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7
Unclergraduate Education
Mathematics departments in colleges and universities today serve
a wide variety of student majors. They train future mathematical
scientists, both for academic work and for work in industry and
government, and, as it has for centuries, mathematical training plays
a key role in the education of physical scientists and engineers. In
recent years mathematical methods have proved to be of increasing
use in the biological and social sciences as well, and so the mathe
matician is now acquiring another large gTOUp of clients. Second
ary and elementary school teachers form yet another increasing
contingent. Some knowledge of mathematics has traditionally been
recognized as a significant part of the general education of a college
student: today this knowledge is being broadened to include some
appreciation of the widely useful tool of computers.
Chapter 3 of the report of our Panel on Undergraduate Edu
cationt surveys in detail these various components of demand for
mathematical training; and, on this basis, Chapter 4 of that report
makes an assessment of the resulting needs for college and university
staff in the mathematical sciences. We give a briefer discussion along
similar lines in the sections below, on total mathematicalscience
course enrollments and on quality and distribution of mathematical
science faculty.
The first thing to be emphasized about the recent history of
undergraduate education in the mathematical sciences is the extent
of its growth and change over the past 25 years. Our Panel on
Undergraduate Education has documented this vividly in Chapter
121
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The 1~lathematical Sciences in Education
2 of its report, presenting eight detailed case histories of changes at
individual colleges and universities. While these institutions repre
sent considerable variety in their educational goals and in their
clientele, they reveal many similar trends over the last quarter cen
tury with regard to the mathematical sciences. The following items
occur repeatedly: very significant increases in mathematicscourse
enrollments; spectacular increases in numbers of mathematics
majors; new undergraduate major programs in the mathematical
sciences; many new advanced courses; increased undergraduate
enrollments in graduate courses; impact of improved high school
curricula on beginning college mathematics courses; special new
courses for statistics, computing, and the social sciences; significant
increases in staff size; difficulties in recruiting new staff.
THE INCREASE IN MATHEMATICS MA TORS
Perhaps the most striking item on the above list is the greatly
increased number of mathematics majors. This is documented more
precisely in Table 1, which shows comparative tabulations and
projections by K. A. Simon and M. G. Fullam of the U.S. Office
of Education.~7
These tabulations show that over the years 19551965, during
which college enrollments roughly doubled, the annual number of
hachelor's degrees granted in mathematics and statistics increased
by a factor of 4.9, while for engineering, the physical sciences, and
the biological sciences the corresponding factors were only 1.6, 1.7,
and 2.8, respectively. The projections shown for the years 1965
1975 call for the annual number of bachelor's degrees ire mathe
matics and statistics to increase by a factor of 3.0, while for engineer
ing, the physical sciences, and the biological sciences the correspond
~ We have rounded off these projections to two significant figures, though, no
doubt, for a year as remote as 1976 it is at most the initial figure that is at all
likely to be borne out. The projections are based on the assumption that the
percentage distribution of degrees by field will continue the 19551966 trends.
This is, we feel, a reasonable basis for projections; but should it turn out to be
only indifferently fulfilled, the projections could, of course, be even more in
error. For instance, projections from the 19~551959 trends on this basis would
hardly have predicted the depression in numbers of engineering bachelor's de
grees that actually occurred in the early 1960's. as shown in Table 1.
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i23
TABLE ~ Earned Bachelor's Degrees, 19551966, with Projections
{0 1976
YEAR
MATHEMATICS PHYSICAL BIOL~ICAL
Ai!lD STATISTICS ENGINE~NG SCIENCES SCIENCE
EARNED, 1955  1966
19541955 4,034 22,589 10,516 9,050
19551956 4,660 26,312 11,672 12,566
19561957 5,546 31,211 12,934 13,868
19571958 6,924 35,332 14,352 14,408
19581959 9,019 38,134 15,460 15,149
19591960 11,437 37,808 16,057 15,655
19601961 13,127 35,866 15,500 16,162
19611962 14,610 34,735 15,894 17,014
19621963 16,121 33,458 16,276 19,218
19631964 18,677 35,226 17,527 22,827
19641965 19,668 36,795 17,916 25,305
19651966 21,190 35,830 18,020 25,680
I'ROJEC~ TO 1976
19631967 24,000 37,000 19,000 28,000
19671968 31,000 43,000 23,000 33,000
19681969 36,000 45,000 24,000 37,000
19691970 38,000 44,000 26,000 38,000
19701971 41,000 43,000 27,000 39,000
19711972 45,000 44,000 28,000 41,000
19721973 49,000 44,000 30,000 44,000
19731974 55,000 45,000 31,000 47,000
19741975 60,000 46,000 33,000 50,000
19751976 65,000 46,000 35,000 53,000
ing projected factors of increase are 1.3, 1.9, and 2.0, respectively.
Admittedly, projections can be no more than rough guides; and, in
fact, discrepancies for 1965 between the above actual figures and the
Once of Education's slightly earlier projected figuresl8 suggest that
the above projected factors of increase for 19651975 might turn
out to be somewhat high for mathematics and statistics and some
what low for the biological sciences. Even so, there seems little doubt
that increased numbers of majors will contribute considerably to
college staffing strains in the mathematical sciences over the next
few years.
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The Mathematical Sciences in Educations
TOTAL ENROLLMENTS IN THE MATHEMATICAL
SCIENCES
The Lindquist and CBMS surveys together show that over the period
Mom academic years 19601961 to 19651966 the total enrollment
in undergraduate mathematicalscience courses increased from
744,000 to 1,068,000, or 44 percent. Since the most closely compar
able general enrollment figures show a 48 percent increase over this
same period, we conclude that for this period the growth in under
graduate mathematicalscience enrollments has been roughly the
same as the growth in college enrollments generally. (See reference
16.)
For the succeeding period from 19651966 to 19701971, however,
we have reason to believe that this situation will change consider
ably. If the projections below, which are a more conservative ver
sion of those of our Undergraduate Panel, are at all near the mark,
then mathematicalscience enrollments will increase by more than
70 percent over this period, while general enrollments will increase
by only around 30 percent. The remainder of this section presents
the case for these projections.
For the academic year 19651966 the CBMS survey found that in
the fouryear colleges and universities of the United States there
were some 1,068,000 undergraduate course enrollments in the mathe
matical sciences. For these colleges and universities, the U.S. Once
of Education predicts a 29 percent rise in total undergraduate
enrollments over the period 19651970. Thus, taking 29 percent of
1,068,000, one might predict a rise over this period of approximately
310,000 undergraduate course enrollments in the mathematical sci
ences. We believe this to be a serious underestimate because of two
factors. The first of these factors is the projected fasterthanaverage
growth in the number of majors in certain fields where it is custom
ary to take a greaterthanaverage amount of mathematical course
work. The second is the fact that majors in several fields are no
beginning to take more courses in the mathematical sciences than
formerly and by 1970 will almost surely be taking significantly more
such courses on the average than in 1965.
Chapter 4 of the report of our Panel on Undergraduate Edu
cation~ has analyzed, for each of the principal fields concerned, the
extent to which one or both of these factors may be expected to
increase mathematicalscience course enrollments over the 196.~
1970 period, above and beyond the 310,000 attributable to the gen
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125
TABLE 2 Projected MathematicalScience CourseEnrollment In
crease, 19651970
FACTOR
General growth
Mathematicalsciences excess
Physicalsciences excess
Engineering excess
Biologicalsciences excess
Psychology excess
Socialsciences excess
Elementaryteacher excess
Introductorycomputing excess
rotary
INCREASE
310,000
81,300
37,000
37,000
9,500
19,500
67,500
60,000
172,000
796,800
evil 29 percent growth predicted for college enrollments over this
period. This analysis is based on the following: (1) the U.S. Office
of Education projections, 7 regarding the number of bachelor's de
grees to be expected in various fields in 1970; (2) information from
the CAMS survey regarding the average number of mathematical
science courses taken by majors in these various fields as of 1965;
and (3) the Committee on the Undergraduate Program in Mathe
matics (C~PM) panel recommendations,~923 as an indication of
the number of mathematicalscience courses to be reasonably antic
ipated for majors in these various fields by 1970.
We and our Panel on Undergraduate Education are very much
aware of the oversimplifications and possibilities for error involved
in making, for several years ahead, any courseenrollment projec
tions whatever. For this reason, the Panel on Undergraduate Edu
cation has, in Chapter 4 of its report, considered the consequences
of varying its projected data and its hypotheses in several reasonable
ways. The prediction of intensification, for several years to come, of
the shortage of qualified college teachers of the mathematical sci
ences is found to be stable under all these variations. Our own
figures, shown in Table 2, provide a further variation. It too pre
dicts an intensified shortage in the supply of qualified teachers over
the next few years. It is for these reasons that we feel some confidence
in this prediction.
With these qualifications and reservations, we present, in Table
2' our best effort to project mathematicalscience courseenrollment
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The Mathematical Sciences in Education
increases from 1965 to 1970. The first item in this tabulation is the
one mentioned earlier: the basic 310,000 increase in mathematical
science course enrollments to be anticipated simply from the U.S.
Once of Education's prediction of a 29 percent general increase in
college enrollments.
For the mathematical sciences, the physical sciences, and engi
neering, the projected excesses shown in Table 2 are those computed
by our Panel on Undergraduate Education. The mathematical
sciences excess arises from the more rapid growth in mathematical
science majors already discussed, together with the fact that such a
major accounts for approximately six course enrollments in his field
during his undergraduate years. The physicalsciences excess arises
from two factors: first, a higherthanaverage projected rise in the
number of majors, who in 1965 typically took approximately three
mathematicalscience courses during their undergraduate years; and
second, the prediction, based on reference 23, that by 1970 these
majors will be taking four rather than three such courses. For engi
neering, the projected 19651970 increase in majors is at 21 percent
instead of the general figure of 29 percent, which by itself would
yield a deficiency rather than an excess in mathematicalscience
course enrollments; but this deficiency is more than offset by the
tendency, already beginning to be seen, for the undergraduate engi
neering curriculum to shift from a fivesemester to a sevensemester
sequence in the mathematical sciences (see reference 24~.
For the biological sciences, psychology, and the social sciences,
the figures shown in Table 2 are just half as large as those in the
report of the Panel on Undergraduate Education. Our figures
correspond to the assumption that, on the average, by 1970 half of
the majors in these fields will be taking, during their undergraduate
careers, one more mathematicalscience course than was typical for
the majors in these fields in 1965. (The Panel on Undergraduate
Education made the assumption that by 1970 all majors in these
fields would on the average be taking one more such course.)
With regard to elementary school teachers, we recognize that a
strong movement is already under way to stimulate the upgrading
of their preservice training in mathematics. The CUPM report23
and regional conferences of elementary teachers sponsored by the
CUPM25 have done much in this respect. As a result, we feel that, on
the average, perhaps two thirds of these teachers will by 1970 be
taking one more mathematics course during their undergraduate
years than in 1965 (whereas the Panel on Undergraduate Educa
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127
lion felt it likely that, on the average, all elementary teachers would
be taking one more such course by 1970~.
It is particularly difficult to make predictions regarding the
rapidly growing and changing field of computing. The recently
published Pierce reports recommends that by academic year 1970
1971 all college students should have an introductory course in
computing. It may turn out that the ideal place for such an intro
ductory course is in high school rather than college; or it may turn
out that introductory computing courses oriented toward various
subjectmatter fields will tend to be taught in various college de
partments, much as elementary applied statistics courses tend to be
today. The most likely assumption, however, seems to be that the
burden of teaching such courses will fall on the college faculty in
computer science and the mathematical sciences generally. In any
event, our Panel on Undergraduate Education has made its projec
tions on the assumption that half of the 5,500,000 undergraduates
in fouryear colleges in 1970 will take, at some time during their
fouryear undergraduate careers, a onesemester college course in
computing, and that half of these are not already included else
where in the tabulation ire Table 2. (This last is certainly not un
reasonable in view of the present distribution of college majors.3
This yields the equivalent of about 172,000 (academicyear) course
enrollments for 1970, and it is this figure that appears in Table 2.
Table 2 yields a total projected courseenrollment increase of
796,800. Dividing this by the courseenrollment total of 1,068,000
for academic year 19651966, we find that, over the 19651970
period, mathematicalscience course enrollments are projected to
increase by approximately 74 percent, which is more than 2.5 times
the 29 percent by which general college enrollments are projected to
increase over this period.
QUALITY AND DISTRIBUTION OF
MATHEMATICALSCIENCE FACULTY
The CBMS survey found that for the fouryear colleges and univer
sitiLes of the United States the 1,068,000 undergraduate mathemati
calcourse enrollments of academic year 19651966 were handled by
a fulltime faculty of some 10,750, aided by a quite small parttime
faculty and, in universities, graduate assistants. This yields a ratio
of one fulltime faculty member to approximately 100 course en
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The Mathematical Sciences in Education
TABLE 3 Highest Famed Degree, 19651966
DECREE
NUMBER
Doctorate (mathematical sciences)
Doctorate (education)
Doctorate (other fields)
Masterts degree
Bachelor's degree
TOTAL
5,000
500
coo
400
10,750
emollients. Now the preceding section estimated some 796,800 mOre
course enrollments in the mathematical sciences for academic year
19701971 than for academic year 19651966. Therefore, this
courseenrollment increase will require some 8,000 more fulltime
faculty in the mathematical sciences for academic year 19701971
than for academic year 19651966.
We now try to assess what the quality of the additional faculty
Is likely to be, using as a rough measure of faculty quality the
proportion of fulltime faculty holding doctorates, as is done in
Cartter's studies.26 ik For academic year 19651966, the CBMS survey
found the 10,750 fulltime faculty in the mathematical sciences to
have highest earned degrees distributed as shown in Table 3.
Thus in academic year 19651966 some 53 percent of the mathe
maticalscience faculty held doctorates in some field, a little over 46
~ Academic degree is, of course, only one dimension in the measure of quality.
it is clear, for instance, that one is qualified to provide instruction in a core
mathematics topic only if he has a thorough grasp of the substantive material of
that topic, of its relation to mathematics at large, and of the manner in which
others will build upon the foundation it provides. By the same token, one is
qualified to provide instruction on a facet of the use of mathematics only if
(again) he has a thorough grasp of the mathematical disciplines he must use, of
the facets of science or technology under study, and of the attitudes and objec
tives that are appropriate to the questions with which he is coping. Thus, not
only must there be an adequate level of competence as measured by academic
degree acquisition, but there must also be a breadth of topical competence
measured in terms of ranges of interest, attitudes, and scientific literacy.
Clearly, in institutions that are only beginning to develop comprehensive prc>
grams, these requirements of quality cannot be met immediately, but they should
be used as guidelines for that development. A minimum first step requires that
the means be found to assist faculty members to maintain contact with the
changes in relevant fields of }knowledge.
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129
percent being in the mathematical sciences. Shortly we shall dis
cuss the distribution of doctorateholding faculty among colleges
and universities, which is far from uniform.
To see how the percentage of doctorates is likely to change over
the next few years, we now need an estimate of the number of new
PhD's who will enter mathematicalscience teaching in fouryear
colleges and universities over the period 19651970. For this we take
the estimate of our Panel on Undergraduate Education, which found
this number to be approximately 3,300 (see Chapter 4 of their
report)). Their analysis followed Cartter's modeler and used his
figure of 2 percent per year for attrition (due to death, retirement,
and net outflow to other professions) of doctorate holders from
college and university teaching. Also, in accord with information
from the csMs survey, they assumed that 70 percent of the newly
produced mathematicalscience PhD's will go into college and uni
versity teaching, as opposed to Cartter's value of approximately 33
percent for all academic fields combined.
To collect our figures, we have estimated that over the period
19651970 some 8,000 new fulltime staff members will be needed in
n~athematicalscience teaching in the fouryear colleges and uni
versities. We also have the estimate of net inflow of some 3,300
PhD's into such teaching during this period. Thus only about 41
percent of the new faculty will have doctorates; hence, if these
estimates are at all close, the percentage of doctorate holders on
mathematicalscience faculties, currently about 53 percent, will
decline.
The distribution of the doctorateholding faculty is far from uni
form, the universities having more than twice the fraction of the
total doctorateholding faculty that liberal arts and teachers' col
leges [have, while the latter carry slightly more of the total under
~aduate teaching load. Specifically, the scams survey reveals that for
academic year 19651966 the distribution of doctorateholding
faculty was approximately that shown in Table 4, which should be
read with the following comments in mind.
[First, it is not surprising that the universities have a much higher
~ With the strong demand from other quarters, it is a distinct possibility that i
the future fewer than 70 percent of the new PhD's will go into academic work.
This could intensify the college teacher shortage predicted here. In any case,
it should be emphasized that the prediction of a growing shortage is stable under
any reasonable variation in the particular percentages used for attrition and for
resew PhD's entering teaching.
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The Mathematical Sciences in Education
TABLE 4 Distribution of DoctorateHolding Faculty, 19651966
PERCENTAGE OF THE PERCENTAGE OF THE
UNDERGRADUATE DOCTORATE HOMING
TYPE OF INSTITUTION
TEACHING LOW FACULTY
Universities
Liberal arts and teachers' colleges
Technological institutes
TOTALS
45
49
6
100
63
30

100
fraction of a limited doctorateholding faculty, since it is univer
sities that carry the bulk of graduate instruction and the direction
of doctoral and postdoctoral research. Second, as far as the first two
undergraduate years are concerned, the proportion of PhD's teach
ing at universities is much more closely comparable with that at
liberal arts colleges and teachers' colleges, because of the wide
employment of graduate assistants in such teaching at universities.
In this connection, the CBMS survey has found that, in the median
university, 40 percent of the freshmansophomore teaching load is
carried by graduate assistants, and, in 38 percent of the universities,
at least half of the freshmansophomore load is carried in this way.
In Chapter 8, we discuss (see page 145) Graduate Student Par
ticipation in Undergraduate Teaching in universities and suggest
improvements for the future. In Chapter 9, in the discussion of The
College Teacher (page 147), we consider what may be done to meet
anticipated shortages in qualified faculty in the mathematical sci
ences, especially in the weaker colleges and in certain critical fields.
THE JUNIOR COLLEGES
There are now more entering freshmen in junior colleges than in
universities, and over one third of all entering freshmen are junior
college students. This statement is impressive even when it is under
stood that the term "junior colleges" here includes twoyear techni
cal institutes, and that the data include parttime students and
students in occupational or general studies programs not chiefly
creditable toward a bachelor's degree. The actual figures, from the
U.S. Once of Education,28 are shown in Table 5. Total enrollment
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131
TABLE ~ FirstTime Enrollments in Fall 1966, Nondegree Credit
and PartTime Students Included
NUMBER
(THOUSANDS
OF STUDENTS)
PERCENTAGE
OF TOTAL
Universities 427 27
Other fouryear institutions 591 38
Twoyear institutions 547 35
TOTALS 1,565 100
figures with a breakdown into fulltime and parttime students are
also instructive and are given in Table 6. These figures show that in
1966 twoyear institutions accounted for 21 percent of all under
graduate enrollments, 17 percent of the fulltime ones, and 30 per
cent of the parttime ones.
Enrollments in junior colleges are geographically heavily con
centrated in certain states, reflecting not only differences in popu
lation density but also differences in state policy regarding the
establishment and expansion of these institutions. Thus 38 percent
of all junior college students are in California, and slightly over 50
percent attend junior colleges in California, Florida, or Illinois.
Against this background of general student enrollments we now
give a few results from the 19661967 CBMS survey of the mathe
TABLE 6 Total Enrollments in Fall 1966, Nondegree Credit and
PartTime Students Includeda
THOUSANDS OF STUDENTS
TOTAI~
FULLTIME PARTTIME
Universities 2,4821,789 693
Other fouryear institutions 2,6261,941 685
Twoyear institutions 1,331739 591
All institutions
6,4394,469 1,969
aFigures from reference 20.
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The Mathematical Sciences in Education
TABLE 7 Distribution of MathematicalScience Course Enrollments
in Fall 1966, by Size of Junior College
INSTITUTIONAL SIZE NO. OF COURSE DISTRIBUTION'
(NO. OF STUDENTS) ENROLLMENTS (%)
5,000 and over 107,000 31
2,000 to 4,999 104,000 30
1,000 to 1,999 55,000 16
Under 1,000 82,000 23
All institutions 348,000 100
matical sciences in junior colleges, the first such survey ever made.
A much more thorough presentation and discussion appears in the
CBMS Survey Committee's report, Volume I, Chapter V. Numbers
of mathematicalscience course enrollments, broken down according
to the size of the institution, are given in Table 7.
Enrollments of entering freshmen in mathematics courses tend
to be at less advanced levels for junior colleges than for fouryear
colleges, as Table 8 shows. In the junior colleges of largest enroll
ment (over 5,000), 55 percent of all mathematics course enrollments
by entering freshmen for the [all of 1966 were below the level of
college algebra and trigonometry. While junior college freshmen
tended to have a generally lower attainment level in high school
mathematics, the differences shown in Table 8 appear to reflect not
so much differences in ability as differences in goals. A considerable
TABLE 8 Percentage Distribution of Entering Freshmen Enroll
ments at Three Mathematics Course Levels
LEVEL
TWOYEAR FOURYEAR
INSTITUTIONS, INSTITUTIONS,
FALL 1966 FALL 1965
Below college algebra
and trigonometry 42% 19%
College algebra,
trigonometry, and equivalent 44% 49%
Analytic geometry,
calculus, and above 14% 33%
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133
fraction perhaps 30 percent of the junior college students aim
for immediate occupations in business and technology rather than
for careers in teaching or other professions. (Most of the remaining
70 percent are students intending to transfer to fouryear colleges
upon graduation.)
Among junior college students, those in occupational curricula
have mathematical needs and abilities somewhat different from
those intending to transfer to fouryear colleges. Some of the
strongly occupationoriented mathematics is, in fact, taught outside
mathematics departments altogether. The most common example
of this is business mathematics taught in a division of business.
Other examples are precalculus technical mathematics and statistics.
It is to be emphasized that industry needs technical aides who are
graduates of strong twoyear college programs. In fact, a spokesman
for a prominent industrial laboratory has indicated that although
salaries and opportunities for advancement are good, welltrained
technical aides are often harder for industry to find in needed num
bers than are those with more advanced professional training.
THE MATHEMATICS FACULTY IN JUNIOR
COLLEGES
In junior colleges in 1967, the fulltime mathematicalscience
faculty numbered approximately 2,700; and taking into account
parttime faculty members, the fulltime equivalent faculty was
approximately 3,100. Their training was overwhelmingly at the
master's level: 84 percent had the master's as their highest degree,
12 percent the bachelor's, and about 4 percent the doctorate. The
field of highest level of training was within the mathematical sci
ences for only 62 percent; for 24 percent it was mathematics edu
cation, and for 14 percent it was in some other field.
To the question, "Do you have difficulty in recruiting and keep
ing an adequate mathematics faculty?" about 73 percent of the
junior college mathematics departments responding to the CBMS
questionnaire said, somewhat surprisingly, that they did not. Prob
ably the principal reason for this is that the better high school
teachers form an enormous and highly available pool of supply.
Another reason is that, even in comparison with the private four
year colleges, the public junior colleges can offer a better median
salary to professors.
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The Mathematical Sciences in Education
Whether this faculty is indeed adequate, and whether it will
prove adequate for the future, may be questioned. The Mathe
matical Association of America, in its resolutions to the Congress
and the National Science Foundation,4 states that the junior col
leges appear to form the weakest link in the chain of higher educa
tion in mathematics. Certainly a junior college teacher now qualified
to teach only the most elementary mathematics courses may in the
future find nothing he can teach; for the preparation of entering
freshmen will undoubtedly continue to improve, and remedial
teaching will increasingly be done by such techniques as programmed
instruction. In its recent report,29 an ad hoc CUPM Panel on the
Qualification of College Teachers of Mathematics states that a
strong mathematics master's degree (what it calls the "first graduate
component") ". . . should represent adequate training for teaching
transfer students in junior colleges, provided the teacher continues
to remain intellectually alive." Although there are no firm percent
ages, many with experience gained in teachinginstitute programs
for college teachers feel that numerous junior college teachers with
master's degrees fail to meet these criteria (see reference 16, Chapter
5) .
In summary, the junior colleges form a fastgrowing but geo
graphically highly nonhomogeneous component of higher educa
tion. In these colleges, a strong and continuing effort will be
needed to raise and maintain faculty professional standards in the
mathematical sciences. The university mathematical community
can contribute to this in two ways, primarily: first, by producing
new junior college faculty with strong mathematical training
through the master's level; and, second, by providing opportunities
for appropriately oriented continuing education for those already
doing mathematicalscience teaching in junior colleges. The first of
these ways underlines the fact that there is need for federal support
for graduate mathematical training that stops at the master's level,
as well as for PhD training. The second is a part of the effort in
continuing mathematical education for college teachers, both
academicyear training and summer training. This too will require
strong federal support.