Investigating pedagogical content knowledge for teaching calculus at university level: a framework and analysis of four cases

Abstract

Within higher education mathematics, the focus on the teaching of calculus is continuing to be highlighted. Researchers have long sought to enhance the quality of calculus teaching at the university level. Even so, there is little clarity regarding the Pedagogical Content Knowledge (PCK) of calculus teachers in higher education. This study seeks to go some way to closing this identified gap by investigating the PCK of calculus teachers. This study proposes a model of PCK for calculus teaching and uses this model to identify how calculus teachers articulate and demonstrate their PCK to achieve their teaching goals, to deliver the building blocks to construct and enable their students’ mathematical understanding, to apply instructional strategies, and to utilise calculus connections with other academic subjects and wider applications. In order to understand the PCK of calculus teachers, this study is situated in higher education in Saudi Arabia. The sample group comprises calculus teachers of first-year university students. This study uses multiple cases and qualitative and quantitative data collected through a triangulated approach using survey, semistructured interview and observation of teaching. The analysis of the data employs a specially developed analytical framework for PCK for teaching calculus. Cross-case analysis identified, in detail, how these teachers articulate and demonstrate their PCK to develop learners' cognition of calculus; address the developmental aspects of the curriculum, apply instructional strategies to deliver their teaching aims and objectives, and to utilise calculus connections. This study's findings are steeped in fine detail and have appropriately addressed the research questions. It is significant in conceptualising, and analysing empirically, the PCK of calculus teachers. The findings identify that all the teachers showed their PCK in relation to how they taught calculus, it was also clear that not all aspects of PCK were equally evident among them. Some focused on specific instructional strategies to target learners' needs, others highlighted students' misconceptions about calculus in different ways. Knowledge of students' thinking about calculus concepts was narrow, while little effort about knowledge of calculus connections was identified. Although the teachers attempted to highlight real-world applications of calculus, identifying real-world connections that the students could understand was lacking. Significantly, calculus relating to other academic subjects was least identified. The findings pave the way for future developments of university calculus teaching and provides a model that can be developed and used widely within the field of calculus teaching in higher education. It is anticipated that this model will support the development of mathematics teaching in higher education in Saudi Arabia and elsewhere.

More information

Identifiers

Catalogue record

Export record