Summary Discussion

Following the on-line student-to-student conversation was like watching students formulate mental rough drafts of their understanding and their misunderstanding. According to Graeber, "one needs to understand students' current knowledge if one wants to amend or extend what they know." [3] While admitting that the quantitative and qualitative analysis of this project produced results that were informative, of most interest and value to the instructors was the window that this activity provided into how students think about solutions to problems. We found that students spend an inordinate amount of time mucking around in the details of a problem. This is neither surprising nor bad. What was surprising to the instructors -- and paralyzing for the students -- was their inability to rise above the details of the problems. They also played elaborate word matching games, trying to conjure up definitions and examples of practice problems that contained key terms found in the stated problem. When student JN writes "…the graph will never go down. It is like the AIDS project. That it was the total number of cases," he is attempting to find meaning in the stated problem that asks students to think about total pollutants given information about the rate of pollution from a problem done much earlier in the term that asked students to think about the growth rate of AIDS given the cumulative total number of AIDS cases over a period of time. Making this link helped him to visualize the graph of pollutant totals but he was unable to make the creative leap from seeing the graph to understanding what it tells him about the situation. The most startling observation was the lack of confidence that students have in themselves and each other. In two of the three problems, there was an early, elegantly written solution that was totally ignored by the rest of the class. The author of the response, unsure of her own work, did not bother to weigh in when others in the class posed incorrect solutions. This lack of confidence, both in themselves and their classmates, made forming consensus an impossible task.

The on-line conversation was also a terrific preparatory exercise for the in-class discussions that followed. This was clearly an unanticipated benefit. Because they had been actively engaged in a collaborative effort, the students were prepared for the in-class conversations that followed the on-line activity. Most students, having suffered through the often confusing thoughts of their classmates, wanted clarification. Our supposition was that motivating them to want a clear resolution of the problems would enhance their understanding of the concepts. The result of their performances on conceptual exam questions (see Table 2) clearly indicates that this was not always the case.

Implementing a web-based student-to-student discussion is a very economic way to learn what students know and what they don't know about a particular mathematical idea. Because the forum is so public, most students feel pressured to think carefully before posting their ideas. For the instructor, watching the conversation unfold provides interesting moments of reflection about students' understandings and misunderstandings. Grading the student responses to the Calculus Conversation problem was very straight forward and took very little time. Finally, using web-based discussions in this manner makes capturing student work for study at a later date effortless.

If the enterprise we call the scholarship of teaching and learning is, at its core, about creating and studying strategies that provide insight to student learning, then I believe this project is on the right track, but with many miles to go. In this first attempt to provide students with an opportunity to deepen their understanding of calculus, the instructors were the ones whose understanding was enhanced. Following a year of intensive work on this problem, we are now poised to improve the Calculus Conversation activity, to frame better questions (Can the nature of a problem prompt students to be more practical or more conceptual? Do students get better at solving the problems? Do they get better at expressing their ideas mathematically? ) and to design a better study. In partial answer to Lee Shulman's question, this is a case of an attempt to improve students' conceptual understanding of fundamental ideas in calculus that resulted in improved teacher understanding of what students know and don't know about key concepts in calculus.