Doctoral Degrees (Mathematics and Computer Science Education)

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

Browse

Now showing 1 - 20 of 82

Mathematics Heads of department experiences of the implementation of technical mathematics: a case study of three technical schools in Pinetown district
(2023) Khoza, Mfundo Mondli.; Ngcobo, Annatoria Zanele.
Curriculum reform in the South African education system has faced a number of changes in the post-apartheid era. The changes in the curriculum aimed to redress the imbalances of the past while addressing the challenges of the future. The major curricular reform that has taken place in the past two decades includes the introduction of the subjects of Mathematical Literacy in 2006 and Technical Mathematics in 2016. Technical Mathematics has been piloted and is being implemented across technical schools in South Africa – which means that Departmental Heads (DHs) face a challenge in being ready to implement it. Therefore, knowledge of the experiences of those who have piloted and implemented Technical Mathematics in their schools is essential. The DHs, as curriculum leaders, must demonstrate an in-depth understanding of the subjects they manage. In this qualitative study, three Technical Mathematics DHs in Pinetown District in KwaZulu-Natal shared the duality of their teaching roles while managing the new subject. Using Samuel’s (2008) Force Field Model, professional forces were used to analyse this study. These forces enabled or constrained the DHs’ implementation, management, and teaching of Technical Mathematics in their respective schools. The conceptual framework used in this study is a modification of Shulman’s (1986) pedagogical content knowledge (PCK) and Ball et al.’s (2008) Mathematical Knowledge for Teaching (MKT), which provided clearer lenses for understanding the study phenomenon. Findings revealed that DHs understand what is needed, but this knowledge is not demonstrated all times; their understanding of their roles is more inclined toward teaching rather than managing the subject. The results of this study showed that suitable teachers, continuous evaluation and support, including practical assessment tasks (PAT), and providing professional development for Technical Mathematics teachers were perceived as enablers. Inhibiting factors were the six-year delay in including PAT, a lack of professional development targeting DH management roles, crumbling content, and learners’ apathy towards the subject. I argue that DH needs targeted support for their role and capacitation in knowledge of implementing and managing Technical Mathematics in order to execute their roles efficiently.
An exploration of how teachers perceive and respond to language demands associated with the learning of mathematical literacy
(2023) Dawson, Quintus.; Bansilal, Sarah
With the introduction of Mathematical Literacy (ML) in schools in 2006, a renewed emphasis has been placed on the teaching of Mathematical knowledge and concepts with the use of real-life context-based education. The intention of ML is to provide learners with the opportunity to learn mathematics that is embedded in real-life context-based settings. Learners are expected to gain competencies such as to reason, make decisions, solve problems, manage resources and interpret information within a mathematical contextual setting. These competencies indicate that language skills have become an integral part of ML. It was found that teachers believe that an understanding of the language of learning and teaching plays a pivotal role in learners’ optimal performance in ML. The teachers implemented several strategies aimed at supporting learners with language demands associated with learning ML. The strategies included identifying key words in ML questions, reading the ML questions aloud, selecting resources carefully, providing support in interpreting the contextual setting before attempting to find the mathematical demand of the question, and using code switching in the classroom. Teachers also suggested ways in which they could be supported in the ML classroom. The purpose of this study was to understand how teachers perceive and respond to the language demands associated with the learning of ML. To investigate how teachers perceive this phenomenon, a qualitative approach was adopted, and the analysis of data was conducted in three stages. First, questionnaires were distributed to 42 ML teachers to elicit their views about the role played by language in teaching and learning. Thirty-three teachers completed the questionnaires. The responses were analysed, and this analysis guided the second stage of the research which was the implementation of more meaningful classroom observations. Of these observed participants, all six agreed to participate further in the study by formulating thought provoking questions to gainmore insights into how teachers perceive and respond to the language demands associated with the learning of ML. The results of the study are limited in relation to generalisability because the sample was small. The findings indicate that teachers do consider the language proficiency of learners when they are teaching ML. The study found that a review of the existing channels of communication between stakeholders of the Department of Basic Education (DBE), and the ML teachers needs to be conducted, along with a review of the quality of the ML examination papers released by the DBE. Also, a research review needs to be conducted on how the DBE views the language demands associated with the learning of ML as reflected in the DBE policy documents on ML and, last, a review on the effect the DBE resources have on teaching and learning in ML classrooms needs to be conducted.
The effect of blended massive open online courses on students’ perceptions of their engagement and learning outcomes.
(2023) Edumadze, John Kwame Eduafo.; Govender, Desmond Wesley.
Massive open online courses (MOOCs) have arisen in recent years and have sparked significant discourse regarding education. MOOCs have become firmly established in the educational landscape and have undergone various adaptations, including the emergence of blended MOOCs integrating campus-based courses with MOOCs, resulting in a more profound learning experience. Blended MOOCs enable the use of various blended pedagogies, which can be used to offset the numerous challenges facing higher education in the global south, such as 1) limited access, 2) inadequacies and obsolescence infrastructure, 3) limited resources, 4) students’ enrolment that is far beyond what they were designed to accommodate, 5) overcrowding in lecture halls, 6) shortage of lecturers, especially research-active and experienced senior academics, 7) poor quality of education to meet the needs of today’s workforce and 8) mismatch between the skills of graduates. This study examines how blended MOOCs relate to students' learning and perceived academic performance. It analyses students' acceptance and use, engagement, and factors influencing their satisfaction and perceived academic performance. This analysis incorporates the revised Community of Inquiry (CoI) framework, including learning presence and the extended Unified Theory of Acceptance and Use of Technology (UTAUT 2) models. An empirical study was conducted with 2875 students from the University of Cape Coast, Ghana, selected via two-stage cluster sampling to accomplish this objective. The respondents were levels 100, 200, 300 and 800 students who were instructed to enrol into MOOCs as open educational resources (OERs) to supplement their campus-based for a semester. Upon the conclusion of the semester, the students were requested to complete a questionnaire. Subsequently, the collected data underwent analysis utilising SmartPLS v4 and SPSS v28 software. In general, the model demonstrates a strong alignment with the gathered data and possesses an adequate level of efficacy in elucidating the phenomenon of students’ usage acceptance (UTAUT 2) of blended MOOCs and its impacts on engagement (CoI), satisfaction and academic performance. The findings have several implications for research and EdTech integration practices among sub-Saharan nations, which were also discussed. Consequently, this study offers significant contributions to the theory and practice of active learning in a blended MOOC environment.
Exploration of Grade 11 learners’ mental constructions and difficulties in learning and solving trigonometric equations: a case of one school in Umlazi district.
(2023) Dube, Njabulo Happyboy.; Ngcobo, Zanele Annatoria.
Trigonometry is a particularly challenging area of mathematics for high school learners. This study investigated South African Grade 11 learners' mental constructions and difficulties when learning and solving trigonometric equations. Participants were selected from an after-school mathematics programme that they attended on a voluntary basis. Data was collected using an activity sheet and semi-structured interviews. Baseline data was collected using an activity sheet from 17 learners Grade 10 learners; data was collected from the same learners a year later using another activity sheet with a different set of problems. Semi-structured interviews were conducted with 7 learners to probe their responses on the activity sheet. Most of the learners were found to be unable to make the necessary mental constructions to solve trigonometric problems at the Grade 11 level. Dubinsky’s (1991) constructivist APOS theory, which describes how learners construct their knowledge of mathematics concepts in stages characterised by action, process, object and schema, was used to analyse the mental constructions of learners. During both phases of data collection, most learners were found to rely on explicit step-by-step calculations to solve problems, indicating that they were operating at the action stage; a smaller number were able to do some of the steps mentally without writing them out, indicating that they had advanced to the process stage. No evidence was found of learners having advanced to the object or schema stages. Moreover, the findings showed that, while the learners perform procedures correctly, they applied rules without giving reasons. Piaget and Garcia’s (1989) triad mechanism were used to analyse the difficulties that hindered learners’ mental construction of concepts. Learners’ difficulties included incorrect conceptions of the equal sign, overgeneralization of rules, and failure to integrate algebra concepts into their construction of trigonometric concepts. Based on the findings, the study recommends that teachers reinforce basic algebraic skills—such as collecting like and unlike terms, using brackets, and addition and subtraction of algebraic terms—before introducing trigonometric concepts. Teachers are urged to explore different methods for teaching trigonometric equations to enable learners to construct knowledge effectively, such as collaborative learning and differentiated classroom activities.
An exploration of preservice teachers’ use of educational technologies as visualization tools when teaching mathematics.
(2022) Zulu, Mzwandile Wiseman.; Mudaly, Vimolan.
This interpretive qualitative study explores the use of educational technologies by preservice teachers as visualization tools during mathematics teaching at secondary schools. Sfard’s commognitive framework and Koehler and Mishra’s technological pedagogical content knowledge theoretical frameworks undergird the study. Data were collected from ten preservice mathematics teachers at a university in the province of KwaZulu-Natal, South Africa. Performance tests, semi-structured interviews, focus group discussions, and observations were employed to collect data, which was analyzed using reflexive thematic analysis. The study found that preservice teachers employed two primary visualization strategies when they engaged in mathematical problem-solving: symbolic mental visualization, which they combined with their understanding of word usage, endorsed narratives and routines to arrive at a solution; and graphic visual mediators, such as diagrams, which they sketched to contextualize the problem statement and verify they solutions and use of mathematical word usage, routines, and endorsed narratives. Participants were found to be unable to solve a mathematics problem if they had not visualized it effectively; using a graphic visual mediator to understand the problem statement did not, however, guarantee success when solving a problem. A relationship was found between the visualization techniques that the participants used in their own attempts to solve mathematical problems and the visualization techniques they used in their lesson planning and teaching of mathematics in the same content area. Participants used innovative strategies to mediate learning, including educational technologies that facilitated visual mediators to enhance learners’ engagement with concepts. Synergies were found between the elements of the commognition and TPACK frameworks as these were used in tandem to analyze data. A model was developed (C+TPACK) that integrates the key elements of these frameworks. Further research is recommended to establish the viability, credibility and generalizability of the model.
A critical analysis of technology adoption in teaching by in-service teachers in Botswana during the Covid-19 pandemic.
(2022) Mafa, Rodnie Kgalemelo.; Govender, Desmond Wesley.
Information and Communication Technologies (ICTs) have been growing immensely for over the years in almost every sector of the economy. Even though this is the case, a lot still needs to be achieved in the education sector. Facilitators in Botswana schools are still depending and relying on the traditional teaching pedagogies and are far from benefiting from the rewards ICTs are bringing to the classroom to enhance teaching and learning. This study critically analysed technology adoption and use in teaching by in-service teachers in Botswana during the Covid - 19 era. It also analysed the ICT infrastructure found and available in Botswana schools and assessed the facilitator's skills, knowledge confidence and their perceptions on ICT integration in teaching and learning. A mixed methods (quantitative and qualitative) approach was adopted and used to collect data for the study. The major findings of the study revealed that the integration of ICT teaching in Botswana schools during the Covid- 19 era showed some challenges in teaching and learning and yet was also beneficial. These challenges were influenced by several factors such as lack of ICT skills, lack of self-confidence in the usage of technology by teachers and lack of appropriate JCT tools for use in schools. The study recommends that all schools (government and public) should be equipped and installed with the right ICTs supporting infrastructure, resources, and that all teachers be trained on the right pedagogies on the integration of ICTs in teaching and learning. In addition, the schools-based curriculums should be designed and developed in a way that accommodates ICT integration in teaching all subjects areas taught in schools.
Students’ and lecturers’ perceptions of factors which influence the creation of a sustainable e-learning environment in a University of Technology.
(2022) Msomi, Alfred Mvunyelwa.; Bansilal, Sarah.
By virtue of their designation, South African universities of technology need to be pioneers in the use of technology in learning and teaching. Many of these universities have knowledge and expertise in establishing and maintaining an ICT-driven environment for instruction. Accordingly, this study is an investigation into how the integration of technology into learning and teaching practices at the University Technology in South Africa, has been experienced by students and lecturers. This study aims specifically to comprehend access and technology usage difficulties, as well as infrastructure and training levels in ICT-challenged environments. In doing so, this study created a conceptual framework for encouraging instructional ICT in universities. The present study conducts a thorough assessment of one historically underprivileged University of Technology in South Africa using a case study methodology. The following individuals made up the study's target population: 835 students across first, second, and third-year levels; and 97 lecturers across the faculties of Engineering, Management Sciences, and Natural Sciences from whom data were collected using questionnaires. In addition, the researcher in the current study, performed semi-structured interviews with Deans of the faculties (2), Heads of Department (5), senior staff members in the Teaching and Learning Development Centre (TLDC) (2) and senior staff members in the Information and Technology Network Department (ITN) (3). Hence data were collected from various sources to gain multiple perspectives regarding the creation of a sustainable eLearning environment within a University of Technology. A mixed-methods approach was used in the current investigation. The use of technology for social empowerment to bridge the digital divide is the focus of a mixed-methods approach, which gathers, analyses, and interprets quantitative and qualitative data in a single study via the theoretical lens of creating a sustainable e-Learning environment. Informed by a continual literature review of the use of emergent learning technologies, the lived experience of e- Learning students, lecturers, and involved stakeholders was utilised as the basis for the first data collection. When choosing participants, deliberate sampling was utilised. From a standpoint of methodological interpretivism and positivist viewpoints as a technique of inquiry, the researcher's function was that of participant observer, interviewer, and human instrument. Thematic analysis was utilised in this study because, in contrast to descriptive statistics and inferential statistics, it is effective in identifying patterns in participant-reported qualitative data. Using the test-retest reliability approach, the instrument's reliability coefficient was calculated using Cronbach's alpha. The data gathered using quantitative approaches had response rates of 84 and 84.3 percent, whereas the data gathered using qualitative methods had a response rate of 100 percent. While processing qualitative data with the ATLAS.ti's package, the quantitative data was analysed using the R statistical computer program, 2020, version 3.6.3, to provide descriptive and inferential statistics. By following the university under study ethical guidelines, the study's ethical component was successfully achieved. The results of this study have shown that integration of technology in an educational institution of higher learning is dependent on many factors. The clearly distinguished factors from this study were: (i) poor access to internet connectivity, (ii) lack of continuous training for both students and lecturers on how technology could be integrated in the teaching and learning practices, and (iii) unaccommodating technology infrastructure. These factors were indicated as having a potential in facilitating or hindering technology integration for lecturers and students. The implications of this finding require the immediate development of capacity building plans and strategies for the adoption and integration of students and lecturers to an e-Learning platform. This study also suggests that for lecturers to effectively incorporate ICT into educational practice, they must have suitable pedagogical abilities in addition to fundamental ICT knowledge and skills. University policies and strategic level agendas are critical for success using the framework developed in this study for ICT-challenged environments. Based on these findings, it is recommended that the university management for the participating institution in the present study understands that the e-Learning system has great potential to improve the teaching and learning environment, provided in-depth ICT policy and strategies are put in place and suitable technology infrastructure is available.
Perfect compactifications of frames.
(2018) Mthethwa, Simo Sisize.; Baboolal, Dharmanand.; Pillay, Paranjothi.
We study the compacti cations of frames. In particular, we study the compacti cations of frames which are perfect. That is, those compacti cations for which the right adjoint of the compacti cation mapping preserves disjoint binary joins. The Stone-C ech compacti cation of a completely regular frame and the Freudenthal compacti cation of a rim-compact frame are known to be examples of such compacti cations. We study the Freudenthal compacti cation of a rim-compact frame with an aim of providing more properties and characterizations of this compacti cation in the context of frames since this is less studied in the literature compared to the Stone-C ech compacti cation of frames. One of the main results that we obtain about the Freudenthal compacti cation of a rim-compact frame is that it is the minimal perfect compacti cation for this class of frames and the maximal -compacti cation. The notion of a full -compact basis is known in the context of spaces. We de ne an analogous concept in the context of frames and show that the Freudenthal compacti cation of a rim-compact frame arises from such a basis. We also establish the one-to-one correspondence between such bases and the -compacti cations of a rim-compact frame. The fact that the compacti cations arriving from such basis are zero-dimensional is also established.It is well known that a frame has the least compacti cation if and only if it is regular continuous. Some conditions under which the least compacti - cation of a regular continuous frame is perfect have been studied by Baboolal in [1] and the study is furthered herein. An N-point compacti cation of a space is any compacti cation whose remainder consists of N points. The N-star compacti cations of frames are known to be the frame analogue of the N-point compacti cations for spaces. It has been shown that the least compacti cation of a regular continuous frame is an example of an N-star compacti cation. We study the conditions under which a 2-star compacti- cation of a regular continuous frame is perfect and we conjecture that the results can be generalized to any N > 1: We prove that, under perfectness, the 2-star compacti cation of a regular continuous frame is the only N-star compacti cation. We also show some results related to the connectedness of the remainder of the 1-star compacti cation. Some contribution to the theory of compacti cations of frames not relating to perfectness has also been made. The concept and the construction of a freely generated frame is well known. We have shown that any compacti cation of a frame L can be realized as a frame freely generated by L subject to certain relations.
A study of optimization problems and fixed point iterations in Banach Spaces.
(2019) Olakunle, Jolaoso Lateef.; Mewomo, Oluwatosin Temitope.
Abstract The study of optimization and xed point problems has remained as an attractive area of research due to its paramount importance in several areas of mathematics and other sciences. It constitutes a beautiful mixture of pure and applied analysis, topology, geometry, statistics and mechanics. It has also found several applications in solving nonlinear phenomena arising in diverse elds such as engineering, economics, biology, management science, transportation, game theory, physics, computer tomography, etc. In this thesis, we present some inertial iterative schemes with strong convergence theorems for approximating solutions of certain optimization problems in real Hilbert spaces. We further analyze a parallel combination extragradient method for nite family of pseudomonotone equilibrium problem and xed point of demi-contractive mappings in real Hilbert spaces. By combining Mann and Krasnolselskii methods with inertial extrapolation term, we propose a new iterative method which converges strongly to a common solution of split variational inclusion problem and equilibrium problem with para-monotone equilibria. More so, we introduce a projection-contraction method for approximating solution of split generalized equilibrium problem in real Hilbert space. We show that our projectioncontraction method converges at a linear rate of convergence. Moreover, we extend the study of projection methods for solving variational inequality problem to re exive Banach spaces. We introduce a projection algorithm and prove a strong convergence theorem for approximating solution of variational inequality problem in re exive Banach spaces and give an application of our result to approximating solution of equilibrium problem in re exive Banach space without prior knowledge of operator norms. Furthermore, we introduce a totally relaxed subgradient extragradient method for approximating a common solution of variational inequality and xed point of quasi-nonexpansive mapping in a 2-uniformly convex and uniformly smooth Banach space. We also study the approximation of solution of variational inequality problem using projection-contraction algorithm in real Hilbert space. Then, we extend the study of split equality monotone inclusion problem to p-uniformly convex and uniformly smooth real Banach spaces. Ultimately, we consider the approximation of common xed points of k-strictly pseudocontractive mappings in a 2-uniformly smooth real Banach space. We introduce a class of N-generalized Bregman nonspreading mappings and propose an iterative method for approximating the common xed points of this kind of mappings which is also a solution of equilibrium problem in a re exive Banach space. Numerical experiments are presented to demonstrate the e ciency and performance of our algorithms in comparison with other existing algorithms in literature. We also achieve strong convergence results using our algorithms for approximating solutions of the underlying problems in each case.
Exploring the use of robotics in the learning of programming.
(2021) Govender, Reginald Gerald.; Govender, Desmond Wesley.
Computer Programming is seen as a valuable skill in the digital era that we presently live in. However, for the novice programmer, it is often accompanied with difficulties resulting in negative reactions. The dawning of the Fourth Industrial Revolution has catapulted many initiatives local and global to promote Computer Programming and Robotics. A major initiative by the South African government is the planning and implementing of a new subject in school to raise the awareness of coding at an early age. The lack of coding exposure and awareness leads to little or no interest in Computer Programming related courses after schooling years. This study focuses on exploring the learning of coding through the use of Robotics among computer registered students with no prior coding knowledge at a University in South Africa. Unlike the traditional use of block-based programming to introduce Computer Programming, which is limited to screen output, the study opted to use a physical manipulative by using a robotic element through prototype building using text-based programming, resulting in live autonomous output of code. The Arduino kit was used as the robot element to acquire knowledge development to the fundamental concepts of Computer Programming using the Python programming language. Participants' coding knowledge was assessed through a series of hands-on online activities. Design Based Research was adopted with the integration of Kolb’s Experiential Learning Cycle, framed within the second-generation Activity Theory. Mix methods were supported as it is in accordance with the pragmatic paradigm favoured by Design Based Research. All data collection took place online through workshops, surveys, questionnaires and a focus group interview. The sample size was 75 achieving a significant partial least squares structural equation model as a minimum of 50 participants was needed based on the ten times rule. The results show that students acquiring a direct learning experience with text-based code with the aid of the robotic element proved to be successful. The robot coding simplified the assimilation of text-based coding as participants could see the execution of their code on the prototype in reality. The eradication of the abstract nature of Computer Programming through Robotics as a physical manipulative solidified the understanding of coding structures. Furthermore, students' belief, interest, motivation, confidence, and Mathematics skill set were found to contribute success in Computer Programming. It was revealed that learning to code in a text-based environment can be made fun. In addition, learning programming with the use of the robot is effective for first time learning of text-based code. The researcher proposes that the introduction of learning programming integrated through the building of prototypes and coding resulting in autonomous robots enhances the learning experience of text-based code.
Some null submanifolds of indefinite nearly Sasakian manifolds.
(2020) Osman, Islam Fawzi Mahgoub.; Massamba, Fortuné.
We investigate some geometric aspects of indefinite nearly Sasakian manifolds. We study specific cases of null submanifolds, namely, invariant, GCR, Screen transversal and Radical screem transversal null submanifolds. Under some conditions there exist leaves that are nearly Sasakian, immersed in the ambient manifold as submanifolds. Furthermore, we give some geometric configuration of the existence of totally geodesic foliations in some distributions.
Optimal control problems constrained by hyperbolic conservation laws.
(2021) Tirab, Mohammed Adam Mahmmoud.; Ngnotchouye, Jean Medard Tchoukouegno.
This thesis deals with the solutions of optimal control problems constrained by hyperbolic conservation laws. Such problems pose significant challenges for mathematical analysis and numerical simulations. Those challenges are mainly because of the discontinuities that occur in the solutions of non-linear systems of conservation laws and become more acute when dealing with the multidimensional case. The problem is formulated as the minimisation of a flow matching cost functional constrained by multi-dimensional hyperbolic conservation laws. The control variable is the initial condition of the partial differential equations. In our analysis of the problem, we review extensively the constraints equation and we consider successively the one-dimensional and the multi-dimensional cases. In all the cases, we derive the optimality conditions in the adjoint approach at the continuous level, which are then discretised to arrive at a numerical algorithm for the solution. In the derivation of the optimality conditions, we replace the non-linear conservation laws either by the relaxation equation or the Lattice Boltzmann equation. We illustrate our findings on examples related to the multi-dimensional Burger and the Euler equations.
Numerical approximations of fractional differential equations: a Chebyshev pseudo-spectral approach.
(2020) Oloniiju, Shina Daniel.; Sibanda, Precious.; Goqo, Sicelo Praisegod.
This study lies at the interface of fractional calculus and numerical methods. Recent studies suggest that fractional differential and integral operators are well suited to model physical phenomena with intrinsic memory retention and anomalous behaviour. The global property of fractional operators presents difficulties in fnding either closed-form solutions or accurate numerical solutions to fractional differential equations. In rare cases, when analytical solutions are available, they often exist only in terms of complex integrals and special functions, or as infinite series. Similarly, obtaining an accurate numerical solution to arbitrary order differential equation is often computationally demanding. Fractional operators are non-local, and so it is practicable that when approximating fractional operators, non-local methods should be preferred. One such non-local method is the spectral method. In this thesis, we solve problems that arise in the ow of non-Newtonian fluids modelled with fractional differential operators. The recurrent theme in this thesis is the development, testing and presentation of tractable, accurate and computationally efficient numerical schemes for various classes of fractional differential equations. The numerical schemes are built around the pseudo{spectral collocation method and shifted Chebyshev polynomials of the first kind. The literature shows that pseudo-spectral methods converge geometrically, are accurate and computationally efficient. The objective of this thesis is to show, among other results, that these features are true when the method is applied to a variety of fractional differential equations. A survey of the literature shows that many studies in which pseudo-spectral methods are used to numerically approximate the solutions of fractional differential equations often to do this by expanding the solution in terms of certain orthogonal polynomials and then simultaneously solving for the coefficients of expansion. In this study, however, the orthogonality condition of the Chebyshev polynomials of the first kind and the Chebyshev-Gauss-Lobatto quadrature are used to numerically find the coefficients of the series expansions. This approach is then applied to solve various fractional differential equations, which include, but are not limited to time{space fractional differential equations, two{sided fractional differential equations and distributed order differential equations. A theoretical framework is provided for the convergence of the numerical schemes of each of the aforementioned classes of fractional differential equations. The overall results, which include theoretical analysis and numerical simulations, demonstrate that the numerical method performs well in comparison to existing studies and is appropriate for any class of arbitrary order differential equations. The schemes are easy to implement and computationally efficient.
A numerical study of heat transfer and entropy generation in Powell-Eyring nanofluid flows.
(2020) Ogunseye, Hammed Abiodun.; Sibanda, Precious.
The heat transfer in non-Newtonian nanofluid flow through different geometries is an important research area due to the wide application of these fluids in biomedical, chemical and thermal engineering processes. The continuous generation of entropy leads to exergy loss which reduces the performance and efficiency of any physical system, therefore, the minimization of entropy generation becomes necessary. In this thesis, we present a numerical study of heat transfer and entropy generation in non-Newtonian nanofluid flows. We study the flow of a Powell-Eyring nanofluid, using models developed from experimental data. The equations that model the flow are, in each case, reduced to systems of nonlinear differential equations using Lie group theory scaling transformations. Accurate, efficient and rapidly converging spectral numerical techniques including the spectral quasilinearizzation, spectral local linearization and bivariate spectral quasilinearization methods are used to find the numerical solutions. The results show, among other findings, that increasing either the nanoparticle volume fraction or thermal radiation parameter enhances the nanofluid temperature, entropy generation and the Bejan number. In addition, we find that the Nusselt number increases with the temperature ratio parameter and thermal radiation. The results from this study may find use in the design of cooling devices to enhance and optimize the performance of thermal systems.
A numerical study of entropy generation in nanofluid flow in different flow geometries.
(2021) Mburu, Zachariah Mbugua.; Sibanda, Precious.
This thesis is concerned with the mathematical modelling and numerical solution of equations for boundary layer flows in different geometries with convective and slip boundary conditions. We investigate entropy generation, heat and mass transport mechanisms in non-Newtonian fluids by determining the influence of important physical and chemical parameters on nanofluid flows in various flow geometries, namely, an Oldroyd-B nanofluid flow past a Riga plate; the combined thermal radiation and magnetic field effects on entropy generation in unsteady fluid flow in an inclined cylinder; the impact of irreversibility ratio and entropy generation on a three-dimensional Oldroyd-B fluid flow along a bidirectional stretching surface; entropy generation in a double-diffusive convective nanofluid flow in the stagnation region of a spinning sphere with viscous dissipation and a study of the fluid velocity, heat and mass transfer in an unsteady nanofluid flow past parallel porous plates. We assumed that the nanofluids are electrically conducting and that the velocity slip and shear stress at the boundary have a linear relationship. We also consider different boundary conditions for all the flow models. The study further analyzes and quantifies the influence of each source of irreversibility on the overall entropy generation. The transport equations are solved using two recent numerical methods, the overlapping grid spectral collocation method and the bivariate spectral quasilinearization method, first to determine which of these methods is the most accurate, and secondly to authenticate the numerical accuracy of the results. Further, we determine the skin friction coefficient and the changes in the heat and mass transfer coefficients with various system parameters. The results show, inter alia that reducing the heat transfer coefficient, the particle Brownian motion parameter, chemical reaction parameter, Brinkman number, thermophoresis parameter and the Hartman number all lead individually to a reduction in entropy generation. The overlapping grid spectral collocation method gives better computational accuracy and converge faster than the bivariate spectral quasilinearization method. The fluid flow problems have engineering and industrial applications, particularly in the design of cooling systems and in aerodynamics.
Numerical studies of nanofluid boundary layer flows using spectral methods.
(2021) Magodora, Mangwiro; Sibanda, Precious.
This thesis is focused on numerical studies of heat and mass transport processes that occur in nanofluid boundary layer flows. We investigate heat and mass transfer mechanisms in the flow of a micropolar nanofluid above a stretching sheet, the squeezed nanofluid flow between two parallel plates and the impact of activation energy and binary chemical reaction on nanofluid flow past a rotating disk. We present an analysis of entropy generation in nanofluid flow past a rotating disk and nanofluid flow past a stretching surface under the influence of an inclined magnetic field. This study aims to numerically determine to a high degree of accuracy, how nanoparticles can be utilized to alter heat and transport properties of base fluids in order to enhance or achieve desirable properties for thermal systems. The heat and mass transfer processes that feature in nanofluid boundary layer flow are described by complex nonlinear transport equations which are difficult to solve. Because of the complex nature of the constitutive equations describing the flow of nanofluids, finding analytic solutions has often proved intractable. In this study, the model equations are solved using the spectral quasilinearization method. This method is relatively recent and has not been adequately utilized by researchers in solving related problems. The accuracy and reliability of the method are tested through convergence error and residual error analyses. The accuracy is further tested through a comparison of results for limiting cases with those in the literature. The results confirm the spectral quasilinearization method as being accurate, efficient, rapidly convergent and suited for solving boundary value problems. In addition, among other findings, we show that nanofluid concentration enhances heat and mass transfer rates while the magnetic field reduces the velocity distribution. The fluid flows considered in this study have significant applications in science, engineering and technology. The findings will contribute to expanding the existing knowledge on nanofluid flow.
A mathematical study of boundary layer nanofluid flow using spectral quasilinearization methods.
(2020) Dhlamini, Mlamuli.; Sibanda, Precious.; Motsa, Sandile Sydney.
Heat and mass transfer enhancement in industrial processes is critical in improving the efficiency of these systems. Several studies have been conducted in the past to investigate different strategies for improving heat and mass transfer enhancement. There are however some aspects that warrant further investigations. These emanate from different constitutive relationships for different non-Newtonian fluids and numerical instability of some numerical schemes. To investigate the convective transport phenomena in nanofluid flows, we formulate models for flows with convective boundary conditions and solve them numerically using the spectral quasilinearisation methods. The numerical methods are shown to be stable, accurate and have fast convergence rates. The convective transport phenomena are studied via parameters such as the Biot number and buoyancy parameter. These are shown to enhance convective transport. Nanoparticles and microorganisms’ effects are studied via parameters such as the Brownian motion, thermophoresis, bioconvective Peclet number, bioconvective Schmidt number and bioconvective Rayleigh number. These are also shown to aid convective transport.
An exploration of the barriers to effective geometric thought in the Further Education and Training phase of selected secondary schools in the Umlazi District.
(2021) Naicker, Kalavani.; Mudaly, Vimolan.
ENGLISH ABSTRACT: Geometry, a branch of mathematics has a history from the study of practical measurement in ancient Egypt to properties of shapes in Greek geometry. Learning and teaching of geometry in South Africa has posed many challenges for educators and learners. In 2011 Euclidean geometry was reintroduced after the removal to redress the inequalities of apartheid and to provide uniform access to mathematics for all learners. This study was therefore conducted in three secondary schools, conveniently selected, to explore the barriers that hamper the performance of Euclidean geometry in the FET phase and to explore ways of eradicating these barriers. Research questions generated data from educators and learners aimed at identifying factors influencing effective geometric thought. This study was guided by the two metaphors, namely acquisition and participation and their impact on the teaching and learning of geometry. The study focussed primarily on the cognitive developmental theory of Piaget and social constructivist theory of Vygotsky both of which supported the acquisition metaphor. It further examined the participative metaphor of Sfard’s theory of commognition. The study adopted the interpretive paradigm with the qualitative design. Purposive sampling was used since the study was conducted with grade 11 learners and educators teaching in the FET phase. Questionnaires, semi structured interviews and focus group discussions were used to collect the data. Lesson observations were used for triangulation purposes. The research instruments were piloted in order to ensure validity and reliability. Data was first organized according to the research questions which was then coded and divided into themes. It was found that many of the barriers to learning geometry are primarily attributed to a lack of engagement of learners in meaningful learning situations. Findings also highlighted language, attitude, the classroom environment, and gaps in educators’ methodology and content knowledge has being some of the factors that contributed to the poor learner participation. Recommendations are made for stakeholders to develop effective geometric thought. This study proposes that greater emphasis be placed on learner-centred methods of teaching. Communication in the form of verbal, written and visual should be considered when teaching geometry. Allowing learners to express their experiences to think and develop new knowledge is essential. Learners need to become active participants of their learning and learning needs to take place in more realistic situations to improve understanding. IQOQA LOCWANINGO: I-Jiyomethri, igatsha lezibalo linomlando osuka ekutadisheni kwesilinganiso esisebenzayo Egibhithe lasendulo kuya azakhiweni zomumo weJiyomethri yamaGrikhi. Ukufunda nokufundisa iJiyomethri eNingizimu Afrika kudale izingqinamba eziningi kothisha nabafundi. Ngo-2011 iJiyomethri yaphinde yafakwa ngemuva kokususwa ukuze kulungiswe ukungalingani kobandlululo kanye nokuhlinzeka ngokulingana ukuthola izibalo zabo bonke abafundi. Ngakho-ke lolu cwaningo lwenziwe ezikoleni zamabanga aphakeme ezintathu, ezikhethwe kalula, ukuhlola imigoqo ephazamisa ukusebenza kweJiyomethri esigabeni seFET nokuhlola izindlela zokuqeda lezi zithiyo. Imibuzo yocwaningo yakhiqiza idatha evela kothisha nakubafundi ehlose ukukhomba izinto ezinomthelela ekucabangeni okusebenayo kweJiyomethri. Lolucwaningo beluqondiswa izingathekiso ezimbili, okungukuthi ukuzuza nokubamba iqhaza kanye nomthelela wazo ekufundisweni nasekufundweni kweJiyomethri. Ucwaningo lugxile kakhulu kumqondo wokuthuthuka wokuqonda kaPiaget kanye nethiyori yezakhi zenhlalo kaVygosky zombili ezazisekela isingathekiso sokutholwa. Iphinde yahlola umbono kaLave noWenger (1991) owawugqamisa ukubaluleka kwesifaniso sokubamba iqhaza futhi okokugcina, umbono kaSfard wokuhlonishwa. Ucwaningo lwamukele ipharadigm yokuhumusha ngomklamo wekhwalithi. Isampuli enenhloso isetshenzisiwe kusukela isifundo senziwa nabafundi bebanga le-11 nothisha abafundisa esigabeni se-FET. Amaphepha emibuzo, izingxoxo ezihleleke kancane kanye nezingxoxo zamaqembu okugxila kuwo kusetshenziselwe ukuqoqa imininingwane. Ukubheka izifundo kusetshenziselwe izinhloso zonxantathu. Amathuluzi zocwaningo ahlolwa ukuze kuqinisekiswe ukuba semthethweni nokwethembeka. Idatha yayihlelwe ngokuya ngemibuzo yocwaningo eyabe isibhalwa ngamakhodi yahlukaniswa yaba ngamatimu. Kunconyelwa ababambiqhaza ukuthi bathuthukise umcabango weJiyomethri osebenzayo. Lolu cwaningo luphakamisa ukuthi kugxilwe kakhulu ezindleleni ezigxile kubafundi zokufundisa. Ukuxhumana ngendlela yokukhuluma, okubhaliwe nokubukwayo kufanele kubhekwe lapho kufundiswa iJiyomethri. Ukuvumela abafundi ukuthi baveze ulwazi lwabo ukuze bacabange futhi bathuthukise ulwazi olusha kubalulekile. Abafundi kudingeka babe ngabahlanganyeli abakhuthele ekufundeni nasezidingweni zokufunda kwabo okumele ukwenzeke ezimeni ezingokoqobo ukuthuthukisa ukuqonda.
Exploring the discourses of preservice mathematics teachers when solving geometry problems=Kuhlolwa ingxoxo yothisha bezibalo abangakasebenzi ngesikhathi bexazulula izinkinga zegeometry.
(2021) Mahlaba, Sfiso Cebolenkosi.; Mudaly, Vimolan.
Research on teaching and learning that aim to improve preservice mathematics teachers’ (PMTs) knowledge of geometry is increasing globally. The current study explored PMTs’ discourses when solving geometry problems. The amalgamation of the commognitive theory and the Van Hiele levels of geometrical thinking theory was used as the theoretical basis for this study. The study uses the difference between ritualistic and explorative discourse as explicated by Sfard (2008) in commognition together with the four Van Hiele levels of geometrical thinking to view and analyse the data. It goes deeper into the theory of commognition to use not only objectification of mathematical discourse but also the four elements of mathematical discourse to reach its conclusion. The current study aimed to answer the main question: How does preservice mathematics teachers’ thinking as evident in their mathematical discourse during Euclidean geometry problem solving relate to their teaching practices in Euclidean geometry? This will be done through answering four subsidiary questions. A qualitative research approach was used to generate rich and descriptive data to answer the posed research questions. Furthermore, the qualitative approach allowed for the collection of data representing participants’ geometry problem solving experiences which was the core of the current study. I purposively and conveniently sampled 6 participants in this study where they completed a task-based and face-to-face interviews. Consent was obtained from these participants prior their participation in the study. Data generated from the two instruments was thematically analysed. Findings from this study revealed that most PMTs use ritualistic discourse when communicating about their geometry problem solving actions. These findings are a consequence of them performing routines for social acceptance instead of generating endorsed narratives. Furthermore, it was observed that others used ritualistic discourse because they rely on scaffolding from others to perform their routines instead of developing their own routines. Despite the dominance of ritualistic discourse participation in the current study, there were instances where PMTs seemed to be using explorative discourse but get stuck somehow and return to ritualistic discourse. The Van Hiele theory revealed that most PMTs still operate within the lower levels of geometrical thinking. The main findings and contribution of this study is that for PMTs to advance their geometrical thinking from level 0 to level 3, they need to transform their discourse participation from ritualistic to explorative. IQOQA Ucwaningo mayelana nokufundisa kanye nokufunda oluhlose ukuthuthukisa inkonzo yokulekelela othisha bezibalo ngolwazi lwe-geometry luyakhula umhlaba wonke. Okufundwayo kuhlola ukusetshenziswa kolimi ngesikhathi kuxazululwa izinkinga zegeometry. Ukuqoqela ndawonye kwethiyori ye-commognitive kanye neVan Hiele yamazinga e-geometrical yenjulalwazi yokucabanga yasetshenziswa njengesisekelo sobunjulalwazi sesifundo. Uhlelokwenza locwaningo ngekhwalithethivu lwasetshenziswa ukuchamusela ulwazi oluchazayo nolunonile ukuphendula imibuzo yocwaningo eyaphonswa. Uma sihlabela phambili, uhlelokwenza ngekhwalithethivu lwavumela ukuqoqwa kolwazi olumele abazibandakanya mayelana nabasebekwazi ukusombulula izinkinga ze-geometry okwakungumgogodla wesifundo. Ngisampule abayisithupha (6) ababamba iqhaza ngenhloso nangokufaneleyo kulesi sifundo njengoba benza umsebenzi kanye nohlelongxoxo ubuso nobuso. Ukuvuma kwababambe iqhaza kwatholakala ngaphambi kokuba basetshenziswe esifundweni. Ulwazi olusengwe emathuluzini amabili lwacutshungulwa ngokobungqikithi. Imiphumela iveza ukuthi iningi lothisha abamukela inkonzo yokusizwa ngokwezibalo basebenzisa inkambiso ethile ngokwezingxoxo uma kukhulunywa ngezinyathelo zokuxazululwa kwezinkinga ze-geometry. Lemiphumela iyizimpendulo zabo zokwenza izinto ezizodwa ukuze bamukeleke ngokwenhlalo esikhundleni sokuphehla izinkulumo ezivunyiwe. Uma sihlabela phambili, kwabhekwa ukuthi abanye basebenzisa inkambiso ngokwengxoxo ngoba bathembele ekufukulweni ngabanye ukwenza inhlalakhona yabo esikhundleni sokukhulisa eyabo inhlalakhona. Ngaphandle nje kokugxila kwenkambiso yengxoxo yababamba iqhaza kulesi sifundo, izibonelo lapho othisha abenkonzo yokusizwa ngokwezibalo kubukeka besebenzisa ingxoxo kusinga kodwa bese beyakhingxeka kwezinye izindawo bese bebuyela enkambisweni ngxoxo. Injulalwazi yeVan Hiele iveza ukuthi iningi lalabothisha abasizwa ngezibalo nanamhlanje badidizela emazingeni aphansi ngokomcabango wegeometry. Okutholakele okunqala kanye nomnikelo walesi sifundo ukuthi labo thisha abasizwe ngokwezibalo uma befuna ukuqhubekela phambili ngokomcabango we-geometry kusukela ezingni leqanda (0) kuya kwelesithathu (3), kuzodingeka bayiguqule ingxoxo yabo yokuzibandakanya kusukela enkambisweni kuya enhlolweni.
An APOS analysis of the teaching and learning of factorisation of quadratic expressions in grade 10 mathematics classrooms.
(2021) Vilakazi, Aubrey Sifiso.; Bansilal, Sarah.
The South African Curriculum and Assessment Policy Statements (CAPS) document, for the Further Education and Training Phase (FET) Mathematics Grades 10-12 (2011) shows that the factorisation of algebraic quadratic expressions or equations pervades the mathematics of the secondary school. As a result, for learners to be successful at mathematics in Grade 12, they need to know a great deal of algebra, particularly the factorisation of quadratics. It is therefore important for us as mathematics educators to identify areas in the factorization of quadratics that teachers and learners are struggling to learn and apply. With this in mind, the study sets to embark on an APOS analysis of the teaching and learning of factorisation of quadratic expressions in Grade 10 mathematics classrooms. Following on from the research questions, this study is located within the principles of the mixed methods case study approach. The combination of methodologies has allowed me to identify broad trends across the groups of learners and those of educators as a whole as well as differences within the participants of the groups themselves. The participants of the study were the groups of Grade 10 learners from the two participating schools, as well as the Grade 10 mathematics teachers from the two circuits of Ilembe District. Five sources of data were used. Firstly, data were generated from 25 teachers from the two circuits who participated in the teachers’ questionnaires. A second data collection instrument was the classroom lessons’ observations of the six teachers. A third data source was the learner group activity and learners’ interviews administered to 12 learners. A fourth data source was the unstructured interviews with six teachers. The final instrument was the analysis of the 205 Grade 10 mathematics 2019 March common paper learners’ scripts. This study was guided by the theory of constructivism and more specifically Action, Process, Object, Schema (APOS) theory which views learning as changes in conception. As an individual engages with a concept, the conception changes from an initial external view towards seeing the concept as a totality upon which other Actions and Processes can act. This study has found that, firstly, teachers and learners tend to rely too much on the use of rules in factoring certain quadratics. In so doing, a prototype of the quadratic expression concept is perceived which consists of isolated and disconnected concepts. As a result, most learners were not able to factor the trinomial quadratic of 𝑎≠1, since they perceived the factoring of 𝑎𝑥2+𝑏𝑥+𝑐 with 𝑎=1 and that of 𝑎≠1 as two different procedures. Secondly, there are also students whose mental constructions (conception) are limited to Action levels in terms of APOS theory. The findings of the study suggest that teachers and learners should be able to consider quadratic expressions as one big idea and follow the fundamental considerations when factoring the quadratic expressions. Furthermore the use of multi-methods in factoring quadratics is encouraged and needed for students to better understand the connections between different methodologies for conceptual development.

Browse

Recent Submissions