The development of arithmetic principle knowledge: How do we know what learners know?

This paper reviews research on learners' knowledge of three arithmetic principles: Commutativity, Relation to Operands, and Inversion. Studies of arithmetic principle knowledge vary along several dimensions, including the age of the participants, the context in which the arithmetic is presented, and most importantly, the type of knowledge assessment (e.g., application of procedures, evaluation of examples). The vast majority of studies utilize single-faceted knowledge assessments, which can lead to incomplete or misleading views of learners' knowledge. Both context and type of knowledge assessment can influence conclusions about learners' arithmetic principle knowledge. However, relatively few studies directly address the possible effects of context or type of knowledge assessment on their results. To move the field forward, research that utilizes multifaceted knowledge assessments is needed. (C) 2009 Elsevier Inc. All rights reserved.