ABSTRACT
The domains of rational number and functions are foundational to many topics in advanced mathematics, and underpin the understanding necessary for participation in the pure and applied sciences (Lamon, 1999). Both domains are also known to be extremely difficult to master. Although many students do eventually learn to perform the basic operations and algorithms that the domains require, their conceptual knowledge remains remarkably weak, as does their ability to tackle novel problems (Carpenter, Fennema, & Romberg, 1993; Harel & Dubinsky, 1992). In this chapter, we discuss a program of research in which we have been trying to model the conceptual understanding that underpins novel performance in these two domains on the one hand, and to design improved curricular approaches for developing this sort of conceptual competence on the other.
