A Radical Approach to a First Course in Calculus

*David M.
Bressoud January, 2008*

The re-evaluation of Calculus at Macalester College began about six years ago with the realization that the traditional sequence of Calculus I and II was not meeting the needs of our students. From questionnaires distributed to the students and from transcript records obtained from our registrar, we learned that about 75% of the students taking Calculus I had studied calculus in high school. For 70% of those who took the course, it would be their last calculus class. With a pass rate of C or higher near 90%, the attrition was primarily because most of these students were in a major, usually either Biology or Economics, that only required a single semester of calculus. It made no sense to teach Calculus I as if these students had never seen calculus before, or as if they were taking it as the first half of a year-long introduction to calculus.

Calculus I also was not serving as a gateway into the major. Of the 80–90 students who took Calculus I each year, at most 3 or 4 would eventually take a junior- or senior-level math course. Of the roughly 25 math majors who graduated each year, we seldom had more than one who had studied Calculus I at Macalester.

So we needed a course that would be fresh and interesting to students who had studied calculus in high school, a course that would stand on its own and be relevant to students for whom this would be their terminal calculus course.

Another important factor in our re-evaluation of calculus was the report of the Biology group in the Curriculum Foundations Workshop that had been sponsored by the MAA’s Committee on Curricular Renewal Across the First Two Years (CRAFTY), held at Macalester in November, 2000. We learned from the assembled biologists that Macalester was typical in that it required two semesters of mathematics for its biology majors. For us, this translated into a semester of calculus and a semester of statistics. But what we were doing did not match the needs of these majors. In the words of the report, “Statistics, modeling and graphical representation should take priority over calculus.”[1, p. 15] The way we were teaching calculus made it almost useless to the biology majors, and the statistics, while useful, was limited to univariate statistics. What biologists really need and use is multivariate statistics.

Inspired by programs such as the one at the US Military Academy
at West Point [2] and with the leadership of our mathematical
biologist, Danny Kaplan, we reconstructed the calculus class from the ground
up, creating a course that we now call *Applied Calculus*. There were
several absolutes that were established at the beginning. First, the emphasis
had to be on differential equations and systems of differential equations.
By the end of semester, students needed to be comfortable reading and interpreting
as well as constructing such equations. While some attention would be paid
to finding exact solutions, numerical techniques and qualitative analysis
of solutions would dominate their study. Second, functions of several variables
would be included as early and as often as was practical. In particular, graphic
representations of functions of two variables would be introduced early in
the semester, and the introduction of partial derivatives would immediately
follow the discussion of the derivative of a function of a single variable.
Third, the last two or three weeks of the semester would be reserved for a
geometric introduction to linear algebra, setting the stage for a subsequent
statistics class that could spend much of its time on multivariate statistics.

Other key aspects of the course emerged as it was field tested over the first two years. The course begins with a review of functions as models of types of behavior. Thus, for example, the sine function is a useful model of periodicity. Time is spent ensuring that students know how to modify the sine (adjusting amplitude, period and location of extrema) to get it to fit a given set of periodic data. The derivative and partial derivatives are introduced at the same time. The point of emphasis is their use as models of rates of growth (or decay).

Nothing conveys what has been stressed within a course as effectively
as the final examination. I have posted the nine final exam questions from
Fall, 2006 (the last time I taught this course) at [3].
Several members of my department are now working on a description of our Applied
Calculus course that will appear in a forthcoming MAA Notes volume being edited
by Glenn Ledder and tentatively titled *Undergraduate Mathematics for the
Life Sciences: Processes, Models, Assessment, and Directions*.

Macalester is too small an institution be able to offer both
a traditional Calculus I and our revamped Applied Calculus. Our traditional
sequence has disappeared. Applied Calculus serves both as a terminal course
and as an introduction to the ideas of calculus. It is also taken by students
who arrive with credit with first semester calculus but who are uncertain
whether or not they want to continue toward advanced mathematics. For those
who do seek preparation for more advanced courses, we offer a single semester
of *Single Variable Calculus* that is appropriate for both the students
who have come out of Applied Calculus and for those who arrive at Macalester
with credit for a semester’s worth of Calculus.

We have not yet done a transcript analysis to parallel the one done at the start of this process, but the informal reports from students and the departments whose majors are served by this course suggest that our change has been very positive. The students feel that they are learning skills and understandings that are truly useful. And we find that there are students whose interest in mathematics is re-kindled by this course. It is feeding students into our Single- and Multivariable Calculus courses.

Macalester’s solution will not work at every college or university, but our basic premise is one that I wish everyone would adopt: We must take a long and honest look at our calculus sequence. What are the backgrounds and needs of the students it is serving? How well does it serve those needs? How can we re-imagine these courses so that they better serve the students we have?

[1] Ganter, Susan, and William Barker, eds.,

[2] US Military Academy’s Applied Calculus is described on the course website at www.dean.usma.edu/departments/math/courses/ma103/

[3] Bressoud, David M., *Report on Calculus at Macalester
College*, presentation at Joint Math Meetings, January 8, 2007. Available
at www.macalester.edu/~bressoud/talks/NewOrleans/AppliedCalculus.pdf

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Illustrative Resources*.

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David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College in St. Paul, Minnesota, and president-elect of the MAA. You can reach him at bressoud@macalester.edu. This column does not reflect an official position of the MAA.

Copyright ©2007 The Mathematical Association of America

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