How Primary School Students' Arguments Develop: Taking Initial Steps in a Deductive Discourse

Abstract

A wealth of studies has shown that students are inclined to use example-based reasoning—empirical arguments—which are insufficient for proving universal statements. Conversely, formal mathematical proofs typically use definitions and axioms and take logical, deductive steps to reach generalised conclusions. While expecting young students to provide formal proofs might be inappropriate, some research has shown that moving students beyond using empirical arguments to prove universal statements is achievable. Nevertheless, it remains unclear how students arrive at particular arguments, and, in the absence of an explicitly expressed deductive proof, what might constitute smaller steps of learning and how young students can be supported to take these steps. A commognitive perspective is taken to examine these gaps in the research. The research focuses on Year 4 students’ discourses about odd and even numbers and their substantiations for claims about the sums of odd and even numbers. Four studies are presented, each of which uses elements of commognitive theory to look closely at students’ discourse. The first study identifies the ways in which students use examples. It finds that students’ example-use can be more nuanced than an empiric-generic dichotomy, and that discursive markers of example-use can be detected in students’ discourse that implicitly signal generality. The three subsequent studies look closely at the discourse of one group of four students, using commognitive tools to reveal the character of the students’ argumentation and examine their first steps in a deductive discourse. The commognitive stance had maintained that learners’ initiation in a new discourse would be socially motivated and process-oriented (ritualised) rather than intrinsically motivated and outcome-oriented (explorative). However, the studies in this thesis show that students were able to take explorative first steps in a new deductive discourse. This has implications for curriculum and task design and pedagogy, as well as for the commognitive perspective on learning mathematics. The findings further suggest that the mechanism of conflict among discourses, commonly suggested as motivation to adopt a new discourse, might be insufficient to prompt learning. Young children in this study required a mathematical reason – not just a social reason – to change their discursive ways.