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

ABSTRACT

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 ofunits 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.

KEYWORDS:

• Multiplicative reasoning
• units coordination
• splitting operation for composite units
• distributive partitioning operation
• reciprocal reasoning

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Notes

¹ Advanced equal sharing problems include a type of equal sharing problems where the result of sharing activity generates a fractional quantity as a share like sharing three pizzas among five people.

² The derived conjectures were mostly about differences among the students at the same level of units coordination, one of which was to investigate students’ ability to coordinate two MUSs in the current paper.

³ The eight students also used their recursive partitioning operations as given prior to activity in other problem situations. The discussion of the details is beyond the scope of this paper.

⁴ All names are pseudonyms. A subscript indicates the group to which the student belongs.

⁵ Comments enclosed in parentheses () describe students’ nonverbal actions or interactions from the researcher’s perspective. Words enclosed in square brackets [] indicate dialogue that was not spoken but we have inferred from the context.

⁶ Note that the multiplicand comes before the multiplier in numerical expression for multiplication in Korea.