Developing Conceptual Understanding of Measurement Division Through Multiple Representations: A Design-Based Research Study

Abstract Measurement division is an essential conceptual foundation for understanding fraction division, yet classroom instruction often emphasizes partitive division, providing students with limited opportunities to develop flexible reasoning about repeated grouping. This study investigated how a redesigned instructional sequence supported Grade 5 students’ conceptual understanding of measurement division through multiple representations and identified instructional design principles for teaching measurement division. The study reports the second iteration of an ongoing design-based research (DBR) project conducted in a Norwegian Grade 5 classroom. Building on findings from the first iteration, the instructional sequence was redesigned to establish measurement division as an explicit object of inquiry before introducing fraction division. Students engaged in exploratory activities using manipulatives, drawings, repeated addition, repeated subtraction, multiplication, and symbolic measurement division, supported by collaborative problem solving and whole-class mathematical discussions. The findings indicate that students developed greater representational flexibility, increasingly coordinated multiple representations of the same mathematical structure, transferred their reasoning across different contexts, and successfully extended measurement division from whole-number to fraction situations. Many students also demonstrated a conceptual understanding that division by a fraction may produce a quotient larger than the dividend, suggesting that the redesigned sequence challenged the common misconception that division always makes numbers smaller. The study contributes three design principles for teaching measurement division: establishing measurement division as an explicit object of inquiry before fraction division, coordinating multiple representations of the same mathematical structure, and preserving that structure while gradually varying contexts and extending the numerical domain from whole numbers to fractions. These findings contribute design knowledge for supporting students' conceptual understanding of measurement division and fraction division.