ABSTRACT
This chapter looks at the role of schemas in assessing rational number knowledge. Particular attention is given to the importance of prior knowledge, the systematic development of coherent schemas, and the necessity of accurate communication about rational numbers. Five situations are described that may serve as the basis for a core set of schemas about rational numbers: part-whole, quotient, measure, ratio, and operator. The chapter briefly addresses each of these and outlines the essential types of knowledge that are associated with them. Examples demonstrate how knowledge of rational number may be assessed by focusing on schema structure.
Schemas have several characteristics that have important implications for assessment. These characteristics force a focus on qualitatively different features of problem solving and understanding. The assessment of schemas requires not so much the posing of different tasks as the asking of different questions. This chapter examines the schema approach to assessment, which focuses on a theory-driven feature that allows the exposure of a multilevel conceptual knowledge of mathematical concepts, or lack thereof. Assessment using a schema structure exposes greater mathematical understanding. The assessment possibilities are more precise, can be corrected more accurately if there is confusion, and more closely identify the students’ overall understanding of a problem. This structure, a derivative of the schema theory, provides the framework against which to compare individual performance.