The Computerized Pump: A Lab-Based Calculus with Precalculus William Mueller Introduction
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 All of which is a little bit puzzling when you consider what it is that computers actually 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 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 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:
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 Certain central aspects of the course should be highlighted:
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, It should also be noted that the algebraic skills of entry-level students are likely to be 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. Conclusions 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. For further information and course materials write: |