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. / Thesis (M.Ed.)-University of KwaZulu-Natal, Durban, 2010.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:ukzn/oai:http://researchspace.ukzn.ac.za:10413/10581 |
| Date | January 2010 |
| Creators | Majengwa, Calisto. |
| Contributors | DeVilliers, Michael. |
| Source Sets | South African National ETD Portal |
| Language | en_ZA |
| Detected Language | English |
| Type | Thesis |
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