Spreadsheet Projects in a Calculus Course

Since the fall of 1999, we have used spreadsheets to help make the connection between math and applications in Differential Calculus with Trigonometry at Virginia Tech. Math 1016 is the first course in a three-semester calculus sequence. Students in the life sciences (including, e.g., Biology, Forestry, and Environmental Science) make up perhaps half the annual enrollment, but many students take the course simply as part of the University's "core" math requirement. The prerequisite is College Algebra. 

To make calculus useful and interesting to the widest possible audience, we consulted colleagues in several of the outside departments. These contacts gave us a few topical suggestions ("Calculate the volume of a tree"), but for the most part they were concerned about their students" skills in using a modest amount of mathematics appropriately in the setting of an applied problem.

To meet these needs, we overhauled the topical outline, switched from a traditional, all-purpose calculus book to the CCHE Applied Calculus (in a custom edition that includes material on trigonometric functions from Functions Modeling Change), and introduced technology through significant "lab" assignments. The labs provide new viewpoints on the subject matter, reinforce the connection between course topics and applications, and introduce some additional topics.

We find that spreadsheets (Microsoft Excel) provide adequate power, are relatively easy to learn, and are readily accepted by the students, who can use this medium elsewhere. The spreadsheets work particularly well with discrete models and data, of course, and they generally give more of a "hands-on" feeling than math packages. (Labs in the subsequent integral calculus course use Mathematica as well.) On the other hand, most of the labs would be difficult or impossible to do with only a calculator.

Virginia Tech has a large computer/ tutoring lab, the Math Emporium, that enables us to work with unconventional class formats. In Math 1016, there are just two weekly class meetings. In place of the third meeting, we offer introductory sessions for the lab assignments eight to twelve times per week, at the Emporium. Faculty, graduate assistants, and/or undergraduate tutors are on hand 74 hours per week to give individual help.

Content of the Lab Assignments
There are nine labs in the semester. The first four build the students' skills in Excel and explore topics we cover in class more deeply. We do not assume prior knowledge of Excel. Our first topic is functions. Students enter some given data and graph it, then answer some questions about functions. Labs Two and Three deal with linear and exponential regression, respectively. We use problems from the text where a set of data points is given. Excel will display an equation for the best-fit on the graph. The students compare the results to rough fits constructed "by hand" from the data. We ask them to make a judgment about their confidence in predictions from the trendlines and from "Forecast" extrapolations.

Lab Four concentrates on more of Excel's capabilities, such as cell referencing and iterative generation of data. The students are ready now for more complicated problems. 

Labs Five and Six are our trigonometry labs. In Lab Five we use two problems from the modeling section in the book. One of these involves a double ferris wheel. The students use information about the rotation times of the rotating arm and of each ferris wheel to calculate the height function for the seat on one wheel. They can then create the graph and determine the overall period. We ask them about the maximum height and a second local maximum. This lab is an excellent challenge for our students.

The second trig lab explores how shifts and stretches affect the functions. We use referenced cells in our formulas; when we change one cell all our data and graphs follow. We can see immediately how a change in amplitude, period, or midline affects the graphs of sine and cosine.

In our derivative lab, we use public debt data from the Internet. This is a great example of an increasing function that first increases at an increasing rate then at a decreasing rate. The students calculate rates of change and the rate of change of the rate of change and then answer several questions. This lab helps them pull all the information on rates of change together before we start using the derivative formulas.

Labs on the Malthus hypothesis and on Exponential vs. Logistic growth are built on spreadsheet projects from the textbook. Both labs involve data gene-rated iteratively from models. Questions for the students include simple observations, comparisons based on numerical experiments, and criticisms of the models. In the Malthus lab, for example, one observes the graphs for food supply, population, and their ratio, and comes up with a precise version of the statement "exponential growth is faster than linear growth."

Practical Issues and Outcomes
One of the goals of teaching in general is to encourage and foster independent learning and responsibility. We all know students fight this very thing they say they want and are ready for. Here are some of our observations and choices.

Working from instructions rather than imitating examples is a significant step. We have found that the students need explicit, clear directions. Questions should be direct, uncomplicated, and specific. We give very explicit help at the lab intro sessions.

Most of our lab problems are from the text for a few reasons. The questions in the text are well written. In addition, students tend to think in terms of processes; using their own book helps make clear the connection between a lab assignment (computers) and ordinary homework (pencils and calculators). 

We hope, in fact, that experience with doing spreadsheet graphs will lead to more use of this tool as a standard homework accessory. Perhaps because our tests at this stage are not given in a computer environment, students rarely do this unless there are specific instructions to use computers. 

We encourage, and sometimes require, working in groups. This is usually a positive action. Students can encourage each other and solve problems together. Group work also cuts down on grading.

There are some negatives to group work. A student may not do the work and yet get credit for it. A student may not understand as well as his/her group members and be embarrassed to speak up. We have detected a few cases where electronic files have been duplicated and handed in to different teachers. To encourage everyone to stay involved, we include questions based directly on the labs in every test. We specify and emphasize Honor Code requirements, and we include a cover sheet with the labs on which group members affirm their active participation and list significant collaborators.

In spite of the mentioned disadvantages of group work, and, although the course structure in general emphasizes individual responsibility, we feel the positives of group work outweigh the negatives. Most students appreciate the opportunity and do the work in the intended way. They give thoughtful answers to the open-ended questions, and they take care with the layout of their papers, often designing striking color printouts.

In one class we collected essays about the labs. The most notable result was the variety of reactions??in preferences for some labs over others and in assessment of what was or wasn't valuable about the projects as a whole. This confirmed our original idea that widening the activity menu allows more individuals to connect. We recommend the lab approach as one way to accomplish this.

Editor's Note: Information and materials from Math 1016 can be accessed from the Math Emporium homepage, www.emporium.vt.edu. Spreadsheet Projects in a Calculus Course Kenneth Hannsgen and Abigail Kohler, Virginia Polytechnic Institute and State University, hannsgen@math.vt.edu and kohlera@math.vt.edu .