Doctoral Degrees (Mathematics and Computer Science Education)

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

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Exploring grade 10 learners’ perceptions of using Microsoft Teams as an online platform for learning trigonometric functions.
(2024) Shandu , Sibusiso Thulebona.; Naidoo, Jayaluxmi.
The poor national mathematics results at the secondary school level, in general, and Trigonometry, in particular, resulted in numerous interventions by teachers and researchers to improve learners’ performance. In terms of teaching and learning, due to the coronavirus disease of 2019 pandemic, there was a shift to blended and online learning. The country's learners are exposed to different learning management systems (LMS). It is the next generation of online classrooms, or learning management systems (LMS) with many educational institutions embracing Microsoft Teams as their online platform of choice. This research illustrates the findings of the first doctoral research conducted that explores grade 10 mathematics learners using Microsoft Teams to learn trigonometric functions. The participants for this research are of 23 committed grade 10 mathematics learners. This qualitative research was conducted in a secondary school situated in KwaZulu-Natal, South Africa. The research was framed by the theories of connectivism and social constructivism. The research covered a variety of data collection, which were observations of classes taking place, a pre-task, learners approach interviews, and post-task, and a reflective journal of the researcher. The data collected for this research was coded and analysed using the ATLAS.ti software tool. Inductive coding facilitated the identification of significant themes from the data, which appeared to be the most significant. Thematic analysis facilitated the extraction of significant insights from the collected data. Research has pointed out that the participants perceive that Microsoft Teams platform is simple and user-friendly. However, to utilise Microsoft Teams efficiently, it is crucial to have faster and more reliable internet connections. This research has comprehensively analysed the findings, consequences, and recommendations pertaining to the views and experiences of blended learning utilising Microsoft Teams for teaching trigonometric functions in mathematics. Although conducted in a classroom setting, this research has broader implications for the use of technology to enhance teaching and learning in higher education. Thus, the research is valuable since it exhibits how using blended learning in educational contexts may enhance teaching learning, and performance in mathematics. Thus, this research has implications for mathematics, mathematics education, and higher education nationally and internationally. This research benefits learners, learners, teachers, teacher educators, academics, researchers, curriculum, and policy developers in mathematics.
An exploration of the learners’ commognition when engaging with geometry in the grade eleven classroom.
(2024) Mulenga, Alexander.; Mudaly, Vimolan.
This study investigates the learner’s commognition when engaging with geometry in a grade 11 class. The theory of commognition, propounded by Anna Sfard, underpinned this study with particular focus on the four characteristics of mathematical discourse; namely, specific purpose word use, visual mediators, substantiated narratives, and routines. The phenomenological qualitative research design approach was utilized to investigate 3 grade 11 learners’ commognition. The study was conducted at a high school in the province of KwaZulu-Natal of the republic of South Africa. Data was collected by means of a written mathematics activity that required participants to use straightedge and compass to construct a circle, followed by proving one Euclidean proof. The participants then participated in a semi-structured group interview based on the written task activity. Data was analysed by means of two analytical tools; the Discourse Profile of the Tangent adapted from Mpofu, and Pournara and others, was used to examine the interview transcripts. The Realization Tree Assessment, initiated by Weingarden with others, and adapted by Haghjoo and his colleagues was used to investigate the written task activity. The examination of the results revealed that the participants word use was problematic and affected their level of objectification. Therefore, it was concluded that the participants discourse was ritualistic. These results suggest that attending to the degree of objectification in the initial stages of transforming ritual to explorations is cardinal before embarking on the process of deritualisation.
Critical analysis of knowledge produced through postgraduate mathematics education research in post-apartheid South Africa: the first decade.
(2024) Goba, Barbara Busisiwe.; Bansilal, Sarah.
The concept of 'knowledge systems' has led to a need for 'research on research'. Therefore, it is crucial to assess the knowledge generated by various knowledge societies, particularly within universities. This rationale drove the Project on Postgraduate Education Research (PPER) to investigate the research output in postgraduate education during the initial decade (1995-2004) following the end of apartheid in South Africa. Within the framework of PPER, this study scrutinized the body of masters and doctoral research conducted at 19 South African universities within the aforementioned period. The study aimed to address the following main research question: What forms of knowledge emerged from postgraduate studies in mathematics education research within South Africa during the years 1995 to 2004? Sub-research questions: 1.1 How were the titles of postgraduate studies in mathematics education (1995-2004) formulated and structured? 1.2 Which phenomena were more frequently explored, and which were less likely to be investigated within these postgraduate studies (1995-2004)? 1.3 What were the predominant research questions, and what assertions were made concerning these phenomena? 1.4 What were the prevalent research paradigms, designs, approaches, and methodologies employed in postgraduate studies on mathematics education (1995-2004)? 1.5 Which theories were commonly utilized and which were less frequently applied in postgraduate studies on mathematics education (1995-2004)? For each of the sub-questions mentioned above, the study delved into the identities of the students responsible for generating this knowledge, along with details about their degrees and affiliations with respective institutions. This study employed Giddens' structuration theory as its theoretical framework. It conducted an analysis of knowledge produced in a corpus of 190 postgraduate studies in mathematics education. This analysis encompassed various elements, starting from titles, research phenomena, questions, claims, methodologies, and extending to theories. The data from these theses was extracted using Suri's Methodologically Inclusive Research iii Syntheses (MIRS). Additionally, interviews were conducted with a sample of supervisors overseeing these theses. The main findings reveal that knowledge in mathematics education predominantly comes from white female postgraduates affiliated with historically white universities (HWUs). This challenges the prevailing notion that white males dominate research in this field. Notably, Wits University emerged as the primary contributor of mathematics education theses in the corpus, with a majority of these studies composed in English. When comparing this result with Khuzwayo's (2005) findings, which emphasized a focus on white education and teacher training in South African mathematics education research from 1948 to 1994, a similarity arises in terms of the demographic producing the majority of research by white postgraduates. However, this does not necessarily imply that white learners were the primary subjects of study. Instead, it suggests another way of 'objectifying' black learners and teachers, as Black Africans are frequently researched by predominantly white researchers. Additionally, the prevalence of white postgraduate students producing more mathematics education theses may result from the apartheid policy on mathematics education, where Verwoerd argued that Black African people did not need to study mathematics as it had no relevance in their communities. Verwoerd's stance remained largely unchanged in the first ten years of post-apartheid South Africa. In the domain of postgraduate studies, the construction of titles employs various linguistic devices, methodological approaches, indication of the results and epistemological stances. Among these linguistic devices, the colon is the most frequently employed. Mathematics education postgraduate studies predominantly centre around five key research phenomena: the affective domain, assessment, knowledge, cognition, and epistemologies, as well as pedagogy and technology/resources. These studies from 1995 to 2004 were primarily small-scale. Each of these phenomena is examined in terms of the research claims posited. Notably absent are studies with a focus on primary mathematics education and rural education. This finding aligns with the results of reviews conducted by Venkat, Adler Rollnick, et. al., 2009; Adler, Alshwaikh, Essack, & Gcsamba, 2017. However, the most recent review of mathematics education journal articles by Morrison, Graven, Venkat, and Vale (2023) indicates an increase in research on primary mathematics education. A prominent theoretical orientation in postgraduate studies is constructivism, particularly within HWUs. Additionally, the preferred approach in mathematics education studies conducted in South Africa between 1995 and 2004 was the case study research design.
A new method with regularization for solving split variational inequality problem in real Hilbert spaces.
(2022) Akutsah, Francis.; Narain, Ojen Kumar
The concept of the optimization problem, fixed point theory and its application constitute the nucleus of nonlinear analysis, which is a major branch of mathematics. Optimization theory, fixed point theory and its applications have a wide range of application in practically every field of science, particularly mathematical sciences. The theory of optimization and fixed point have received great attention from authors around the world and these areas will continue to receive such great attention. The theory has been well developed by well-known researchers in these areas. However, there are still a lot of work to be done. The goal of this thesis is to advance the theory of optimization and fixed point in the framework of Hilbert and Banach spaces. The substance of this thesis is separated into two parts. The research efforts of the first part of this thesis (Chapter 3 to Chapter 6) has to do with introducing some new iterative methods for approximating the solution of a variational inequality problems, split variational inequality problems, equilibrium problems, split monotone variational inclusion problem, split generalized mixed equilibrium problem and fixed point problems in the framework of a Hilbert and Banach spaces. In addition, we introduce a new class of bilevel problem in the framework of real Hilbert spaces and a new regularization technique, and inertial terms for approximating solutions of split bilevel variational inequality problems. Furthermore, we establish that the proposed iterative methods converges strongly to the solution of the aforementioned problems as the case may be. Then, we present some numerical experiments to show the efficiency and applicability of our proposed methods in comparison with other state-of-the-art iterative methods in the literature. The second part (Chapter 7) of this thesis deals with developing iterative algorithms and introducing some nonlinear mappings in the framework of the Hilbert and Banach spaces. First, we present a modified (improved) generalized Miteration with the inertial technique for three quasi-nonexpansive multivalued mappings in a real Hilbert space. In addition, we present some fixed point results for a general class of nonexpansive mappings in the framework of the Banach space and also proposed a new iterative scheme for approximating the fixed point of this class of mappings in the framework of uniformly convex Banach spaces. Finally, we apply our convergence results to certain optimization problems, integral equations, and we present some numerical experiments to show the efficiency and applicability of the proposed method in comparison with other existing methods in the literature.
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 = Uphenyo ngezifundo eziningi ezivulelekile ze-inthenethi ehlanganisiwe yokufunda nokusebenza kwabafundi.
(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. Iqoqa. Izifundo Eziningi Ezivulelekile Ze-inthenethi (ama-MOOC) zivele eminyakeni yamuva nje futhi zivuse izinkulumo ezibalulekile mayelana nemfundo. Ama-MOOC asezinze aqina emkhakheni wezemfundo futhi abhekane nezimo ezihlukahlukene, okuhlanganisa ukuvela kwama-MOOC ahlanganisa izifundo ezisekelwe ekhampasini nama-MOOC, okuholela ekuhlangenwe nakho okujulile kokufunda. Ama-MOOC ahlanganisiwe anika amandla ukusetshenziswa kwezifundo ezihlukahlukene ezixubile, ezingasetshenziswa ukulungisa izinselelo eziningi ezibhekene nemfundo ephakeme eningizimu yomhlaba, njengokuthi 1) ukufinyelela okulinganiselwe, 2) ukuntuleka kwengqalasizinda kanye nesasetshenziswa, 3) izinsiza ezilinganiselwe, 4) abafundi ukubhaliswa okungaphezu kwalokho obekuklanyelwe ukukuthatha, 5) ukuminyana emahholo okufundisa, 6) ukushoda kwabafundisi, ikakhulukazi izifundiswa eziphezulu ezimatasa nokucwaninga, 7) izinga eliphansi lemfundo ukuze kuhlangatshezwane nezidingo zabasebenzi banamuhla kanye 8) nokungafani phakathi kwamakhono abafundi abathweswe iziqu. Lolu cwaningo luphenya ubudlelwano phakathi kwama-MOOC ahlanganisiwe kanye nokufunda kwabafundi nokusebenza kwezemfundo ngokuhlaziya umthelela wokwamukela ukusetshenziswa kwabafundi nokuxhumana nabo kanye nokuhlola izici ezithinta ukusetshenziswa kwabo kwangempela, ukuzibandakanya, ukwaneliseka kanye nokusebenza kwezemfundo kusetshenziswa inhlanganisela yomphakathi obuyekeziwe (CoI) okuhlanganisa ukuba khona kokufunda kanye nenjulalwazi Ehlanganisiwe Enwetshiwe Yokwamukela Nokusetshenziswa Kobuchwepheshe (UTAUT 2). Ucwaningo olunobufakazi lwenziwa nabafundi abangu-2875 abavela enyuvesi yaseGhana, abakhethwe ngesampula yezigaba ezimbili ukuze kufezwe le nhloso. Abaphendulile bekungabafundi bezinga le-100, 200, 300 kanye ne-800 ababeyalelwe ukuthi babhalise kuma-MOOC njengezinsiza zemfundo ezivulekile (ama-OER) ukuze bangezelele ekhempasini yabo ngesemester. Ekupheleni kwesemester, abafundi bacelwa ukuthi bagcwalise uhlu lwemibuzo. Kamuva, idatha eqoqiwe yahlaziywa kusetshenziswa isoftware ye-SmartPLS v4 kanye ne-SPSS v28. Ngokuvamile, imodeli ibonisa ukuqondana okuqinile nedatha eqoqiwe futhi inezinga elanele lokusebenza kahle ekucaciseni ukwenzeka kokwamukela ukusetshenziswa kwabafundi (i-UTAUT 2) yama-MOOC ahlanganisiwe kanye nomthelela wawo ekubandakanyekeni kwe-CoI, ukwaneliseka nokusebenza kwezemfundo. Okutholakele kunemithelela eminingana yocwaningo kanye nezinqubo zokuhlanganisa ze-EdTech phakathi kwezizwe eziseNingizimu yeSahara, nazo okwaxoxwa ngazo. Ngakho-ke, lolu cwaningo lunikeza iminikelo ebalulekile kuthiyori nokusebenza kokufunda okusebenzayo endaweni ehlanganisiwe ye-MOOC.
Exploration of Grade 11 learners’ mental constructions and difficulties in learning and solving trigonometric equations: a case of one school in Umlazi district = Ukuhlolwa kwezincazelokuzakhela engqondweni kanye nezinkinga zokufunda nokuxazululwa kwama-equation esifundo seminxamithathu kubafundi beBanga 11: ucwaningo kwesinye sezikole esifundeni saseMlazi.
(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. Iqoqa. Isifundo seminxamithathu siyingxenye eyinselelo enkulu ezibalweni ze-mathematics kubafundi bamabanga aphezulu. Lolu cwaningo lwahlola izincazelokuzakhela engqondweni kanye nezinkinga uma abafundi beBanga 11 befunda nalapho bexazulula izinkinga zesifundo seminxantathu eNingizimu Afrikha. Ababambiqhaza bakhethwa kulabo abasohlelweni lokufunda izibalo emva kokuphuma kwesikole abaziyela kulo ngokuzinikela. Kwasetshenziswa ishidi lemisebenzi kanye nezingxoxo ezingahlelelwe ukuze kuqoqwe imininingo. Imininingo eqoqwa ekuqaleni kocwaningo yatholakala kubafundi abayi-17 beBanga le-10 kusetshenziswa ishidi lemisebenzi; imininingo yaqoqwa kubo labo bafundi ngonyaka olandelayo kusetshenziswa ishidi leminye imisebenzi elinezibalo ezihlukile. Kwabanjwa izingxoxo ezingahleliwe nabafundi abayi-7 ukuthola izimpendulo zabo mayelana neshidi lemisebenzi. Iningi lalaba bafundi kwatholakala ukuthi abakwazi ukuqhamuka nezincazelokuzakhela emakhanda ukuze bakwazi ukuxazulula izinkinga zesifundo seminxantathu ezingeni lomsebenzi weBanga 11. Ukuhlaziya izincazelokuzakhela ezingqondweni zabafundi kwasetshenziswa injulalwazi ye-constructivist APOS kaDubinsky (1991) echaza ukuthi abafundi balwakha kanjani ulwazi lwabo lwemiqondomsuka yezibalo ngokwamazinga/ngokwezigaba ahambisana nokwenza, inqubo, okufundwayo/okuxazululwayo, kanye ne-schema. Kulezi zigaba ezimbili zokuqoqwa kwemininingo kwatholakala ukuthi abafundi abaningi bathembela ekubaleni isinyathelo ngasinye ukuze baxazulule inkinga. Lokhu kukhombisa ukuthi basezingeni lokwenza; abafundi abambalwa bakhombisa ukukwazi ukulandela izinyathelo ezithile ngekhanda bengabhalanga phansi, okukhombisa ukuthi babethuthukile kancane sebesezingeni lenqubo. Abutholakalanga ubufakazi obuveza ukuthi abafundi base bethuthukele emazingeni okufundwayo noma e-schema. Okunye futhi ukuthi imiphumela yakhombisa ukuthi abafundi balandela izinyathelo ngendlela efanele; basebenzisa imithetho ngaphandle kokunikeza izizathu. ukuhlaziya izinkinga ezithikameza ukwakha ezingqondweni izincazelokuzakhela zemiqondomsuka. Izinkinga zabafundi zazihlanganisa ukungaqondisisi ngendlela uphawu lokulingana, imithetho yokucacisa ngokwedlulele, kanye nokwehluleka ukuhlanganisa imiqondomsuka ye-algebra nokusebenzisana kwayo neyesifundo seminxantathu. Ngokwemiphumela yocwaningo kuphakanyiswa ukuthi othisha baqinise amakhono e-algebra ayisisekelo – njengokuhlanganisa amatemu afanayo kanya nangafani, ukusetshenziswa kwabakaki, kanye namatemu e-algebra okuhlanganisa nokusuza – ngaphambi kokuqalisa ngemiqondomsuka yesifundo seminxantathu. Bayanxuswa othisha ukuba bahlole izindlela ezihlukahlukene zokufundisa ama-equation esifundo seminxantathu ukuze abafundi bakwazi ukuzakhela ulwazi ngendlela efanele. Lezi zindlela kungaba ukufunda ngokubambisana noma ukunikezwa kwemisebenzi ehlukene ngokwamazinga abafundi.
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.

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