Students’ and experts’ schemes for rate of change and its representations

• We describe students’ and experts’ schemes for rate of change.
• We characterize the use of multiple representations of rate of change.
• We identify four emergent schemes that distinguish experts’ and students’ conceptions of rate of change.
• We describe the importance of schemes for function for understanding of rate of change.
• We propose specific meanings that promote coherence among representations of rate of change.

The purpose of this paper is to articulate students’ and mathematicians’ schemes for rate of change and describe how those schemes affected their conception of rate of change in graphical, tabular, and algebraic representations. We argue that experts’ schemes for rate of change allowed them to conceive of rate of change as similar across representations while students’ conceptions of rate of change depended on a single representation resulting in their inability to conceive of rate of change represented in multiple ways. We describe the difficulties that students had in conceiving the rate of change represented in multiple ways and contrast those with the coherence that mathematicians possessed. Lastly, we discuss the implications of a coherent scheme of meanings for the use of multiple representations. We propose that exposure to multiple representations requires identification of the meanings, which we intend the students to develop before considering various ways in which to represent that meaning.