Teaching of division in Grade 3 and Grade 4: disruptions to coherence

Abstract

In South Africa, Reddy (2005) and Schollar (2004) have investigated learners’ work on division tasks. While these studies showed that learners struggle with the solving of division tasks at the primary school level, these studies did not investigate how sequences of signifiers were assembled by teachers when they work with sharing and grouping situations and bald calculations. This focus came to the centre in this study because of evidence of different degrees of coherence in the signifier sequences produced by teachers. The focus on coherence was coupled with attention to issues identified in earlier research in the field of primary school mathematics: a lack of progression in signifier modes of representation from more concrete to more abstract forms, limited progression in the sophistication of division calculation strategies and limited number range during the teaching and learning of division. Given previous research findings that teaching division is difficult, this study sought to investigate the nature of disruptions to coherence in the teaching of division and how division situations, signifiers, division calculation strategies and number range featured in teachers’ solving of division tasks. Three Grade 3 and three Grade 4 mathematics teachers participated in this study. A total of 22 lessons on the teaching of division were observed and video recorded. Grounded theory framed the analysis of these lessons and led to the development of a framework that combined a focus on sequences of signifiers and a look at progression in signifier use, division calculation strategies and number range, considered in relation to coherence. Theoretically, the contribution of the study lies in its conceptualization of the nature of coherence in the context of evidence from lesson episodes ranged along a continuum from coherent to incoherent within the signification pathways framework. A coherent categorization of instruction relating to a division task episode indicated a pathway involving links between signifiers that were mathematically justifiable at all points within the episode. Two thirds of the coded episodes in this study played out coherently. In "coherent with limitations" episodes (14/119 episodes), while teachers’ work with signifiers showed a coherent chaining across the pathways with new signifiers connecting with previous signifiers in the pathway and linking with the signified situations, some signifiers in the chain applied in localised ways to the episode, rather than in generalizable ways. In episodes coded as containing "some ambiguity" (20/119), signification pathways involved instances where signifiers linked to more than one signified in contradictory ways. In incoherent episodes (2/119), teachers work with signifiers showed mathematical errors. These findings point to relatively widespread occurrence of episodes in division teaching in the early grades in South Africa, with the research design also pointing to more limited range across division task types in the more disadvantaged schools. Bald calculations were used commonly in both grades, and featured more often than division-situation based episodes in the coherent category. Regardless, of the initial situation (sharing or grouping) or bald calculations, tasks were more frequently enacted with group-based distributions. This language of description has been useful for identifying and developing more coherent teaching