Calculus students’ fraction and measure schemes and implications for teaching rate of change functions conceptually

Abstract This paper investigates the implications of three students’ fraction and measure schemes for their understanding of rate of change functions. Steffe and Olive's (2010) descriptions of students’ reasoning about fraction and measure and Thompson's (1994) descriptions of students’ reasoning about rate of change serve as the theoretical foundation of the paper. Two of the students’ fraction and measure schemes obstructed their ability to make sense of rate of change functions. These two students assimilated with two levels of units and had additive fraction schemes. The student who assimilated with three levels of units and had multiplicative fraction schemes efficiently developed a productive meaning for rate of change functions. This study provides additional evidence from 2049 students showing difficulty with fraction and measure is typical for calculus students. Teachers implementing conceptual calculus instruction should consider how students who have only additive fraction schemes will make sense of rate of change functions.