Learners' participation in the functions discourse

Abstract

This study investigated learners’ mathematical discourse on the hyperbola using commognitive theory, with particular focus on the use of words, narratives, routines and visual mediators. Data was collected by means of task-based interviews with nine Grade 10 and 11 learners from a township school in Johannesburg, South Africa. An analytical tool, named the Discourse Profile of the Hyperbola, was adapted from the Arithmetic Discourse Profile of Ben-Yahuda et al (2005) and was used to analyse learners’ mathematical discourse. The study focused on three representations of the hyperbola, namely, the formulae (equation); the graph and the table. Learners’ views and definition of the asymptote, in relation to the graph, emerged as a central theme in the analysis. The analysis also focused on the mismatch between what is said and what is done by learners, for example most learners sketched the graph of a hyperbola showing a vertical asymptote yet talked as if there is no vertical asymptote. Most routines were ritualized, for example learners failed to link iconic and symbolic mediators they had used in responding to tasks. However, there were traces of exploratory routines from a few learners, evidenced by links between equations, and identifying the hyperbola from unfamiliar tasks. While a few learners used literate words, colloquial word use was dominant. The discourse of learners was found to be visual. For example, some reasoned that an equation with a fraction represents a hyperbola while an equation not expressed in standard form does not represent a hyperbola. Some learner narratives are not endorsed by the community of mathematicians.