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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 147

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148 The Mathematical Sciences in Education 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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Special Educational Issues 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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150 The Mathematical Sciences in Education 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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Special Educational Issues 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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152 The Mathematical Sciences in Education 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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Special Educational Issues 153 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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154 The Mathematical Sciences in Education 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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Special Educational Issues 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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156 The Mathematical Sciences in Education 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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Special Educational Issues . . 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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158 The Mathematical Sciences in Education 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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Special Educational Issues 159 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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160 The Mathematical Sciences in Education 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~.