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9
Special Eclucational Issues
THE COLLEGE TEACHER
Our Panel on Undergraduate Education has given detailed con-
sideration to a number of problems of college teachers of the mathe-
matical sciences: see especially Chapters 4, 5, and 6 of their report."
Foremost among these problems is the shortage of qualified college
faculty, discussed in Chapter 4 of reference 1 and in Chapter 7 of
this report. The principal way of training new qualified teachers to
meet this shortage is to sustain graduate mathematical education,
especially PhD production, at a vigorous level. Over the five-year
period ending in 1966, PhD production in the mathematical sciences
increased at an average annual rate of approximately 18 percent (see
Table 10, page 139~. This table also indicates that an average annual
rate of only approximately 10 percent has been projected for the
succeeding five-year period ending in 1971. It is also anticipated that
it may not be possible to sustain the closely correlated support of
basic research in the core areas at more than this level (see discussion
of The Core on page 197~. Much could be done to alleviate the
shortage of qualified college faculty in the mathematical sciences if,
for the five-year period ending in 1971, an annual rate of increase of
PhD production near 18 percent instead of near 10 percent could
be maintained. Also, certain measures could be taken to avoid
wastage of mathematical talent at the graduate level (see the section
on Wasted Mathematical Talent, page 159~.
Along with the training of new faculty, however, strong and con-
tinuing efforts are needed to upgrade and update the qualifications
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of the present college mathematical faculty. Developments in the
applications of mathematics over the period 1961-1966 show sharply
the changed requirements in faculty competence if students in un-
dergraduate colleges are to be offered relevant mathematical educa-
tion. The mathematical disciplines in which enrollments have grown
most rapidly in this period are linear algebra, probability and statis-
tics, and computer-related mathematics. This situation requires a
substantial retraining program for the college mathematics faculty if
the relevant mathematical education is to become widely available in
a reasonable length of time.
The rapid growth in demand for instruction in the fields of linear
algebra, probability and statistics, and computer-related mathematics
from 1960-1961 to 1965-1966 is indicated in Table 11. The data are
taken from the Lindquist and CBMS surveys.
TABLE 11 Growth in Certain Mathematics-Course Enrollments
FIELD 1960 - 1961 1965 - 1966 LACTASE (I)
Linear algebra 4,000 19,000 375
Probability and statistics 23,000 44,000 91
Computer-related mathematics 4,000 20,000 400
Numerical analysis 3,000 5,000 67'
Analytic geometry and calculus 184,000 295,000 60
Precalculus mathematics 430,000 554,000 29
All undergraduate mathematics 746,000 1,068,000 44
During this period, mathematics-course enrollments as a whole
grew at about the same rate as did undergraduate enrollments gen-
erally. College enrollment in precalculus courses grew at a much
lower rate, calculus enrollment grew at a substantially higher rate,
but the enrollments in linear algebra, probability and statistics, and
computer-related mathematics grew at a rate spectacularly greater
than did undergraduate enrollments as a whole.
In many colleges the availability of suitable courses, or even any
courses, in these subjects for undergraduate students is very low,
quite apart from the quality of the instruction. The CBMS survey
indicates that out of the country's four-year colleges and universities
there are perhaps 750 that teach mathematics to a significant number
of students and yet offer no course in linear algebra. As for comput-
ing work, even more colleges offer no course in computer program

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149
ming,. while only a negligible fraction offer any computer-related
mathematics beyond elementary programming.
Effective action to meet this situation will have to be on a massive
scale. One retrained faculty member per department will hardly be
sufficient to provide the necessary nucleus if a department is to move
forward to the desired level. Progress toward meeting the needs could
be made with a three-year program in which, by retraining one
faculty member per year, 750 mathematics departments could ac-
quire a faculty member in each of the three areas alluded to whose
retraining at least qualified him to teach strong elementary courses
in this area. A new program of this breadth and magnitude would be
expensive upwards of $9 million per year, it has been estimated.
Chapter 5 of the report) of our Panel on Undergraduate Educa-
tion eloquently points out that in the most seriously underdeveloped
colleges the needs for faculty retraining are almost overwhelmingly
severe and will require special measures. A number of concrete sug-
gestions are made. We also commend to the attention of college
administrations, federal agencies, and private foundations the dis-
cussion in that report (Chapters 4, 5, and 6) of other pressing prob-
lems of the college teacher: geographic and intellectual isolation
from the mainstream of mathematical activities, the need for new
curricula and new teaching methods, the multiplicity of demands on
the college teacher's time, and the difficult working conditions and
inadequate facilities with which he often has to cope.
APPLIED MATHEMATICS
The specialproblems of education in applied mathematics were the
subject of an extensive recent conference sponsored by the Society for
Industria, and Applied Mathematics. The proceedings of this con-
ference gave been published in detail,35 and we refer to these pro-
ceeding~ for a full discussion, from several points of view, of some of
the matters we discuss here.
It has been recognized for many years that applied mathematics
does not attract a sufficient share of talent from the younger genera-
tior~ Many more students could enter various fields connected with
# It least not in mathematical-science departments. In some colleges or univer-
sides where no mathematical-science department offers a course in computer pro-
~ramming, such a course may be available within a school of engineering or
ousiness or in some other department.

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applications and pursue satisfactory and useful professional careers.
Weyl's 1956 reports states:
The most serious problems besetting the conduct of applied mathematical
training programs at present, are in order of their importance:
(1) the lack of qualified students;
(2) the difficulty of finding applied mathematicians qualified and in-
terested to accept faculty appointments; and
(3) the questions of relative size and administrative relations of faculty
Organizations.
The shortage of top students in applied fields (except in computer
science) is sometimes blamed on the allegedly introspective and
monastic tendencies of modern mathematicians, which are said to
have a pernicious effect on the teaching of mathematics. The follow-
ing statement is from a letter by an applied mathematician:
Many mathematics instructors in colleges and universities proceed as if all
of their students are destined to become mathematicians. They do not con-
vey to the students the scientific origins of mathematical ideas or the
possibilities of applying mathematics to natural and social sciences. They do
not foster the skills necessary for future practitioners of those sciences. In-
stead they inculcate in the students a snobbish conviction that only pure
mathematics is intellectually respectable.
This criticism may be partly justified. On the other hand, another
applied mathematician wrote:
Good students are attracted by great teachers. The trouble with applied
mathematics education in this country is the paucity of applied mathe-
maticians in universities who deserve the admiration of the best students
and who are able to articulate the intellectual excitement of Heir subject.
At any rate, we consider that the future of applied mathematics,
in the broadest possible sense, depends largely on the educational
efforts made today and in the near future. Two things are seeded:
broadening of the education of young scientists and mathematicians
generally and education in applied mathematics as a discipline with
its own objectives, attitudes, and skills. We believe that both efforts
should begin during the undergraduate years.
Concerning the general education, we believe that not enough
is taught to students in their undergraduate work about applications
of mathematics, both in traditional subjects and in the very modern
and new fields. There should be mathematics courses, available to

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151
all, that stress the heuristic process by which mathematical models
of scientific questions are arrived at and that emphasize strongly the
character of mathematical thought that leads to results from which
one can infer the answers to such scientific questions. This is not the
same detailed material that most efficiently leads young mathematics
students to the frontiers of professional mathematics, and it re-
quires attitudes not necessarily compatible with the attitudes of such
mathematics courses. For some special students, summer sessions
especially devoted to mathematics in the service of science would be
a useful institution. As an example, the presentation of the range of
possibilities inherent in computing machines to both undergraduate
and graduate students would be timely and useful.
Such presentations would require, of course, a reservoir of pro-
fessors. It would be useful to organize the preparation of a sufficient
number of persons who could teach and inspire students in this
direction. It also seems to us desirable to initiate, in a number of
institutions, possibilities of exchange courses and lectures. What we
have in mind are mathematicians teaching courses on some phases
of new mathematical theories in the departments of physics, biology,
or economics, and vice versa. Physicists and others could lecture to
mathematicians on the essence of problems arising in the new de-
velopments in their own fields. In both cases, the lecturers should
present the methods and the problems in the forefront of their
sciences.
As far as the preparation of professional applied mathematicians
is concerned, we note that this task is accomplished in various ways
in various countries. In Great Britain, students enter the university
either as applied mathematicians or as pure mathematicians. In the
Soviet Union and in Germany, specialization takes place later in a
student's career. In this country, no single system has been used! and
many American applied mathematicians were originally trained as
engineers or physicists. This Committee believes that one could not
and should not prescribe a method for educating applied mathe-
maticians to be used in all or most colleges and universities. On the
contrary, widespread experimentation should be encouraged. Never-
theless, we want to recommend for special attention and support a
plan for undergraduate education in applied mathematics which
has recently been evolving at Harvard and MIT. This plan and its
underlying philosophy are described in C. C. Lin's address in the
proceedings35 of the Aspen Conference, mentioned above.
Concrete educational needs in physical mathematics and other

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parts of applied mathematics are discussed further in Chapter 14.
We emphasize, however, that unless there is a sizable core of devoted
applied mathematicians who are to be judged both in professional
standing and in university duties as applied mathematicians per se,
and who are supported for their activities in applied mathematics,
efforts to encourage applied mathematics are doomed to failure.
Without a group of devoted, talented, and competent people, most
of the crucial suggestions will not materialize. (We note, with some
apprehension, that most of the specific recommendations we make
concerning applied mathematics already appear in the 1956 survey.9)
COMPUTER SCIENCE
As of the summer of 1967 there were approximately 40,000 auto-
matic computers in the United States, excluding special military
computers. In Parts I and II we have pointed out the penetration of
computing into almost every facet of technical, scholarly, and eco-
nomic activity. If these computers are to serve society as well as they
should indeed, if they are even to solve the complex problems so-
ciety already counts on them to solve-there is a vital need for
persons educated at all levels in computer science. The Pierce re-
port3 calls for an extremely large increase in the funds for education
of college and university students in computing and for computing
costs incidental to such education. Such increased activity cannot
occur without a correspondingly large increase in well-educated man-
power devoted to teaching computing in colleges and universities
and leading the development of the subject. The total requirement
for those educated in computer science certainly demands a program
of massive magnitude.
These new computer scientists must be educated in our universi-
ties. The universities have heard the challenge, and departments
~ . . . . .
at computer science are springing up at an ever-lncreaslng rate.
Already bO-odd departments exist at universities in the United
States and Canada. In a major university not now possessing such a
department there is usually a plan to create one in the near future.
Although each university should certainly organize itself to best
serve its own goals and constituency, it appears to us that the au-
tonomy of a separate department offers one reasonable way to permit
computer science to develop as it must. A position dominated by an-
other science or by a branch of engineering may provide too many

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constraints to the development of the subject and the education of
the new generation.
Once organized, departments of computer science are typically
engaged in three kinds of activities:
1. Research in computer science, which may or may not involve
an expensive computer laboratory;
2. Education of three classes of students: (a) future specialists in
computer science at the bachelor's, master's, and/or doctor's level;
(b) university students who need computing as a research tool in
their university or postuniversity careers; (c) general students who
wish to learn about computers as an important part of the world
they live in;
3. Service to the university community. Computer scientists are
typically interested in trying to help others use computing in their
research, both through direct consultation and through leadership
and technical advice to the university computation center.
The supply of leaders in computer science is critically low. One
evidence of this is the difficulty experienced by most departments in
finding suitable faculty. Another is the fact that computer science
has roughly one third of the active professional workers in the
mathematical sciences, yet only one twentieth of the new PhD's, and
an even smaller proportion in earlier years.
What can be done about this? Clearly no ideal solution can be
found for the next few years, but we see a refeed for a strenuous pro-
gram to improve the quantity and quality of leadership as rapidly
as this can be accomplished. Since it takes good faculty to produce
good graduates, and good graduates to become new faculty, a
classical chicken-egg problem must be solved. Since some 50-odd
departments of computer science now exist, the first step would seem
to be to make sure that these departments can attract as many good
faculty members as can be located. With suitably high salaries and
good research environments, many could be found in industry.
Others can be attracted from such other university departments as
mathematics and engineering, though they usually require further
· . .
ec ucatlon in computer science.
In the better universities, at least, the new departments of com-
puter science are finding themselves swamped by applications for
admission frown excellent students. (At Stanford, for example, it was
possible for 1966-1967 and 1967-1968 to admit only approximately

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25 students each year out of over 200 well-qualified applicants; and
even so, in 1966-1967 there were more graduate students in com-
puter science than in mathematics.) Though more reliable data are
needed, it appears that in several of the best departments of com-
puter science, the real limiting factor in the production of PhD's
is not students but buildings, facilities, and faculty.
Educational programs in computer science have been the subject
of several papers published in the Communications of the Associa-
tion for Computing Machinery. See especially references 36 and 37,
which give extensive details of undergraduate and graduate pro-
grams in computer science, respectively. The preliminary curricula
discussed in reference 36 were planned carefully to fit in with those
of the Committee on the Undergraduate Program in Mathematics.
In reference 37 are given syllabi for doctoral examinations in one
university, as well as a general philosophy of education in computer
science.
Forsythe37 explicitly calls attention to the need for a suitable
university computation facility for use in connection with educa-
tional programs in computer science. The necessary systems include
flexible, fast-acting compilers and other resident systems of a sort
rarely furnished by the manufacturers of hardware. Another need
is a sufficient budget for the computer time used in education; satis-
factory methods of government support have not yet been worked
out. Often, government accounting procedures prevent students
from using university computers that are standing idle. Solving these
accounting problems is important, but not easy, since they are asso-
ciated with nationwide accounting policies. E. A. Feigenbaum and
Courtney S. Jones have outlined the problem (see Appendix D).
The problem and some proposals for its solution are also discussed
in reference 38 from a different point of view.
In these early years of computer science, teaching even an ele-
mentary course is likely to be a research experience for the instructor.
This implies that formal teaching loads must be lighter in computer
science than in better established disciplines. This indicates needs
during these formative years for (a) special support for developing
and updating courses in computer science, (b) support, from all
sources of funds, for faculty research during the academic year, and
(c) even more faculty additions than would otherwise be needed.
Moreover, since much computer-science research is validated by the
successful operation of computer programs rather than by the suc

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155
cessful proof of theorems, special attention has to be given to the
form and character of assessment of research quality.
Because of the need to furnish computer-science students with
some computing time for their work, an additional budget is needed.
We make the rough estimate that graduate students in computer
science each need $1,000 per year of computing time.
The most critical shortage in the computer world is for well-
educated systems programmers and persons who can teach systems
programming. (For example, to provide the large amount of student
computing service implied by recommendations of the Pierce re-
port,3 without intolerable costs, will require the resourcefulness of
many more systems programmers than are now available.) An essen-
tial ingredient of education in systems programming is the oppor-
tunity to experiment with the control programs of a modern
computer. Such experiments cannot be tolerated on a computing
machine dedicated to continuous reliable campus-wide service, be-
cause of the probability of interrupting the service. Hence, in addi-
tion to the 51,000 of computing time estimated above, advanced
students specializing in systems programming will require access to
a substantial computer separate from the central campus computer
utility, so that systems experiments can be safely encouraged. If the
experimental machine can be shared with other research projects,
we estimate that the additional costs of systems experimenting may
approximate $5,000 per year for the advanced student of systems pro-
gramming. For comparison, we note that it is not uncommon for a
student of systems programming to use $15,000 worth of computing
time for thesis research.
One very important problem in computer-science education is
the appropriate organization of a university in regard to computing.
Because computing is such a new field, one typically finds computing
groups arising in several parts of a campus for example, in elec-
trical engineering, in the graduate school of business, in experi-
mental physics, in the medical school, in industrial engineering, in
operations research, in the school of arts and science. Furthermore,
a university may have several computation facilities. Such dispersion
of effort, unless most carefully coordinated, tends to produce in-
compatible computer systems, duplicated faculties, and other prob-
lems. Some unpleasant campus controversies have already arisen
over competition for the major role in computer-science education.
We can only suggest that university administrations save money and

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manpower by strong coordination of efforts. The relation of the
computer-science department to the university computation facility
is especially important. We believe they should be sufficiently closely
related to exert desirable influences on each other, and yet their
distinct functions of research and education on the one hand, and
university service on the other, require clearly separated adminis-
trative settings.
In today's tight fiscal situation it is extremely hard to add a
burgeoning new field to an already overcrowded and overcommitted
university. The new departments of computer science are typically
suffering from shortages in space, in faculty, in research funds, in
budget, in computer resources, and in some cases in research
assistantships and traineeships. The problems seem to require mas-
sive infusions of new money.
STATISTICS
As has been the case for 20 years,39 there is a wide demand for ade-
quately trained statisticians. The diversity of the needs and of the
kinds of statisticians who can help to meet them is briefly considered
in the further discussion of Statistics on page 203. Most needed are
statisticians who are motivated by mathematics, on the one hand,
and by the challenge of dealing with uncertainty or revealing what
their data are trying to say, on the other.
Unfortunately, the number of PhD's being completed in mathe-
matical statistics in the United States today is relatively small-
fewer than 10 percent of the roughly 800 PhD's per year being pro-
duced in the mathematical sciences generally, according to estimates
by our Panel on Graduate Education. This is not because of too few
graduate trainin,, opportunities: the country has about 70 universi-
ties that offer PhD degrees in statistics, and even those with quite
strong programs are not getting many able students. Attracting more
first-rate men to do PhD-level work in the field is a genuine problem.
The major difficulty lies at the undergraduate level. In statistics,
as in all the mathematical sciences, preparation for research and
professional activity is naturally well under way before a student
completes his undergraduate years. Fewer than one half of the uni-
versities with PhD programs in statistics also offer undergraduate
statistics majors. Almost no other institutions offer either such

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157
mayors or mayors combining statistics and mathematics. Only a
small fraction of U.S. undergraduates have appreciable access to
statistics as a subject. It is true that undergraduate preparation for
majors in mathematics, with its traditional emphasis on core mathe-
matics, provides an excellent foundation of knowledge for potential
graduate students in statistics. It does not, however, provide nearly
enough students with either motivation to study statistics or an un-
derstanding of the extramathematical aspects of statistics.
All those things that can be done at the undergraduate level to
expose more students to a proper understanding of statistics and its
role are urgently needed. Including mathematical statistics in a
conventional mathematics major can help; the students will at least
learn that there is such a subject as statistics, but they are unlikely
to gain any feeling for its nonmathematical aspects. Introducing
mathematics students to statistics, taught so as to communicate its
extramathematical aspects, while using mathematics freely where
appropriate, can help much more.
Even this does not meet the needs of the many students who might
find research in statistics attractive but who happen to be repelled by
the attitudes conveyed in courses of a conventional mathematics
major. Majors in many fields, particularly scientific ones, have pro-
vided pathways to statistics for outstanding individuals. We cannot,
however, expect these routes to meet the need. If an open road to
graduate study in statistics is to be provided, the minimum is a joint
program in mathematics and statistics, in which statisticians who
have or can adopt an extramathematical attitude have a substantial
share in setting tone and attitudes. Where faculty skills and atti-
tudes permit and appropriate courses can be given, an undergradu-
ate major in statistics offers greater possibilities.
The whole problem of undergraduate routes to graduate school is
vital for statistics. Those departments now giving the PhD degree
are already diverse in attitudes and course content, and will become
increasingly so as they strive to meet the needs of an ever more
.
heterogeneous student clientele.
Both to attract Food students and to give the training they need,
courses In statistics must gain excitement and realism by avoiding
the impressions (all too frequent in today's courses) that: (a) the
model is already available; (b) the data and the problem are small-
scale and easily treated by a standard approach; (c) anything large
and complex would be beyond the reach of these methods and there

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fore outside of statistics; (d) data are nice and clean; (e) all statis-
tics is mathematics; (f) all statistics is based on probability (whereas
much has nothing to do with it) .
If this is to happen on a nationwide scale, as it should, both teach-
ers and students will require access to new material, providing un-
derstandable accounts of enough different instances of: (a) model
building; (b) large-scale investigations involving both large-scale
data processing and statistical thinking; (c) "dirty" investigations
skillfully handled; (d) material on the nonmathematical, nonprob-
ability parts of statistics. As such material becomes available, there
will be need for a substantial effort to make its effective use possible
in many more institutions than now provide their undergraduates
with effective access to statistics.
RESEARCH VERSUS TEACHING
During recent years, it has become fashionable to complain that
government sponsorship of scientific research has led to poorer teach-
ing, especially on the college level, because of separation of leading
scientists from contact with undergraduates. On the other hand,
many professionals believe that the teaching of the mathematical
sciences in American colleges and universities is better today than it
was, say, 30 years ago. In particular, the reduction in formal teach-
ing hours that has taken place during that period has often resulted
in replacing routine teaching of traditional courses by more am-
bitious, modern, and creative pedagogical efforts.
While it is difficult to obtain objective evidence of this, we have
made an effort to ascertain whether and to what degree the leading
research mathematicians are, in fact, removed from teaching in gen-
eral and from undergraduate teaching in particular. At the Com-
mittee's request, the American Mathematical Society has identified
about 300 distinguished American mathematicians to whom a short
questionnaire concerning their teaching duties was sent. The list
includes those mathematicians who have delivered invited addresses
at AMS meetings or at International Congresses of Mathematicians
since World War II, members of the National Academy of Sciences,
and recipients of NSF Senior Postdoctoral Fellowships, Sloan Fellow-
ships, Guggenheim Fellowships, and various other prizes.
It was found that, on the average, during the three academic
years 1962-1965 all but eight of the 283 respondents taught at least

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three hours a week, with the typical teaching load being six hours;
and all but 30 of the 283 taught some classes for or open to under-
graduates. In assessing these figures, which we believe compare
favorably with those in other sciences, one should remember that
a research mathematician usually does more teaching in seminars
and personal consultations with students than during formal instruc-
tion hours. Thus, there is considerably more teaching by such
scientists than is indicated by numbers of classroom teaching hours.
WASTED MATHEMATICAL TALENT
We believe that a large number of mathematically gifted young
men and women with potential to become productive mathemati-
cal scientists and college mathematics teachers fail to do so for
socially determined reasons. Waste of mathematical talent occurs
at two points in particular, and we feel that preventive action can be
taken. The problems touched on here are clearly not peculiar to
the mathematical sciences.
As noted in Chapter 7, over 70 percent of liberal arts colleges
have, at most, one staff member who is a PhD in a mathematical
science, and approximately 40 percent have none at all. In such
institutions, talented students will in general not receive proper
guidance and will almost certainly not receive undergraduate edu-
cation sufficient for successful graduate careers. The situation is
aggravated by the fact that in many cases students attending such
institutions come from economically and culturally deprived regions
and population strata.
A fuller discussion of this problem is given in Chapter ~ of the
report) of our Panel on Undergraduate Education. This Panel has
proposed various measures to improve the situation. If these
measures are undertaken and are successful, they will contribute
toward relieving the shortage of mathematical scientists and college
teachers and will also contribute in a modest way toward the solu-
tion of an acute social problem. We endorse the recommendations
of our Panel and have singled out one (Recommendation 15, page
97) that could be implemented immediately. It aims to provide a
"fifth undergraduate year," which would enable exceptionally
talented students with weak preparation to begin graduate work at
more competitive levels of preparedness.
Our Panel on Undergraduate Education has also pointed out, in

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Chapter 5 of their report, that at present our society is not utilizing
fully the intellectual capacity of women. Social attitudes, family
responsibilities, and out-dated nepotism rules prevent many talented
young women from beginning or continuing graduate study and
from continuing professional work after having received a PhD
degree. A modest recommendation to help alleviate the situation
has been given in Recommendation 13 (page 26~.