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Abstract
The purpose of this study was to investigate how preservice teachers understandings of multiplication and division support and constrain their understandings of ratios and proportional relationships in terms of quantities. Six preservice 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. Two semi-structured interviews were conducted with each pair. The results revealed that preservice teachers understandings of partitive and quotitive division, and reliance on only multiplicative relationships shaped in critical ways how they reasoned about ratios. Preservice teachers who did not explicitly identify different meanings for division were found to have difficulty maintaining one of two distinct perspectives on ratio. The results also showed that additive relationships such as repeated addition and the phrase for every might have constrained teachers from keeping the two perspectives distinct.