Rate of change involving two covarying quantities: Grades 9–12 students' understanding and difficulties

This study explored high school students’ understanding of rate of change in physical and functional situations involving two covarying quantities, and the nature of difficulties the students exhibited as they reasoned, represented, and made connections between various representations of rate of change in two covarying situations. Ninety-nine students from Grades 9–12 enrolled in Algebra I through AP-Calculus courses in a public high school in the United States, took part in the study. The participants completed a researcher-developed Diagnostic Test on Rate of Change (DTRC) that assessed their understanding of various representations of rate of change involving two covarying quantities in physical and functional situations. A follow-up semi-structured interview was conducted with 16 of the participants to gain more insights into the students’ conceptual understanding and nature of difficulties they have with the topic. Overall, the results indicated that the participants’ understanding of rate of change was weak, did not vary significantly across grade levels, and did not build consistently in a manner that could enable them to develop a solid mathematical structure of functions by the end of Precalculus. Implications of the results for instruction and limitations of the study are provided.