Two Distinct Perspectives on Ratios: Additive and Multiplicative Relationships between Quantities

Abstract

The purpose of this study was to investigate how pre-service teachers’ formation of additive
and multiplicative relationships support and constrain their understandings of ratios and proportional
relationships in terms of quantities. Six pre-service teachers were selected purposefully based on their
performances in a previous course. An explanatory case study with multiple cases was used to make
comparisons within and across cases. A semi-structured interview was conducted with pairs of pre-service
teachers. The results revealed that pre-service teachers’ heavy reliance on additive or multiplicative
relationships critically shaped their reasoning about ratios from the two perspectives. Pre-service teachers
who only attended to multiplicative relationships were found to have a robust understanding of
proportional relationships. Pre-service teachers, who did attend to additive relationships, even if they used
multiplicative relationships, struggled to form appropriate proportional relationships from the two
perspectives.