Learner mathematical errors in introductory differential calculus tasks : a study of misconceptions in the senior school certificate examinations

D.Phil.The research problematised the learning of mathematics in South African high schools in a Pedagogical Content Knowledge context. The researcher established that while at best, teachers may command mathematics content knowledge, or pedagogic knowledge, that command proves insufficient in leveraging the learning of mathematics and differentiation. Teachers' awareness of their learners' errors and misconceptions on a mathematics topic is critical in developing appropriate pedagogical content knowledge. The researcher argues that the study of learner errors in mathematics affords educators critical knowledge of the learners' Zones of Proximal Development. The space where learners experience misconceptions as they attempt to assign meaning to new mathematical ideas to which they may or may not have obtained semiotic mediation. In their Zones of Proximal Development learners may harbour concept images that are incompetition with established mathematical knowledge.Educators need to study and understand those concept images (amateur or alternative conceptions), and how learners come to have them, if they are to help learners learn mathematics better. Besides the socio-cultural v1ew, the study presumed that the misconceptions formed by learners in mathematicsmay also beexplained within a constructivist perspective of learning. The constructivist perspective of learning assumes that learners interpret new knowledge on the basis of the knowledge they already have. However, some of the knowledge that learners construct though meaningful to them may be full of misconceptions. This may occur through overgeneralisation of prior knowledge to new situations. The researcher presumed that the ideas that learners have of particular mathematical concepts were concept images they construct. Though some of the concept images may be deficient or defective from a mathematics expert's point of view, they are still used by the learners to learn new mathematics concepts and to solve mathematics problems. The lack of success in mathematics that results in the application of erratic concept images ultimately leads to unsuccessful learning of mathematics with the danger of snowballing if there are no practicable interventions. Differentiation is a new topic in the South African mathematics curriculum and most teachers and learners have registered problems in teaching and learning it. Hence it was imperative to do research on this topic from an angle of learner errors on that topic. The significance of the study is that this research isolated the differentiation learner errors and misconceptions that teachers can focus on for the improvement of learning and achievement in the topic of introductory differentiation. The research focused on the nature of errors and misconceptions learners have on introductory differentiation as exhibited in their 2008 examination scripts. It sought to identify, categorise (form a database) and discuss the errors and their conceptual links. A typology of errors and misconceptions in introductory calculus was constructed. The study mainly used qualitative methods to collect and analyse data. Content analysis techniques were used to analyse the data on the basis of a conceptual framework of mathematics and calculus errors obtained from literature. One thousand Grade 12, Mathematics Paper 1 examination scripts from learners of both sexes emanating from diverse social backgrounds provided data for the study. The unit of analysis was students' errors in written responses to differentiation examination items