By Lynn Steen
The documents and commentary provided below offer an overview of the published work of CUPM since it's beginnings in 1953 as well as the educational and mathematical context in which it did its work. The materials are meant to serve both as an historical resource and as a tool for consideration of today's undergraduate program in mathematics. (Note: Some referenced documents are available via live links; some others have been digitized and may be found on the web via academic library subscriptions; the reminder, primarily longer documents, are available only on paper.)
The history of CUPM is only partly revealed in the reports that it published. The people who contributed to its panels and subcommittees, the conferences and presentations it sponsored, the arguments and compromises that led to its recommendations—all this and more are required for a full appreciation of this important chapter in American collegiate mathematics. A richer history (and pre-history) with these kinds of details can be found in CUPM, The History of an Idea by William (W. L.) Duren, Jr. which appeared in the January 1967 Fiftieth Anniversary Issue of the Amer. Mathematical Monthly (Vol. 74:1 Part 2, pp. 23-37). What follows here is the skeleton of that history focused on the published record—the reports, pamphlets, and recommendations of CUPM from its founding until the early 1990s.
In 1953, "before Sputnik and before the computer had run wild," reports Duren, MAA president E.J. McShane appointed a special committee on the undergraduate program. Following a quick survey, this ad hoc committee reported "widespread dissatisfaction" with the college mathematics program and recommended a national "program of 'doing' to overcome the inertia" of the "enormously ponderous structure" that sustains "all the deficiencies" of the present program. Action by the MAA's Board of Governors in relation to CUP—the Committee on the Undergraduate Program, as it was known in its early years—is documented in the report of the 36th Annual Meeting of the MAA (Amer. Mathematical Monthly, 60:2 (Mar. 1953) 214-218):
The Board voted to approve the appointment by the President of a committee to study the possible establishment of an employment bureau, of a joint committee (with the National Council of Teachers of Mathematics) on teacher education in mathematics, of a committee on the Undergraduate Mathematical Program, and of a joint committee (with the National Council of Teachers of Mathematics) to explore the possibility of publishing a mathematical journal for high school students.
Throughout its history, CUPM has undergone changes both in the context in which it worked as well as in the structures within MAA that supported its work. As the first MAA committee on the undergraduate curriculum (appointed when the MAA headquarters were still located at the University of Buffalo), CUPM operated via quasi-independent "panels" on various topics needing urgent attention. Gradually these panels became separate committees, and then in the early 1990s MAA gathered all these independent panels and committees into a new "council" structure. These changing patterns are reflected in this document by a division into seven sections following an approximate chronological order.
During the five year period 1953-57 at the beginning of the Eisenhower administration and after the end of the Korean War, the American Mathematical Monthly manifested mathematicians' dissatisfaction with collegiate mathematics in these early pre-Sputnik years of the "Cold War." A string of reports, reviews, and arguments focused on the need for, in the words of William Duren, a "much-talked about revolution in the mathematics program." The papers linked below, listed chronologically, represent the ferment of this period:
Soon after the Committee on the Undergraduate Program (CUP) began its work, it compiled a litany of issues facing undergraduate mathematics, including
Undergirding "this complex of inertial elements," according to the new CUP, was "the desertion of elementary teaching by the best mathematicians, old and young."
Arguing that in school and college mathematics new ideas have usually been "largely overpowered by the self-propagation of the traditional but retrogressive stock," CUP's first major decision was "to seek one, universal freshman course for all reasonably qualified students." The Committee reasoned that
... only by means of such a universal course can the best principles of liberal education be served. Only in this way can we avoid the error of forcing the immature student, upon entering college, to make choices which will seriously restrict his freedom of development in later years.
This new course CUP called Universal Mathematics. The full rationale and a proposed outline for this course can be found in the Committee's first report to the Association by W.L. Duren, Jr., et al. [Amer. Mathematical Monthly, 62:7 (Sept. 1955) 511-520.]
Since the purpose of CUP was to promote rather than implement reform, its action on this matter was to encourage (rather than to create) this proposed new course. ("This Committee will not get into the textbook business and will not approve or disapprove any textbook.") Two summer writing groups associated with (but not appointed by) CUP took up the challenge of Universal Mathematics. One, meeting at the University of Kansas during the summer of 1954, focused on the first half of the CUP outline. A mimeographed preliminary edition—not a text, but "a book of experimental text materials"—entitled Universal Mathematics, Part I: Functions and Limits was published by the University of Kansas in September, 1954 (see the announcement by Carl B. Allendoerfer in the Monthly, 61:12 (Dec. 1954) p. 726.)
A second Writing Group met at Tulane University during the following summer. This team focused on sets and axioms—the second half of the CUP outline. A mimeographed preliminary edition, Universal Mathematics, Part II: Elementary Mathematics of Sets, Robert L. Davis, editor, was published by Tulane University in September, 1955 (see this announcement in the Monthly, 62:7 (Sep. 1955) p. 500.) Three years later, in 1958, MAA published a revised version of this volume under the title Elementary Mathematics of Sets With Applications.
The 1954 Summer Writing Group reports in the Preface to Universal Mathematics that in order to make the course "available to the maximum number of students," it was designed for "all first-year college and university students" with "at least two, and preferably two and one-half, units of high school mathematics, including intermediate algebra." This preparation the authors described as "normal" high school preparation. (For students of engineering, they suggested a supplementary "Technical Laboratory" to develop the problem–solving skills needed by such students.)
As the Preface notes, the book's "most striking feature" is its "dual presentation in two parallel lines of discourse." One offers "essentially all of its own" definitions and assumptions, but contains "only intuitive indications of proofs." The other, a formal presentation is "quite difficult" and is "not intended as a text to be followed continuously" but as a resource from which the instructor can "select those proofs which are appropriate for his students."
The 1955 Summer Writing Group at Tulane continued the dual presentation strategy from Universal Mathematics, Part I. Although Part II was originally conceived as a second semester course to follow Part I, the Tulane authors designed Part II to be independent of that course. Consequently, Part II "could be the first semester course in the first college year, without serious omissions."
In contrast to existing texts that treat set theoretic ideas "entirely in a logical and somewhat recondite manner," Universal Mathematics seeks to connect "the ideas of sets with a wide variety of subjects in science, engineering, behavioral studies, as well as pastimes and games." In so doing, it "lays a foundation for statistics "without entering into statistics proper." In their Preface, the authors of Part II note that students' lack of experience with sets seriously limits their ability "to grasp the ideas of choice and chance, and of statistics." In turn, this "lack of training in these modes of thought is a handicap in many aspects of science, business and government."
Notwithstanding their sense of urgency, the Tulane authors note with caution that "The effort to present a variety of mathematics subjects in a framework of a theory of sets is frankly experimental. Its value remains to be tested."
A review of Universal Mathematics, Part I by Herbert (H.P.) Evans of the University of Wisconsin appeared in the Monthly in 1956 (pp. 196-199) followed by a report (pp. 199-202) by CUP chair William (W.L.) Duren, Jr., on a "mass trial" with 750 first year students at Tulane University. Evans commends the authors for making a "serious effort "to present real mathematics at an elementary level" and for treating a classical subject "from a fresh and modern viewpoint." He worries, however, that "even the intuitive treatment will be difficult for most students to follow and that the instructor will not always find it easy to present the material effectively."
Duren's report of the Tulane trial confirmed Evans' worries. "The book ... caused considerable difficulty to students and instructors. The main trouble is that students cannot read it." Universal Mathematics is "not adaptable to students whose high school background in mathematics is scant." Duren cites many factors that contribute to these difficulties, one of which is the "unusual method by which it was uncompromisingly written." Only after the formal account was complete was the intuitive version prepared, "written to follow [the formal account] closely in the same order." This approach—when "mathematical theory determines the formulations of concepts and their order of introduction—requires of instructors "much more imagination to find appropriate motivations and interpretations for young students."
(It is probably worth noting here that William Duren was chair of the Committee on the Undergraduate Program (CUP) that developed Universal Mathematics, was a member of the 1954 Summer Writing Group at the University of Kansas that developed Part I, and is the author of this report on the difficult experience of using this draft for first year students at Tulane.)
A few months after this review and report on Universal Mathematics appeared, the Monthly also published a contrasting strategy for first-year college mathematics in a report entitled College Mathematics for Non-Science Majors prepared by a special subcommittee of the California Committee for the Study of Education. The final section of this report offers comments about the CUP proposal for Universal Mathematics. While the California committee supports the CUP goal of a single introductory course for students interested in the "technical mathematics" of "natural science, social sciences or the arts," it sees these students as quite distinct from those for whom a course in mathematics for general education is intended:
We feel that it would be inadvisable to give the technically able students, with superior mathematical background of high school mathematics, the same course as average liberal arts students pursuing non-scientific courses, who have had little training in secondary mathematics.
To meet a perceived demand for early specialization, CUP suggested a curricular division in mathematics following Universal Mathematics: one course for physical science and engineering majors and a quite different one for those more interested in the biological and social sciences. In 1958 a Writing Group at Dartmouth College produced Modern Mathematical Methods And Models, a two-volume set of experimental text materials intended for the second of these CUP-proposed courses. The first volume, Multicomponent Methods, begins with matrix methods which are then used to study functions of several variables. The second volume, Mathematical Models, introduces probability, order relations, Markov chains, and mathematical models.
Recognizing that many of the ideas in this course "would previously have been considered too difficult for a sophomore course," the authors nonetheless argue that students of biological and social science need to learn "appropriate mathematical tools early enough to use them," and therefore these mathematical ideas "must somehow be brought to a more elementary level." The first volume received a favorable brief review in the Monthly in March, 1959 (pp. 246-247).
Both Universal Mathematics and Modern Mathematical Methods And Models were developed by writing groups in support of CUP's call for innovation in undergraduate courses because for these two courses there were few, if any, existing texts. In contrast, there were many texts available that covered the topics in CUP's other sophomore course for students of physical science and engineering. So no special writing group was needed for this course, although CUP did put forward as a model of excellence George B. Seligman's notes on a course taught by Emil Artin to the top ten or fifteen percent of freshmen at Princeton University. According to the Monthly review, A Freshman Honors Course in Calculus and Analytic Geometry Taught at Princeton University (as these notes were titled) "should certainly be on the active bookshelf of every mathematician interested in teaching gifted students."
In summary, during its five years of work from 1953 through 1958, MAA's Committee on the Undergraduate Program (CUP), supported and distributed (without charge) five volumes intended to spark innovation in freshman and sophomore mathematics:
Since these volumes fulfilled the committee's original mandate, CUP also prepared several archival reports of its work:
In 1958 the National Science Foundation awarded MAA a grant to convene a broadly representative group of mathematicians and users of mathematics "to study some of the problems created by the revolution in mathematics and to seek solutions for them." These problems included rapid increases in enrollments, unprecedented demand for mathematics teachers, the need to modernize textbooks, and better means of assimilating mathematical research into "the body of common knowledge." The conference was expected also to consider the role, organization, and operation of the MAA.
MAA President G. Baley Price convened this conference in Washington, DC on May 16-18, 1958. Subsequently, a report entitled The Washington Conference was published in the Monthly (65:8 (Oct. 1958) 575-586]) Among the many issues taken up by this group was a request by CUP that it be discharged as of September 1, 1958 with the expectation that it "be reorganized on a larger scale." In response, the conference passed a resolution in which it urged that the work of CUP "be continued with all possible vigor and on an expanded scale."
Consequently, on Nov. 15-16, 1958, MAA president Price convened another conference in Washington "to assist in formulating plans and policies for future work" of the now-discharged CUP was called "to re-examine the assignment of the Committee" and "to take steps to establish a new Committee with adequate funds, personnel, and program. Subsequently, a report entitled Conference on the Committee on the Undergraduate Program was published in the Monthly (66:3 (Mar. 1959) 213-220).
The first recommendation of this conference was to create a Committee on the Undergraduate Program in Mathematics (CUPM) with "power to delegate its various activities to subcommittees." Among the expanded priorities suggested for the new committee was to publish statements of minimal standard for teachers of mathematics in schools and colleges, as well as recommendations concerning desirable preparation for graduate programs in various fields and career areas. The new CUPM was duly appointed and held its first meeting at the end of 1958, just six weeks after the CUP Conference.
With support from the relatively new National Science Foundation, CUPM set up an office in California to support the work of the several panels and subcommittees that were established to carry out its expanded mandate. It held conferences and published newsletters to stimulate interest in various topics in twentieth century mathematics that were in high demand in the expanding post-war economy. The part of its work that had the most widespread and lasting impact was a series of curriculum guides covering virtually every aspect of undergraduate mathematics. For the most part, these guides were small stapled pamphlets, typed rather than typeset, that were distributed without charge using funds from the NSF grant.
The pressures on undergraduate mathematics to which CUPM was responding in these early post-Sputnik years were part of a broader pattern of crisis in mathematics from primary school through research universities. In 1963 a group of leading mathematicians undertook a serious examination of the school curriculum for the purpose of establishing long-term goals for K-12 mathematics education. Although the report of this "Cambridge Conference" is described by its authors as "tentative and highly provisional," it generated considerable controversy among both mathematicians and school teachers.
Shortly after the Cambridge Conference, the National Academy of Sciences convened a special committee to examine the status and needs of the mathematical community. Because of the distinctive role played by collegiate mathematics, this committee convened a special panel on undergraduate issues, chaired by John Kemeny (one of the first members of CUPM). Reports from the committee and the panel were published in two separate lengthy volumes (one at 256 pages, the other 114 pages). This was the first major national report to begin using the appellation "mathematical sciences" rather than "mathematics." (It was another 15 years before CUPM reports began using this same language.)
As the economy changed and employment prospects for mathematically trained graduates declined in the early 1970s, the NSF changed priorities and withdrew support for the type of programs that supported CUPM. To complete the funded phase of CUPM's work, MAA published a two-volume Compendium containing many of its more recent reports. Now this Compendium has been scanned to make the reports in it available online in pdf format. (Some other CUPM reports are also online but available only through certain library systems.)
The inventory on the following pages lists the CUPM curriculum guides and recommendations that were published in this NSF-funded phase of CUPM's work. The links to on-line versions of certain reports are generally not to the original versions, but to the slightly updated versions that were subsequently published in the Compendium.
At the end of CUPM's NSF grant, the MAA published in two volumes A Compendium of CUPM Recommendations, subtitled "Studies, Discussions, and Recommendations by the Committee on the Undergraduate Program in Mathematics of the Mathematical Association of America." This Compendium contained selected reports published by CUPM since 1965 arranged by topic in seven sections, each preceded by a preface. What follows below are slight adaptations of the prefaces to each section followed by links to the contents of that section.
The Committee on the Undergraduate Program in Mathematics (CUPM) was established as a standing committee of the Mathematical Association of America (MAA) in 1959. (A detailed history of CUPM can be found in CUPM, The History of an Idea by W. L. Duren which appeared in the American Mathematical Monthly, 74:1 Part 2 (Jan. 1967) 23-37.) With financial assistance from the National Science Foundation, CUPM in 1960 began to engage in several projects and activities related to improvement in the undergraduate curriculum. These projects often involved the publication of reports, which were widely disseminated throughout the mathematical community and were available from the CUPM Central Office upon request. Since a change in the funding policy of the United States government makes the continuing production and free distribution of such reports extremely unlikely, the MAA has decided to publish in permanent form the most recent versions of many of the CUPM recommendations so that these reports may continue to be readily available to the mathematical community and may conveniently be kept on the reference shelves of mathematics libraries.
This Compendium is published in two volumes, each of which has been divided into sections according to the category of reports contained therein. These CUPM documents were produced by the cooperative efforts of literally several hundred mathematicians in the United States and Canada. The reports are reprinted here in essentially their original form; there are a few editorial comments which serve to update or cross-reference some of the materials. The editorial work for the Compendium was started by William E. Mastrocola during his term as Director of CUPM and completed after his return to Colgate University. He was assisted in the early stages by Andrew Sterrett and Paul Knopp, Executive Directors of CUPM during 1972 and 1973. Preparation of the final manuscript for the printer was the joint work of William E. Mastrocola and Katherine B. Magann. The considerable efforts of these individuals is deserving of special recognition.
CUPM's interest in the training of mathematics teachers has pervaded its activities throughout the Committee's existence. The Panel on Teacher Training, one of the original four panels, began its work at a time when mathematics instruction in elementary and secondary schools was undergoing significant changes. Throughout the years since its original report was issued, the Panel's recommendations and ongoing activities have had a profound influence on the education of elementary and secondary school teachers.
The 1961 Recommendations for the Training of Teachers of Mathematics identified five levels of mathematics teachers:
To complement the 1961 recommendations, CUPM also published Course Guides for the Training of Teachers of Elementary School Mathematics and Course Guides for the Training of Teachers of Junior High School and High School Mathematics. When it was proposed, the Level I curriculum received widespread attention and approval. It was approved formally by the Mathematical Association of America and it was endorsed by three conferences held by the National Association of State Directors of Teacher Education and Certification (NASDTEC) and the American Association for the Advancement of Science (AAAS). It formed a part of the Guidelines for Science and Mathematics in the Preparation Program of Elementary School Teachers, published by NASDTEC-AAAS in 1963.
In the years 1962-66 CUPM made an intensive effort to explain its proposed Level I program to that part of the educational community especially concerned with the mathematics preparation of elementary school teachers. Forty-one conferences were held for this purpose. Participants in these conferences, who came from all fifty states, represented college mathematics departments and education departments, state departments of education, and the school systems. The details of CUPM proposals were discussed and an effort was made to identify the realistic problems of implementation of the recommendations. As a result of these conferences and of other forces for change, there was a marked increase in the level of mathematics training required for the elementary teacher.
Level II and III conferences similar to those held for Level I were deemed unnecessary because the Level II and III guidelines had apparently been accepted by the teaching community through distribution of the recommendations and course guides. One indication of this acceptance has been the publication of numerous textbooks whose prefaces claim adherence to the CUPM guidelines. Throughout the decade of the 1960's, CUPM continued to expend considerable effort on the problems associated with the preparation of teachers. Minor revisions of the original recommendations were produced in 1966, and the course guides for Level I were similarly revised in 1968.
In 1965 CUPM published A General Curriculum in Mathematics for Colleges (GCMC) as a model for a mathematics curriculum in a small college. GCMC became a standard reference in other CUPM documents. The shortage of mathematicians, already severe by the late 1950's, had seriously impaired the ability of many colleges to implement CUPM recommendations, including GCMC. Qualified new faculty members were extremely difficult to obtain, and many established teachers were so overloaded with teaching responsibilities that they could not keep abreast of developments in their field.
By 1965 the time was obviously ripe for CUPM to see what could be done to alleviate this problem. An ad hoc Committee on the Qualifications of College Teachers of Mathematics was appointed to study and report on the proper academic qualifications for teaching the GCMC courses. Simultaneously, CUPM established a Panel on College Teacher Preparation and instructed it to study a number of related topics: existing programs for the preservice and inservice training of college teachers, opportunities for support of college teacher programs by government and foundations, the supervision and training of teaching assistants, supply and demand data, etc.
In 1967 the Qualifications Committee issued its report, Qualifications for a College Faculty in Mathematics. The report identifies four possible components in the formal education of college teachers and describes teaching duties suitable for individuals with academic attainment equivalent to a given component. It also makes suggestions concerning the composition of a small undergraduate department.
Immediately upon publication of the qualifications report, the Panel on College Teacher Preparation fell heir to several tasks. One of these was the responsibility for a series of regional conferences designed to bring together mathematicians and college administrators to discuss some of the issues raised by the report. Another was the task of preparing a detailed description of a graduate program modeled after the "first graduate component" defined in the qualifications report. This latter project was undertaken by the Graduate Task Force, a group with membership drawn from the Panel and from CUPM. Its report, A Beginning Graduate Program in Mathematics for Prospective Teachers of Undergraduates, was issued in 1969.
Meanwhile, other members of the Panel conducted a study on the supervision and training of teaching assistants in mathematics. Their findings were reported in a newsletter published in 1968. The need for a "companion volume" for the qualifications report was established when the Panel on Mathematics in Two-Year Colleges issued its 1969 report A Transfer Curriculum in Mathematics for Two-Year Colleges. CUPM felt it was necessary to describe the qualifications for persons to teach the courses in the Transfer Curriculum, and for this purpose it appointed an ad hoc Committee on Qualifications for a Two-Year College Faculty in Mathematics. This group's recommendations are given in the document Qualifications for Teaching University-Parallel Mathematics Courses in Two-Year Colleges, published in 1969.
Publication of the several reports mentioned in the preceding paragraphs completed CUPM's original plan of providing course guides for each of the five teaching levels defined in 1961. By 1967, however, the pressure for further change was already beginning to be felt. A minor revision (1968) of the Level I course guides contained the statement:
The five years that have elapsed since the preparation of the Course Guides have seen widespread adoption of the ideas of the new elementary school curricula, not only of the work of such experimental or quasi-experimental groups as the School Mathematics Study Group (SMSG) or the University of Illinois Curriculum Study in Mathematics (UICSM), but also of many new commercial textbook series which incorporate such ideas. In addition, there have been attempts to influence the future direction of elementary school mathematics by such groups as the Cambridge Conference. In the near future, the Panel believes, it will be necessary to examine our courses to take account of these developments. We hope in the next couple of years to begin the sort of detailed, intellectual study of current trends in the curriculum and of predictions of the future which will be necessary in order to prepare teachers for the school mathematics of the next twenty years.
During the years 1968-72 the Panel on Teacher Training continued this promised study. It sought to understand current trends and future possibilities through a variety of means:
The Panel concluded from this study that a revision of the CUPM recommendations and course guides for Levels I, II, and III was indeed required. Its 1971 report, Recommendations on Course Content for the Training of Teachers of Mathematics, was a result of that decision.
During the early seventies the Panel on College Teacher Preparation continued its interest in the role and preparation of teaching assistants. A 1972 newsletter, "New Methods for Teaching Elementary Courses and for the Orientation of Teaching Assistants," contains a statement by the Panel on teaching experience as part of Ph.D. programs. In 1972 the Panel also issued a booklet entitled Suggestions on the Teaching of College Mathematics whose purpose was to disseminate some ideas about practices that are believed to have contributed to successful teaching of mathematics in colleges and universities.
The Panel on Mathematics in Two-Year Colleges was formed in 1966 following some preliminary study of the need and potential in this area for the kind of activities which CUPM had successfully pursued in other areas. The members of the Panel were chosen from two-year colleges, four-year colleges, and universities so that extensive experience in various phases of education would be available. The Panel initially sponsored a series of meetings at which representatives of a wide spectrum of two-year colleges provided much detailed information about local variations, supplementing the Panel's studies of the national scene. The Panel also participated in several other activities related to the problem, such as meetings of the National Science Foundation Intercommission Panel on Two-Year Colleges, meetings of various organizations of two-year college mathematics teachers, individual visits to institutions, and a wealth of personal contacts.
During this study phase the Panel was divided into subpanels concentrating on three topics: mathematics for general education, mathematics for technical-occupational programs, and mathematics for four-year college transfer programs (in all disciplines). Many two-year college teachers who consulted with the Panel expressed the opinion that guidance was most needed on the first two topics. However, it became increasingly clear as the study progressed that considerable overlap existed in the problems in these three areas and that an initial concentration on the third topic was most natural, both logically and from the viewpoint of CUPM's customary methods of operation. Thus, the Panel decided to concentrate its initial efforts on the construction of a program for university-parallel mathematics courses in two-year colleges. Its report, A Transfer Curriculum in Mathematics for Two-Year Colleges, was issued in 1969.
Concurrent with the decision of the Panel to restrict itself to the university-parallel curriculum, CUPM appointed an ad hoc Committee on Qualifications for a Two-Year College Faculty in Mathematics, whose membership overlapped that of the Panel. The report of this Committee, which appears in this Compendium's section on Training of Teachers, discusses the qualifications of teachers of university-parallel mathematics courses and makes some general remarks concerning two-year college mathematics faculties.
The Transfer Curriculum report is essentially an adaptation of the first part of A General Curriculum in Mathematics for Colleges to the particular circumstances of those students in two-year colleges who intend to transfer to a four-year institution. That report intentionally deferred the consideration of lower-level or non-university-parallel courses as a matter for further study. In 1970 CUPM appointed a Panel on Basic Mathematics to consider the first of these two areas: courses at a level below that of Mathematics A in the Transfer Curriculum. Among the members of this new Panel were persons from the Two-Year College Panel and representatives from developing institutions. The Panel felt that it would be possible to replace many of these courses by a single flexible course which involved a mathematics laboratory and was innovative in its approach. Its recommendations, together with an outline and commentary on the proposed course, appear in the 1971 publication A Course in Basic Mathematics for Colleges.
In 1971 CUPM issued A Basic Library List for Two-Year Colleges. This list was compiled by an ad hoc committee, with the assistance of many teachers from two-year colleges, four-year colleges, and universities.
Having offered suggestions for the improvement of university-parallel and basic mathematics programs, CUPM then turned to the much more complicated area of mathematics for technical-occupational programs in two-year colleges. A reconstituted Panel on Mathematics in Two-Year Colleges laid plans for producing materials designed to improve mathematics instruction for students in these fields. Due to lack of funds, it has not yet been possible to bring these plans to fruition.
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.
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.
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.
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.
Following the conclusion of the sustaining NSF grant to CUPM and publication of the Compendium of its work under that grant, CUPM continued as a standing committee of the MAA with one major part of its original portfolio split off into a separate standing committee: CTUM, the Committee on the Teaching of Undergraduate Mathematics. At the same time changing circumstances in school and college education created considerable ferment not least among school, college, and university mathematicians. To set the context for CUPM's subsequent work, we list here a representative sample of reports that greatly influenced this work. A few of these reports were published by MAA, most not; many benefited from extensive advice from members of MAA:
During this same period, CUPM continued to publish reports on areas of vital importance to collegiate mathematics. First came a major updating of its earlier recommendations on the undergraduate major. Whereas formerly the focus of attention was on students intending to attend graduate school in mathematics or a mathematically intensive field, changing demographics of undergraduates caused colleges, and hence CUPM, to develop principles for a mathematics major for students with much broader goals.
CUPM also issued reports focused on the elementary part of the curriculum where the vast majority of students encounter college mathematics. Two of these looked at the experiences of college students who take the lowest level courses in "mathematics appreciation" or courses ordinarily taught in secondary school. Three others begin to examine issues concerning the nature of introductory college mathematics in the emerging computer age—that is, continuous vs. discrete mathematics. As calculus seeped into the standard secondary school academic track, and student interest tilted towards computer science, major issues at this interface became increasingly problematic. By 1989 CUPM once again pulled together its major recent reports in a compendium entitled Reshaping College Mathematics. (As with the previous compendium, links are to the republished versions rather than to the originals.)
In 1989 several new forces emerged that began to disrupt the landscape of mathematics education. Nationally, President George H.W. Bush and Arkansas Governor Bill Clinton in his capacity as chairman of the National Governors Association convened the nation's governors for an unprecedented Education Summit to set forth national goals for education under the slogan "America 2000." One of these goals was that "by the year 2000, U.S. students will be first in the world in science and mathematics achievement."
At the same time, the National Council of Teachers of Mathematics, published the first version of proposed national standards for school mathematics. Shortly thereafter, the National Research Council produced two reports on the urgency of revitalizing the mathematical sciences—one about issues, the other about people and demographics. Here is a sample of some of the major reports from sources other than MAA that carry consequences for undergraduate mathematics:
During this same period MAA's work on policy for collegiate mathematics was divided among several committees, panels, and task forces. Many were only loosely associated with CUPM. (As noted earlier, MAA subsequently set up a Council structure to group committees in related areas.) In 1992 the MAA published yet another "compendium" of reports concerning the undergraduate curriculum, this one containing both reprints and new reports from innovative email "focus groups" whose goals were to record a variety of helpful ideas rather than to make specific consensus-based recommendations.
Here is a list (with some links) of MAA publications on the undergraduate program in mathematics from this period, only a few of which emerged as reports by or of CUPM:
Forty years after MAA's began the Committee on the Undergraduate Program in Mathematics two conferences were held to review the history and evolution of the core curriculum in collegiate mathematics. The opening essay surveyed the evolution of CUPM's recommendations; the entire volume, published in 1998, contains papers touching on those mathematics courses taken by the vast majority of U.S. college students.
Beginning in the mid-1990s, reports from CUPM and other MAA committees dealing with the undergraduate curriculum were generally published on-line (and sometimes only on-line). Access to these more recent reports is available from the CUPM Web Page.
As a reference tool for those who may be interested in certain aspects of CUPM's work, we list here the references cited above, grouped by topic. Topics are arranged in an order that moves from more general and elementary to more specialized and advanced; within topics, references are listed in chronological order. (In a few cases a citation is listed twice if it fits naturally under two different topic headers.) These reference lists include reports that bear on the agenda of CUPM whether or not they were sponsored by CUPM or its various associated subcommittees. Quite a few are studies by other groups or individuals, many of whom had close ties to CUPM or MAA. All were mentioned earlier in this document.
Since many CUPM reports have been reprinted in one of three compendia, we use these abbreviations in citations:
As above, although CUPM citations refer to the original publication, most links go to the version found in the compendia rather than to an on-line copy of the original.
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