Mathematical Knowledge for Teaching Fraction Multiplication

TLDR

The study compares the mathematical knowledge of two sixth‑grade teachers with Connected Mathematics Project materials, focusing on tasks that use drawings to represent fractions as length or area. By analyzing teachers’ explanations and students’ reasoning across extended lesson sequences, the authors develop a theoretical framework linking teachers’ unit structures to their pedagogical use of drawings. The findings indicate that reasoning with three levels of quantitative units is necessary but not sufficient; teachers must also construct flexible three‑level unit structures supported by drawn distributive property representations to develop generalized numeric methods for fraction arithmetic.

Abstract

The present study contrasts mathematical knowledge that two sixth-grade teachers apparently used when teaching fraction multiplication with the Connected Mathematics Project materials. The analysis concentrated on those tasks from the materials that use drawings to represent fractions as length or area quantities. Examining the two teachers' explanations and responses to their students' reasoning over extended sequences of lessons led to a theoretical frame that emphasizes relationships between teachers' unit structures and pedagogical purposes for using drawings. In particular, the present study builds on the distinction made in past research between reasoning with two and with three levels of quantitative units and demonstrates that reasoning with three levels of units is necessary but insufficient if teachers are to use students' reasoning with units as the basis for constructing generalized numeric methods for fraction arithmetic. Teachers need also to assemble three-level unit structures with flexibility supported by drawn versions of the distributive property.