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OCR for page 19
~ 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
OCR for page 21
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.
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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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Representative terms from entire chapter:
mathematics courses
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
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
