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Doctoral Degrees (Mathematics and Computer Science Education)

Permanent URI for this collectionhttps://hdl.handle.net/10413/7140

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Browsing Doctoral Degrees (Mathematics and Computer Science Education) by Subject "Calculus--Study and teaching."

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    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.
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    First-year engineering students' concept development of integral calculus at a South African university of technology.
    (2015) Ndlazi, Nokwethemba Jubilee.; Brijlall, Deonarain.
    This thesis reports on a study to explore the development of the concept of integration among the first year engineering students at a South African university of technology. The study focused on concept definitions that were evoked through symbolic as well as visualisation of integrals. It further explored various concept images evoked the techniques of integration. A framework combining the Action-Process-Object-Schema (APOS) and the Three-Worlds of Mathematics (TWM) theories was adopted as a tool to analyse students’ concept formation of an integral. This was a qualitative case study that consisted of two phases. Firstly, a pilot phase was introduced as Phase 1 of the study to uncover issues that could be probed more deeply when the study was rolled out to a larger group of students. The activity sheet was administered and interviews were conducted with seven students who were willing to participate in the study. Secondly, as Phase 2 of the study, the modified activity sheet was then administered to 22 first year students who also volunteered to be in the study. The intention was to provide comprehensive investigation of concept development of integral calculus. Students were also organised into focus groups in order to explore emerging mental constructions during the discussions among the students. The findings of the research indicated that students operated mainly at an action level of cognition for integral calculus. Their definition of an integral was restricted to the notion finding an integral with no association to the area below the graph of a function. Students mainly conceptualised an integral as an anti-derivative. With regard to techniques of integration, students relied on rules and algorithms without reflecting on objects and processes embedded within the rules. Cases of inadequate perquisite schemas for integral calculus such as basic algebra, inverse trigonometric functions and some aspects of differentiation were also noted. Although there were notable strengths in skills such as completing a square and resolving fraction into partial fractions, there was little understanding of the underlying concepts. This study contributed by presenting a genetic decomposition for integration that is premised on APOS and TWM theories. While the action level of APOS was dominant, the proceptual-symbolic was the main prevalent world of mathematics learning.