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OCR for page 35
3 Majors in Mathematics and Statistics
· Attrition from the mathematical sciences pipeline is approximately half per
year after the ninth grade.
· The number of degrees awarded annually at each of the three degree levels
increased sharply after 1960, peaked in 1970, and then declined sharply to about
the mid1960s level and about the average levels of the past 40 years.
Many students decide to major in mathematics after entering college, and the
circumstances In graduate and undergraduate programs are closely connected,
being simultaneously subject to the same forces.
Relatively few women, blacks, and Hispanics receive degrees in the mathematical
sciences, especially at the graduate level.
Graduate students, nearly onehalf of whom are not U.S. citizens, are mostly
supported by teaching assistantships.
Introduction
The prebaccalaureate "pipeline" population in mathe
matics and statistics is elusive as students move in and out
of degree programs. The attrition rate for this population
is certainly not uniform from year to year, but, for recent
years, assuming an attrition rate of 50% per year from the
ninth grade onward gives surprisingly close estimates of
the actual number of bachelor's, master's, and doctoral
degrees awarded in mathematics and statistics (Figure
4.1).
As indicated in Figure 4.1, in 1972 there were an~roxi
", ,
Halving twice more on the way to the master's degree
yields 3,500 degrees (there were 2,700 reported in 1982),
and three more halv~ngs yield 437 doctoral degrees in
1985, which closely approximates the 400 U.S. citizens
who received doctorates in 1986. (It is noted that the
assumption of five years from the baccalaureate to the
doctorate in the model is short of the average of six years
of registration, and there are no adjustments for nonU.S.
students below the doctoral level.)
The losses of mathematical talent are not evenly distrib
uted among racial and ethnic groups or the sexes. At each
critical juncture in the pipeline more women and minority
students drop out than do white males. In the eighth grade
the mix of students is roughly equal to their representation
in the population. Fifteen years later this representation is
highly skewed in favor of nonHisDanic white males. with
mately 3.6 million U.S. students in the ninth grade. An
attrition rate of 50% per year yields about 225,000 college
freshmen in 1976. An attrition of 50% in each of the next
four years of college yields about 14,000, approximately
the number of bachelor's degrees awarded in mathematics 78% of doctoral degrees being earned by 42~o of the
and statistics in 1980. (The actual number reported for population (Figure 4.2). The highest losses in the pipeline
1980 was 11,000 (NCES, 1988a), but this figure was near occur among blacks and Hispanics, two segments of the
the minimum reported since 1960 end has increased since.) population that are increasing.
35
OCR for page 35
A Challenge of Numbers
10,000,000
1,000,000
100,000
1 ~
: =
10,000
1.000
100
Ninth Graders
. (3.6 i~IrO~~
Frestmen
\~(294~000)
~l,<
B.S.I
\_(11
~
C=;
..... .. ~ :
r degrees
1 000)
_ _ M.S. pegrees ~
. ~_. ~r~s
1972 1976 1980 1982 1986
FIGURE 4.1 Students in the mathematical sciences pipe
lineabout half are lost each year. SOURCES: National
Center for Education Statistics (NCES, 1987a), Cooperative
Institutional Research Program (CIRP, 1987b), and Ameri
can Mathematical Society (AMS, 1986 to 1988~.
The numbers of mathematical sciences degrees awarded
annually to U.S. citizens and others at the bachelor's,
master's, and doctoral levels have shown similar patterns
since 1950 (Figure 4.3~. Steep increases occurred from the
mid 1 950s to peak production in the early 1 970s. Equally
steep decreases then occurred until the early 1980s, and
moderate increases have occurred since. Although similar
patterns exist in other areas of science and engineering, the
numbers of mathematical sciences degrees have been the
slowest to rebound from the declines of the 1 970s and early
1980s, and degree production is still very low even when
compared to that for the other sciences and for engineering.
Current numbers of degrees awarded annually are approxi
mately the same as those for the early 1960s and very near
the averages for the past 40 years: 15,000 bachelor's
degrees, 3,000 master's degrees, and 800 doctoral degrees.
These numbers are each 4045% lower than the analogous
numbers for peak production in the early 1970s. This
supply appears to be slightly short of current demand, and
significantly higher demands at all three levels are pro
jected.
The composition of the population of degree recipients
has changed over the past 15 years. The most dramatic
change has been in the newdoctoralde=,ree population, in
which the fraction of U.S. citizens has tumbled from four
f~fths to less than onehalf. The other significant change
36
has been in the representation of women among degree
holders, particularly at the baccalaureate level. The per
centage of women among the annual recipients of bache
lor's degrees has increased steadily so that current levels
almost represent parity. This is not the case, however, at
the master's degree and doctoral degree levels.
Recent increases in the number of bachelor's degrees
appear to be fueled by students changing to a major in
mathematics after entering college, since the number of en
tenng freshmen expressing an intent to major in mathemat
ics remains very low, at less than 1%. Consequently, pro
jections of future numbers of degrees based on the current
number of planned majors are tentative. That current num
40
MY
White Females
(nonHispanic)
8
17
8th 12th B.S. M.S. Ph.D.
Grade Grade a Bachelor's b Master's b Doctorate b
High school record indicates possible choice of mathematics
as a college major
b Degrees in mathematics
FIGURE 4.2 A representation of U.S. students in the mathe
matics pipeline. SOURCES: Bureau of the Census (BOC,
1982 and 1986), National Center for Education Statistics
(NCES, 1987a), and American Mathematical Society (AMS,
1986 to 1988~.
OCR for page 35
Majors in Mathematics and Statistics
30000
25 ~O
2n nno
10.000
onn
n
6.000
1 2t)0
1950 1956 1962 1968 1974 1980 1986
1950 1956 1962 1968 1974 1980 1986
1950 1956 1962 1968 1974 1980 1986
FIGURE 4.3 Number of mathematical sciences degrees awarded by U.S. institutions, 1950 to 1986. Left: Bachelor's degrees.
Middle: Master's degrees. Right: Doctoral degrees. SOURCE: National Center for Education Statistics (NCES, 1988a).
her does not predict significant increases in degree produc
tion in the near future (Figure 4.4~.
There is also a striking similarity in the trends in the
numbers of degrees awarded at the three levels in that no
noticeable phase lag exists from bachelor's to master's to
doctoral degrees. This suggests that the attitudes about ma
joring in mathematics originate in the colleges and univer
sities and that the undergraduate and graduate programs
are simultaneously subject to the same forces.
Undergraduate Majors
The number of baccalaureate degrees awarded in mathe
matics rose from 11,000 in 1960 to a high of 27,000 in
1970, declined steadily throughout the 1970s and early
1980s, and by 1986 had rebounded slightly to the current
level of 16,000 (Figure 4.53. The fraction of freshmen
anticipating a mathematical sciences major has dropped
sharply, from 3.3% in 1970 to 0.6% in 1987, and the share
of degrees awarded to women has increased steadily, from
about oneth~rd in the mid 1 960s to almost onehalf (46%)
in 1986.
Over the past 20 years total mathematical sciences
enrollments and the number of mathematical sciences
majors have reflected disparate trends. For example, the
number of bachelor's degrees awarded per 1,000 mathe
matical sciences course enrollments in fouryear irlstitu
tiorls has varied from 7 to 23, and from 4 to 15 if twoyear
college enrollments are included (Table 4.11.
At the undergraduate level there have been close ties
between the mathematical sciences and computer science
(see Box 1.1~. The decline in the number of degrees in
mathematical sciences since 1970 has occurred at the same
time as a comparable increase ire the number of degrees in
computer science (Figure 4.6~. Joint majors in mathemat
ics and computer science and overlapping employment op
portunities are additional indications of the interconnec
tions.
When or why students decide to major in mathematics
is not clear, although evidence suggests that many make
TABLE 4.1 Mathematical sciences bachelor's degrees per 1,000 mathematical sciences enrollments, 1965 to 1985
1965 1970 1975 1980 1985
Fouryear enrollments only 20 23 15 7 9
All enrollments 15 15 9 4 6
SOURCES: Adapted from Conference Board of the Mathematical Sciences (CBMS, 1987) and National Center for
Education Statistics (NCES, 1987b).
37
OCR for page 35
A Challenge of Numbers
these decisions after entering college. Comparing the
number of fulltime freshmen who anticipated a mathe
matics major with the actual number of mathematics de
grees conferred four years later shows clearly that recently
many students have decided to major in mathematics after
entering college. The expected number of bachelor's
degrees in the mathematical sciences is correlated with
freshmen interest, and decreases at the same rate as that
level of interest, since freshmen enrollments have re
mained relatively stable.
Several studies have shown, however, that broad deci
sions about science versus nonscience careers are made by
the time a student graduates from high school, and high
achievement in mathematics at the secondary school level
is predictive of a mathematicsrelated career (NAS, 1 987a).
Knowledge of mathematics has been said to be central in
separating scientists from nonscientists, and its signifi
cance is increasing in areas of business and management.
Major losses, as high as 50%, take place during the college
years in the general science and engineering talent pipe
line, but recently it appears that college students have been
slowly migrating toward, rather than away from, mathe
matics as a field of study.
Prior to 1975 many more entering students planned a
mathematics major than actually received a mathematics
degree. But in each year since the early 1980s the number
5
4
3
%
\~
:
2
1
1966 1973
1980 1987
FIGURE 4.4 Percentage of entering college freshmen expect
ing to major in mathematics. SOURCE: Cooperative Insti
tutional Research Program (CIRP, 1987b).
38
30,000 
25.000 
2O,OOO
5,000
1O,OOO
5,000 1
1970 1972 1974 1976 1978 1980 1982 1984 1986
FIGURE 4.5 Bachelor's degrees awarded in the mathemati
cal sciences, 1970 to 1986. SOURCE: National Center for
Education Statistics (NCES, 1988a).
Of students who have entered college intending to major in
mathematics has actually been lower than the number
receiving a bachelor's degree in mathematics four years
later. The interest of entering freshmen in mathematics as
a probable major has been very low in the 1 980s, but degree
production has increased (Figure 4.7~. Thus the antici
pated major of college freshmen as measured by the Coop
erative Institutional Research Program's survey (CIRP,
1987b) is not a very good forecast of the number of mathe
matics degrees awarded four years later. The discrepancy
between expected and actual major and the recent chang
ing migration of talent into rather than out of the mathemat
ics pipeline make any discussion of persistence in the
choice of major somewhat speculative.
High achievers, as would be expected, are more likely
to persist in their freshman choice of a mathematics major.
A 1985 study of students who were college freshmen in
1981 showed that most freshmen who had planned to major
in mathematics and had achieved an "A" grade point
average after four years of college did in fact major in
mathematics. Students with lower grade point averages
were less likely to continue with their freshman choice of
a mathematics major and to switch instead to other fields,
most often to the social sciences or business (Table 4.21.
OCR for page 35
Majors in Mathematics and Statistics
One notable trend, mentioned above, is the increased
representation of women among recipients of baccalaure
ate degrees in mathematics. In 1950 less than onefourth
of the 6,000 mathematics degrees were awarded to women,
and in 1986 almost half of 16,000 were. This progress in
the representation of women at the baccalaureate level,
however, has not translated to corresponding progress at
the graduate level. Even though the fraction of women
among baccalaureate degree recipients has been at more
than 40% since 1975, only 27% of the current fulltime
graduate students enrolled in mathematical sciences doc
torategranting institutions are women. This is a priority
issue of the Association of Women in Mathematics: "The
widespread and successful participation of women in
undergraduate mathematics must be better understood in
order to counteract the attrition that occurs at other stages
of the educational process, most notably in adolescence
and in the graduate years" (AWM, 1988, p. 9~.
Blacks and Hispanics receive few of the bachelor de
grees awarded in the mathematical sciences. The ethnic
and racial composition of U.S. citizens receiving bache
lor's degrees in 1985 in the mathematical sciences (and in
all fields, for comparison) is given in Table 4.3. NonU.S.
citizens received only a small fraction (SYo) of these
degrees.
In 1982, according to an opinion survey, most senior
college and university academic administrators (not de
partmental chairs) did not think that the quality of under
40 000
30,000
20,000
10,000
Computer Science ~
I_ Mathematical Sciences
. _.
, . i . . . . . . . . . . .
1970 1974 1978 1982 1986
FIGURE 4.6 Number of bachelor's degrees awarded in
mathematicaland computer sciences, 1970 to 1986. SOURCE:
National Center for Education Statistics (NCES, 1988a).
TABLE 4.2 Changes in mathematical sciences majors by
undergraduate grade point averages,1981 freshman cohort
Undergraduate grade point averages
A A or By B or less
Percent persisting in
freshman choice
Percent defecting to
Biology
Physical sciences
Social sciences
Engmeenag
Business
Education
Technical fields
Other
87
N
13
N
N
N
N
N
N
22 5
N
N
28
N
19
8
19
N
3
14
20
14
~2
15
17
a N means no significant number.
SOURCE: Holmstrom, E. I., unpublished data.
graduate science and engineering students had declined
during the previous five years. The majority felt that there
had been no change in quality, and onefour thought
there had been improvement. These administrators also
thought that the most able students who were shifting fields
were shifting toward rather than away from science and en
gineering (ACE, 1984~. However, a more recent survey of
mathematical sciences departments regarding the quality
of undergraduate majors in mathematics indicates that lack
No of quality is a major concern.
The 19851986 CBMS Survey asked mathematical
sciences departments to rate the lack of quality and the lack
of quantity of undergraduate majors as problems in their
programs (CBMS, 1987~. About half of the responding
departments rated the lack of quantity as a major problem,
and a slightly higher percentage rated lack of quality as a
major problem (Table 4.4~.
Undergraduate mathematics majors are the primary
source of students for graduate programs in the mathemati
cal sciences. As is true for doctoral degree holders in the
physical sciences and engineering, almost threequarters
of those who receive doctorates in mathematics also have
39
OCR for page 35
A Challenge of Numbers
~_ _
a bachelor's degree in the same field. Nearly 20% of recent
mathematics graduates have enrolled as fulltime graduate
students, but often in areas other than mathematics or
statistics (see section headed "Doctoral Degree Recipi
ents"~. Those not continuing their studies have found
favorable employment opportunities.
In 1986 a low general unemployment rate (2%) and a
high science and en~ineenng employment rate (74%)
characterized the labor market for recent mathematics
bachelor's degree recipients. This compared with rates for
bachelor's degree holders in all fields of 4% and 64% for
general unemployment and for science and engineering
employment, respectively. Industry employed more than
half of recent mathematics graduates, onefourth found
teaching positions, and the remainder worked for govem
ment. On average, of every five mathematics graduates
employed in a science or engineering field, two found work
in a mathematics or statistics field, two in computer sci
ence, and one in engineering, economics, or some other
field. The 1986 median annual salary of $24,100 for
mathematics bachelor's degree holders was just below the
TABLE 43 1985 bachelor's degrees awarded in mathematical sciences
average of $25,000 for all science and engineering fields,
trailed the average for engineering (530,000) and for
computer science ($28,000), but topped that for all other
science fields. Salaries were higher by about $7,000 in
industry and government than they were in educational in
stitutions.
Degrees for Secondary School Mathematics Teachers
Two different but related trends are primarily respon
sible for the current low number of degrees being awarded
to prospective secondary school mathematics teachers.
Interest in the study of both mathematics and education
declined sharply in the 1970s and early 1980s. This drop
in popularity has translated to fewer degrees in both fields
(Figure 4.83. The following is ~ discussion of the numbers
arid some characteristics of persons receiving degrees as
preparation for secondary mathematics teaching. A related
discussion of school mathematics teachers in the workplace
is in Chapter 5.
The loss of interest in teaching degrees dipped to its
WhiteBlack HispanicAsianIndian Total U.S. NonU.S. Total
Total, math. sci. 12,162766 25788059
Percent
14,124
761 14,885
By race in the U.S. 86. 1% 5.4% 1.8% 6.2 So 0.4% 100%
By citizenship 95% 5%100%
Men 54%
Women 46%
Total, all fields 979~477
Percent
By race in the U.S. 88.0~o 6.1% 2.8% 2.7% 0.4% 100%
By citizenship 97% 3%
Men
Women
SOURCE: ACES, unpublished data.
40
100~o
49%
51%
OCR for page 35
Majors in Mathematics and Statistics
TABLE 4.4 Summary of responses on quality and quantity of undergraduate majors
Mathematics
UniversitiesPublic Four Year Private Four Year
l
Statistics 
Universities
Lack of quality
Lack of quantity
38% major
15% no/minor
39% major
18% no/minor
62% major
6% no/minor
54% major
20% no/minor
39% major
7% no/minor
42% major
9% no/minor
31~o major
9% no/minor
22% major
21% no/minor
NOTE: The possible responses were 0, 1, 2, 3, 4, or 5 with 0 meaning "no problem" and 5 meaning "major problem," and the
others indicating gradations between these. The percentages given above for "major" represent the 4 and 5 responses, while
"no/minor" represents the 0 and 1 responses. By subtracting the sum of these percentages from 100, one can get the percentage
of 2 and 3 responses.
SOURCE: Conference Board of the Mathematical Sciences (CBMS, 1987~.
lowest level in 1982 arid has rebounded slightly since. The
reasons for the loss of appeal of teaching are varied and
complex, but they include low salaries, broader career
options for women, and lack of prestige associated with
teaching (OTA, 1988b). Additionally, market conditions
for teachers throughout the 1970s were not favorable
because of a perceived general oversupply that actually
was present in some disciplines but not necessarily in
mathematics. The recent turnaround is supported by in
creased public awareness, higher salaries, and a more
favorable market. However, increases are not expected to
be sufficient to meet anticipated demand.
The number of baccalaureates conferred In education
and the number in mathematics have declined as dramati
cally as the interestlevels of entering, freshmen. From 1971
to 1985, education degrees awarded annually decreased by
50%, and mathematics degrees decreased by 39%. Fifteen
years ago, one in five baccalaureate recipients majored in
education; today, the number is less than one in ten.
Most new teachers were education majors in college,
but some were singlesubject (such as mathematics) ma
jors who did supplementary work in education. The single
subject majors appear to be increasing,. This switch from
education to subject majors reflects the current questioning
of the usefulness of an education decrees "The utility of the
education major is under serious consideration at the
moment and several groups have proposed a wideranging
overhaul of teacher edllcatiorl'' (OTA, 1988b, p. 54~.
Only a small portion of those majoring in education
have specialized in mathematics education and hence are
likely to become high school mathematics teachers. About
1 To of the bachelor's degree holders in education, 0.5~o of
the master's degree holders, and 0.4~c of the doctoral
60,000
50~000
40.000
30,000
20,000
1 0,000  _
OI I I I I
1 1 ~1 ~ ~
1971 1973 1975 1977 1979 1981 1983 1985
Number of freshmen entering four years earlier and intending
~\ Exnected Nllmber of Decrees*
\
\'
· _
;
Actual Number of Degrees
to major in mathematics.
FIGURE 4.7 Expected versus actual number of bachelor's
degrees in mathematical sciences. SOURCES: Cooperative
Institutional Research Program (CIRP, 1987b) and National
Center for Education Statistics (NCES, 1987a).
41
OCR for page 35
A Challenge of Numbers
degree holders have specialized in mathematics. Since the
early 1980s, education departments have produced be
tween 1,000 and 1,500 candidates for high school mathe
matics teaching each year. Recently, there has been an
upswing, in the number of bachelor's degrees awarded in
mathematics education (see Appendix Table A4.4~; how
ever, these figures are small when compared to the 16,000
school districts that need mathematics teachers.
In addition to education majors specializing in mathe
matics, the other main source of high school mathematics
teachers is undergraduate mathematics majors. A survey
of recent mathematics graduates conducted by the NSF in
1986 found that about onequarter of bachelor's degree
recipients taught in educational institutions, presumably
high schools, and onethird of master's degree holders also
taught, some in high schools and others in college (NSF,
1987a).
Standardized test scores for students in the field of
education are lower, on the average, than are those for
students in other general areas of study, but data on the
relative qualifications and abilities of students attracted to
education are limited. The SAT mathematics and verbal
scores for those planning education majors are lower than
the average scores for all SAT takers. But no breakdown
of these scores is available by intended level or field of
teaching, and so it is not possible to determine how those
planning to teach secondary school mathematics compare
TABLE 4.5 1987 SAT scores by intended college major
Mathematics Verbal
Business and commerce
Educationa
~ · .
ngmeerlng
Language and literature
Mathematics
Physical sciences
TOTAL
459
437
554
518
602
576
476

a Includes elementary and secondary education majors.
SOURCE: College Entrance Examination Board (CEEB,
1987~.
42
TABLE 4.6 1986 GRE scores by undergraduate and intended
graduate major
~
Undergraduate
major
Educationa
En~glneer~ng
Math. sciences
Physical sciences
Other humanities
Total
Quant.Verbal
451441
674464
657484
637505
517343
540489
Intended graduate
mayor
Quant. Verbal
464 457
671 461
657 481
639 504
523 542
540 489
a Includes elementary and secondary education majors.
SOURCE: Educational Testing Service (ETS,1987~.
with others, both in education and mathematics. The 1987
SAT scores by intended major for selected fields are given
in Table 4.5.
Graduate Record Examination (ORE) scores for 1986
for students planning graduate work show that students
who majored in education scored below the national aver
ages, almost 90 points on the quantitative portion and 47
points on the verbal portion (Table 4.61. Those who
planned graduate work in education also had belowaver
age scores in both the quantitative and verbal components
of the ORE.
The above discussion reflects a serious lack of informa
tion about students preparing for careers in teaching.
There is little information that distinguishes entrants into
the various teaching fields, and there is a wide variety of
degree programs indicative of the differing certification
requirements across the country.
408
408
456
537
475
507 Both the quantity and quality of graduate students are
430 considered problems by mostmathematical sciences gradu
ate departments. In 1986 most of the approximately
18,000 mathematical sciences graduate students were en
rolled in doctorate:,ranting institutions. Over 40% of
those enrolled fulltime were norlU.S. citizens (Figures
Graduate Students
OCR for page 35
Majors in Mathematics and Statistics

4.9 and 4.10), and that fraction was greater in the top
graduate (Group I) institutions (Figure4.1 1~. Fewer women,
blacks, and Hispanics enroll in graduate programs than
their share among bachelor's degree holders would pre
dict. The dominant mode of support is graduate teaching
assistantships, with the result that nearly onehalf million
students~ne of twelve each year are taught solely by
graduate assistants. Threefourths of the graduate students
have undergraduate degrees in mathematics.
Of the 18,000 graduate students enrolled in mathemati
cal sciences programs in 1986, almost 12,000 were en
rolled fulltime in doctorategranting institutions, and the
remainder were either parttime students or were in mas
ter'sgrantin;, institutions. From 1975 to 1986 the total
number of students enrolled fulltime increased by about
1~700. or 17%. This total gain of 1,700 students actually
represents a loss of U.S. students ( 1,400) and a gain of non
U.S. students (3,100~. The number of U.S. students en
rolled reached a low in 1981 and has increased since to the
current level. The number of nonU.S. students has almost
tripled. The change in the composition of graduate stu
dents has been significant in the mathematical sciences,
even when compared to other science and en~ineenn~
fields experiencing the same general trends (Figure 4.91.
In the mathematical sciences the number of U.S. stu
dents enrolled was actually less in 1986 than in 1975
(Figure 4.10~. This drop in U.S. students combined with
the increased numbers of nonU.S. students has had an
impact on both teaching and learning. Since most oradu
ate students in the mathematical sciences are teaching
assistants, this increasing reliance on nonU.S. students
has affected undergraduate teaching and is believed to be
affecting the choice of mathematics or mathematicsre
lated majors.
Because undergraduate education in many countries is
much more specialized than it is in the United States, non
U.S. students have often been exposed to much more
mathematics arid thus come to graduate school with a more
sophisticated understanding of the field than do their U.S.
counter ts. For instance, in China a baccalaureate de
gree in mathematical sciences is comparable to the tradi
tional master's degree in the United States (MAA, 19831.
Such countries have opted for depth of specialization over
the breadth of a liberal arts education.
Besides the decreasing number and proportion of
Americans, the other swilling characteristic of mathemati
(thousands of decrees)
10 ~
8
6
O
Education
Mathematics
Am_
1 1 · · · ' I
1966 1970 1975 1980 1985 1987
150
100
50
o
1971
\Education
Mathematics
1975 1980 1986
FIGURE 4.8 Left: Interest in mathematics and education among entering college freshmen. Right: Degrees in mathematics
and education among exiting college seniors. SOURCES: Cooperative Institutional Research Program (CIRP, 1987b) and
National Center for Education Statistics (NCES, 1987a).
43
OCR for page 35
A Challenge of Numbers
cat sciences graduate students in doctorategrantin5, insti
tutions is that almost threequarters are men. However,
there are about 1,000 more women currently enrolled full
time in doctorategranting institutions than there were in
1975, and the fraction of women has increased from 21%
in 1975 to 27% in 1986 (see Appendix Table A4.6~. Small
numbers of nonAsian minority students are enrolled in
graduate programs; however, in terms of percentages,
more such students are enrolled in master' sgrantina insti
tutions than are enrolled in doctorategranting institutions
(see Appendix Table A4.9~.
Mathematical sciences graduate students depend heav
ily on institutional teaching assistantships for support dur
ing their studies. In 1986, 70% listed institutional support
as theirmajor source of income, 17% were selfsupportina,
8% reported having federal SUppOIt, and 5% listed other
outside support (Figure 4.121. There were gender differ
er~ces in the sources of major support for fulltime graduate
students in doctorategrantin~ institutions. Women were
more likely to have institutional and selfsupport than were
men, and they were less likely to receive federal and other
outside support than were the male graduate students.
The various types of support for graduate study include
fellowships, traineeships, research assistantships, teaching,
assistantships, and other types of support. Of the graduate
students in the mathematical sciences in 1986, 59% sup
1 9.000
o~ooo
8.000
6~000
4 000
2.000
o
1 975 1 977 1 979
50  ~
40
30
20
10
O
_. ~ ~ 1975
_· ~
Mathematical Physical Engineering Total
sciences sciences
FIGURE 4.9 Percent offulltime graduate students in doctor
ategrantin~, institutions who are nonU.S. citizens, 1975 and
1986. SOURCE: National Science Foundation (NSF, 1988a).
ported themselves with teaching assistantships, To had
research assistantships, 7% hadfellowships, 1% had~ainee
ships, and24% reported other types of support. Nearly half
the teaching assistants taught their own classes, while the
others conducted recitations, tutored, and graded papers.
Compared to physical sciences and engineering, graduate
students, mathematical sciences graduate students are much
more likely to teach and much less likely to have research
assistantships. OnequaIter of all science and engineering
graduate students have research assistantships, and despite
increases in recent years, Drily 9% of those in mathematics
do. Although about onequarter of all science and engi
neenng students have teaching assistantships, fully three
so
Total _

~
%
40
20
10
) ~
1 98 1 1 983 1 985 1 97,
~ [Doctoral
r ~ ~
~ _
it_
I I I I . . ~· .
~. . . .
1 979 1 98 1 1 983 1 983
O Group I
· Masterts
FIGURE 4.10 (Left) Mathematical sciences graduate students enrolled fulltime in doctorategranting institutions, 1975 to
1986. (See Appendix Table A4.5.) FIGURE 4.11 (Right) Percent of nonU.S. citizens as mathematical sciences graduate
students by type of institution, 1977 to 1986. (See Box 3.2 or note on Appendix Table A5.13 for explanation of Group I.)
SOURCES: National Science Foundation (NSF, 1988a) and Conference Board of the Mathematical Sciences (CBMS, 1987~.
44
OCR for page 35
Majors in Mathematics and Statistics
. .
TABLE 4.7 Summary of responses on quality and quantity of graduate students
Mathematics
Public fouryear colleges
Universities
 Statistics 
Universities
Lack of quality
Lack of quantity
50~o major
2% no/minor
52% major
14% no/minor
44% major
21% no/minor
53% major
24% no/minor
56% major
14% no/minor
55% major
20% no/minor
NOTE: The possible responses were 0, 1, 2, 3, 4, or 5 with 0 meaning "no problem" and 5 meaning "major problem," and the
others indicating gradations between these. The percentages given above for "major" represent the 4 and 5 responses, while
"no/minor" represents the 0 and 1 responses. By subtracting the sum of these percentages from 100, one can get the percentage
of 2 and 3 responses.
SOURCE: Conference Board of the Mathematical Sciences (CBMS, 1987~.
fifths of those in mathematics do. Some of the differences
in support among, fields are due to the nature of the field,
and particularly to whether laboratory work is significant
or not it has not been significant in the mathematical
sciences. "Other types of support," a category that includes
selfsupport, was highest among engineers at 38%, lowest
among physical scientists at 9%, and in between among
mathematical sciences graduate students at 24% (Figure
4.13~. The percent of graduate students with fellowships
was about the same in the mathematical sciences, the
100% ~
80~c
60%
40~G
.20C/o
0%
Male Female
Self support
Other
outside
support

111115 Federal
Institutional
FIGURE 4.12 Source of major support for mathematical sci
ences graduate students in doctorategranting institutions,
1986. (See Appendix Table A4.7.) SOURCE: National
Science Foundation (NSF, 1988a).
physical sciences, and engineerin ,.
Almost threequarters (74%) of prospective graduate
students who take the GRE and state an intention to do
doctoral work in mathematical sciences have an under
graduate major in mathematics (ETS, 1987~. Another 22%
have majored in another science or engineering field, and
4% are conscience majors. Among those intending to do
graduate work below the doctoral level, the large majority
(68%) have a baccalaureate degree in mathematics, 26%
have majored in another science or engineering field, and
6% have a baccalaureate degree in a conscience field.
Even though most students who pursue graduatelevel
work in the mathematical sciences have backgrounds that
include degrees in the same field, the quality of incoming
students is often not what graduate programs expect. The
19851986 CBMS Survey asked responding mathematical
sciences departments to rate the lack of quality and the lack
of quantity of graduate students as problems in the pro
~rams (CBMS, 1987~. More than half of the responding
graduate departments rated both the lack of quantity and
the lack of quality as major problems (Table 4.71.
Master's Degree Recipients
The production of mathematical sciences master's
degrees has followed the same pattern as that for both
45
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A Challenge of Numbers
bachelor's and doctoral degrees, peaking in 1970 after a
steady climb and dropping since then. The number of
degrees steadily increased from about 1,000 in 1950,
peaked at S,600 in 1970, and has decreased since with a
slight rebound in the last few years. In 1986 approximately
3,200 master's de;,rees were awarded. The current level
of production is the same as that in the mid 1 960s, repre
sents a drop of 44% from the peak production of 1970, and
is about equal to the average for the past 40 years (Figure
4.14).
Perhaps what is most notable about recipients of mas
ter's degrees is just how little is written or known about
them. There are several major surveys that track doctoral
degree holders, but these include few details on master's
degree recipients. Little is known about how master's
degree holders fit into the broader mathematical commu
nity. Since most master's degree recipients about two
thirdswork in nonacademic settings, the utilization of
the master's degree in the workplace is very different from
the utilization of the doctorate.
Total, Sciences and
Engineering
Total. Sciences
Engineering
Physical Sciences
Mathematical Sciences
% 20 40 60 80 100
Other types of support ~ Traineeships
Fellowships ~ Research assistantships
~ Teaching assistantships
FIGURE 4.13 Types of major support for graduate students
in doctorategranting institutions, 1986. (See Appendix Table
A4.8.) SOURCE: NationalScienceFoundation(NSF, 1988a).
46
6.000
Coon
4,000
3,000
2 30
........
........ ;
.......
. ....
..................
.............
...............
...
. .
.....
.. ...
·..................
.... ...
. . . .. .
.....................
... .....................
.............................
_
970
FIGURE 4.14 Master's degrees awarded, mathematical
sciences. SOURCE: National Center for Education Statistics
(NCES, 1988a).
Most institutions that offer doctoral degree programs in
mathematics or statistics also offer programs for master's
degrees. In mathematics, in addition to the 155 institutions
offering doctoral degrees, another 273 institutions grant a
master's degree as the highest decree (see Box 3.21. In sta
tistics, there are approximately 217 master's degree pro
grams at 172 institutions.
In 1986 threefourths of the 2,700 mathematical sci
ences graduate students in master' sgranting (not doctor
ategranting) institutions were enrolled parttime. Mas
ter'sgranting institutions included a higher proportion of
women and U.S. citizen graduate students, than did doc
torategranting institutions. However, the proportion of
nonU.S. students enrolled fulltime in master' sgranting
institutions increased from 13% in 1977 to 34% in 1986, or
by about 700 students. The gender composition of the
master's degree recipients changed from 20% women in
the mid1960s to 35% women in 1986. The majorincrease
in the representation of women occurred dune=, the period
from 1965 to 1975; since then the number and proportion
of degrees awarded to women have been relatively steady
at about 1,000 and 35%, respectively (see Appendix Table
A4.11.
OCR for page 35
Majors in Mathematics and Statistics
100
80
60
%
40
20
o
. , _
_
_
_
........... ~_
11'
Total of 3,200 Master's
Degrees in 1986
Other
Pure
Applied
HI Statistics
and Actuarial
General
FIGURE 4.15 I aster's degrees in mathematical sciences,
distribution by subfield. SOURCE: National Center for
Education Statistics (NCES, 1988a).
By subfield, twothirds (65%) of the 3,200 mathemati
cal sciences master's degrees awarded in 1986 were in gen
eral mathematics, about 16% were in statistics or actuarial
science, and 12% were in applied mathematics (Figure
4.159. This distribution has not changed much in recent
years except for increases in applied mathematics since the
early 1980s.
The ethnic and racial composition of U.S. citizens
receiving master's degrees in 1985 was almost identical to
the composition of graduate students enrolled fulltime in
doctorategranting institutions in 1986. Although non
Asian minorities were more highly represented in part
time enrollments and in master' sgranting institutions,
their shares of degrees awarded were similar to their frac
tions of fulltime graduate students in doctorategranting
institutions. NonU.S. students received onequarter of al
master's degrees awarded in 1985 (Table 4.81; just four
years earlier in 1981, their share of the master's degrees
was 18%.
Mathematical scientists, like engineers attain doctoral
degrees after master's degrees at a relatively low rate
compared to that for other scientists (see Chapter 2~. But
a significant proportion of master's degree recipients in the
mathematical sciences do continue on to study for a doctor
ate. Based only on the numbers of decrees awarded at the
three levels, about one in five bachelor's recipients con
tinue on for a master's degree, and also one in five master's
degree recipients continue on for a doctoral degree. Less
than one in ten women continue on for a doctorate from a
master's degree, but nearly one in four men do (Table 4.91.
Among the alreadyreduced attainment rates for mathe
matical science students compared to students of the other
sciences, the graduate degree attainment rate for women is
exceptionally low. The attrition of women aloe=, the path
from the bachelor's to the doctoral degree is significantly
higher in the mathematical sciences than in other science
fields (NBC, 1983~.
Demand for mathematical scientists at the master's
degree level has increased since 1976. According to a
TABLE 4.S 1985 master's degrees awarded in mathematical sciences programs
U.S. Citizens
White Black Hispanic Asian Indian Total U.S. Foreign Total
Total 1,873 53 49 164 7 2,146 685 2831
Mer1 1,170 34 28 108 4 1,344 499 1843
Women 703 19 21 56 3 802 186 988
Percent
By race in the U.S. 87.3
By citizenship
SOURCE: NCES, unpublished data.
2.5
7.6
0.3
100
76
24 100
47
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A Challenge of Numbers
TABLE 4.9 Attainment rates of master's and doctoral degrees
Percent Master's
to Bachelor's
(2year lag)
All natural sciences
and engineering, 22
Mathematical sciences
Men
Women
Percent Doctorate
to Master's
(5year la;,)
21
21
24
17
19
24
9
SOURCE: Appendix Table A4.11 and National Science Foundation (NSF, 1987b).
survey by the NSF, recent mathematics master's degree lOOO
recipients had a very low general unemployment rate
(1.5%) and a high employment rate in science and engi
neering occupations (90%) (NSF, 1987a). This compared
with a general unemployment rate of 2. loo for all recent 800
science and engineering master's graduates and an em
ployment rate of 84% in science and engineering. Median
annual salaries for 1984 and 1985 mathematics master's
graduates showed men receiving $4,000 more than women 600
in 1986, and industries on the average paid graduates
$7,000 more than did educational institutions. Close to
half of master's graduates worked in the business and
industry sector, twofifths worked in educational institu
tions, and the remainder worked in government or other
sectors.
Doctoral Degree Recipients
The number of doctorates awarded by U.S. universities
in the mathematical sciences rose from about 200 in 1950
to a high of about 1,000 in 1973 and has fallen since to
about 750 (Fi Sure 4.161. In 1973 nearly four of five of these
doctorates were earned by U.S. citizens. In 1988 fewer
than half were. In the period from 1977 to 1987, more than
onethird of the approximately 9,000 doctoral decrees
awarded in the mathematical sciences went to nonU.S.
cltlzens.
48
Percent Doctorate
to Bachelorts
(7year la;,)
4
5
1.5
.......
. ~
. ~\
. ~
200
...
Mu S. ~j 738
. I.. Pda;Ies. 1~
1 . . ~ ~ ~ ~ _ ~ _
11 l l ~ 1 ~
1 ~ 11 ~ 11 ~ 1111@11
1 i l 1B 111
1 ~ ! l ~ 15
. l ~ ~ . . , en ~
73 77 82 87
,76
a@:
i73
FIGURE 4.16 Ph.D. degrees in mathematics. (See Appendix
Table A4.16.) SOURCE: American Mathematical Society
(AMS, 1976 to 1987).
OCR for page 35
Majors in Mathematics and Statistics
TABLE 4.10 Ratio of new doctorates in mathematics to new doctorates in selected other fields, 1970 to 1985
Chemistry Physics/Astronomy
Biology
Engmeenng
1970
1975
1980
1985
0.55
0.65
0.48
0.37
0.74
0.88
0.76
0.64
0.36
0.33
0.20
0.18
0.36
0.38
0.30
0.22
SOURCE: Edward A. Connors. Original data were taken from National Science Foundation (NSE, 1988c).
Certain of the trends in the number of mathematical
sciences doctorates awarded are not unique; some of these Other comparisons point out that the decline in mathe
same patterns have been observed in the total number of matical sciences doctorates since the mid1970s is more
exceptional. Edward A. Connors notes that each of the
ratios of the number of new doctorates in mathematics to
the numbers in chemistry, physics and astronomy, biology,
and eng~neenng has declined when computed for the years
1970, 1975, 1980, and 1985 (Connors, 1988~. For ex
ample, in 1970 there were approximately half as many
degrees in the mathematical sciences as in chemistry, but
this had dropped to onethird as many in 1985 (Table 4.101.
A possible reason for this decline ~ ratios is the decision
of those holding baccalaureates in mathematics to switch
to doctoral study in other areas, and data are given to
support the contention. The shifts increased dune=, the late
1960s and through the 1970s but appear to have leveled off
in the 1980s (Table 4.11).
The number of women U.S. citizens earning doctorates
subfield of mathematics.
research doctorates awarded by U.S. universities. This
total was about 10~000 in 1960, rose to a maximum of
nearly 34,000 in 1973, and then fell slightly until it stabi
lized at about 31,000 ire 1978. The percent of U.S. citizens
among the recipients was nearly 85% in the early 1970s and
had fallen to 72% in 1986. The number of doctorates
awarded in the physical sciences, including the mathemati
cal sciences, declined dunog the 1970s but increased
dunug the 1980s. Statistics was an exception to the decline
as degree production in that field was approximately stable,
but in mathematics the decline was extreme. Mathematics
is further exceptional in that it has been the slowest to stem
the decline of the 1970s. The percent of U.S. citizens
among, new doctorates awarded in the physical sciences
stood at 63~o in 1986, about 12 points higher than in the
TABLE 4.11 Mathematics majors going on to doctoral study in other areas of science and engineering, 1960 to 1985
Mathematics majors earning ~ = ~
doctorates in science/engineennga doctorates in mathematics Ratio
1960 290 207 0.71
1965 639 484 0.76
1970 1362 924 0.68
1975 1310 883 0.67
1980 1139 609 0.53
1985 959 506 0.53
a Includes mathematics and the social sciences.
SOURCE: Edward A. Connors. Original data were taken from National Science Foundation (NSF, 1988c).
49
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A Challenge of Numbers
TABLE 4.12 Ethnic representations among new mathematical sciences doctorates, U.S. citizens, 1977 to 1986
Number of new U.S. citizen Percentage of new Approximate
doctorates in mathematical U. S. Citizen percentage of
sciences Doctorates U.S. population
AsianAmericans 224 4.3 2
Blacks 80 1.5 12
Hispanics 34 0.6 7
Native Americans 26 0.5 0.5
Total, U.S. citizens 5,249
NOTE: For changes since 1974, see Appendix Table A4.12.
SOURCE: American Mathematical Society (AMS, 1976 to 1988~.
each year in the mathematical sciences has been essentially
constant since 1973, usually in the range 80 to 90. The
percent of U.S. citizens receiving doctorates that is ac
counted for by this group has risen from 10~o to 20%
because the total number of U.S. citizens receiving mathe
matical sciences degrees each year has fallen from 774 to
362. Overall, among both U.S. and nonU.S. citizens,
women constituted 10% of the new doctoral degree hold
ers in 1973 arid 17% in 1987. Among all doctorate
recipients in 1986, women were better represented, havin,
received 35% of the degrees; but in the physical sciences
women were awarded only 16% of the new doctoral
degrees and in engineering only 7~G.
Blacks and Hispanics receive inordinately few of the
doctorates awarded in the mathematical sciences. AMS
100%
Annual Survey results showed that during the tenyear
period from 1977 to 1986, these two groups combined
received about 237c of the doctorates awarded to U.S.
citizens (Table 4. 121.
Analogous data (NRC, 1987) for all research doctor
ates awarded by U.S. universities in the tenyear period
from 1977 to 1986 are given in Table 4.13. Additionally,
since the NRC report includes information on the visa
status of nonU.S. citizens, a second comparison is al
lowed. Including those nonU.S. citizens with permanent
visa status changes the representations very little except
when AsianAmericans are included. Then the change is
dramatic. AsianAmericans constituted 46% of the non
U.S. citizens with permanent visas who received a doctor
ate from a U.S. university between 1977 and 1986.
TABLE 4.13 Ethnic representation among all new research doctorates, U.S. citizens and permanent residents, 1977 to 1986
Number of Percentage of
doctorates to U.S. citizen
U. S. citizens doctorates
Number of
doctorates to
U. S. citizens or
permanent residents
Percentage of
U.S. citizen or
permanent resider
doctorates
AsianAmericans 4,579 1.910,404 4.2
Blacks 9,903 4.210,821 4.3
Hispanics 4,970 2.15,697 2.3
Native Americans 790 0.3792 0.3
SOURCE: National Research Council (NRC, 1987~.
50
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Majors in Mathematics and Statistics
100
80
60
%
40
20
o
Male Female
O Applied
mathematics
Topology

Analysis
· Probability/
Statistics
Algebra
FIGURE 4.17 Doctoral degrees in mathematical sciences,
distribution by subfield and sex. SOURCE: National Science
Foundation (NSF, 1983); see Appendix Table A4.15.
The five major subfields in the mathematical sciences
are algebra, probability and statistics, analysis, topology,
and applied mathematics. These five subtields accounted
for 65% of all doctorates awarded to both men and women
in the period from 1960 to 1982. The most popular
subfields in the mathematical sciences? as measured by the
number of degrees awarded, have not been the same for
men and women (Figure 4.17~. In the two decades begin
ning in 1960, the three most popular subfields for women
were algebra, probability and statistics, and analysis. For
men the most common choices were analysis, probability
and statistics, and applied mathematics. Women were
more than twice as likely as men to opt for either algebra,
probability and statistics, or analysis rather than for applied
mathematics.
A broad interpretation of the mathematical sciences
includes programs in other departments, such as operations
research, computer science, statistics, and mathematics
education. For example, operations research might be
taught in departments of mathematics, engineering, or
business. In the last decade, the total number of doctoral
decrees awarded annually in the broadly interpreted area of
mathematical sciences has been relatively constant at about
1 ,400 (Figure 4. 18~. The trend has involved student shifts
away from mathematics and education and toward com
puter science, with the numbers choosing statistics and
operations research remaining steady.
The median time from entry into graduate school to
receipt of the doctoral degree was 7.3 years for the 1986
mathematical sciences doctoral degree recipients and has
been essentially the same since 1958. The median for all
fields was 10.4 years in 1986. Generally, this median is in
the 7 to 9year range for the sciences and engineering and
in the 9 to 13year range for the social sciences, the
humanities, and education. The median registered time,
which is the time spent from entry until completion of
enrollment in graduate school, was six years for those who
received mathematics doctorates in 1986.
Threequarters of the 1986 mathematics doctoral de
gree recipients had their bachelor's degrees ire mathemat
ics, and the same fraction (73%) had master's degrees. In
all the physical sciences, 73% of those with doctorates had
bachelor's degrees in the same field, but only 52% had
master's degrees. For all fields these percents were 55%
and 79%, respectively.
About 200 postdoctoral appointments in mathematical
sciences were made at doctorategraIlting institutions in
1986. In recent years the number of appointments has
ranged from a low of 110 in 1981 to a high of 225 in 1985
(NSF, 1988a). An NRC survey showed that the number of
doctoral degree holders with postdoctoral study plans
increased from about 10% to 23~c of the total in the past tern
1,600
1,400
1,200
1,000
800
600
400
200
O r ' ~ ~ ~ ~ . . . . _
1976 1978 1980 1982 1984 1986
C1 Mathematics
Education
1~ Operations
Research
Statistics
· Computing
Mathematics
FIGURE 4.18 Number of doctorate recipients in broadly
interpreted mathematical sciences. (See Appendix Table
A4.14.) SOURCES: National Research Council (NRC, 1987)
and American Mathematical Society (AMS, 1976 to 1987).
51
OCR for page 35
A Challenge of Numbers
years (NRC, 1987~. This is stillfar belong the 40~o in all the
physical sciences who planned postdoctoral study. The
NRC survey asked new doctoral degree recipients to check
one of five reasons as the most important reason for either
taking or deciding against a postdoctoral appointment. In
this survey, two dominant reasons were given for deciding
against postdoctoral study in mathematics: "No postdoc
toral available" was given by 38%, and "attractive employ
ment" was given by 42%. The other three choices were
"little or no benefit," "inadequate stipend," and "other."
The "no postdoctoral available" response was given most
often in mathematics (38%) and the next most often in the
humanities (31%~; the response for all fields was 20%. The
dominant reason given in favor of postdoctorate by those
with mathematics doctorates was gaining additional expe
rience, the reason given by 69% of those planning postdoc
toral study. This reason too, led in all fields, with the
overall percentage at 56%.
Excluding those who have postdoctoral positions, about
threequarters of doctorate mathematical scientists are
currently employed in academe. The fractions taking
52
nonacademic employment are higher for applied mathe
matics and for statistics and lower for mathematics. There
have not been major changes in the employment patterns of
doctoral degree holders over the years. Approximately
20% find employment in business and industry arid the rest
are employed in government. An increased demand in aca
demia for mathematical scientists is likely because of the
current low supply and an increase in retirements.
Patterns and Prospects
At all degree levels, several patterns are apparent. The
numbers of degrees awarded annually have decreased
significantly since peaking in the early 1970s but have
shown some increases recently. The attrition rates in
degree programs are high. Relatively few women, blacks,
and Hispanics receive degrees, especially graduate de
grees, and nonU.S. citizens are close to achieving a
majority among graduate students. There are increased
demands for mathematical sciences degrees in the
workplace, and these increases are projected to continue.