On “Understanding” Children's Developing Use of Inversion

Abstract

The principle of inversion, that a + b − b must equal a, is a fundamental property of conventional arithmetic. Exploring how children use and understand the principle of inversion can provide important insights about the development of mathematical thinking and about ways of optimizing instruction. Research on children's use and understanding of inversion has been focused primarily on whether they use inversion, with much less attention placed on what this understanding comprises and how it develops. To remedy this situation, we propose a framework in which understanding inversion is represented in terms of a matrix of possibilities. This framework is useful for highlighting the diverse ways in which children can show their understanding, for describing individual differences, for tracking changes in understanding, and for prompting investigations on the mechanisms that contribute to conceptual development.