Learners' conceptualization of the sine function with Sketchpad at grade 10 level.
This study investigated how Grade 10 learners conceptualise an introductory activity to the sine function with The Geometers' Sketchpad. In a study by Blackett and Tall (1991), the initial stages of learning the ideas of trigonometry, are described as fraught with difficulty, requiring the learner to relate pictures of triangles to numerical relationships, to cope with ratios such as sinA = opposite/hypotenuse. A computer approach might have the potential to change this by allowing the learner to manipulate the diagram and relate its dynamically changing state to the corresponding numerical concepts. The learner is thus free to focus on specific relationships, called the principle of selective construction, as stated by Blackett and Tall (1991). The use of this educational principle was put to test to analyse the understanding of Grade 10 learners' introduction to the sine function. Data was collected from a high school situated in a middle-class area of Reservoir Hills (KZN) by means of task-based interviews and questionnaires. Given a self-exploration opportunity within The Geometers' Sketchpad, the study investigated learners' understanding of the sine function only within the first quadrant: A) as a ratio of sides of a right-angled triangle B) as an increasing function C) as a function that increases from zero to one as the angle increases from 0° to 90°. D) as a relation between input and output values E) the similarity of triangles with the same angle as the basis for the constancy of trigonometric ratios. The use of Sketch pad as a tool in answering these questions, from A) to E), proved to be a successful and meaningful activity for the learners. From current research, it is well-known that learners do not easily accommodate or assimilate new ideas, and for meaningful learning to take place, learners ought to construct or reconstruct concepts for themselves. From a constructivist perspective the teacher cannot transmit knowledge ready-made and intact to the pupil. In the design of curriculum or learning materials it is fundamentally important to ascertain not only what intuitions learners bring to a learning context, but also how their interaction with specific learning experiences (for example, working with a computer), shapes or changes their conceptualisation. The new ideas that the learners' were exposed to on the computer regarding the sine function, also revealed some errors and misconceptions in their mathematics. Errors and misconceptions are seen as the natural result of children's efforts to construct their own knowledge, and according to Olivier (1989), these misconceptions are intelligent constructions based on correct or incomplete (but not wrong) previous knowledge. Olivier (1989), also argues that teachers should be able to predict what errors pupils will typically make; explain how and why children make these errors and help pupils to resolve such misconceptions. In the analysis of the learners' understanding, correct intuitions as well as misconceptions in their mathematics were exposed.