Students’ understanding of the associative property and its applications: noticing, doubling and halving, and place value

Understanding the properties of multiplication is a critical precursor to students’ thinking algebraically. However, these properties have not been the focus of extensive rigorous research, particularly the associative property. In this study, we report on follow-up interviews with 25 year 5–6 students who had completed five items taken from an assessment of mental computational fluency. This assessment required students to reason from the perspective of a fictional student (Emma), who had applied the associative property in various ways to solve multiplication problems. In the interviews, students had to explain their understanding of Emma’s thinking. Coding of this interview data revealed distinct continuums of understanding of each of the three applications of the associative property (noticing, doubling and halving, and place value), which teachers could use to inform their planning. The findings reveal that students best understood the noticing application, closely followed by doubling and halving. By contrast, the place value application was not well understood, which we attribute to students relying on truncation procedures applied with little or no conceptual understanding.