The conceptual overlap between arithmetic and algebraic referential mapping

Abstract

To investigate whether arithmetic and algebra referential mapping entail similar or different representational processes, 135 13-year-olds participated in a three-phase study. In Phase 1 (the pre-algebra phase) students solved arithmetic and algebraic equations and mapped arithmetic equations (e.g., 5 + = 9) onto simple arithmetic word problems. Nine months later (Phase 2), with the assistance of problem-solving hints provided by an interviewer, students mapped symbolic equations onto algebraic word problems. After a further nine months (Phase 3), the students solved equations and completed an algebraic word problem solving test. The results showed that algebraic equation problem-solving and the hints required for mapping symbolic equations onto word problems (Phase 2) both predicted subsequent unassisted algebraic word problem equation mapping (Phase 3) and retrospectively predicted arithmetic word problem mapping abilities (Phase 1). The findings highlight the ways in which representational ability contributes to algebraic reasoning and suggest that arithmetic representational processes may scaffold algebraic referential mapping. The pedagogical implications of the association between representational abilities in the two domains are discussed in terms of approaching arithmetic word problem-solving from a more representational perspective.