Doctoral Degrees (Mathematics and Computer Science Education)

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

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An APOS exploration of conceptual understanding of the chain rule in calculus by first year engineering students.
(2011) Jojo, Zingiswa Mybert Monica.; Brijlall, Deonarain.; Maharaj, Aneshkumar.
The main issue in this study is how students conceptualise mathematical learning in the context of calculus with specific reference to the chain rule. The study focuses on how students use the chain rule in finding derivatives of composite functions (including trigonometric ones). The study was based on the APOS (Action-Process-Objects-Schema) approach in exploring conceptual understanding displayed by first year University of Technology students in learning the chain rule in calculus. The study consisted of two phases, both using a qualitative approach. Phase 1 was the pilot study which involved collection of data via questionnaires which were administered to 23 previous semester students of known ability, willing to participate in the study. The questionnaire was then administered to 30 volunteering first year students in Phase 2. A structured way to describe an individual student's understanding of the chain rule was developed and applied to analyzing the evolution of that understanding for each of the 30 first year students. Various methods of data collection were used namely: (1) classroom observations, (2) open-ended questionnaire, (3) semi-structured and unstructured interviews, (4) video-recordings, and (5) written class work, tests and exercises. The research done indicates that it is essential for instructional design to accommodate multiple ways of function representation to enable students to make connections and have a deeper understanding of the concept of the chain rule. Learning activities should include tasks that demand all three techniques, Straight form technique, Link form technique and Leibniz form technique, to cater for the variation in learner preferences. It is believed that the APOS paradigm using selected activities brought the students to the point of being better able to understand the chain rule and informed the teaching strategies for this concept. In this way, it is believed that this conceptualization will enable the formulation of schema of the chain rule which can be applied to a wider range of contexts in calculus. There is a need to establish a conceptual basis that allows construction of a schema of the chain rule. The understanding of the concept with skills can then be augmented by instructional design based on the modified genetic decomposition. This will then subject students to a better understanding of the chain rule and hence more of calculus and its applications.
Exploring university students' mental constructions of the limit concept in relation to sequences and series.
(2019) Chagwiza, Conilius Jaison.; Maharaj, Aneshkumar.; Brijlall, Deonarain.
The present thesis refers to some first semester calculus 1 university students’ mental constructions of the limit concept in relation to sequences and series. A plethora of research on the limit concept is available and suggests that the concept is on record of being difficult for students to learn and comprehend. However, in Zimbabwe, there is inadequate research on mental constructions made by students of the limit concept in relation to sequences and series. This research aims at filling this gap in the literature. This study utilized the Action-Process-Object-Schema (APOS) theory in exploring conceptual appreciative displayed by students when dealing with limits of sequences and series. The study proposes the genetic decompositions on how students might construct the mental constructions in learning the sequences and series through the use of Activities-Classroom discussions –Exercises (ACE). Collection of data was done by the use of a methodology that used practical teaching. All the thirty students who took calculus 1 accepted to participate in this study and answered the limit test questions. The students’ written responses were analyzed using APOS theory. Ten students were selected for interviews through purposive sampling. Two declined to take part leaving eight to take part in the process. The APOS theory was used to analyze the interview results. The revision of preliminary genetic decomposition was done basing on the analyzed data. The instructional method employed, facilitated the appreciation of the limit concept in relation to sequences and series by the students. Nearly all students showed that they operated at the Action level, a good number showed that they operated at least at the Process level and more than half of the students showed that they operated at the Object level. Three out eight interviewed students indicated that the managed to operate at the Schema level on some of the test questions. However, there is need for the establishment of a conceptual basis that promotes and allows the construction of the limit concept schema in relation to sequences and series. Furthermore, interviewed students’ responses paralleled the chronological improvement of the limit concept as reported in literature. Historical analysis of the development of concepts needs to be reflected upon when preparing and designing instruction. This would help the lecturer to foresee the challenges that lay ahead and address students’ difficulties during the learning process. The implementation of APOS Theory is recommended for the learning of other mathematical aspects, which cause difficulties in students’ learning. Moreover, other constructivist learning methods can be fused together with the APOS Theory to obtain improved results on students’ performance in mathematics.
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.

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Browsing Doctoral Degrees (Mathematics and Computer Science Education) by Author "Brijlall, Deonarain."