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.

Preparation of this article was supported by a grant from the National Sciences and Engineering Research Council of Canada to the first author, a graduate scholarship from Alberta Ingenuity to the second author, a Dr. Jane Silvius Graduate Scholarship in Child Development to the third author, and a postdoctoral fellowship from the Social Sciences and Humanities Research Council of Canada to the fourth author. The authors are also grateful to Anna Matejko, who provided valuable advice and support, and to the editors and A. Baroody, who provided helpful feedback on an earlier draft.

Notes

¹Alternatively, success on such problems could reflect an altered order of calculation, a + b − c = a + (b − c) (Robinson et al., 2006).

²Caution is required when adopting hypotheses about inconsistency, however, because distinguishing inconsistency from measurement error (arising from guessing, for example) can be difficult.

³ Sherman and Bisanz (2007) originally wrote about induction based on counting, but here we broaden the hypothesis to include other forms of quantitative experience.