How do teachers of mathematics introduce differential calculus?

This thesis reports on the findings of a qualitative study investigating how teachers of mathematics in English schools introduce differentiation in elementary calculus. Understanding how teachers introduce the derivative is crucial to uncovering more meaningful and effective ways for helping students understand differential calculus. The study adopts the commognitive framework and investigates the teaching of the derivative by examining the word use and narratives, the visual mediators and the routines in the teachers’ pedagogical calculus discourse. Interviews with the teachers and observation of their introductory lessons on the derivative were used to collect qualitative data for the study. For the analysis of the qualitative data, the study introduces a new approach that is described as a commognitive thematic discourse analysis, which is a combination of the commognitive framework and thematic analysis. The commognitive thematic discourse analysis was then used to deconstruct the teachers’ pedagogical calculus discourse on the derivative and to identify the overarching themes in the research data, which are presented as a narrative of the findings. The study found that teachers used multiple visual mediators such as numerical, algebraic and graphical representations in constructing the definition of the derivative. Using dynamic geometry software such as GeoGebra and Autograph enhanced the teachers’ construction and substantiation routines in teaching the definition of the derivative. The study also found that teachers were able to construct the definition of the derivative without using the formal definition of limits; instead, the teachers took what is described as a ‘quasi-limit’ approach. The study also uncovered some inconsistency and ambiguity with word use and calculus symbolism in the teachers’ pedagogical calculus discourse in the transition between gradient (for straight line graphs) and the gradient function (for curved line graphs). This study is aimed at contributing to research that seeks to understand and improve the teaching of elementary calculus.