"You Have to Count the Squares": Applying Knowledge in Pieces to Learning Rectangular Area

This article extends and strengthens the knowledge in pieces perspective (diSessa, 1988, 1993) by applying core components to analyze how 5th-grade students with computational knowledge of whole-number multiplication and connections between multiplication and discrete arrays constructed understandings of area and ways of using representations to solve area problems. The results complement past research by demonstrating that important components of the knowledge in pieces perspective are not tied to physics, more advanced mathematics, or the learning of older students. Furthermore, the study elaborates the perspective in a particular context by proposing knowledge for selecting attributes, using representations, and evaluating representations as analytic categories useful for highlighting some coordination and refinement processes that can arise when students learn to use external representations to solve problems. The results suggest, among other things, that explicitly identifying similarities and differences between students' past experiences using representations to solve problems and demands of new tasks can be central to successful instructional design.