A power meaning of multiplication: Three eighth graders’ solutions of Cartesian product problems

• Identifies three multiplicative concepts for a power meaning of multiplication.
• Compares multiplicative concepts for linear and power meanings of multiplication.
• Identifies schemes for students’ solution of Cartesian product problems.

This article examines data from five teaching episodes with three eighth grade students who were participants in a 3-year constructivist teaching experiment. The five teaching episodes were a transition point in the teaching experiment: the students were beginning to work in contexts that were deemed to support their development of a meaning for squaring quantities—a power meaning of multiplication. Prior to these teaching episodes, the students had worked in contexts that were deemed to support their development of a linear meaning of multiplication.This paper focuses on the novel cognitive operations and multiplicative concepts that the students developed to solve Cartesian product problems, problems that were deemed could support students to establish a power meaning of multiplication. The findings from the study contribute to prior research by (1) examining an appropriate use for Cartesian product problems with middle grades students, and (2) identifying similarities and differences in the multiplicative concepts students constructed to solve linear-meaning multiplication problems and power-meaning multiplication problems. Implications for teaching are considered.