The development of distributive partitioning operations

Abstract

Distributive partitioning operations are used in solving tasks such as sharing three items among five people. Data from six ninth grade students who participated in a year-long teaching experiment is used to elaborate the constraints and affordances of three distinct distributive partitioning schemes — a distributive sharing scheme, a distributive partitioning scheme, and a reversible distributive partitioning scheme. Variation in the students’ fractional operations and their abilities to reason with multiple levels of units structures are used to explain differences amongst these three schemes. While reasoning with two levels of units is sufficient for constructing a distributive sharing scheme, three levels of units are necessary for constructing both the distributive partitioning scheme and the reversible distributive partitioning scheme. Further, while part-whole comparisons are sufficient to construct a distributive sharing scheme and a distributive partitioning scheme, constructing a reversible distributive partitioning scheme requires the fractional awareness derived from an iterative fraction scheme.