Is the derivative a function? If so, how do students talk about it?

Abstract

This study explores how calculus students talk about the derivative as a function based on the concepts of function at a point and function on an interval. Twelve Calculus I students took a survey and explained their solution process during interviews. The analysis of students’ discourse with the commognitive lens focusing on their use of words and visuals showed that most students did not appreciate the derivative at a point as a number and the derivative function as a function. Moreover, one of the common descriptions of the derivative as a tangent line implies that they considered the ‘derivative’ as a point-specific object but also a (linear) function defined on an interval. This description was closely related to their use of the word, ‘derivative’ for both ‘the derivative function’ and ‘the derivative at a point.’

Keywords:

• derivative
• function
• calculus