This study addressed the inherent quandary of misconceptions that impacts on performance as reported in the National Diagnostic Report of the learner performance in the National Senior Certificate (NSC) 2011 examinations. In order to gain insights into learner performance, Sfard’s (2008) commognitive framework was used as a theoretical lens to examine learners’ mathematical thinking about functions. This study first described components of function concept from a constructivist perspective and further redescribed these in discursive terms i.e. from a focus on learners’ use of terminology to words/word use; from representations to visual mediators; from competencies to routines; and from concept definition to endorsed narratives. Data was collected through written tests and interviews of Grade 11 learners in one of the multilingual schools in one Province in South Africa. The research approach was first quantitative .Twenty six learners were given tasks involving functions that would highlight their errors. The study then moved to an interpretive qualitative approach based on Sfard’s commognitive theory. The qualitative study had five participants. A multiple methods strategy of data collection was employed during this stage: learners’ interview transcripts, written work and researcher’s field notes. The quantitative study confirms that learners were making errors on functions. The analysis of the qualitative study revealed that learners’ discourse included a combination of colloquial and mathematical discourse as expected. Interestingly, while all the features of mathematical discourse were present in learners’ mathematical discourse, their routines and words were linked to errors.

Piaget and Freudenthal advocated thinking levels. In the 1950's the van Hieles developed a five level model of geometric thought. Judith Land adapted the model in 1990, utilising four levels to teach the concept of functions. These four levels have been considered here in the formation of concepts of sequences and series. The origin and relevance of sequences and series have been studied and the importance of visualisation, patterning and generalisation in the instructional process investigated. A series of lessons on these topics was taught to a group of six higher grade matriculation students of mixed ability and gender. Questionnaires related to student progress through the various levels were answered, categorised, graphed and analysed. Despite the small number of students, results seem to indicate that emphasising visualisation, exploring patterns and generalisation and teaching the topics as a reinvention had made a positive contribution towards progress through the various thought levels. / Mathematics Education / M.A. (Mathematics Education)

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