"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 (diSessa, 1988(diSessa, , 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.This case study applies core components of an epistemological perspective referred to as knowledge in pieces (diSessa, 1988(diSessa, , 1993 to answer an instance of the following research question: How can students coordinate their understandings of problem situations with those of external representations when learning to solve problems? 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

The Emergence of 3D Geometry From Children's (Teacher-Guided) Classification Tasks

“…Other learning science research on children's mathematical doings imposes theoretical concepts (e.g., particular patchworks of knowledge in pieces) to explain what is happening, for example, in a set of lessons on learning rectangular area (Izsák, 2005). Our approach is different, being concerned as it is with providing an ethnographically adequate description, which means that the "ethnographer must articulate the same hesitant and momentary contexts that the natives are displaying to each other and using to organize their concerted behavior" (McDermott, Gospodinoff, & Aron, 1978, p. 246), of a sorting task in a second-grade mathematics classroom in the course of which students are held accountable for their actions.…”

Commentary 2: knowledge integration in mathematics learning: the case of inversion

“…The degree of knowledge fragmentation decreased with learners' increasing experience with a domain (Clark, 2006;Straatemeier, van der Maas, & Jansen, 2008). Most of this research has been conducted in the domain of physics, but at least two empirical studies (Izsák, 2005;Wagner, 2006) demonstrated fragmented knowledge about mathematics as well.…”

The Construction of Causal Schemes: Learning Mechanisms at the Knowledge Level

“…KiP4: Learning on a small time-scale: Befitting a smaller grain size and a richer space of potential structural transformations, Knowledge in Pieces researchers skew toward microstudies, often focusing on describing learning at multiple stages, in particular contexts, and learning that takes place in episodes on the order of just a few minutes in duration (Izs ak, 2005;Parnafes, 2007;Wagner, 2006). Opposing views, partly because they usually have a more restricted space of theoretical possibilities marked out (e.g., "theories"), tend toward before and after studies of instructional treatments.…”