A Closer Look at Teachers' Proportional Reasoning

Teachers' mathematical knowledge has important consequences for the quality of the learning environment they create for their students to learn mathematics. Yet relatively little is known about how teachers reason proportionally, despite the fact that proportional reasoning is foundational for several mathematics concepts and that ratios and proportional relationships constitute a major component of the middle school mathematics curriculum. In this study, we investigated how teachers reasoned proportionally on a nonroutine ratio task and the extent to which their proportional reasoning was able to predict their overall understanding of the relevant concepts: ratios and proportional relationships. Using data collected from 238 US mathematics teachers, we found that teachers' proportional reasoning could be grouped into four categories: incorrect, additive, relative, and proportional reasoning. Our results also indicated that teachers' overall knowledge of ratios and proportional relationships aligned with the way they reasoned proportionally, meaning that teachers who used incorrect reasoning on a separate task received the lowest scores on average on the ratios and proportional relationships measure, whereas those who reasoned proportionally had the highest mean scores on average. Implications of the study include the need to shift attention to the way teachers reason in relation to the two elements of proportional reasoning (covariance and invariance) to capture the nuances in their understanding of ratios and proportional relationships.