Reasoning with Two Types of Multiplicative Units Structures in Solving Middle-Grade Mathematics Problems

In the present study, we illuminate students' multiplicative reasoning in the context of their units-coordinating activity. Of particular interest is to investigate students' use of three levels of units as given material for problem-solving activity, which we regard as supporting a more advanced level of multiplicative reasoning. Among 13 middle school students with whom we have conducted clinical interviews, this study focuses on eight students who conceived fractions with three levels of units and reports their units-coordinating activities in solving diverse middle-grade level problems. The result of the study indicates that the ability to coordinate two multiplicative units structures was an evident characteristic by which we classified them into two distinctive groups. Specifically, two students outperformed the other students in solving advanced equal sharing and bar transformation problems by the coordination of two explicitly multiplicative units structures and in representing a multiplicative relationship between two unknown quantities by the coordination of two implicitly multiplicative units structures. The findings suggest that the ability to use the coordinated result of two multiplicative units structures needs to be attended to as a sophisticated form of multiplicative reasoning, which we believe potentially undergirds students' advanced mathematical learning beyond elementary school.