An Evidence-Based Learning Progression for Multiplicative Thinking with Whole Numbers: Identifying Transition Barriers in Years 3 to 6

Abstract

Multiplicative thinking (MT) is a key foundation for fractions, proportional reasoning, and algebra, yet many students experience persistent barriers in transitioning from additive to MT. This study analysed responses from 253 Australian students in Years 3 to 6 to the Diagnostic Assessment of Multiplicative Thinking (DAMT), a purpose-designed assessment developed to elicit students’ reasoning across multiplication and division tasks involving equal groups, arrays, multiplicative comparison, and Cartesian product situations. Using an evidence-based approach that integrated Rasch partial credit modelling with examination of students’ coded responses and solution strategies, a five-stage learning progression was derived. The learning progression distinguished qualitatively different forms of reasoning, ranging from one-to-one correspondence to property-based MT. Findings identify critical transition points, particularly in division, multiplicative comparison, Cartesian product, and two-digit-by-two-digit multiplication. Students’ reasoning varied systematically across multiplicative situations, numerical complexity, and operation types, revealing persistent transition barriers. These findings have implications for interpreting assessment evidence and supporting the development of robust MT in primary classrooms.