UC San Diego

The use of prior knowledge has been a much-researched topic. For example, it has also been well established that prior knowledge can act as a foundation on which new knowledge is constructed (Smith, diSessa, & Rochelle, 1993). Other studies have looked at the forgetting of prior knowledge (Ebbinghaus, 1885) and at the inhibition of retrieval of prior knowledge (e.g., Estes, 1987).What has remained understudied in mathematics education research is how prior knowledge changes as new knowledge is built upon it. However, a limited amount of transdisciplinary research (particularly in linguistics) has examined changing prior knowledge as a case of backward transfer (i.e., as the generalization of recently -constructed knowledge "backwards" onto longer-held knowledge). This study extends that work by examining backward transfer in a mathematics classroom environment. Specifically, this study examines how previously-held linear functions knowledge changes as middle school students construct beginning conceptions of quadratics functions. Using a design-based instructional intervention on quadratic functions, this study aimed to create backward transfer that productively influenced students' linear functions knowledge. Qualitative analysis of pre- and post-interviews and the intervention, using a mixed methods approach (i.e., a priori and inductive codes; Miles & Huberman, 1994), revealed three major findings. First, productive backward transfer was produced. In particular, evidence was found that students' understanding of linearity was deepened as a result of the quadratics instructional intervention (e.g., students reasoned proportionally with the changes in the independent and dependent variable). Second, evidence was found supporting the claim that backward transfer is, in fact, a particular kind of transfer (i.e., three relationships between backward and forward transfer were found). Third, the process of noticing was shown to provide explanatory power for the occurrence of backward transfer. The potential significance of this study is that it (a) addresses the relationship between prior knowledge and new knowledge from a new and potentially informative angle; (b) contributes to the emergence and theoretical conceptualization of alternative transfer perspectives; and (c) provides general principles by which to address backward transfer and informs the teaching of linearity and quadratics within Algebra 1 courses