The Inverse Relation of Addition and Subtraction: A Knowledge Integration Perspective

A number of mathematical concepts and computational procedures are linked to the inverse relation of addition and subtraction on an abstract mathematical level. In this discussion article for the special issue on subtraction-related principles, we suggest that the mainstream of research on inversion is conducted from a Knowledge Dissociation Perspective in which researchers show that children often fail to see abstract relations in the domain. Implicit rationale of the studies is that seeing these inter-relations occurs naturally and that we have to find the cognitive processes that can explain exceptions from this rule. Based on a review of key findings from cognitive research on knowledge acquisition we argue that a Knowledge Integration Perspective would be more adequate in which children acquire different concepts and procedures in different situations. Only experts but not children who are new in a domain can see their abstract mathematical inter-relations. Thus, research should shift its focus from merely describing dissociations between children's concepts or procedures to looking for causal mechanisms and instructional approaches that help children to integrate their knowledge by seeing the underlying deep structures of mathematical principles and problems.