Algebraic Representations of a Splitting Task: Implications of Numerical Reasoning

Abstract

ABSTRACT This qualitative research study uses middle-grades students’ numerical reasoning to model their symbolic representations of the relationship between two multiplicatively related unknowns on an algebra task. Students in sixth grade through ninth grade participated in clinical interviews that assessed their numerical reasoning using the Number Sequences framework, their interpretation of the equal sign, and their algebraic reasoning in the context of writing equations. Analysis used the mental structures that define their numerical reasoning to explain their algebraic capabilities, and changes in their reasoning about the equal sign in different contexts. Findings show that splitting, one mental structure related to the students’ numerical reasoning, is related to students’ symbolic representations of multiplicatively related unknowns. Furthermore, students who had not constructed a splitting operation tended to use numerical examples in place of reasoning about unknowns, and to reason operationally about the equal sign to compensate for this limitation on an algebra task.