Preservice Mathematics Teachers’ Understanding of and Abilities to Differentiate Proportional Relationships from Nonproportional Relationships

This study investigated preservice middle school mathematics teachers’ (PSTs) understanding of proportional and nonproportional relationships and their abilities to differentiate these relationships from each other. The PSTs’ abilities to interpret and represent proportional and nonproportional situations and solution strategies were also investigated. Forty PSTs who attended a mathematics education course on fractions, ratios, and proportions participated in the study. The data included the PSTs’ written responses to four open-ended problems and semi-structured interviews conducted with six of the PSTs. The analysis of the data showed that the PSTs mostly attended to the simultaneous increases or decreases and constancy of the rate of change when determining relationships. The PSTs’ over attention to these specific features constrained their understanding of the proportional and nonproportional relationships. Therefore, they had difficulty differentiating proportional relationships from nonproportional relationships even after instruction on these relationships. Moreover, some PSTs still identified nonproportional relationships as proportional after determining correct additive relationships. In addition, the PSTs had difficulty representing and interpreting proportional and nonproportional relationships and relied on cross-multiplication and across-multiplication strategies when solving the given problems.