An investigation of grade 11 learners' understanding of the cosine function with Sketchpad.

Abstract

This study investigated how Grade 11 learners from a school in KwaNdengezi, near Pinetown, in Durban, understood the cosine function with software known as The Geometer’s Sketchpad. This was done on the basis of what they had learnt in Grade 10. The timing was just before they had covered the topic again in their current grade. The researcher hoped, by using The Geometer’s Sketchpad, to contribute in some small way to teaching and learning methods that are applicable to the subject. This may also, hopefully, assist and motivate both teachers and learners to attempt to recreate similar learning experiences in their schools with the same or similar content and concepts appropriate to them. In this research project, data came from learners through task-based interviews and questionnaires. The school was chosen because of the uniqueness of activities in most African schools and because it was easily accessible. Most learners do not have access to computers both in school and at home. This somehow alienates them from modern learning trends. They also, in many occasions, find it difficult to grasp the knowledge they receive in class since the medium of instruction is English, a second language to them. Another reason is the nature of the teaching and learning process that prevails in such schools. The Primary Education Upgrading Programme, according to Taylor and Vinjevold (1999), found out that African learners would mostly listen to their teacher through-out the lesson. Predominantly, the classroom interaction pattern consists of oral input by teachers where learners occasionally chant in response. This shows that questions are asked to check on their attentiveness and that tasks are oriented towards information acquisition rather than higher cognitive skills. They tend to resort to memorisation. Despite the fact that trigonometry is one of the topics learners find most challenging, it is nonetheless very important as it has a lot of applications. The technique of triangulation, which is used in astronomy to measure the distance to nearby stars, is one of the most important ones. In geography, distances between landmarks are measured using trigonometry. It is also used in satellite navigation systems. Trigonometry has proved to be valuable to global positioning systems. Besides astronomy, financial markets analysis, electronics, probability theory, and medical imaging (CAT scans and ultrasound), are other fields which make use of trigonometry. A study by Blackett and Tall (1991), states that when trigonometry is introduced, most learners find it difficult to make head or tail out of it. Typically, in trigonometry, pictures of triangles are aligned to numerical relationships. Learners are expected to understand ratios such as Cos A= adjacent/hypotenuse. A dynamic approach might have the potential to change this as it allows the learner to manipulate the diagram and see how its changing state is related to the corresponding numerical concepts. The learner is thus free to focus on relationships that are of prime importance, called the principle of selective construction (Blackett & Tall, 1991). It was along this thought pattern that the study was carried-out. Given a self-exploration opportunity within The Geometers' Sketchpad, the study investigated learners' understanding of the cosine function from their Grade 10 work in all four quadrants to check on: * What understanding did learners develop of the Cosine function as a function of an angle in Grade 10? * What intuitions and misconceptions did learners acquire in Grade 10? * Do learners display a greater understanding of the Cosine function when using Sketchpad? In particular, * As a ratio of sides of a right-angled triangle?

* As a functional relationship between input and output values and as depicted in graphs? The use of Sketchpad was not only a successful and useful activity for learners but also proved to be an appropriate tool for answering the above questions. It also served as a learning tool besides being time-saving in time-consuming activities like sketching graphs. At the end, there was great improvement in terms of marks in the final test as compared to the initial one which was the control yard stick. However, most importantly, the use of a computer in this research revealed some errors and misconceptions in learners’ mathematics. The learners had anticipated the ratios of sides to change when the radius of the unit circle did but they discovered otherwise. In any case, errors and misconceptions are can be understood as a spontaneous result of learner's efforts to come up with their own knowledge. 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 the errors learners would typically make. They should explain how and why learners make these errors and help learners to correct such misconceptions. In the analysis of the learners' understanding, correct understandings, as well as misconceptions in their mathematics were exposed. There also arose some cognitive conflicts that helped learners to reconstruct their conceptions.