Young children’s development of multiplicative thinking: making the transition from models to symbols in learning multiplication and division

This thesis is a report of an investigation into young children’s learning of multiplication and division, more specifically their development of multiplicative thinking. An impetus for this study was a concern that many children have difficulty moving beyond the use of models to solve multiplicative problems and the implications for further learning when they are reliant on these and on additive thinking. It was a mixed method research study that fitted appropriately with a social constructivist perspective on learning. A design research (Hjalmarson & Lesh, 2008) methodology was utilised within a primary school classroom, the purpose of which was two-fold. First, to enable the researcher to observe and interact with students in order to gain insights into their thinking; and second to consider the learning experiences and instructional approaches that best support students’ transition from additive to multiplicative thinking. The study involved 26 Year 3 students (8 and 9 year olds) from two primary schools in suburban Melbourne, who were interviewed on four occasions using task-based clinical interviews, over an eight-month period. Within this time frame, 13 of these students (referred to as the Experimental group) were involved in two classroom interventions on multiplication and division, respectively. Data collection methods utilised during these interventions included semi-structured interviews, observations, field notes, artefacts, and teacher journal. A product of the in-depth analysis of individual student interview data was the development and formation of a Hypothetical Solution Strategy Framework for Whole Number Multiplicative Word Problems. This has the potential to assist teachers and researchers in identifying changes and/or patterns in student strategy choice over extended periods of time, in order to gauge students’ developing understanding of multiplication and division and transition from additive to multiplicative thinking. Five key findings pertaining to students’ strategy choice were evident from this study. 1.There was a relationship between strategy choice and level of task difficulty: the more difficult the task the more sophisticated the strategy choice; the less difficult the task, the less sophisticated the strategy choice. 2.Students solved problems relating to different semantic structures using abstracting strategies. 3.There was little difference in students’ approach to partition and quotition division tasks. 4.A preference for multiplication to solve division problems. 5.Students’ recording of a division equation matched the problem structure. The first two findings suggest that the engagement of students in tasks that involve numbers outside their range of experiences, across the different semantic structures, supports students’ transition from additive to multiplicative thinking. Findings three to five suggests that rather than emphasising the forms of division, teachers need to engage students in a range of problem types and ensure that the model they use matches the questions being asked. Findings from this study make a significant contribution to the current body of literature pertaining to young children’s development of multiplicative thinking and the instructional practices and learning experiences that best support this development.