A microgenetic study of the multiplication and division inversion concept.

This microgenetic study investigated the discovery and development of the multiplication and division concept of inversion. Little is known about multiplicative concepts relative to additive concepts, including the inversion concept. Grade 6 participants (mean age = 11 years 6 months) solved multiplication and division inversion problems (e.g., d x e/e) for several weeks. In the final week they solved inversion, modified inversion (e.g., e x d/e), and lure problems (e.g., d/e x d) to investigate transfer of knowledge. Despite years of formal arithmetic instruction and repeated exposure to inversion problems, over a third of the participants failed to discover the inversion-based shortcut whereas another third used the shortcut almost exclusively. Almost all participants had difficulty appropriately generalising the inversion concept. Current theories of mathematical understanding may need to be modified to include the developmental complexities of multiplicative concepts.