Partitioning: The Emergence of Rational Number Ideas in Young Children

Abstract

The study sought to trace the emergence and differentiation of partitioning as a process that leads young children to construct ideas of rational numbers. The method involved clinical sessions with children as they manipulated materials while solving tasks designed to reveal partitioning processes. The data analysis led to a theory concerning the emergence of partitioning capability, and conceptual structures that undergird the capability were identified. A five-level theory of the development of the partitioning process is presented. Key position papers by Kieren (1976, 1980) have created a new theoretical context for inquiring into the child's acquisition of rational number concepts. Recent work within the new perspective (Kieren & Nelson, 1978; Kieren & Southwell, 1979) has lent support to Kieren's position. Basic in Kieren's perspective is the process of dividing a whole into parts, a process variously referred to as partitioning, subdivision, or dissection. The purpose of the present study was to trace the emergence and differentiation of the process of partitioning as revealed in children's attempts to subdivide a continuous whole into equal parts. METHOD The method involved a clinical interaction with a child as he or she grappled with tasks specifically designed to reveal partitioning capabilities. As the child grappled with the tasks, the researchers grappled with the even more complex task of attempting to make sense of the child's behavior. We hoped to discover underlying patterns that would help us trace the emergence and differentiation of the partitioning process. The method can be characterized as a clinical interaction technique set within a discovery paradigm. The interactive encounters were designed to yield insights rather than to verify hypotheses; validity, as opposed to generalizability, was our concern. The interaction was characterized by flexibility in questioning. The initial question for each task was standard, but the subsequent questions, although following a general pattern, were varied, as were the numbers in the problem, depending on the behavior of the child. The method permitted probing,