Exploring the role of metonymy in mathematical understanding and reasoning: The concept of derivative as an example

In this paper we examine the roles that metonymy may play in student reasoning. To organize this discussion we use the lens of a structured derivative framework. The derivative framework consists of three layers of process–object pairs, one each for ratio, limit, and function. Each of the layers can then be illustrated in any appropriate context, for example graphically as slope, verbally as rate of change, or kinesthetically as velocity. We will illustrate three main cases of metonymy using data from student reasoning with derivative–paradigmatic metonymy, individual metonymy and part–part metonymy. Metonymies may function both in powerful and problematic ways for students as they come to understand and work with a complicated concept such as the derivative. It is the goal of this paper to illuminate and clarify how metonymy may function in student reasoning in the hopes of providing insights to teachers and researchers.