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CUP and Universal Mathematics

The First Project in MAA's Response to This Ferment

Soon after the Committee on the Undergraduate Program (CUP) began its work, it compiled a litany of issues facing undergraduate mathematics, including

  • "the rigid sequential organization of the traditional program with each course depending heavily on technical prerequisites,"
  • "the growing tendency to repeat high school courses in college," and
  • "[today's] emphasis upon cultural and liberal aspects in education."

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:

  • Universal Mathematics, Part I: Functions and Limits. F.A.I. Bowers, Jr., R.N. Bradt, C.E. Capel, W.L. Duren, Jr., G.B. Price, W.R. Scott. University of Kansas, 1954.
  • Universal Mathematics, Part II: Elementary Mathematics of Sets. R.N. Bradt, C.E. Capel, W.L. Duren, Jr., J.G. Kemeny, E.J. McShane, D.R. Morrison, G.B. Price, A.L. Putnam, W.R. Scott, & A.W. Tucker. Tulane University 1955. Revised as Elementary Mathematics Of Sets With Applications. R.N. Brandt & R.L. Davis. Mathematical Association of America, 1958.
  • A Freshman Honors Course in Calculus and Analytic Geometry Taught at Princeton University. Emil Artin & George B. Seligman. Committee on the Undergraduate Program, Mathematical Association of America, 1957; University of Buffalo, 1958.
  • Modern Mathematical Methods And Models, Volume I: Multicomponent Methods. E.J. Cogan, R.L. Davis, J.G. Kemeny, R.Z. Norman, J.A. Snell, & G.L. Thompson. Committee on the Undergraduate Program, Mathematical Association of America, 1958.
  • Modern Mathematical Methods And Models, Volume II: Mathematical Models. Cogan, E.J., Kemeny, J.G., Norman, R.Z., Snell, J.L., & Thompson, G.L. Committee on the Undergraduate Program, Mathematical Association of America, 1958.

Since these volumes fulfilled the committee's original mandate, CUP also prepared several archival reports of its work:

  • Report of the Committee on the Undergraduate Mathematical Program. W.L. Duren et al. Amer. Mathematical Monthly, 62:7 (Sep. 1955) 511-520. An interim report describing the work of the committee and highlighting experimental courses that address some of the issues on the committee's agenda.
  • Collected Reports of the CUP. Committee on the Undergraduate Program, W.L. Duren, Jr., editor. University of Virginia, 1957.
  • Transition Report and Outline of Recommended Courses. R.L. Davis. University of Virginia. 1958.