Browsing by Author "Likwambe, Botshiwe."

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A case study of the development of A.C.E. students' concept images of the derivative.
(2006) Likwambe, Botshiwe.; Christiansen, Iben Maj.
This research focuses on the development of the concept images of the derivative concept of students enrolled in the in-service programme 'Advanced Certificate in Education' at University of KwaZulu-Natal, Pietermaritzburg campus. In addition, two qualified teachers not enrolled in the programme were included. A theoretical framework which describes the derivative as having three layers - the ratio, limit and function layers - that can be represented by a variety of representations - graphical, rate, physical and symbolic - is used to analyse the development of the students' concept images. This framework was adopted from previous research, but expanded to allow for situations where a student's concept image did not fall into any of the layers or representations. In those cases, the concept image was classified into the non-layer section or the instrumental understanding section. The findings of this research show that of the five ACE students who were interviewed, only one had a profound concept image in all the three layers of the derivative, with multiple representations as. well as connections among representations within the layers. This one student also passed the calculus module with a distinction. The other four students had the ratio layer and graphical representation profound in their concept images, while the other layers and representations were pseudostructural with very few connections. Two of these students passed the calculus module while the other two failed. All the students showed progression in their concept images, which can only be credited to the ACE calculus module. However, it is clear that even upon completion of this module, many practicing teachers have concept images of the derivative which are not encompassing all the layers and more than one or two representations. With the function layer absent, it can be difficult to make sense of maximization and minimization tasks. With the limit layer absent or pseudo-structural , the concept Itself and the essence of calculus escapes the teachers - and therefore also will be out of reach of their learners.
Exploring mathematical activities and dialogue within a pre-service teachers’ calculus module: a case study.
(2018) Likwambe, Botshiwe.; Naidoo, Jaqueline Theresa.
Local and international research findings have shown that high school learners, university students, as well as some of the practicing educators, struggle with calculus. The large numbers of unqualified or under-qualified mathematics educators are a major contributing factor to this problem. Many researchers agree on the fact that profound subject content knowledge is one of the contributing factors to effective teaching. Thus, this study seeks to explore what is counted as mathematics teaching and learning, what is counted as mathematics, as well as the nature of dialogue in a calculus lecture room. The Mathematics for Teaching framework and the Cognitive Processes framework informed this study, in order to explore what was counted as mathematics teaching and learning in the calculus lecture room. The Mathematical Activities framework and the Legitimising Appeals framework informed this study, in order to explore what was counted as mathematics in the calculus lecture room. The Inquiry Co-operation Model also informed this study, in order to explore the nature of dialogue within the calculus lecture room. The findings of this study showed that there are various mathematical activities that develop the students’ higher order thinking which is required for problem solving. These activities include mathematical activities that promote conjecturing, proving, investigations, the use of multiple representations, the use of symbols, the use of multiple techniques, as well as activities that promote procedural knowledge through conceptual understanding. These activities also keep the students’ cognitive demand at a high level. The findings of this study also showed that the types of questions that are asked by the lecturers have a positive impact on the development of the students’ high order thinking, as well as in terms of keeping the students’ cognitive demand at high levels. The study has also shown that the lecturers exhibited a variety of mathematics for teaching skills and this is done both explicitly and implicitly. It has also been revealed that introducing the rules of anti-differentiation as the reverse of differentiation is an alternative way to introducing the concepts of integral calculus. Based on these findings, it was recommended that students who enrol for the calculus module with low marks in mathematics, ought to use the derivative concept and the rules of differentiation as a foundation to build on the rules of anti-differentiation.