Exploring mathematical activities and dialogue within a pre-service teachers’ calculus module: a case study.
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Local and international research findings have shown that high school learners, university students, as well as some of the practicing educators, struggle with calculus. The large numbers of unqualified or under-qualified mathematics educators are a major contributing factor to this problem. Many researchers agree on the fact that profound subject content knowledge is one of the contributing factors to effective teaching. Thus, this study seeks to explore what is counted as mathematics teaching and learning, what is counted as mathematics, as well as the nature of dialogue in a calculus lecture room. The Mathematics for Teaching framework and the Cognitive Processes framework informed this study, in order to explore what was counted as mathematics teaching and learning in the calculus lecture room. The Mathematical Activities framework and the Legitimising Appeals framework informed this study, in order to explore what was counted as mathematics in the calculus lecture room. The Inquiry Co-operation Model also informed this study, in order to explore the nature of dialogue within the calculus lecture room. The findings of this study showed that there are various mathematical activities that develop the students’ higher order thinking which is required for problem solving. These activities include mathematical activities that promote conjecturing, proving, investigations, the use of multiple representations, the use of symbols, the use of multiple techniques, as well as activities that promote procedural knowledge through conceptual understanding. These activities also keep the students’ cognitive demand at a high level. The findings of this study also showed that the types of questions that are asked by the lecturers have a positive impact on the development of the students’ high order thinking, as well as in terms of keeping the students’ cognitive demand at high levels. The study has also shown that the lecturers exhibited a variety of mathematics for teaching skills and this is done both explicitly and implicitly. It has also been revealed that introducing the rules of anti-differentiation as the reverse of differentiation is an alternative way to introducing the concepts of integral calculus. Based on these findings, it was recommended that students who enrol for the calculus module with low marks in mathematics, ought to use the derivative concept and the rules of differentiation as a foundation to build on the rules of anti-differentiation.