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The Computerized Pump: A Lab-Based Calculus with Precalculus

William Mueller
Department of Mathematics, Statistics & Computer Science
Mount Holyoke College
South Hadley, MA 01075


Several years ago, after a long evening grading calculus labs, I found myself scanning the late-night television spectrum, searching for distraction. I happened upon a comedy sketch parodying those old college quiz shows; called "Night School High-Q", it featured an especially hapless assemblage of contestants. One woman would pound eagerly on her buzzer after hearing only the first few words of any question, pause with a sudden shyness, and then offer the same uncertain answer every time: "Computers??" I laughed; I thought it was perfect: incomprehension, naive trust, and, for those who chose to watch, a certain sardonic satisfaction in seeing that trust so misguided. In short, the trials of a computer-based undergraduate mathematics course.

Trials indeed. Students, afraid of the less structured modeling problems that computers allow, quick to turn that fear into panic, and then turn that into pleas for the more familiar discipline of neatly circumscribed problems whose algorithms march, in a soothing monotony, to predictable conclusions. Colleagues, uncertain of computers themselves, quick to accede, feeling some sort of Orwellian corruption of young minds has probably been averted. Both of them wondering how computers ever insinuated themselves into a mathematics curriculum in the first place.

All of which is a little bit puzzling when you consider what it is that computers actually do: tedious computations, very quickly. There is clearly no heuristic purpose served by, for example, repeatedly performing long divisions, once the idea has been mastered in grade school. (Indeed, what computer skeptic would be willing to part with their pocket calculator?) Yet when it comes to computers freeing students from the routine computations in something like calculus—that sacred initiation into the master/slave relationship between teacher and student—there is resistance.

It is the potentially interested student who stands to lose. Consider, as an example, the important introductory topic of interpreting the behavior of a function by examining its graph. The old scold "Don't just plot points!" can't help but sound hollow once students have seen plotting software do just that, quickly and successfully. Instead of emphasizing the tiresome computation of critical points, bounds, asymptotes, periods, symmetries, etc., it is now quite reasonable to just plot points, and then infer the information from the graph. Connections between the geometric properties of the graph and the algebraic and differential properties of the function can still be drawn, but one is no longer limited to looking at only a few thoroughly worked examples. With the assistance of algebraic and symbolic differentiation software, it is easy to consider functions far too messy for the traditional approach. Investigating something like the influence of varying parameters on a family of functions—tedious if not impossible using traditional methods—is quick and intuitive with the aid of a plotter.

What computers do, then, is ultimately something much more than efficient computation. They allow students to quickly manipulate their mathematical intuitions. They provide an opportunity for teachers to create a curriculum based on exploration, discovery, and the real joys of doing mathematics. By eliminating unnecessary tedium, they allow students to travel to mathematical places that would otherwise be beyond their abilities. They make that heady rush of connection, which research mathematicians have always struggled to communicate, somehow seem accessible.

If there is genuine desire to pump more students into mathematical fields, particularly those students who have seen little to recommend those fields in the traditional curriculum, then the use of computers seems a natural starting point. The usual agendas for entry-level courses, filled to bursting with inscrutable topics that somehow absolutely must be covered, have always misrepresented the soul and passion of the subject. Good students will come to see the importance of εδ-proofs, and theorems on continuity, once they are hooked, but much of this material is probably best left to a first analysis course. What we must do, first, is make the challenges of doing mathematics seem worth the investment.

A Start

Approaches to a more computer-based mathematics curriculum will vary, necessarily, from place to place and from teacher to teacher. Even with some consensus concerning which topics should be covered in a calculus or precalculus course, making those topics interesting and relevant will always depend to some degree on the strengths and interests of the teacher, and on the particular group of students. This, I believe, is as it should be. It seems counterproductive, therefore, to present any new approach as some kind of complete package that must be imported wholesale to another site. I will not attempt to do so in the course description that follows. I will recommend, instead, that interested readers pick and choose among the ideas presented, with the understanding that their own involvement with the material is essential.

We have been trying out a new calculus-with-precalculus sequence at Mount Holyoke over the past year that is meant to incorporate some of the beneficial aspects of using computers discussed above. Funded by a grant from the Charles Dana Foundation, the sequence was conceived as a way of finding new and better ways of stimulating and supporting interest in mathematics among women, particularly those women (grouped without much analysis by race) who have been most conspicuously absent from mathematical fields in the past. (At Mount Holyoke, of course, all of the students are women.)

We began with a few simple ideas. First of all, we decided that the course would be sold as a two semester Calc I/Calc II sequence, and would not, in any way, carry the usual "remedial" connotations of precalculus. It would, nevertheless, prominently feature the incorporation of relevant precalculus material. (Cox, in his article in this volume, describes a course at Amherst that served in many ways as a starting point.) A student uncertain of her background or abilities might thus be reassured that the course would address these concerns.

To make room for the precalculus material, we would consolidate the topics into a few major themes each semester, as follows:

Semester 1

  • Mathematical modeling: simple modeling exercises (introduction to mathematical word processor)

  • Functions as models of causality: representations of functions as graphs, tables, and expressions; linear and quadratic functions; quadratic formula; completing the square related to shifts of the graph (introduction to graphing software)

  • Derivatives as models of rates: algebraic manipulations related to discovery of differentiation rules; solving equations and approximation of roots related to max/min (introduction to algebraic and symbolic differentiation software)

  • Simple differential equation models: linear approximation and Euler's method (introduction to Euler's method software)

Semester 2

  • More differential equation models: review of first semester algebra and Euler's method (extensive use of Euler's method software)

  • Exact solutions of differential equations and introduction to integration: trig, exponential, log, inverse functions (use of previous software)

  • Riemann sums and the Fundamental Theorem: sum formulas and related algebraic manipulations; induction (introduction to integration software)

The more open structure afforded by fewer topics was also meant to address very specifically the preference of female students, related in several studies, for a less resolutely linear, and more inwardly spiraling approach to central ideas. This was to be implemented, finally and crucially, by regular computer-based laboratory assignments in which students would explore modeling problems until some appropriate mathematical formulation was discovered.

The class met in a regular classroom setting three times each week, where we discussed homework assignments (of the traditional, routine, hand-worked variety), worked examples illustrating new techniques, and sought to make connections with the current laboratory assignment. Students were given an average of two weeks to complete each lab, occasionally with a week's oasis to introduce new topics (and sometimes to quell incipient taxation revolts). There were two regularly scheduled periods during which students could meet in the computer lab and receive additional assistance, but the room was open throughout the week, and students were expected to make use of it at other times. Students worked in groups of two to three, with the goal of producing a "journal quality" lab report using a mathematical word processor. (See Davis et al., this volume, for a good description of a similar laboratory setting.) Help was available outside of class through generous office hours and regular evening help sessions staffed by teaching assistants.

Materials for the course were, of necessity, a hybrid. We were able to find compatible approaches in the two-volume Calculus in Context series from the Five College Calculus Project and the North Carolina School of Science and Math's Contemporary Precalculus Through Applications, from which we took reading assignments, homework problems, and ideas for many of the labs. Labs were a bit ad-hoc the first time through, assembled from a variety of pre-existing sources (Davis et al., extended exercise sequences from the texts, some material from Duke's Project CALC), and the lack of uniformity was a source of some distress for the students. The labs, at the core of the course, are thus the focus of our current developmental efforts. This gradual transformation of the themes of the course into a coherent body of assignments must, after all, be the defining activity of any laboratory-based course.

Certain central aspects of the course should be highlighted:

  • The emphasis on modeling. Generally speaking, students seem to find mathematical methods most engaging when they are introduced in terms of their applications. Modeling problems provide a setting in which traditional computational tools, and the powerful computational abilities of the computer, can best be seen working together.

  • The emphasis on writing. Modeling, and the idea of formulating a problem in some way other than simply writing down an equation, will be new, and often puzzling, to most students. Writing, rewriting, and critiquing the lab reports of classmates helps students understand the essential mathematical task of saying things clearly, and helps them understand when English, and when mathematical symbolism, will serve that purpose best. (I often have lab groups exchange and comment on each other's reports before they hand them in.) Word-processed final drafts, which look very professional, make students feel that they have done something important.

  • The emphasis on mechanical computation. Some of the other computer-based approaches with which I'm familiar swing too far, I think, from this traditional aspect, and at a great disservice to the students. Exploring and formulating ideas is certainly an exciting part of mathematics, and probably the part most likely to capture the imagination of entry-level students, but it quickly becomes a hollow exercise if students do not have the skills to carry through basic computations. Moreover, the computer loses its function as computational tool, and becomes more of a magic box, when confidence in employing algorithms is low. Much of mathematics, after all, is fundamentally recursive, and students often yearn, understandably, for the earthly comforts of procedure. Routine homework assignments, and traditional quizzing and testing, must play a role.

Classroom Notes

There are certain inescapable difficulties associated with running a class such as the one outlined above. (I have run other such courses and these seem to be constants.) The first is that there will be, no matter what you do, great resistance, bordering on rebellion, when students are asked to write lab reports. This is simply not what they have come to think of as mathematics. It seems (at first) to take much more time than they are willing to put in to it, and they are quick to convince themselves that they are being cheated out of the "real" solution to problems. Why don't you just tell us what to do? they will complain. Of course you do have to be quite clear what it is that you are asking them to do, or else everything will be worse than it has to be. Lab assignments must give a clear statement of objectives, and carefully steer students in the direction that you want them to go, leaving them only to "discover" things that have been, for the most part, put in front of their noses. They need to know what it means to "write a lab report". I hand out a sheet of general purpose tips on mathematical exposition (largely common sense things that would apply equally well to English reports) and xeroxed examples of articles from popular mathematics magazines. The good news—and prospective first-time teachers of such an approach should listen well, and take heart—is that this resistance does melt away, once students realize that the reports are going to be a regular part of the course, that they will be rewarded for the time they put into writing them carefully, and that, despite their initial complaints, they are indeed learning mathematics.

It should also be noted that the algebraic skills of entry-level students are likely to be very bad. This cannot be underestimated, or overemphasized. Drill, in an unintimidating setting, is essential if weaker students are to survive. Encouraging students to work in groups and exchange ideas, without teacher intervention, helps build confidence. I give students their classmate's e-mail addresses at the beginning of the term and send them class notes and assignments so that they log on to the system. I found that this generated much discussion among students who otherwise might not have talked to one another (in addition to developing their computer literacy generally). Evening help sessions, run by student's peers (math majors), became an indispensable part of the course.

Finally, it is worth noting the obvious: that a general class atmosphere of openness and respect for each other is the only way to bring out students who would otherwise be frightened away. I tried hard to listen to students in class and in lab, and get their classmates to do the same, even when someone was heading off in an apparently orthogonal direction. The absence of an overloaded syllabus helped make this possible. I always kept my office door open, and had some of my most rewarding moments when otherwise uninquiring students would peek shyly around the threshold. Training the teaching assistants to be patient, and to suggest directions and encouragement rather than answers, was important for making the help sessions work in a way that complimented the rest of the course.


For a first time through, with all the bugs and frustrations, I think the course went well. There were disappointments, of course, and some of them remain inexplicable, but the number of success stories was encouraging.

The main problem we had was attrition: after beginning with 39 students (17 minority), we ended the first semester with 28 (10 minority), and then rode through the second semester with 14 devotees (7 minority). I believe this was largely due to the unwillingness of many students to accept a new approach. The extent to which their conception of mathematics had already solidified in high school seemed like an impenetrable barrier in many cases. Many of these same students, it seems, also lacked the maturity, in their first semester of college, to take any responsibility for their own education, and this is certainly something that our approach demanded. Admittedly, all of this was exacerbated by the developmental nature of the course, which sometimes showed through the optimistic wrappings.

Of the students who did stick with it, the level of enthusiasm and the ability to work independently by the end of the course was really remarkable, especially given the trepidation with which they had approached mathematics in the beginning. Some of the final lab reports were so well written, and so confident in style, that they required no comments other than praise. Most of the students who remained asked about further mathematics courses; one of them even received one of the departmental awards for outstanding first year students. It is difficult to quantify such things. I hope, at the very least, that these students will remember some enjoyable mathematical encounters, and that they have left the course with something more satisfying than a head full of formulas.

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