The role of abstraction in learning about rates of change

Fourteen Year 11 advanced mathematics students participated in individual teaching interviews designed to investigate how they learnt various rate of change concepts. The theoretical framework compared two models of abstraction: the empirical abstraction model of Mitchelmore and White and the nested RBC model of Hershkowitz, Schwarz, and Dreyfus. Examples of learning were found that fitted the nested RBC model, but none that fitted the empirical abstraction model. It was concluded that the nested RBC model is valuable for understanding student learning of the concepts of average and instantaneous rate of change, but that empirical abstraction is likely to be more valuable in understanding how students develop a global concept of rate of change earlier. In the past six years or so, two models of abstraction in mathematics have been advanced. One, the empirical abstraction model, assumes that students make abstractions as the result of recognising underlying similarities between superficially different contexts that are familiar to the student (Mitchelmore & White, 2004). The other, the so-called nested RBC model, assumes that students construct new abstractions by reorganising existing abstract concepts (Hershkowitz, Schwarz, & Dreyfus, 2001). To date, there hav