#### Pre-Graduate Training

The Panel on Pregraduate Training was appointed in 1959 to study the needs of, and to recommend programs for, undergraduate students who intend to study mathematics at the graduate level. The initial efforts of the Panel were concentrated upon the construction of an ideal curriculum for students of outstanding ability. Course outlines designed to lead the undergraduate rapidly toward the frontiers of mathematical research and the Ph.D., purposely overlooking local problems which might be caused by inadequate preparation at the secondary level or by lack of staff at the college level, appeared in the 1963 publication *Pregraduate Preparation of Research Mathematicians*.

Despite many misunderstandings regarding its assumptions and intent, this report served effectively as a basis for discussion and planning at many institutions. It was reprinted in 1965, together with some additional comments on constructive use of the booklet and admonitions to the effect that misinterpretation of the spirit of the outlines might result from a lack of knowledge of the Panel's basic assumptions and objectives. Perhaps the report's chief value lies in showing what is regarded as ideal preparation for graduate study in pure mathematics by a very distinguished group of mathematicians.

Having completed its work on an ideal undergraduate program for the future research mathematician, the Panel turned to the urgent practical task of recommending specific undergraduate curricula for colleges which would be unable, for any of a variety of reasons, to achieve quickly the goals of its original report. These recommendations were drawn up after consultation with representatives of about 25 of the leading graduate mathematics departments. They appear in the 1965 document *Preparation for Graduate Study in Mathematics*, together with outlines for upper-division courses in Abstract Algebra and Real Analysis.

#### Statistics

In 1968 CUPM appointed a Panel on Statistics for the purpose of providing guidance to departments of mathematics at smaller colleges and universities on instruction in statistics. Two concerns of general interest were identified for study by the Panel: a program to prepare students for graduate study in statistics and a basic service course in statistics for students who have not studied calculus. The Panel pointed out that these two topics represent curricular extremes for statistics instruction in most undergraduate programs and that many students' program of study will lie between these extremes.

The Panel's first report, *Preparation for Graduate Work in Statistics*, was issued in 1971. This document describes the type of training and experiences which undergraduates contemplating graduate study in statistics ought to have. It outlines a basic one-year course in probability and statistics, indicates those mathematics courses which are valuable for pregraduate preparation in statistics, and comments on computer requirements and experience with data.

The Panel's second project involved a study of the introductory, non-calculus statistics courses which are offered by practically every college and taken by students in a wide variety of fields. Prompted by the fact that many of the existing courses are unsatisfactory for a variety of reasons, the Panel developed a set of objectives for such a course and made concrete suggestions for realizing the objectives. A detailed list of topics for a conventional course in introductory statistics, as well as some suggested alternate approaches, appears in the 1972 publication *Introductory Statistics Without Calculus*.

#### Computing

During the decade beginning in 1962, CUPM made a continuing effort to advise college mathematics departments on curricular matters related to the tremendous growth in the use of the computer and the pervading influence which the computer has come to exert on society. Initial steps in this direction were taken by the Panel on Physical Sciences and Engineering, which issued its *Recommendations on the Undergraduate Mathematics Program for Work in Computing* in 1964. Taking account of the significant changes which had recently occurred in the relationship of mathematics to computing and to computing machines, the Panel proposed a program designed to prepare students whose careers were likely to be intimately connected with high-speed computing. The program included reference to three types of courses: (1) mathematics courses of a general nature which should be available for the prospective specialists in computer science; (2) technical courses in computer science; and (3) an introductory course in computer science.

Two years later CUPM commissioned R. W. Hamming of Bell Telephone Laboratories, Inc., to prepare a monograph on *Calculus and the Computer Revolution*. Published in 1966, this book describes and illustrates briefly some aspects of computing as they are related to the beginning calculus course.

A task force was appointed in 1966 for the purpose of advising CUPM on a future course of action with regard to computing. The task force suggested the creation of a Panel on Computing, which would work closely with various computing organizations and would have several charges related to the impact of the computer on mathematics education. Such a panel was appointed in 1967. One of the Panel's projects was to gather and disseminate information regarding the use of computers in introductory calculus courses. A newsletter entitled *Calculus With Computers*, issued in 1969, contained general observations and summaries of statements from various institutions that had instituted computer-oriented calculus courses.

The Panel's primary aim was to develop a systematic approach to the impact of computers on undergraduate mathematics programs, rather than to address itself to the training of computer scientists *per se*. (The latter topic had already been considered by the Association for Computing Machinery in its report *Curriculum 68—Recommendations for Academic Programs in Computer Science*.) The Panel formulated a specific undergraduate program in computational mathematics, combining courses in mathematics, computer science, and computational mathematics—complete with course outlines and suggestions for implementation. This course of study is presented in the 1971 publication *Recommendations for an Undergraduate Program in Computational Mathematics.*

The main concern of this report is for the education of mathematicians who wish to know how to use and to apply computers. The report of the Panel on Computing attacked a significant problem: the need for new, innovative courses directly concerned with computational mathematics and computer science. Remaining to be considered, however, was another important question: How should the computer affect traditional mathematics courses? To study this question and related points, CUPM in 1971 appointed a Panel on the Impact of Computing on Mathematics Courses to succeed the Panel on Computing. The new Panel's investigations culminated in the publication of *Recommendations on Undergraduate Mathematics Courses Involving Computing* in 1972. This document includes outlines for lower-division courses in elementary functions, calculus, discrete mathematics, and linear algebra with stress on algorithms, approximations, model building, and the nature of the entire problem-solving process.

#### Applied Mathematics

The importance of applications of mathematics to other areas was recognized by CUPM early in its existence. Among the original four panels were a Panel on Mathematics for the Physical Sciences and Engineering and a Panel on Mathematics for the Biological, Management, and Social Sciences, each charged with the task of making recommendations for the undergraduate mathematics program of students whose major interest lay in one of the stated fields.

The Panel on Physical Sciences and Engineering concentrated its efforts on the training of engineers and physicists, issuing its first report, *Recommendations on the Undergraduate Mathematics Program for Engineers and Physicists*, in 1962. The demand for this document was so great that it was necessary to have it reprinted in 1965. Significant developments which occurred during the mid-sixties prompted the Panel to revise its recommendations and issue a new report in 1967.

In the meantime this Panel had also developed CUPM's first definitive statement regarding the role of the computer in undergraduate mathematics. Its 1964 report *Recommendations on the Undergraduate Mathematics Program for Work in Computing* contained outlines for introductory and technical courses in computer science and a description of a program for mathematics majors planning to enter the field of computing. (This document is not being reproduced in this *Compendium* because it has been superseded by more recent CUPM documents discussed above in the section on Computing.)

Another document, *Mathematical Engineering--A Five-Year Program*, was issued by the Panel in 1966 to provide a means of alleviating what was then a drastic shortage of engineers having a substantial background in mathematics. Described as "a suggestion, rather than a recommendation," this report gives several outlines for options in operations research, orbit mechanics, and control theory.

The Panel on Mathematics for the Biological, Management, and Social Sciences, confronting problems which were less well defined, issued its *Tentative Recommendations for the Undergraduate Mathematics Program for Students in the Biological, Management, and Social Sciences* in 1964. Primarily concerned with the mathematics curriculum for prospective graduate students in those fields, the report was meant to serve as a basis for discussion and experimentation. As a result of several issues raised in reaction to this document, CUPM decided in 1967 to concentrate on individual disciplines and, as a first step, appointed a Panel on Mathematics in the Life Sciences, charged with making recommendations for the mathematical training of the undergraduate life science student, whether or not he goes on to graduate school. The term "life science" here referred to agriculture and renewable resources, all branches of biology, and medicine. This Panel worked closely with the Commission on Undergraduate Education in the Biological Sciences, and its investigations culminated in the publication of *Recommendations for the Undergraduate Mathematics Program for Students in the Life Sciences—An Interim Report* in 1970. Although it was anticipated that a final form of this report would eventually be issued, this project was never undertaken due to lack of funds.

Appointed in 1964, the CUPM *ad hoc* Subcommittee on Applied Mathematics was charged with suggesting appropriate undergraduate programs for students planning careers in applied mathematics. The Subcommittee's recommendations for such a program, together with suggestions for implementation and course descriptions, appeared in the 1966 report *A Curriculum in Applied Mathematics.* During the years 1967-69 an Advisory Group on Applications kept CUPM informed on current developments in applied mathematics.

The extremely rapid development of applications of mathematics, particularly in fields outside the physical sciences, together with a renewed interest in applications among mathematicians, led CUPM to appoint in 1970 a Panel on Applied Mathematics, whose duty was to reconsider some of the questions which the Subcommittee had studied earlier, and to draw up new recommendations in line with the nature and methods of applied mathematics. The Panel's suggestions, which emphasize the role of model building, are given in the 1972 report *Applied Mathematics in the Undergraduate Curriculum*. This report contains detailed outlines of three options for a course in applied mathematics, each of which utilizes the model-building approach.