Teachers' Understanding of Ratios and Their Connections to Fractions.

In this study, we considered how middle school teachers understood the relationship between fractions and ratios. We used two instruments to collect data from 11 teachers and relied on Knowledge in Pieces as a lens for considering understandings teachers have and how coherent those understandings are. From our analysis, we developed three main findings: participants did not have a single definition for ratios; they used specific vocabulary when discussing ratios; and their language evoked additive strategies rather than multiplicative relationships. Further, we concluded that they each had a number of knowledge resources, but that those resources may not yet be wellconnected to each other. This has implications for professional development.

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