External Influences on U.S. Undergraduate Mathematics Curricula: 1950-2000 - Calculus Reform

Calculus Reform

Some advocates of discrete mathematics became fairly explicit that calculus needed to be nudged a bit to make room for discrete ideas in the lower division mathematics curriculum.  For example, Ralston urged inclusion of discrete mathematics in a public debate with Ronald Douglas, a proponent of the primacy of calculus [54, p. 11].  More provocatively, the Williamstown Conference of 1982 included the paper “How to Cure the Plague of Calculus” [51].  The desire for more discrete mathematics seems to be a durable theme:  in 2007, years after the Williamstown Conference, one finds it expressed in Bressoud’s Launchings column of November 2007 on the MAA website [5]. 

This photograph of Ronald G. Douglas, early advocate of calculus reform, was taken by Paul Halmos in 1966, when the two were colleagues at the University of Michigan. It appears in the Paul Halmos Photograph Collection here in Convergence.

Ron Douglas at a workshop on Hilbert Modules and Complex Geometry at the Oberwolfach Conference Center in Germany in 2009. (Photo by Renate Schmid from the Oberwolfach Photograph Collection (MFO) is used under the terms of Creative Commons License Attribution-Share Alike 2.0 Germany.)

By 1986, advocates for calculus started to formulate a program of improvement for calculus at the Tulane Conference [11].  But it is a mistake to view the Tulane Conference and the subsequent calculus reform movement as entirely a defensive response to insurgent discrete mathematics enthusiasts.  Calculus had been an ongoing target of curricular change in the second half of the 20^th century as can be seen in the excellent background notes in [54].  For the earlier history of calculus teaching see [53].  Calculus reform became perhaps the most widely pursued curricular initiative of the 1950-2000 era, ultimately producing numerous reports, textbooks, meetings and conference sessions, and involving at least 127 NSF projects.  In 2001 it was estimated by Susan L. Ganter that “More than 500 mathematics departments at postsecondary institutions nationwide are currently implementing some level of calculus reform” [15].

As for what brought about calculus reform, one thing seems indisputable:  it was not weak enrollments.  In the Fall of 1985, around the time calculus reform started up, the number of students in mainstream calculus was, according to CBMS, 2.54 times the number in all forms of calculus in the Fall of 1960. This ratio accords well with the increase in the number of students in college from 1960 to 1985. For a comprehensive discussion of conditions and attitudes leading to calculus reform, see [54, pp. 7-22].  Of the factors mentioned there, we repeat the following which have external origins: NSF funding; changing national needs in regard to economic competitiveness; technology; a “more diverse, less selective spectrum of students coming to college”; and declines (of largely mysterious origins) in the number of mathematics majors, leading to more concern for average students.  No concrete evidence was given for these allegedly causative factors, but the authors rendering these judgments were highly respected and very well positioned leaders in the mathematical community, so their judgments are surely more reliable than most armchair speculations.

Not much mentioned in [54] as an animating concern of calculus reform, perhaps because good data was not then available, was the desire to give a larger role to modeling and applications.  In thinking about this issue of identifying the “real” motivations of calculus reform (or any large scale project), it is worthwhile to consider that those involved may not all have had exactly the same ideas and that ideas may change over time.  As an example, Douglas, the prime initiator of calculus reform, did not mention applications in his lead essay for the Tulane Conference [11].  However, the content workshop of that conference, a group of eight mathematicians, did mention it as a goal, on a par with Douglas’ two main goals of using technology and increasing conceptual depth [11].  Likewise, one can see applications in a prominent role in the implementation projects for calculus reform, which followed the conference by about two years.  According to [15], showing applications of calculus was the second most commonly mentioned goal in the 127 calculus reform projects supported by NSF between 1988 and 1994, appearing as one of the goals in 73% of the proposals.  It appeared in more proposals than enhancing conceptual understanding, or using multiple representations, or trying new pedagogical approaches such as discovery learning, etc.  Only “use of technology” was more frequently mentioned as a project goal.  Another evidence of the importance of applications in calculus reform can be seen in a survey of a random selection of 62 colleges trying out calculus reform in the mid 1990s:  Table 8 of [54] shows that incorporating applications was a more important goal than any other except for using technology.