Why Calculus?

A much delayed sequel to the previous post...


My investigation of the prominence of calculus in mathematics education starts in the American Mathematical Monthly in the 1950s. I learned from Klein[1] that in the 1950s the mathematics curriculum in the U.S. changed to include calculus in the high school curriculum, at least for some students. I chose to start looking for primary sources with the Monthly because I am familiar with the journal, and because it seemed likely to have some articles dealing with mathematics teaching at the college level, and possibly at the high school level.

Benjamin Finkel founded the American Mathematical Monthly in 1894 as a journal aimed primarily at teachers of mathematics. The Monthly has been in constant (though not quite monthly) operation ever since, but it soon shifted focus to collegiate mathematics.[2] Most articles in it, both now and in the 1950s, discuss mathematics itself, either by way of exposition or as presentations of original research and novel proofs. However, articles discussing mathematics education, the role of mathematics in society or the liberal arts, biographical notes, and descriptions of conferences and meeting are not uncommon, particularly in earlier volumes. In the 1950s, the Notes section of the Monthly had a subsection for Mathematical Notes and another for Classroom Notes, with the latter often discussing approaches to teaching particular topics but sometimes giving a more general description of a particular program or course.

I looked at volumes 57-63 of the Monthly, which were published between 1950 and 1956, inclusive. The articles focused on mathematics education mostly focused on mathematics education at the college level, with particular interests in education of future primary and secondary school teachers and whether colleges ought to run mathematics courses for first year students not planning to do further work in mathematics or science (and what such a course should include). Several articles addressed perceived defects in mathematics teaching in primary and secondary school, mostly as a problem which colleges should address by better preparing teachers of mathematics at those levels. Two articles in particular called for specific changes in the high school curriculum.

"Mathematics in School and College," excerpted from the 1952 book General Education in School and College, appeared in the Monthly in June-July 1953. The book was the result of a study conducted by Andover, Exeter, Harvard, Lawrenceville, Princeton, and Yale, and provided both a survey of the present curricula of the institutions involved and recommendations for modifications that would allow the preparatory schools involved to better prepare their students to take advantage of college instruction.

The study concluded that the school mathematics curriculum was ripe for change. The contemporary curriculum consisted of, for most students, two years of algebra, one of plane geometry, and one of trigonometry and solid geometry. The fourth year was optional, and some students in some schools ("less than one in five") progressed more quickly and finished high school with a year of calculus. The committee conducting the study believed that the basic direction of this curriculum was appropriate, but that it included several inessential topics, and some topics which were studied at too great a length. After a few snarky remarks[3], they recommended for "condensation or omission" the topics of solid geometry (the "greatest single offender"), complex numbers, determinants, the geometry of the circle, and logarithmic solution of triangles. With the time saved, the committee suggested that students should learn calculus, statistics, or both.

The committee had many reasons to recommend that schools expand their calculus program. Compared to solid geometry and the logarithmic solution of triangles, calculus is an inspiring subject; the study claims that "the student who has once grasped the meaning of differentiation and integration sees the world after in a larger and more significant way." For scientists and engineers, calculus is a fundamental tool, and even some students who would not become scientists were expected to take physics courses in college, which could be improved if the students had some prior exposure to calculus. The argument also had the weight of tradition; calculus was "the standard freshman course in the best of our colleges" and so the logical course to follow the rest of the high school mathematics program. Finally, some schools already had small calculus programs, and so certainly had qualified teachers.

The committee found the case for teaching statistics in some respects stronger than the case for teaching calculus, citing that statistical notions "are among the fundamentals of modern social measurements" and that "an awareness of the real meaning of these notions is a protection to the consumer as well as a necessity for the producer of information." Furthermore, statistical reasoning teaches students "that mathematics has uncertainty and that uncertainty can be mathematically treated." However, time constraints and concerns about teacher preparation led the study to conclude that most schools "should move toward a curriculum in which the basic 12th grade course is the introduction to the calculus."

"Mathematics in the Secondary Schools for the Exceptional Student" appeared in the Monthly in May 1954. This article discussed the results of what appears to have been a follow-up study to the one discussed in the previous article. The new study was conducted by twelve colleges and twenty-seven high schools. The study was the School and College Study of Admission with Advanced Standing, intended to provide a curriculum which schools could offer for advanced students who would then be able to receive college credit for courses which followed the curriculum.

This second study suggested an integrated math curriculum for the first three years of high school, after which the exceptional students for which the program was designed would be ready for "a substantial introduction to differential and integral calculus, with enough applications to bring out the meaning and to illustrate the fundamental importance of this subject." By passing a standardized exam on this course, they would then be able to receive college credit. The program for the last year is still familiar, and for good reason-- this study was the pilot program for what is now the Advanced Placement program.

Although the curriculum proposed in the study was designed for especially bright or driven students, the committee noted that the material covered in the first three years would be suitable for most high school students. They suggested that schools either run courses covering the same material more slowly for those students not aiming to take the calculus course or provide them with alternative courses for the fourth year of study, such as statistics or solid geometry. The committee did not recommend any means of granting college credit for some courses, however. In the case of solid geometry, this was probably because the course was already common at the high school level and not necessary to progress through the college curriculum, and in the case of statistics the concerns from the previous study about the number of qualified teachers presumably indicated that demand for such a mechanism would be low.[4]

From here, it looks like following the emphasis on calculus as a final math course in high schools would be much like tracking the development of the Advanced Placement program as a whole. I won't be doing that study any time soon, though, since I am no longer regularly in a research library. Expect my next post to be on something completely different!

[1]: http://www.csun.edu/~vcmth00m/AHistory.html
[2]: http://www.maa.org/pubs/ammhistory.html
[3]: My favorite is, "Of course it is possible to design problems of bewildering complexity in every subject from long division to trigonometry; it is also a waste of time."
[4]: Wikipedia indicates that the AP Statistics exam was first administered in 1996.