Comparing contrasting instructional approaches: a way for research to develop insights about backward transfer

Backward transfer, defined as the influence that learning about new concepts has on individuals' reasoning about previously learned concepts, is a relatively new topic in mathematics education. Our exploratory study compared the backward transfer produced in a mathematics enrichment programme that emphasised covariational reasoning to that produced in real algebra classrooms that did not emphasise a particular type of reasoning. Quadratic and linear functions were the new concepts being learned about and the concepts previously learned about, respectively. One theme that emerged from our comparisons was that quadratic instruction that emphasised covariational reasoning had less, but more consistent, backward transfer than quadratic instruction that did not have a strong emphasis on a particular type of reasoning. This and other findings led us to new insights about backward transfer, as well as tentative new ideas about backward transfer more generally, that can be tested in future studies..