Tommy Ratliff

Group projects can be a very time-consuming and stressful part of a calculus course. In the past when I assigned projects, I was generally happy with the process and the results, but the students were somewhat nervous about writing papers for a math class, and I was nervous about administering and grading papers. However, I made a few adjustments this spring when I assigned three group projects in my Calculus II course and in the second semester of a combined Pre-Calculus/Calculus I sequence. I was so pleased with the entire experience that I plan to assign group projects in all of my calculus courses from now on.

I have three main reasons for assigning projects:

- I want the students to work on more difficult and open-ended problems than I could assign in regular homework sets.
- I want the students to improve their mathematical communication skills by working in groups and by writing their results in self-contained papers.
- I want to keep the course fresh and interesting by varying the structure of the class meetings and by giving the students the opportunity to be creative in their papers.

The second reason for the success of the projects this spring was that each project was written as a letter from a fictitious character to the students asking for their advice on some problem. This clearly defined the target audience for the paper and gave the students an idea of the mathematical background that they should assume of the reader. The plot lines in the projects were a little bit goofy, although not imprecise, which helped relax the students and gave them the opportunity to be creative when writing their papers.

For example,
in a Calculus II project on infinite series, Wile E. Coyote writes to
the students that he has
a recurring nightmare that he and the Roadrunner are standing at
opposite ends of a road that is 1 kilometer long. He can move toward
the Roadrunner at a speed of 1 meter/second, but after each second the
road stretches uniformly and instantaneously by 1 kilometer. He wants
the students to tell him if he ever catches the Roadrunner, and if he
does, how long it will take. With some help, the students made a few
specific calculations and recognized the pattern that the distance
between the Coyote and Roadrunner after *n* seconds is

Since the harmonic series diverges, they
determined that the Coyote will eventually catch the Roadrunner, and
they used the integral test to approximate that it will take
*2e^999-1* seconds.
Some of the best papers I received all semester tried
to place
this length of time in perspective.
One compared to the time that the earth had been in
existence (although the group members disagreed on whether this
was 4 billion or 10,000 years), and another explained the length of
time in a poem called ``Ode to Coyote'' that concluded that it was
``Just too many millennium to comprehend.''

I used several projects from Student Research Projects in Calculus [Cohen, Gaughan, Knoebel, Kurtz, MAA, 1991] (although I took a few liberties with the plot lines), and I wrote several on my own. (If you are interested in seeing copies of the projects or the checklist, I have placed them on my World Wide Web homepage http://www.stolaf.edu/people/ratliff/, now http://tratliff.webspace.wheatoncollege.edu/index.html.) The classes met three times a week for 55 minutes, and I handed out the projects at the end of one meeting and gave the students the next meeting to work on the projects in groups of two or three. Each group turned in one paper about a week later that usually ranged from four to seven pages and counted for 10\% of their final grade. I allowed the students to pick their own groups, and while some students did shift groups after the first project, I did not have any serious problems with any of the groups. I will need to keep a closer eye on the dynamics of the groups in the future, probably by asking the students to evaluate the contributions made by each member of the group.

I did not assign homework on the days that they worked on the projects, but I did not reduce their homework in any other way. I gave three exams and a comprehensive final as I usually do, and I was pleasantly surprised that the students did not feel overwhelmed. On their end-of-semester evaluations, only 5 of the 33 students who responded said that the course required much more effort than their other courses, while 24 said that the effort was somewhat more or about the same. A number of the students said that the projects helped them understand the concepts of the course. As one student put it, ``The group projects, although somewhat time-consuming, were a great learning experience for relating what we were learning in the book to real life problems.'' In retrospect, the projects were a very rewarding, and very manageable, addition to the course, for both the students and me.