12 - Teaching for Understanding: A Case of Students' Learning to Use the Uniqueness Theorem as a Tool in Differential Equations

Summary

Students in many undergraduate mathematics courses tend not to readily and appropriately use theorems as tools for making arguments and solving problems (Schoenfeld, 1989; Hazzan & Leron, 1996). Students' reluctance to use theorems as tools is a problem that is not only cognitive in nature (that is, the difficulty is in how students conceptualize particular mathematical ideas), but also social in nature (that is, the nature of class discussion, the interpretation of tasks and ideas, etc.). In this chapter we highlight results from a classroom-based research program in differential equations that has resulted in some positive progress on the problem of students' reluctance to use theorems as tools for reasoning and solving problems.

The main result of the analysis of student learning and use of the uniqueness theorem for first order differential equations is the delineation of four interrelated cognitive and social factors that help account for why students actually made progress in using the uniqueness theorem as a tool for making arguments and solving problems (Rasmussen, 2004). The intention is that readers might, after understanding the details of this particular case, find the four factors useful more generally as an orienting framework for thinking about ways in which they can promote their students' use of theorems as tools for reasoning in other content areas. Thus, even if one does not regularly teach differential equations, this chapter intends to offer useful information for those who want their students to develop and use formal mathematics with understanding.