlap in the memberships of these professional societies. The four college and university mathe- matics societies, AMATYC, AMS, MAA. and SIAM, have a combined membership of approximately 46,000 people, and the two statistical societies, ASA and IMS, have a combined membership of approximately 17,000 people. In addition to these seven, there are other organizations that have specialized interests in college and univer- sity mathematical sciences. These include the Association for Symbolic Logic (ASL), the Association for women in Mathematics (AWM), the Biometric Society, the Econometric Society, the Fibonacci Association, the National Association of Mathematicians (NAM, which is concerned with the interests of blacks in mathematics), the Operations Research Society of America (ORSA), the honorary society Pi Mu Epsilon, and The Institute of Management Sciences (TIMS). Four confederations of professional organizations have some of the above organizations as members. The Joint Policy Board for Mathematics (JPBM) represents the AMS, MAA, and SIAM. The Conference Board of the Mathematical Sciences (CB MS) represents the following 15 professional societies: AMATYC, AMS, ASA, ASL, AWM, IMS, MAA, NAM, NCTM, SIAM, the Association of State Supervisors of Mathematics, the National Council of Supervisors of Mathematics, ORSA, the Society of Actuaries, and TIMS. The Council of Scientific Society Presidents includes representatives from 33 organizations. including, AMATYC, AMS, CBMS, MAA, NCTM, and SIAM. The Commission on Professionals in Science and Technology includes the AMS, MAA, and SIAM in its membership of 16 professional societies. 21

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A Challenge of Numbers the 35% of all the world's scientific and technical articles produced by U.S. scientists and engineers. However, the U.S. share of mathematics articles has dropped sign~fi- cantly since 1973, when the share was 48%. The current share, 37%, is below the analogous fractions for clinical medicine (41~o), earth and space sciences (41%), engi- neenng and technology development (40%), and biomedi- cine (39%) arid above those for biology (37%), physics (27%), and chemistry (21%~. The share for all these fields has either dropped or is the same as in 1973, but the drop for mathematics has been the largest. This drop in the U.S. share of mathematics articles is probably reflected in the growing percentage of references in U.S. articles to articles from other counmes. That percentage increased from 16% in 1974 to 29% in 1984. All other fields had analogous increases over this period, but, again, the increase for mathematics was the largest. Also, the influence of U.S. articles as measured by citations in the world's literature dropped slightly between 1973 and 1982. Even though the drop was slight, a much larger drop of 25% occurred in the rate at which U.S. articles were cited in non-U.S. articles. The citation rate in the world's literature was bolstered by a 23% increase in 525 500 475 450 425 the rate at which U.S. articles were cited in U.S. articles. There is evidence that international collaboration and university-industry collaboration are increasing in mathe- matics research. University-industry coauthored papers, as a percent of all industry mathematics papers, increased from 28% in 1973 to 42% in 1984. Internationally coau- thored papers, as a percent of all mathematics papers with authors from more there one institution, increased from 34% in 1973 to 48% in 1984 (NSB,1987~. Mixed Precollege Indicators While the achievements of U.S. research mathemati- cians compare well internationally, as does the preparation of top U.S. students, the achievements of most U.S. stu- dents at the high school level do not. Recent international comparisons of achievement test scores in precollege mathematics place U.S. students well below those in coun- tries that are now major economic competitors of the United States. For example, in a 198 1-1982 test of students from 13 countries, the most able U.S. students taking the test (the top 1 percent) scored the lowest in algebra among the analogous cohorts of all 13 countnes and among the 20 _ I. . . ~ ~ 1967 1971 1975 1979 1983 1987 19 4 \ Male \ Total Female c . , ~, . 1J I ~I I I I r I I I 1973 1978 1983 1988 FIGURE 3.1 SAT mathematics scores, 1967 to 1987. FIGURE 3.2 ACT mathematics scores, 1973 to 1988. SOURCE: College Entrance Examination Board as reported SOURCE: American College Testing Program (ACT, 1989). in Digest of Education Statistics (NCES, 1987b). 22

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College and University Mathematical Sciences . . . . lowest in calculus. The algebra achievement of the U.S. top 5% was lower than that of the corresponding competi- tors from all but one country. The most able Japanese students scored higher than their counterparts in the other countries, and the average Japanese student outperformed the top 5% of the U.S. students in college preparatory mathematics (IAEEA, 1987~. Another more recent international assessment of mathe- matics and science skills places U.S. students last in mathematics performance overall among 13-year-old stu- dents in five countries and four Canadian provinces. On each of the six topics measured, U.S. students scored 10 to 20 percentage points below the top scorers. Yet when asked if they were good at mathematics, two-thirds of the American students felt they were, compared with one- fourth of the South Korean students, who were the top scorers (ETS, 1989~. Attitudes in the United States toward mathematics are mixed, and there are continuing myths about both the nature of mathematics and how one learns mathematics. Many believe that mathematics is a static subject and that success depends more on talent than on effort. Attitudes reported by U.S. students in grades 8 and 12 (see Appen- dix Table A3.1) reveal that almost two-thirds of U.S. students at both grade levels think mathematics and its subtopics are important, but only half of these students have an ease with mathematics, and less than half like mathematics. These attitudes toward mathematics at the high school level translate to fewer college freshmen who are interested in and prepared to pursue studies in mathe- matics. Students are not reaming enough mathematics in high school to prepare themselves for college-level courses or for the future workplace. White males are still leaving high school better prepared than either females or minority group members; however, this gap has been closing in recent years. The average number of 1 -year course credits in mathematics completed by the high school seniors of 1982 was 2.5. For males the average was 2.6; for females, 2.5; for whites, 2.6; for blacks, 2.4; for Hispanics, 2.2; for Asians or Pacific Islanders, 3. 1; and for Native Amencans, 2.0. Half the students earned less than two credits in college preparatory courses (which include algebra 1, 2, and 3; geometry; trigonometry; analytic geometry; linear algebra; probability and statistics; and calculus). Nearly 1 in 20 earned less than one mathematics credit (NCES, 19851. Those students who plan to get a bachelor's degree do take more mathematics credits (3.1) than does the average high school graduate, and almost 90% of college freshmen have taken at least three years of mathematics. Mathematics preparation in high school is a major factor in determining how well students perform on college achievement tests. After a general decline in the 1 970s and early 1980s, average mathematics scores for the Scholastic Aptitude Test (SAT) have risen slightly and for the Ameri- can College Testing (ACT) have leveled off in recent years (see Figures 3.1 and 3.2~. Scores showing both gender and ethnic and racial differences have fueled the controversy over whether the tests are biased against women and minorities. Males have consistently scored higher than females by about 50 points on the mathematics section of the SAT for the last two decades; mathematics scores for blacks and Hispanics showed steady improvement during this same period but were below the national average (see Appendix Table A3.21. The reversal of the steady slide in mathematics scores is cause for some optimism, but the improvements are not substantial enough. Students are still not well prepared for hi=,her-level mathematics courses, and, according to the National Assessment of Educational Progress (NAEP), the prowess that has been made is in lower-level skills. The NAEP has measured achievement in mathematics by U.S. students of ages 9, 13, and 17 in 1978, 1982, and 1986 and has extrapolated the assessment back to 1973 from previ- ous NAEP analyses (see Appendix Table A3.4~. The high- lighted summary from the 1986 report includes the follow- ing (ETS, 19881: Recent national trends in mathematics perform ance are somewhat encouraging, particularly for stu dents at ages 9 and 17. Subpopulations of students who performed comparatively poorly in past assess ments have shown significant improvement in aver a=,e proficiency since 1978: at all three ages [9, 13 23

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A Challenge of Numbers and 17], black and Hispanic students made appre- ciable gains, as did students living in the Southeast. While average performance has improved since 1978, the gains have been confined primarily to lower-order skills. The highest level of performance attained by any substantial proportion of students in 1986 reflects only moderately complex skills and understandings. Most students, even at age 17, do not possess the breadth and depth of mathematics proficiency reseeded for advanced study in secondary school mathematics. The lack of improvement in precollege mathematics preparation and the increased enrollments in college mathe- matics courses have led to difficulty for students in meeting the expectations of traditional college courses. This has resulted in fundamental and extensive changes in college and university mathematics. Troublesome Transitions from High School to College In general, there appears to be a mismatch between the articulated expectations of colleges and universities and the preparation of entering freshmen. Colleges and univer- sities have not clearly articulated and enforced standards and have tried to accommodate extremely diverse back- grounds. A high school graduate, regardless of courses taken, can usually find a place in some college or univer- sity. This clash between expectation and preparation is evident in public and private statements and in the growing overlap between material covered in college courses and that covered in high school courses. In no discipline is this more apparent than it is in mathematics. Several authors and studies have addressed this issue, and problem resolu- tion is generally considered a shared responsibility as reflected in the followin, selected positions (CARN, 1983, pp. 7-9~: Right now. the colleges are genuine in their feel- ~ngs that too many students are not adequately pre- pared for higher education. On the other hand, if the colleges had a modicum of conscience they must 24 know that their own shift in standards and require- ments had something to do with the situation that faces the schools today. The fact is that across the board, not just at com- munity colleges, college entrance requirements place little, and in some cases no, emphasis in the substan- tive content of what high school students should have mastered as the necessary prerequisite to col- lege study. There is no common body of knowledge, no specific set of intellectual skills against which students can measure their own readiness or on which colleges can base admission and placement ~ . . Decisions. SuIprisingly, the courses a student takes are not important in getting into most colleges although they may be critical to success once a student is there. Half the colleges set no specific course requirements at all and only about one-fourth consider courses the students took in making the decision for or against admission. When specific courses are identified, the most frequently required courses are English (the usual requirement being four years), mathematics (two years is the average requirement) and the physi- cal sciences (one year). The transition from high school to college mathematics is especially troublesome for many students and institu- tions. Students enter college with widely varying levels of mathematical preparation: some are not competent in computation at a sixth grade level, and others have taken calculus in high school. Initial placement in college mathematics is complicated by this imbalance in the prepa- ration of students and the several possible entry points for beginning students. Students begin mathematics in col- leges with courses ranging from arithmetic to courses that assume mastery of calculus. It is not difficult to describe a dozen or so possible first courses within this range, and many institutions have as many as a half-dozen entry courses. Placement programs have become more common and sophisticated over the past decade. In the mid- l 970s the Mathematical Association of America (MAA) began its

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College and University Mathematical Sciences BOX 3.2 AMS-MAA Survey Reports The American Mathematical Society (AMS) has sponsored and conducted surveys of college and university mathematics programs and departments each year since 1957, and was joined in sponsorship by the Mathematical Association of America (MAA) in 1987. These surveys, initially covenn, only salanes, have expanded to include course enrollments; numbers and characteristics of faculty members; numbers, employment, and characteristics of new doctoral degree holders; faculty mobility; and nonacademic employment. The AMS-MAA survey reports are published annually in the Notices of the American Mathematical Society (AMS, 1976 to 1988~. The survey population of the AMS-MAA ~llrvPv~ is n~rtiti~n~A Hal A-crr=~c aster ;- the ~~+I~ ~:~1 sciences into groups as follows: ._J_ TV AWAY ~11_ ~vy ~ 1A~ ~vvCL~= 11A LllG 111aL11~111aL1~1 Groups I, II, and III: These are departments that offer doctoral degrees in mathematics and that have been placed into one of three cate=,ones by their ranks in a 1982 assessment of research-doctorate programs in mathematics by the Conference Board of Associated Research Councils. Group I consists of the 39 top-ranked programs (those with an assessment rating between 3.0 and 5.0); Group II, the next 43 (those with a rating from 2.0 to 2.9); and Group III, the remaining 73 programs. It should be noted that many of the dena~ment~ that Offer ~ 1_ A ~ ~ I ~ 1 ~ _ .1 ~ us uu`;torm programs In ma~nemarlcs contain programs in other areas of the mathematical sciences, mostnotably in statistics; and almost all have bachelor's and master's degree programs in mathematics. The AMS-MAA survey results cover all mathematical sciences programs in these departments, not just the doctoral programs in mathematics. Groun IV: This group consists of 69 departments (or programs) of statistics, biostatistics, or biometrics that offer a doctoral program. Group V: This group consists of 57 departments (or programs) in applied mathematics, applied science, operations research, or management science that offer a doctoral program. Group VI: This group consists of 28 Canadian departments (or programs) that offer a doctoral program. Group M: This group consists of 273 departments in the mathematical sciences granting a master's decree as the highest degree. Most of these offer bacheior'c an nrnor~mc trig ~: ~r ~ ~ t~ A ~^ ~ V ~ ~ ~- ~_ I, _ _ _ Group is: l his group consists of 950 departments in the mathematical sciences granting ~ her' Adore as the highest degree. '^ . ~ ~. . ~ . ~ , ~ . . ~_ ~t _ __A ^_A ~ A v ~^ __ 1L ITS noted mat some paws or me AMb-MAA surveys have included Canadian institutions, and some of the AMS-MAA survey results include counts of Canadian degrees, which usually amount to 6-8% of the total of U.S. and Canadian degrees. Because the intent of this report is to describe the circumstances in U.S.institutions, the Canadian data are not included, when feasible. Althou ,h programs in two-year colleges have not been included in recent AMS-MAA surveys, they were included in surveys conducted from 1977 to 1980. 25

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A Challenge of Numbers Placement Test Program, which produces packages of tests for use in placing students in the initial college mathemat- ics course. Scores on placement tests, SAT or ACT scores, the high school record, the college major, and student attitudes are variables that are used to determine initial placement. Some institutions have compulsory placement, but most are advisory. The following summary of the preparation of college freshmen in New Jersey demon- strates the magnitude of the placement problem and the challenge it presents to institutions (CARN, 1983, p. 12~: In New Jersey, all freshmen entering public col- leges and universities are tested in basic skills. Ofthe approximately 30,000 students who took the tests in BOX 3.3 CBMS Surveys The Conference Board of We Mathematical Sciences (CBMS) has sponsored five surveys of undergraduate programs in the mathematical and computer sciences, one every five years beginning in 1965. The surveys have sampled programs in universities, four-year colleges, and two-year colleges to project total enrollments in various courses and the numbers, responsibilities, and characteristics of faculty members. In the 1985 survey, computer science programs were treated separately, and attempts were made to separate the data on computer science pro- grams that are located in mathematical sciences departments. Recent surveys have also asked for opinions on issues judged to be important by departments. The 1985-1986 CBMS Survey population is based on the 1982 NCES classification of 157 universities (95 public and 62 private), 427 public four-year colleges, 839 private four-year colleges, and 1,040 two-year colleges. In this system of classification, universities are institutions that place considerable emphasis on graduate instruction. There were 156 institutions so classified in 1986. This group of institutions has a large overlap with the 155 institutions that offer doctoral degrees in mathematics. This latter group is used as a subpopulation in the AMS-MAA surveys, and for most purposes, assuming that these groups are the same causes no difficulty. Many departments in the mathematical sciences contain programs in various areas: mathematics, applied mathematics, statistics, computer science, operations research, and others. Separate data on computer science were not commonly available until about 1980. In this report an attempt is made to separate the descriptive data into at least that for mathematics and that for statistics and to omit the data on computer science except as it affects mathematics and statistics. The 1985-1986 CBMS Survey used the term "mathematics department" when referring to the unit in the mathematical sciences in an institution that might or might not have separate statistics or computer science depart- ments. This "mathematics department" might have programs in all three areas-mathematics, statistics, and computer science-and in other areas. The CBMS survey population indicates that among the 157 universities, 40 have separate statistics departments and 105 have separate computer science departments. Among the 1,265 four-year colleges, 291 have separately surveyed computer science departments and only 5 have separately surveyed statistics departments. Very few of the 1,040 two-year colleges have separate units in either computer science or statistics. This indicates that most departmental units in the mathematical sciences teach the three disciplines of mathematics, statistics, and computer science. Of course, as the ASA list of statistics programs indicates, programs in statistics occur in several different units in colleges and universities, and many of these are not in the CBMS survey population. Similar circumstances exist in academic programs in computer science. 26

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College and University Mathematical Sciences BOX 3.4 Minorities and Women Fewer blacks, Hispanics, and women study mathematics and choose mathematically based careers than one would expect from their fraction of the total population. Asian-Americans choose mathematically based careers at rates higher than one would expect from their fraction of the total population. The number of Native Americans choosing such careers is very small, but the rate nearly equals the fraction this group fonns of the total U.S. population. The tow numbers of blacks and Hispanics hold throughout the system at all levels of the educational pipeline and in the workplace. The loss of women is more acute at He higher levels of the educational pipeline and in the workplace. These circumstances are well documented in the literature and by the data given in this report (see Chapter 4), but the low numbers are not described repeatedly for each group or educational activity. Instead, unusual patterns and significant situations pertaining to minorities and women are pointed out. Since this report is principally descriptive, no recommendations for increasing these low numbers are offered. The issue is articulated and some samples of intervention programs are described in Boxes 3.5 to 3.8. The data indicate clearly the seriousness of these circumstances and the consequences of continuing the current pattems. 1981, only 38 percent were fully proficient in com- putation at a sixth grade level, and 35 percent failed to demonstrate competence at this minimal level. Most discouraging of all, even the 7,000 students who had taken college preparatory courses in mathe- matics algebra, geometry, and advanced algebra- did poorly. Only 4 percent of these were judged fully proficient in algebra and nearly two-thirds failed that portion of the test. Results for the test of verbal skills were hardly more encouraging. Of all the students who took the tests, 28 percent were rated as profi- cient, about 44 percent were lacking in one area (reading, vocabulary, grammar, writing) or another, and 28 percent failed in all areas. Adequate preparation in mathematics in high school has been said to be the greatest single ticket to admission to and success in science and engineering careers. On the other hand, inadequate preparation in mathematic restricts major career choices and complicates an already difficult transi- tion. There is evidence that this transition may cause many students to drop out of the pipeline toward mathematically based careers. One longitudinal study (1972 to 1979) showed that two of three students who were in the science and engineering pipeline at the end of the twelfth grade, that is, according to their planned college major, had dropped out of that pipeline by the junior year of college (NAS, 1987a). For blacks, the loss amounted to three of four students, and for Hispanics, seven of eight. Box 3.4 describes the general situation that exists concerning par- ticipation by minorities and women in the mathematical sciences. If mathematics is the ticket to success, then in an increasingly technological world, lack of it will be a stamp of exclusion. Efforts are being made to reverse this situation through intervention prolgrarns (see Boxes 3.5 to 3.8~. The dual nature of mathematics, which is both an academic competency and an academic subject, provides students with tools and concepts. The tools are necessary to capture a problem from any field in the proper quantita- tive terms; the concepts are what make mathematics an exciting discipline, and their mastery is a mark of an educated person. Thus "students need to be exposed to both faces of mathematics. They reed to see that mathe- matics as an academic subjectboth depends on and stren;,th- ens mathematics as an academic competency; the content of the two aspects of mathematics should be in harmony" (CEEB, 1985a, p. 151. 27

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A Challenge of Numbers Mathematics is a subject that builds on past knowledge, and further study in the field requires mastery of certain academic competencies. In addition to sharing forward logical progression with the other sciences, mathematics is unique in that once a student reams material in mathemat- ics, it is assumed that that knowledge is retained forever. Thus it is very important that standards and expectations be unifonn throughout the system and that there be no signifi- cant changes in the transitions from one type of school to another. Remediation in College The difficult transition from high school to college has affected students' attitudes about the role of college. In general, there appears to be a growing reliance on college to improve basic skills. More than 40% of entering, freshmen in 1985 reported that an important reason for their attending college was to improve their reading ability and study skills; 70% said they were going to college to be able to make more money (CIRP, 1987b). Feelings of being ill-prepared for college mathematics and reliance on remedial courses have been increasing among students. Almost one in ten (9%) entering freshman have already had some type of special tutoring or remedial work in mathematics. This percentage is much higher than that for any of the other fields; the next highest are for English (boo) and reading (5%~. Additionally, a full one- fourth (25%) of all entering freshmen anticipated that they will need special tutoring or remedial help in mathematics, more than twice as many as those that predicted they will need help in English ( 12%), science (9%), reading (5 %), or some other field. Although the same proportion of men as women reports past remediation or tutoring in mathemat BOX 3.5 Intervention Programs Special programs in science, engineenng, and mathematics are offered to encourage suldy by women and non- Asian minorities. The indications are that some of these intervention programs do work. Some of the key characteristics of academic-based intervention programs include the following (AAAS, 1984, p. 15~: Academic component focused on enrichment rather than remediation; Highly competent teachers; Emphasis on applications and careers rather than on theory; Integrative approach to teaching; Multiyear involvement with students: Strong leadership; Stable, long-tenn funding, base; Recruitment of participants, University, industry, and school cooperative program; Opportunities for in-school and out-of-school learning experiences; - Parental involvement and community support; Specific attention to removing educational inequities; Development of peer support systems; Role models; Student commitment to "hard work"; Evaluation, long-term follow-up, arid careful data collection; and "Mainstreaming" of program elements into the institutional programs. . 28

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ics, more women (27%) than men (22%) anticipated that they will need further special tutoring or remedial help (CIRP, 1987b). Not only do students anticipate that they will need remedial courses, but they do also in fact enroll in these courses. As many as one-fourth of all college freshmen are taking remedial courses in mathematics (BOC, 1988a). Enrollments in certain remedial courses arithmetic, high school algebra and geometry, and general mathematics- have climbed steadily and steeply since the 1960s, muck more so than enrollments in other mathematics courses (Figure 3.3~. More students are needing and taking high- school-level mathematics courses in college, raising the much-discussed question: Should students expect to par- ticipate in higher education without the requisite back- ground, and to what extent should colleges and universities try to accommodate these students? Those students ca- pable of entering college should be provided with as much preparation in mathematics as possible before leaving high school, but many such students are not (CEEB, 1985a). The current wisdom is that all students should study mathematics on an academic track each year during high school (NRC, 19891. Remedial courses, tutoring, and other supplements to normal college-level instruction became commonplace in colleges and universities during the period from 1970 to 1985. Today almost all two-year and four-year colleges offer remedial instruction or tutoring. In mathematics, remedial instruction increased dramatically from 1970 to 1985, with the expansion slowing down from 1980 to 1985. There are signs that the trend may be reversing, both in philosophy and practice. In fall 1970, college enrollments in remedial courses constituted 33% of the mathematical sciences enrollments in two-year colleges end by 1985 had increased to 47%. In four-year colleges and universities. remedial enrollments constituted 9% of the mathematical sciences enrollments in 1970 and hadincreasedto lobby 1985. For fall 1985, these percentages translate to nearly three-fourths of a million enrollments 251,000 in four-year institutions and 482,000 in two-year institutions. The need for reme- dial instruction was ranked as the most serious problem in College and University Mathematical Sciences . . BOX 3.6 The Texas Prefreshman Engineering Program The Texas Prefreshman Engineering Program (TexPREP) was started in 1986 as a statewide expan- sion of the successful San Antonio PREP program, begun in 1979 by Manuel P. Berriozabal. The purpose of TexPREP is to identify potential future scientists and engineers by identifying high school and middle school students of high ability and to provide these students with academic reenforcement to pursue science and engineering fields. The pro- gram operates at seven different locations through- out Texas. Of the 2,000 students who have participated in TexPREP, more than three-quarters have been mi- nonty students and half have been women. Of those participants who are college-a~e, most (88%) plan to attend college or have graduated from college. A large share (68%) of TexPREP graduates major in science or engineering fields. TexPREP has a strong academic component, with courses in logic, algebra, engineering, computer science, physics, arid techni- cal writing. Other activities include field tnps, guest speakers, and practice SAT examinations. SOURCE: Information supplied by Manuel P. Ber- riozabal, University of Texas at San Antonio. two-year mathematical sciences programs in the 1980 CBMS Survey and was still top-rated, aloe=, with the need to use temporary faculty for instruction, in the 1985 survey (CBMS, 1987~. Departments in four-year colleges and universities rated remediation as a problem at a level that corresponded to the amount of remedial teaching required. Remediation was rated as a major problem by 39% of universities, by 66% of four-year public colleges, and by 45 C%C of four-year private colleges. No responding statistics department rated remediation as a major problem (CBMS, 19871. 29

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A Challenge of Numbers . . . . Remedial courses for most students are not considered to be preparatory for a mathematics-based college curricu- lum and are at best repeats of material usually taught in high school. There is little evidence of a significant effect on the supply of talented people in the pipeline aside from the knowledge and skills obtained through the content of these specific courses. Reversing an earlier pattern of low achievement in mathematics is too difficult for normal remedial programs. Mere repetition of material frequently results in duplication of previous failures to learn the necessary concepts (NRC, 1989~. Service Courses In the 20 years from 1965 to 1985, total mathematical sciences enrollments in colleges and universities approxi- mately doubled, increasing from 1.5 million per term to al- most 3 million (Figure 3.4a). This included undergraduate and graduate enrollments in two-year and four-year insti- tutions. Undergraduate mathematics enrollments in four- year colleges and universities increased by more than 60%, rising from about 1 million in 1965 to 1.6 million in 1985 (Figure 3.4c), while two-year college mathematics enroll- ments almost tripled, rising from 350,000 to slightly over 1 million (Figure 3.4d). In addition to remedial enroll- ments, there were large enrollments in service courses for other disciplines and in courses for general education. Ap- proximately half of all enrollments were in courses below the level of calculus and half were in courses at or above that level. For all enrollments, including those in two-year colleges, about two-thirds were below the level of calculus and one-third were at or above that level. The two-year college data included enrollments in mathematics, statistics, and some computing and data processing (Figure 3.4d). The large increases were almost all in remedial enrollments (accounting for nearly half the 1985 total) and in "other," which consists primarily of Total, Math Enrollments College Algebra and Tngonometry High School Algebra and Geometry Anthmetic/General Mathematics 1 . . 1 1 0% 50% 100% 150% 200% 250% FIGURE 3.3 Percent increase in enrollments in selected mathematics courses in colleges and universities, 196~ to 1985. SOURCE: Conference Board of the Mathematical Sciences (CBMS, 1987~. meets in four-year colleges and universities were in reme- dial, precalculus, and calculus-level courses. The calculus enrollments shown in Figure 3.4 include those in differen- tial equations and in linear algebra. Advanced course enrollments, those above the level of calculus, have in- creased only slightly, with much of the increase having occurred since 1980. Most students who study the mathematical sciences in colleges and universities for general educational purposes enroll in courses that are designed to serve particular needs in various curricula that use mathematics. Among these courses are ones in college algebra and trigonometry, finite mathematics, calculus, statistics, and liberal arts mathe- matics. Only the liberal arts mathematics course, and possibly the statistics course, are likely to have been designed with the general education of the student in mind. Enrollments in liberal arts mathematics courses peaked in 1975 at 175,000 and have dropped dramatically since then, to 70,000 in 1985 (CBMS, 1987~. Enrollments in elementary probability and statistics courses increased significantly from 126,000 in 1975 to 180,000 in 1985. Additionally, computer science courses became generally available and popular in the 1970s. Since many such courses have no college mathematics course as a prerequi- site? these are likely alternatives to mathematics courses for specialized vocational courses, many of which are called technical mathematics. These courses generally have a low-level content, some being ari~metic-based and some general education. Enrollments in elementary computer being algebra-based. science courses were estimated at more than 250,000 in fall The increases in under;,raduate mathematics enroll- 1980 and at over 400,000 in fall 1985 (CBMS, 1987~. 30

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Mathematical sciences departments provide "hard" service courses for many college curricula, most notably in the traditional areas of the physical sciences and en~ineer- ing. "Hard" service courses are those with specified content that will be needed in students' later studies, as opposed to "soft" courses, which have few or no restric- tions on content. "Hard" service courses with large enroll- ments have become much more common in recent years for students in business, the social sciences, the life sciences, and preprofessional curricula. One critical service course area provided by mathemati- cal science departments is for students in teacher education programs. These students include prospective secondary school mathematics teachers (see Chapters 4 and 5) and prospective elementary school teachers. The courses that prospective secondary school mathematics teachers take range from college algebra through advanced undergradu- ate courses; requirements vary widely across the country. Prospective elementary school teachers are likely to take one or two mathematics courses especially designed for them, but most do not take any other college-level mathe- matics courses (OTA, 1988b, p. 65~. Survey data on courses taken by college students from 1980 to 1984 indicate that an average college graduate takes 8.4 semester hours of mathematics, which translates to between two and three mathematics courses in college for the average bachelor's degree holder. The range includes a high (for nonmathematics majors) of 21 hours (or about six to seven courses) for computer science majors and a low of 2 hours (less than one course) for fine arts and English majors. Although definitive data are not available, there are indications that the amount of mathematics stud- ied by college students has increased in the past decade (see Appendix Table A3.7~. The fraction of college students who take mathematics courses and their success in those courses compared to their other courses give an indication of the difficulty students have with mathematics. To determine course- taking pattems, the Department of Education conducted a national longitudinal study based on an analysis of the college transcripts of 1972 high school graduates who attended college. A significant share, two of five, took no College and University Mathematical Sciences _ .. . . mathematics courses at all in college, and another one of five took only one course, that is, one to three credits in mathematics. (Even though these data are old, more recent data are neither available nor expected until 1992.) Stu- dents found mathematics courses difficult as evidenced by lower grade point averages (GPAs) in mathematics courses BOX 3.7 Professional Development Program The Professional Development Program (PDP) at the University of Califomia, Berkeley, houses the Charles A. Dana Center forIr~novation in Mathemat- ics and Science Education, which is directed by Philip Uri Treisman. The Dana Center program pro- motes achievement in mathematics courses for minorities by providing an environment where stu- dents can learn and by fostering productive study habits. The study group approach, modeled on Asian study groups, incorporates many of the key charac- teristics of successful intervention programs: high expectations of competence, a strong academic component, capable and appropriate instruction, co- operative learning, and commitment from students. A critical feature is an assumption of competence, the Dana Center program being regarded as an hon- ors program as opposed to a remedial one. This program has been associated not only with successful completion of calculus courses by more of the minority students who participate, but also with high retention and graduation rates. The pro- gram has been expanded to include other California universities, and Treisman is currently working on a high school program in mathematics for minorities. SOURCE: AMS Notices, "Research Mathematics in Mathematics Education," Volume 35, Number 8, October 1988, American Mathematical Society, Providence, R.I., pp. 1 123-1 131. 31

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A Challenge of Numbers than in other courses. Only about 10% had an overall GPA of less than 2 (on a scale of 4), but for mathematics courses 35~c had a GPA of less than 2; half had overall GPAs between 2 and 3, but only one-third fell in this range for mathematics courses; and 35% of students in the sample had GPAs at the top end of the range, between 3 and 4, but only 29~c fell within this range for mathematics courses. Mathematics as an Academic Competency and Subject As stated above, mathematics is both an academic (thousandsj 2.500 2,000 1.500 1,000 5()0 1965 (thousands) 1 In 1.400 1,200 1,000 800 600 400 200 Advanced ,~,, ., .; ; i i.; , >~ i. ~ ... _> ~ ~ :~ By; i~ ~ Remedig 970 1975 (a) competency and an academic subject. In recent years, the demand for mathematics as a competency-in service courses has increased dramatically. At the same time, the number of students choosing mathematics as a major has decreased, creating, among other things, an increasing demand for mathematics teaching and a decreasing supply of mathematics teachers. Chapters 4 and 5 give data on majors in mathematics and statistics and on the utilization ofthese majors in the workplace. Teachers, for both school and college, will be a principal subject because of the increasing demand for mathematics education. (thousands) _ 200 ~ . 150 ~ . ~- 1 lilll 1 ~ 1 At: At' 100 - . it: . , . ~. .... 1980 1985 Advanced (thousands) ~ . __ ~ _ ~_ .. ,., ,, .~.,., , ~, ~ ~ - . . .<- . ~ ; Remed OCR for page 19
College and University Mathematical Sciences _ _ . _ . . . BOX 3.8 The Mathematics, Engineering, Science Achievement Program The Mathematics, Engineering, Science Achievement (MESA) Program was established with the goal of increasing the number of black, Hispanic, and Native American students completing bachelor's degrees in California in Me fields of mathematics, science, or engineering. Begun in 1970, the program is based at Lawrence lIal1 of Science in Berkeley, California, and operates under the auspices of the University of California at Berkeley. Because of its success in recruiting and training minority students at the junior high, high school, and undergraduate levels for science and engineering decrees, the MESA prog ram in California has served as a model for other states. Intemships, field tnps, incentive awards, counseling, freshman orientation and guidance, financial aid and scholarships, and student study groups are some of the activities provided by the program Students are encour- aged through MESA's Pre-College Program to take preparatory classes in mathematics and science in junior and senior high school. These courses, although usually optional for students, are critical to their remaining in the science and engineering pipeline. Most of the high school graduates participating in MESA have pursued mathematics-based majors. The retention rates in college of MESA participants are considerably higher then Nose for nonparticipants. SOURCE Office of Technology Assessment, Educating Scientists and Engineers: Grade School to Grad School, p. 39 (OTA, 1988a). 33

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