THE CALCULUS TRAP*
ADDRESS: Department of Mathematics, Dartmouth College, Hanover NH 03755 USA.
ABSTRACT: Evaluation of a five-year mathematics initiative at Dartmouth showed that while a term of introductory college calculus often left weak or uninterested mathematics students discouraged about mathematics, when such students enrolled in interdisciplinary mathematics and humanities courses they had a more productive mathematical experience. This paper suggests that more diverse introductory mathematics offerings at the college and even high school level might preserve the mathematical interest of a greater number of students than the present emphasis on calculus for all students.
KEYWORDS: Calculus, interdisciplinary, math avoidance, evaluation.
A DIALOGUE
Q: "Why did you decide to take calculus?"
STUDENT: "I'm still trying to figure that out. I'm not really a science-oriented person. I'm more creative arts. But I felt I had to prove myself in the science-math department. I took calculus last year, also, and that's why I took it, I guess."
Q: "Did taking calculus change your level of interest in math?"
STUDENT: "Yes."
Q: "In which direction?"
STUDENT: "The negative direction. I went to study sessions, I had a tutor, I did everything, yet I didn't really succeed in the class. It made me feel I wouldn't be very successful in math."
THE PROBLEM
During my five years as evaluator for the Dartmouth Mathematics Across the Curriculum project, this student's story became depressingly familiar. About 20% of the 108 students I have interviewed from seven calculus classes-a randomly drawn sample from a pool of about 1100 students-- related some close variant of this story. Like most students entering elite colleges, they had taken calculus in high school, although many felt they emerged with an insecure understanding of the subject. And like the student above, most identified themselves as "not a math person" or "not a fan of math." They chose calculus to meet their quantitative requirement partly out of inertia (continuing a subject from their final high school year), partly out of pride (believing bright students should take challenging courses), and partly out of a perceived absence of alternatives. Lacking interest in the mathematics itself or in another field where calculus is an important tool, a fast-paced term of college calculus eroded their interest in mathematics altogether. This paper takes up the question of how these capable students, and hundreds more like them, were transformed from indifferent math students into math-avoiders. In this light, it proposes alternate mathematical paths to revive their flagging enthusiasm.
The MATC at Dartmouth offers an apt laboratory for investigating this question. Because the project created or revised mathematics courses ranging from advanced calculus for specific sciences through introductory calculus and discrete mathematics to mathematics and humanities courses, the evaluation documented the perspectives of the widest possible range of students in a full spectrum of course offerings. This comprehensive approach had the effect of placing each course in the context of all others, allowing us to understand not only how students with different backgrounds and interests responded to the same course (the usual evaluative strategy) but, as importantly, how students with similar backgrounds responded to different courses. We not only interviewed students like the one above who chose introductory calculus, we also documented the experiences of his counterparts in background and interest who enrolled in the experimental mathematics and humanities courses. In this process it became clear that there was a significant subset of students who were "not fans of math," their interest in mathematics having been subverted by high school mathematics, often calculus. When these students continued on in college calculus, their disavowal of mathematics was virtually assured. In notable contrast, however, when they enrolled in a mathematics and humanities course, a great many rediscovered an interest in the subject.
It is well to remind ourselves before continuing that 80% of those interviewed, whether they took the calculus course with computer-based realworld applications or the standard course, were not soured on mathematics by their calculus experience. For some students these calculus courses were eye-openers, revealing for the first time the beauty of mathematics and its power to solve real and pressing social and scientific problems. Others were happy to have achieved some mastery of a tool they needed for success in a science or economics major. Some appreciated calculus as part of a complete liberal education, or as an intellectual experience valuable for honing analytic skills. And some, like the student who compared his calculus experience to that of being a cadet at West Point, wore their calculus credits as a badge of honor, proud simply to have proved that they could do it. It is also well to be aware that among those who were not soured on mathematics in general were a number who did not find the calculus experience itself satisfying. While in these cases the practical value of calculus for their major outweighed dissatisfactions with the course, this is no reason to declare the need for calculus reform to have been met.
But our concern here is the 20% for whom calculus was so unsatisfying as to color their response to mathematics altogether. This is not a negligible fraction, and certainly too large to sacrifice to a life of math-- avoidance. What mathematics might be "useful" to a student like the one quoted above? It is clear that answering interesting questions in some fields requires a knowledge of calculus; in others, statistics or discrete mathematics has greater value. As students achieve competence in these specific areas of mathematics, they also enhance their general mathematical reasoning, confidence and aptitude. These epiphenomena are often as important to later quantitative success as the specific course content. Even liberal arts students preparing themselves for careers in which mathematics has no clearly defined role need strong quantitative reasoning abilities and a good sense of how mathematics is done. For the last decade policy-makers have been at pains to emphasize the importance of quantitative abilities to success in all arenas, many yet unimagined [5, 6]. This paper proposes that the goal of mathematical competence for the greatest number might be better served if calculus were not the exclusive gateway to the fundamental set of mathematical capabilities we expect our undergraduates to acquire. Given the wealth of interesting mathematics, it is probably more productive to ask what mathematics might impart these capabilities to this set of liberal arts undergraduates than to ask how we might make calculus more appealing to them.
A SOLUTION
The Dartmouth MATC developed nine courses linking non-calculus mathematics such as group theory, number theory, infinity, chaos theory and the fourth dimension with art, music, history, literature and philosophy.1 Some were offered as first-year seminars, attracting strong mathematics students. Others, which were open to all students, without pre-requisites, drew both competent and insecure math students. While not all courses were unqualified successes, particularly early on, the success of the later courses demonstrates-in addition to a steep faculty learning curve-that mathematics usually reserved for post-calculus students can be presented effectively at an introductory level. Table 1 shows that students in the mathematics and humanities courses revealed changes in their beliefs and attitudes about mathematics, as measured by the pre-post mathematics survey, that were consistently (and usually significantly) more positive than that of calculus students. The "Overall Index," constructed by dividing an individual's total post-survey score by the total pre-survey score, provides a gross measure of change in his/her attitudes about math over the interval of a course. Indices greater than 1.00 show an overall gain in desirable attitudes; those less than 1.00 show an overall loss. The "Ability," "Interest," "Personal Growth" and "Utility" indices are similarly constructed from the four scales derived through factor analysis from the survey data and reference, respectively, students' perception of their mathematics ability, their interest in math, their belief in its importance for their personal growth, and in its usefulness in their professional lives.
