Covariational reasoning and invariance among coordinate systems

• We investigate pre-service secondary teachers’ covariational reasoning.
• We characterize their covariational reasoning in multiple coordinate systems.
• Different images of change play a role in their activity.
• Covariational reasoning is important for representational activity.
• Polar coordinates offer a context for promoting covariational reasoning.

Researchers continue to emphasize the importance of covariational reasoning in the context of students’ function concept, particularly when graphing in the Cartesian coordinate system (CCS). In this article, we extend the body of literature on function by characterizing two pre-service teachers’ thinking during a teaching experiment focused on graphing in the polar coordinate system (PCS). We illustrate how the participants engaged in covariational reasoning to make sense of graphing in the PCS and make connections with graphing in the CCS. By foregrounding covariational relationships, the students came to understand graphs in different coordinate systems as representative of the same relationship despite differences in the perceptual shapes of these graphs. In synthesizing the students’ activity, we provide remarks on instructional approaches to graphing and how the PCS forms a potential context for promoting covariational reasoning.