How representations support young students’ conceptual understanding of multiplication? A study on exploring third-grade students’ representational reasoning

One typical challenge in learning multiplication is that students apply multiplication algorithms without understanding why they work. To better enable students to develop a conceptual understanding of multiplication, the distributive property of multiplication over addition can be introduced, allowing students to comprehend and justify computational processes by emphasizing the underlying structure of multiplication algorithms. As a strategic contribution to understanding the distributive property, this study adopts representational reasoning, where mathematical concepts and principles are developed through engaging with and connecting various representations. Cuisenaire rods and arrays were employed to help students grasp the idea of the distributive property, and a qualitative analysis was conducted to investigate students’ representational reasoning. Results indicated that Cuisenaire rods and arrays were effective in helping students understand the decomposition of multiplication factors and in applying partial products to solve problems. However, students often modeled multiplication by interpreting it additively or struggled to connect the Cuisenaire rods and arrays with numerical equations. These findings suggest that while representational reasoning with Cuisenaire rods and arrays can enhance students’ conceptual understanding of multiplication, it does not necessarily ensure proficiency in transitioning between different representations. The study offers pedagogical implications for using representations to support students’ understanding of multiplication.