82

The Role of Applications Mathematics Curriculum Peter Hilton*

in the Undergraduate

In the summer of 1977 an ad hoc Committee on Applied Mathematics Training was formed under the auspices of the National Research Council (Washington), with gener- ous support from the Sloan Foundation. In August, 1979, the Committee published its report under the title indi- cated above.

The proposal to the Sloan Foundation drew attention to "the mismatch between college mathematics curricula and opportunities for employment of mathematicians." It went on " . . . the time is ripe for a closing of the gap between abstract and applied mathematics; the move to- wards incorporating more training in the standard college mathematics curriculum is appropriate intellectually as well as pragmatically." The Committee was to make spe- cific recommendations on expanding the applied mathe- matics curriculum in the undergraduate mathematics pro- grams of U. S. colleges and universities.

The Committee, consisting of 13 mathematicians - pure mathematicians, applied mathematicians and indus- trial mathematicians I - met frequently over a two-year period. The opportunity was taken to consult members of the mathematical community, both on an individual basis and by means of presentations at national meetings of the American Mathematical Society, the Mathematical Associa- tion of America, and the Society for Industrial Applica- tions of Mathematics. It is now to be hoped that the re- commendations of the Committee may form the basis for widespread discussion of the issues raised in the report.

Early in the deliberations of the Committee the view- point emerged clearly that its concern should be broadly with the mathematical sciences and that its recommenda- tions should be directed towards enabling all students to have the experience of applying mathematics. Thus, while the Committee regarded the design of a "major" as one of its principal - though, by no means its exclusive - con- ceres, there was no attempt to develop guidelines for an "applied mathematics major", to be contrasted with a "pure mathematics major". Rather, it was agreed that we should be thinking of a broad major in the mathema- tical sciences.

* This abridged account of the Report of the Committee on Applied Mathematics Training was prepared by the chairman of the committee. Any bias introduced by the abridgment is solely the responsibility of Peter Hilton.

1 See Appendix for the list of members.

This major should offer flexibility of choice, to suit the needs of different students, and should include an applied component for all students. However, this flexibility was

not to be purchased at the price of the quality of the mathematical curriculum; indeed, the Committee's pro- posals seem likely to involve a strengthening of the curri- culum. Those students intending to concentrate in their subsequent work or studies on applications of mathemat- ics would be expected to choose their electives accord- ingly. The same principles apply of course to students of mathematics majoring in other disciplines.

The Committee organized its thinking round five main themes. These are:

A. Attitudes B. The Unification of the Mathematical Sciences C. Curriculum D. Applications - Experience Programs E. Societal Aspects

The Committee became increasingly aware, as its delib- erations and consultations proceeded, that the principal problem was one of attitude: "We have been made aware of the widespread existence of attitudes that must be sub- stantially modified if students are to be encouraged to view positively the prospect of a c a r e e r . . , in science, in industry or in government service." However these are not the only attitudes that militate against mathematics occu- pying its rightful place in the life of the community. Some people - even within our universities and colleges - have a negative attitude toward any sort of academic mathematics; some students have a negative attitude toward any form of nonacademic employment; and, within the mathematics community itself, there are attitudes of suspicion, even mistrust, which are very frequently mutual.

The vast changes and developments within mathemat- ics itself, and the - partly consequential - proliferation of mathematical activity across the campus are discussed in Theme B. Broadly, the Committee's conclusion is that these developments could be helpful to the study of ma- thematics if the student were not obliged, as today he or she tends to be, to make premature choices of major be- tween various departments serving the mathematical sciences. There must be full collaboration between the faculties of these departments to ensure the availability of

the best possible courses and the maintenance of the stu- dent's options.

The discussion of curriculum questions by the commit- tee is entirely non-prescriptive. Detailed curricula are being developed by a panel of the Committee on the Undergrad- uate Program in Mathematics of the MAA, under the chair- manship of Professor Alan Tucker, and the report of this panel should be available in 1980; the Committee shared its own thinking with Professor Tucker and established com- mon ground on general curricular questions.

The Committee did make certain very significant re- commendations with respect to curriculum. One (which commanded the assent of a strong majority of the Com- mittee but was not adopted unanimously) allotted a cen- tral role to the study of differential equations in the core curriculum; another proposed that the major in the mathe- matical sciences should be accompanied by a strong minor in a mathematics-related discipline. The Committee also explicitly advocated that all students should become familiar with computing and should acquire some basic understanding of probability and statistics.

The Committee devoted considerable attention to the question of incorporating into the student's regular pro- gram some experience of actually applying mathematics. The report cites examples of such programs at various uni- versities and colleges (Antioch, Drexel, Clempson) and draws attention to the fact that, with respect to applica- tions, it is not sufficient to learn about them through standard lecture courses. Where circumstances may render the introduction of an internship program (involving local industry) difficult to achieve, it should be possible to simulate such actual experience through a series of case- studies or some form of senior research project.

The final theme of the report, devoted to social aspects of the problem of effective mathematical education and mathematical activity, was not given the same attention as the other themes, not because it was less important but because it involved the resolution of issues that do not lie within the control of the mathematical community, and because detailed elaboration of such issues does not lie within the competence of the Committee. Indeed, the Committee was always concerned to make practical recom- mendations and not to indulge in pious platitudes; thus it

83 faced squarely the necessity to increase the number of credit hours for the major and to obtain federal funding to implement a program of summer institutes to enable faculty to become familar with new applications of mathe- matics and the new mathematics thrown into prominence by applications.

The report is available, free of charge, on application to

Office o f Mathematical Sciences National Research Council, 2101, Constitution Avenue Washington, DC 20418 USA

Members of the Committee:

Peter Hilton, Case Western Reserve University and Battelle-Seattle Research Center, Chairman

David J. Benney, Massachusetts Institute of Tech- nology

Hans J. Bremermann, University of California, Berkeley

Felix E. Browder, University of Chicago Victor Klee, University of Washington, Seattle Michael C. Reed, Duke University I. Richard Savage, Yale University Herbert E. Scarf, Yale University Stephen Smale, University of California, Berkeley Elias M. Stein, Princeton University Jean Taylor, Douglass College, Rutgers University Daniel H. Wagner, Daniel H. Wagner, Associates,

Paoli, Pennsylvania Shmuel Winograd, IBM Thomas J. Watson Research

Center

Case Western Reserve University Dept. of Mathematics and Statistics Cleveland, Ohio 44106