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

Abstract

This article extends and strengthens the knowledge in pieces perspective 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. This case study applies core components of an epistemological perspective referred to as knowledge in pieces DiSessa developed the knowledge in pieces perspective to explain emerging expertise in Newtonian mechanics. The perspective holds that knowledge elements are more diverse and smaller in grain size than those presented in