There is a strong push from within mathematics education reform to incorporate representations in math classrooms (Behr, Harel, Post, & Lesh, 1993; Kieren, 1993; NCTM, 2000). However, questions regarding what representations should be used (for a given topic) and how representations should be used (such that students gain a deep understanding of that topic and a deep understanding of the representations) remain largely unanswered. Hence, we need a well-specified and general theoretical treatment of how students co-develop domain and representational competence.

In this dissertation study, I use design-based research (DBR) to investigate and support growth and change in students' knowledge of rational number operations. "Among all the topics in K-12 curriculum, rational numbers arguably hold the distinction of being the most protracted in terms of development, the most difficult to teach, the most mathematically complex, the most cognitively challenging, and the most essential to success in higher mathematics and science" (Lamon, 2007). In order to shed some light on the domain of rational number operations, I designed a learning environment centered on the Area Model for Fraction Multiplication (AM-FM) representation, a computer-based tool intended to help students develop a deep understanding of fraction multiplication.

Data for the dissertation were collected from an urban school with a racially and socio-economically diverse student population. I met with ten students once a week for four weeks. During the first and last session students were asked to think-aloud through a pretest and posttest. The second and third sessions consisted of semi-structured clinical interviews during which students were asked to solve fraction multiplication problems using the AM-FM representation. All sessions were videotaped and transcribed. Two students were chosen to serve as cases of knowledge growth and change.

Findings indicate that both students followed a particular learning trajectory for making sense of fraction multiplication when using the AM-FM representation and their emergent knowledge was context sensitive. Futhermore, DBR is predicated on (a) design refinement and (b) local theory development (diSessa & Cobb, 2004; Schoenfeld, 2006). With respect to design, the AM-FM representation and the clinical interview protocol was refined based on analysis of the data. With respect to local theory, I offered a decomposition of competence with fraction multiplication (i.e., domain competence) and the AM-FM representation (i.e., representational competence). Local theory was also refined based on an analysis of the data.