Masters Degrees (Mathematics and Computer Science Education)

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Exploring lecturers’ interpretation and implementation of the intended NCV mathematics curriculum at a TVET college in KwaZulu- Natal Durban.
(2022) Ngidi, Lungile.; Mthethwa, Themba Mngomeni.
The aim of this study were to explore lecturers’ interpretation and implementation of the NCV Mathematics curriculum at a TVET college in KwaZulu Natal Durban. The researcher employed an interpretivist paradigm using a qualitative research approach so as to allow greater understanding of the lecturers’ interpretation and implementation of the curriculum. Data was generated using document analysis and semi-structured interviews. Document analysis was carried out in order to determine what should be taught in NCV Mathematics classroom, and how it should be taught. The semi-structured interviews addressed questions about lecturers’ understanding of the curriculum, their views about the purpose of the curriculum and the challenges they encounter implementing the curriculum. Due to COVID 19 pandemic, interviews were held online via Zoom. The analysis of the curriculum document revealed that NCV Mathematics curriculum is a structure of applied Mathematics, including problem solving and Mathematical modelling, with various contexts including real-life context. It is an integrated-type of curriculum that encourages learnercentred approach. The interviews have shown that lecturers harbour different views about the curriculum, where lecturers reported encountering challenges during its implementation. NCV Mathematics emerges as something unclear in terms of its purpose. Most of the lecturers argue that they don’t understand the purpose or the intentions of the curriculum designers, since this involves aspects that are difficult for students.
A case study: the use of GeoGebra to alleviate learner difficulty in learning the similarity of triangles in a South African grade 9 classroom.
(2022) Mpanza, Nompumelelo.; Shongwe, Michael Bafana Mthembiseni.
The mixed methods study investigated the use of GeoGebra as a dynamic geometry software (DGS) to alleviate the learning similarity of triangles. A mixed methods philosophical framework in the form of an exploratory case study was used to conveniently and purposively select a sample of 60 Grade 9 learners enrolled at Sondelani Full-Service School (pseudonym), a township school in the Pinetown district in KwaZulu–Natal province, South Africa. During this research, GeoGebra and the concept of similarity of triangles were introduced to the participants. Then, participants answered several (Euclidean geometry) Similarity Achievement Test (SAT) questions prescribed by the National Mathematics pacesetter for Grade 9 and 10. A 10-item Likert scale questionnaire intended to elicit participants’ attitudes about GeoGebra and its impact on Euclidean geometry and mathematics was administered to these participants. The questionnaire also included four open-ended items, asking participants to reflect on the application of GeoGebra. The analysis of SAT data revealed that performance was higher after GeoGebra instruction (𝑀 = 22.50) than during traditional instruction, which did not feature GeoGebra (𝑀 = 11.65). Thus, it was found that the use of GeoGebra is an appropriate tool to increase achievement in learning geometry concepts; to promote accuracy, visualization; learner participation; and to create enjoyment and learner interest towards learning mathematics. It is recommended that mathematics teachers need to use GeoGebra for effective teaching and learning of similarity of triangles.
Grade 10 learners’ experiences of the teaching strategies in algebra used by their mathematics teachers: a case study of two schools in the Pinetown District.
(2022) Amaefule, Remigius Nnadozie.; Goba, Busisiwe.
This study makes use of social constructivism theory as the lens to explore Grade 10 learners’ experiences of learning algebra in two selected High Schools in one township in Pinetown Education District, KwaZulu-Natal Province. The purpose was to examine learners’ perceptions of strategies that their teachers use in teaching Grade 10 algebra and the way it supports or inhibits their learning of the topic. algebra is considered to be an important topic in mathematics. In High School mathematics curriculum, algebra is particularly important at Grade 10 because, it is at this time the Grade 10 learner moves to higher concepts and advance in their learning of algebra. However, observations indicated that among other areas that posed challenges to High School learning of mathematics in the district, learners’ perception of algebra as difficult topic was a major factor. This prompted interest in the present study that used qualitative research approach and a case study design to explore in-depth, the learners’ experiences in order to better understand the issues in teaching and learning Grade 10 algebra in the township schools. Questionnaires were developed for purposes of individual interviews given the limitations the COVID-19 pandemic restrictions imposed on conducting physical contact interview and focus group methods originally designed. Hence the remote use of qualitative questionnaire instruments was employed to collect the data. Participants were purposively selected through written correspondence assisted by school administration in both schools. The sample comprised of 20 Grade 10 learners, 5 male and 5 female learners, each from the two schools. Choice of the schools was decided based on convenience because of their nearness to the researcher. The data was analysed using thematic analysis and triangulated at data source level. Findings show that some of the Grade 10 learners found algebra as a boring topic because they do not understand the algebraic concepts due to the teachers teaching approach. It was also found that these experiences impacted them negatively and positively in their perceptions of learning algebra. The findings implications for teaching Grade 10 algebra are that (a) given that element of algebra is important in high school mathematics, finding ways of improving teachers’ ability to use effective strategies to teach for learners’ understanding of the topic at this crucial level is important for the township schools, (b) in addition, in the teaching algebra concepts, teachers should consider how to effectively differentiate for those learners that struggle to understand the teaching in English language. Deliberate efforts should be made to explain concepts in ways, and using examples and in language levels that connect with these learners. Therefore, this study recommends the following: (a) the Grade 10 teachers are encouraged to always use variations of teaching strategies to teach the Grade 10 algebra so that learners can appreciate the learning; (b) Home language need to be involved in teaching mathematical concepts at Grade 10 curriculum so that both languages, i.e., home and second language would be simultaneously used to enable the learners consolidate their learning.
On the geometry of locally conformal almost Kähler manifolds.
(2020) Khuzwayo, Ntokozo Sibonelo.; Massamba, Fortuné.
In this work, we study a class of almost Hermitian manifolds called locally conformal almost Kähler manifolds. These are almost Hermitian manifolds which contains an open cover fUtgt2I and a family of C1 functions ft : Ut ! R such that each conformal metric gt on Ut is an almost Kähler metric. Locally conformal almost Kähler manifolds also falls under a class of locally conformal symplectic manifolds. More precisely, locally conformal almost Kähler manifolds are manifolds whose fundamental 2-form is locally conformal symplectic. We first recall some of the existing geometric properties of almost Hermitian manifolds. Then further use these properties to derive those of locally conformal almost Kähler manifolds. A new example of a locally conformal almost Kähler manifold is given. We further investigate the relationship between the covariant derivative and the Nijenhuis tensor on a locally conformal almost Kähler manifold. The equivalence of the Nijenhuis tensor de ned on each Ut and the one de ned globally is also proven. The relationship between the curvature tensors induced by the two conformal metrics on a locally conformal Kähler manifolds are considered. In particular, we show that a locally conformal almost Kähler manifold is an almost Kähler manifold under some curvature conditions. To achieve our goal, we first prove the relation between scalar curvatures and together with the corresponding scalar-curvatures and of a locally conformal almost Kähler manifold. Moreover, among other results, we also investigate the canonical foliations of locally conformal almost Kähler manifolds. Accurately, we give necessary and sufficient conditions for the metric on a locally conformal almost Kähler manifolds to be a bundle-like for foliations F.
On the numerical solution of the Lane-Emden, Bratu and Troesch equations.
(2020) Sithole, Phumla Remember.; Sibanda, Precious.; Goqo, Sicelo Praisegod.
Many engineering and physics problems are modelled using differential equations, which may be highly nonlinear and difficult to solve analytically. Numerical techniques are often used to obtain approximate solutions. In this study, we consider the solution of three nonlinear ordinary differential equations; namely, the initial value Lane-Emden equation, the boundary value Bratu equation, and the boundary value Troesch problem. For the Lane- Emden equation, a comparison is made between the accuracy of solutions using the finite difference method and the multi-domain spectral quasilinearization method along with the exact solution. We found that the multi-domain spectral quasilinearization method gave a better solution. For the Bratu problem, a comparison is made between the spectral quasilinearization method and the higher-order spectral quasilinearization method. The higher-order spectral quasilinearization method gave more accurate results. The Troesch problem is solved using the higher-order spectral quasilinearization method and the finite difference method. The solutions obtained are compared in terms of accuracy. Overall, the higher-order spectral quasilinearization method and multi-domain spectral quasilinearization method gave the accurate solutions, making these two methods to be the most reliable for these three problems.
On multivariate overlapping grid spectral quasilinearization methods for problems in cavity flow.
(2022) Nzama, Sibonelo.; Sibanda, Precious.; Goqo, Sicelo Praisegod.
We investigate fluid flow in cavities with different boundary conditions. Three cavity flow problems of varying complexity are investigated in this study. In the first problem, a flow filled with a porous medium, and with adiabatic and impermeable walls is considered. The left wall is heated. For the second problem, we investigate free convection in an enclosed square with porous medium and nanofluid. We assume that the side walls have a high fixed temperature and a lower fixed temperature for the horizontal walls. The third problem is more complex, and it involves investigating a square enclosure with porous medium, a top moving wall, and the side walls heated with a sinusoidally varying temperature. We analyze the effect of fluid parameters on the fluid flow characteristics such as the streamline distribution, isoconcentration, isotherms, local Nusselt number, skin friction, and the local Sherwood number. The flow equations are solved using two recent numerical techniques, namely the multivariate overlapping grid spectral quasilinearization method (MOGSQLM) and the multivariate spectral quasilinearization method (MSQLM). The MOGSQLM is an extension of the MSQLM with improved accuracy. Using the two methods we determine the solution, the residual solution errors and the computational time to achieve a converged solution. The MOGSQLM is found to be more accurate, and for this reason, only the MOGSQLM is used to numerically solve the third problem. The MOGSQLM was found to be the better method in terms of convergence, accuracy, and CPU time. The changes in the Rayleigh number alter the flow pattern from circular to elliptic with stronger circulation in the core region.
On zero-dimensionality of remainders of some compactications.
(2022) Nogwebela, Gugulethu Manase.; Mthethwa, Simo Sisize.
A compactication of a topological space is a dense embedding of the space into a compact topological space. We study dierent methods of compactifying a topological space with the focus on zero-dimensionality of the remainder. Freudenthal compactication is known as a maximal compactication with a zero-dimensional remainder and is guaranteed to exist for rim-compact spaces. It is shown that this compactication can be characterized using proximities. In fact, there is a one-to-one correspondence between compactications and proximities and, in particular, between compactications with zero-dimensional remainder and zero-dimensional proximity. Almost rim-compact spaces are spaces that are larger than the rim-compact spaces and they are shown to also have a compactication with a zero-dimensional remainder. But these do not exhaust spaces that have a compactication with a zero-dimensional remainder, for example, recently it was found that spaces that lie between the locally compact part and its Freudenthal compactication also have a zero-dimensional remainder. It is known that the Freudenthal compactication is also perfect, we study the relationship between maximum compactications with a zero-dimensional remainder and the perfectness of these compactications.
Exploring Grade 9 mathematics learner performance in schools that participated in the 1+9 programme.
(2021) Ngcobo, Edward Sazi.; Ngcobo, Annatoria Zanele.
The main aim of this research was to explore Grade 9 learners’ mathematics performance in schools where teachers participated in the 1+9 programme, which was designed to equip teachers with strategies to teach mathematics in the Senior phase(Grade 7-9). This study is guided by the qualitative method within an interpretive paradigm. The conceptual framework adopted by the study is a compressive model of the education system (Shalveson, 1989). This study took place in three schools selected from uMgungundlovu District in KwaZulu-Natal, using two data collection methods, namely the document review where the researcher took the marks schedules of the grade nine learners for the year 2016, 2017 and 2018. The researcher then randomly selected that marks of 42 learners in each of the three school, put them in tables and graphs to compare the performance in each of these years. In the second method the researcher conducted the interviews with ten learners and three teachers that were part of the 1+9 programme. Guided by the aim of the study, the following objectives underpinned this research: (i) To determine the extent of improvement in learner performance in mathematics post the 1+9 programme; and (ii) To determine the Grade 9 learners’ competence to solve mathematical problems after their teachers took part in the 1+9 programme. In order to unpack the objectives and carry out an in-depth analysis of the phenomena of this study, which is exploring Grade 9 learner performance post the introduction of the 1+9 programme, throughout this study the researcher asks the following research questions: (a) How did the implementation of 1 + 9 programme improve learner performance in mathematics at grade 9 level?and (b) What do Grade 9 learners’ written responses reveal about their competency in mathematics after the implementation of the 1+9 programme? The research findings reveal that the development programme for teachers must be continuous, to ensure that they are up to date with the current status of mathematics, so that they will implement the current knowledge for teaching. The findings that have resulted from this study will serve to recommend to professional development organisations, the Department of Basic Education and other stakeholders that professional development of teachers takes place to ensure that they use the trackers, lesson preparation plans and pace setters, and have a common assessment each term in Grade 9, to ensure that the teaching and learning of mathematics is taken seriously. The findings of this research could serve to encourage the Department of Education and the Departmental Head to carry out thorough monitoring of the educators’ work, to ensure that it is done properly.
Exploring pre-service teachers’ understanding of similarity and proofs in Euclidean geometry.
(2022) Mbatha, Mduduzi Mhlengi.; Bansilal, Sarah.
This qualitative study explores pre-service teachers’ understanding of similarity and proofs in Euclidean geometry in one South African university from KwaZulu-Natal (KZN) province. Such insight is vital for addressing pre-service teachers’ geometric knowledge, which has been found lacking. The research participants were 34 pre-service teachers (PSTs) in their first year of study towards a Bachelor of Education degree specialising in mathematics at the FET phase. A pen and paper test and semi-structured interviews were employed in gathering the required data for this study. The Van Hiele levels of geometric thought were used as a theoretical framework, which formed the basis for the analysis and discussion of findings. The findings indicated that most pre-service teachers performed adequately on familiar items but struggled with those unfamiliar, which were not typical grade 12 examinable questions. A follow-up of semi-structured interviews was conducted with seven PSTs of mixed abilities to probe the originality of their written responses. Although all interviewed PSTs indicated an improvement when responding to research items verbally than in writing, they did not reach the expected acquisition necessary to teach geometry effectively. Overall, this study found that many PSTs displayed poor levels of understanding similarity and proofs, including (1) limited understanding of the definition of similarity to triangles; (2) poor understanding of how to prove two figures are similar; (3) the haphazard use of geometric theorems in devising proofs; (4) a display of higher Van Hiele levels of understanding for familiar items but lower levels of understanding for unfamiliar items. These findings raised concerns about this group of PSTs teaching geometry, especially if certain concepts require more complex skills that are slightly beyond the secondary school curriculum. It is recommended that professional teacher education training offered to pre-service teachers should include aspects such as (1) Improving PSTs’ geometry content knowledge, (2) Teaching geometry for understanding and (3) Improving PSTs’ written mathematical responses. These factors may be pivotal in improving pre-service teachers’ geometric knowledge beyond the scope of the secondary school curriculum.
Exploring the effectiveness of parent engagement in the teaching of foundation phase geometry.
(2021) Hopkins, Siobhan Kerry.; Mudaly, Vimolan.
Learners need mathematics to fulfil academic and vocational dreams, to learn to think in a particular manner and to survive in a world where so many mathematics skills are prevalent. Learners show improvement when they understand the mathematics they are doing, and it is not merely seen as a set of rules. When you connect the dots meaningfully for the learners using diagrams, technology, physical objects, and everyday examples, they start to really understand and problem solve. However, in my experience, the one area where many learners were not showing significant improvement and seem to lack understanding, was geometry. This sparked a sincere interest in studying the cause of the geometry struggle and means of addressing it. The more I looked into it, the more research was pointing to the fact that geometry understanding has to start at a foundation level. You cannot expect learners to engage in complex geometric proofs involving difficult deductive reasoning when they do not know and understand the basics leading up to this. Thus, this master’s thesis explores the teaching of foundation phase geometry and how intervention can happen at the grass roots in order to see long term benefits. One of the essential ingredients in developing correct concept formation at foundation phase, is having access to hands on activities through adult-guided play. The reality in South Africa, is that the ratio of learners to teachers is too high to allow this to happen in a meaningful way in the classroom. Too little time is assigned to geometry in the foundation phase curriculum as more important numeracy concepts and learning to read and write, are prioritized. However, lockdown brought to the foreground, the important role that parents can play in improving the education of children. Although not all parents were effective teachers, surprisingly, many very effective in assisting in the educational process of their child when asked to do so. This study therefore looks at parents’ input as an interventive means of assisting in the process of teaching foundation phase geometry. Although we all know the ideal solution is a highly qualified teacher in a small classroom with all the necessary resources, this is only a reality for about 3% of the South African population. This study is seeking intervention for the other 97% who do not have the privilege of the ideal. This study was a case study using qualitative methodology. Video analysis of one-on-one time with parents and their children was used to analyse the effectiveness of parent involvement. Teacher interviews were used to assess how space and shape is currently being taught and parent questionnaires were used to gather data on how parents felt about being involved in helping their child with space and shape learning. This study showed that through simple communication with parents, regardless of what socio-economic background they came from, effective activities can be designed to bring about meaningful scaffolding in geometry learning. Although video-analysis revealed very positive findings, parents felt that many other parents would not help their own children due to circumstantial constraints.
Teaching computer applications technology in an under-resourced school: a teacher’s self-study.
(2021) Memela, Balungile Pinky.; Masinga, Lungile Rejoice.
My personal history self-study research focus was on teaching Computer Applications Technology in an under-resourced school. This study aimed to ascertain what I can learn from my personal history, to improve my CAT teaching practice in an under-resourced school by exploring effective teaching strategies. I was frustrated by my CAT learners, who were performing poorly in CAT due to the ineffective implementation of the CAT curriculum, thus producing the poor quality of CAT results. Adopting a Sociocultural perspective on teaching and learning assisted me in understanding that learning is constructed through social and cultural interaction and learners learn more efficiently by using tools to solve problems by using resources available to them in their environment. My first research question was: What can I learn from my personal history about teaching Computer Applications Technology in an under-resourced school? This question allowed me to revisit and re-examine my learning and teaching experience from when I first fell in love with technology at home and through doing practical subjects in high school and college studying computer studies. And as a novice teacher, I have a better understanding of strategies I can employ to improve my teaching of CAT in an under-resourced school. Throughout this self-study research journey, I worked closely with my critical friend, who was also a master’s student. I used various data sources to generate data for this self-study, such as my reflective journal, photographs, memory drawings, and a collage. My second research question: How can I improve my teaching of Computer Applications Technology in an under-resourced school? To respond to this question, I worked with my college friend, who is also my colleague, deputy principal, and a former CAT learner, as my participants. The different discussions and activities we had for this study helped me understand how they perceived and received my CAT teaching in an under-resourced school. I used multiple self-study practices that helped me generate data for my research. I used artefact retrievals such as photographs, a stiffy disk, and collage making. From my personal history selfstudy and contribution from my participants, I identified four significant learnings regarding teaching and learning of CAT in an under-resourced school: (i) CAT teacher as a resource manager, (ii) Time allocation according to policy document versus disadvantaged school and (iii) Collaborative learning as a response to limited computer resources. As a CAT teacher, I learned that I serve as a human resource that can connect other resources through interacting and collaborating with others to help me facilitate valuable teaching support to learners.
Exploring the challenges experienced by Grade 8 learners when learning angles associated with parallel lines in geometry.
(2021) Dlamini, Xoliswa.; Naidoo, Jayaluxmi.
This research study explored challenges experienced by Grade 8 learners when learning angles associated with parallel lines in geometry. This research study was conducted in a school situated in the city of Pietermaritzburg, KwaZulu-Natal, South Africa to understand factors that shape learner’s experiences towards learning angles associated with parallel lines in geometry. This research study was framed within the social constructivist theory which guided and examined the systematic explanation of the experiences of Grade 8 learners when learning angles associated with parallel lines in geometry, by following certain principles outlined by the social constructivist theory. This research study was situated within the interpretivist paradigm and focused on an explanatory qualitative case study to, explore learners’ experiences and get an in- depth understanding of the challenges they are experiencing when learning angles associated with parallel lines in geometry. This research study used inductive data analysis and evaluation process to sort data generated from the 36 purposely selected Grade 8 learners. The process of data analysis and evaluation was done through the use of Curriculum Assessment Policy Statements (CAPS document), questionnaire, worksheet and the semi-structured interview schedule. The findings of this research study confirmed that there are challenges experienced by 8 learners when learning angles associated with parallel lines in geometry. It was recommended that this research study be conducted at a larger number of diverse schools across the country, to explore more challenges experienced by Grade 8 learners when learning angles associated with parallel lines in geometry.
An exploration of a visualization intervention in a Grade 7 mathematics classroom in the Pinetown District.
(2021) Bakare, Onozare Mercy.; Mudaly, Vimolan.
With the growing research on visualization in mathematics, it is important to understand how visualization intervention strategies impact learners' solving and success of mathematical word problems. This study focused on exploring a visualization intervention in a grade 7 mathematics classroom in the Pinetown District of KwaZulu-Natal. The methods used by learners and their effectiveness in solving word problems were investigated as this formed the basis of this study. Their understanding of the methods and strategies chosen was revealed through an interview, leading to a visual intervention on how they (learners) can become better visualizers. Boonen, Van der Schoot, Van Wesel, De Vries, and Jolles, (2013, p. 57) asserted that the difficulties learners encounter in solving word problems emancipate from lack of understanding of the problem text, identifying solution-relevant components, the relations between them, and making a complete and clear representation of the situation described in the problem. Good problem solvers ought to have a good understanding of the text and strategies required for every given problem, and for this to take place, learners are required to think visually. Visualization and its importance in mathematics or in solving mathematical word problems cannot be overstressed. It is a skill that learners ought to possess to become good problem solvers. Therefore, it is not enough for learners to possess these skills and form visual images, but they also should be able to use the skills when required and for analytical reasoning. Hence, the reason for conducting this study is for learners to be taught these skills and strategies through an intervention process and determine the effectiveness of the intervention given to them. Data was gathered using a qualitative research method. An interpretative approach was used, which helps to understand what is being understudied. Learners were given word problems to solve, and a one-on-one open-ended interview was conducted on randomly selected learners from the class. This research was conducted in the naturalistic setting of theparticipants; the sample was purposive and convenient. The conclusion drawn from the investigation findings has shown that learners do not have a natural inclination to use diagrams or any visualization form before the intervention strategies were introduced. Secondly, evidence suggested that learners' strategies in completing the initial task administered were not all effective. Finally, there was a significant improvement in learner’s performance, their use of visuals, and the accuracy of their methods after the intervention process.
An exploration of the extent Grade 9 mathematics teachers engage with learners’ errors in the teaching and assessment of mathematics.
(2021) Frimpong, Tawia Iddrisu.; Ndlovu, Annatoria Zanele.
This study explores the extent to which Grade 9 mathematics teachers engage with learners’ errors in the teaching and assessment of mathematics. This qualitative study adopted an exploratory case study design and an interpretivist focal lens. The sampling was purposive, and the participants were three Grade 9 mathematics teachers, one from each of three high schools in the Harry Gwala District of KwaZulu-Natal province. The instruments used to generate data were semi-structured interviews and classroom observation. The findings of the study revealed that the level or the extent to which teachers engage with learners’ errors depends on 1) teachers' understanding of remedial teaching, which informs 2) teachers’ ability to deal with learners’ errors, and 3) teachers’ mathematical knowledge, which is the content and pedagogical knowledge of teaching mathematics. Teachers also engage with learners’ errors for the following reasons: 1) to provide remedial teaching, 2) to correct learners' mistakes or errors, 3) to provide feedback to learners, and 4) to promote peer learning. The findings of the study further revealed that teachers have limited time to engage with learners’ errors, since they are time-bound to curriculum coverage or to finish the Annual Teaching Plan However, it is important to note that the teachers who participated in the study do not have much knowledge about remedial teaching; therefore, they do not depend on or use remedial teaching to engage with learners’ errors. The rationale for why teachers engage with learners’ errors that emerged from the study includes to correct learners’ errors, to provide feedback to learners, and to enhance remediation. It is recommended that teachers must be educated on remedial teaching, how it is done, and its importance in helping learners to learn better. Also, the Department of Basic Education should design the Annual Teaching Plan to include remedial teaching for at least two to three hours, to be carried out before teaching moves on to the next topic. These three hours can be split further into lessons where a teacher can spend at least 10–15 minutes to explain errors before moving on to the next concept.
Exploring teachers’ use of visualisation tools when teaching Grade 9 problem-solving in mathematics. a case of Umlazi District Dinaledi schools in South Africa.
(2020) Shoba, Makhosazana Faith.; Naidoo, Jayaluxmi.
This study focused on exploring Grade 9 mathematics teachers’ use of visualisation tools when teaching problem-solving in their classrooms. This issue has been a challenge in South Africa, particularly in matric and grade 9 mathematics Annual National Assessment. The use of visualisation when teaching problem-solving in the mathematics classroom has been viewed as critical to learner’s performance, in response to the abstract nature of mathematics. However, problem-solving and the importance of the use of visualisation is emphasised in the Curriculum Assessment Policy Statement for the Senior Phase. Moreover, it is also included in every topic of the learners’ Grade 9 mathematics workbook for everyday classroom activities. Therefore, this study aimed to answer the questions of what visualisation tools teachers use and how they use these when teaching problem-solving. Lastly, why do they use them during their lesson in their classroom? Polya’s 4-step problem-solving and Activity theory was used as a theoretical framework for this study. A qualitative case study of two Dinaledi Comprehensive Technical High School in Umlazi District was conducted to explore the use of visual tools by five grade 9 mathematics teachers during their teaching of problem solving. Teacher’s questionnaire, classroom observations, and semi-structured interviews for teachers were used to generate data. The findings revealed that mathematics teachers do teach problem solving in their Grade 9 classrooms as stated by the policy document. However, the use of visualisation tools in the mathematics classroom seems to be infrequent. Therefore, the teachers highlighted the lack of resources and understanding of what problem-solving is, as a challenge to their use of visualisation tools. However, the study suggested that the department of Kwa-Zulu Natal education should provide in-service training for Grade 9 teachers on the effective use of visualisation tools when teaching problem-solving. It was also suggested that schools should provide resources that can enhance problem solving, and mathematically related resources for their mathematics lessons. It was further suggested t schools to have a mathematics classroom, which will provide a mathematics atmosphere with relevant mathematics resources for effective and efficient teaching and learning of mathematics.
Application of bivariate spectral quasilinearization method to second grade fluid flow equations.
(2020) Dlongolo, Simphiwe Gloria.; Sibanda, Precious.; Goqo, Sicelo Praisegod.
In this study, the steady flow of a second grade magnetohydrodynamic fluid in a porous channel is investigated. We further investigate the hydromagnetic flow of a second grade fluid over a stretching sheet. The partial differential equations that describe the flows are solved numerically using the bivariate spectral quasilinearization method. The method is extended to a system of non-similar partial differential equations that model the steady two dimensional flow of Falkner-Skan flow of an incompressible second grade nano fluid. The work is also concerned with heat and the mass transfer from the electrically conducting second grade magnetohydrodynamic fluid over a stretching sheet. The sensitivity of the flow characteristics with respect to the second grade fluid parameter, magnetic field parameter, thermal radiation parameter, and the chemical reaction parameter are investigated. The accuracy of the numerical method is determined using the residual error analysis.
An exploration of the integration of technology by mathematics teachers: the case of 10 schools in KwaZulu-Natal under Umlazi District.
(2020) Zulu, Mzwandile Wiseman.; Mudaly, Vimolan.
This interpretive qualitative study sought to explore the integration of technology by mathematics teachers in Umlazi district of KwaZulu-Natal province. A purposive non-probability sample of ten teachers from ten different schools in Umlazi district participated in the study. Data was collected from the participants using a questionnaire, interview, classroom observation schedule and document analysis, while the thematic analysis method was employed to analyse the data. The findings of the study revealed that the South African Department of Basic Education advocates for the integration of technological tools for all teaching and learning processes in basic education. However, the study showed that teachers underutilise the technological resources they have at their disposal and that they mainly rely on the teaching of mathematics using traditional methods of teaching. The findings established possible factors contributing to the under utilisation of technology by mathematics teachers in Umlazi district. These factors include: (1) lack of training to confidently integrate technology; (2) lack of technology pedagogical content knowledge; (3) limited access to technological tools; (4) crime (break-ins); (5) overcrowded classrooms; (6) lack of technical support for software updates; (7) electric power failure during the teaching and learning hours; and (8) lack of exposure to government policies advocating for technology integration in teaching. Lastly, the study found that though teachers are not actively utilising the technological tools in the teaching of mathematics, they however demonstrated a positive attitude towards its use. The results further showed that if the factors above are well addressed, the mathematics teachers in Umlazi district would be utilising their tools effectively in their teaching practices.
Exploring Grade 11 mathematics learners learning of trigonometric identities: a case study of one school in KwaZulu-Natal.
(2019) Khuzwayo, Nontobeko Zamanguni.; Naidoo, Jayaluxmi.
The purpose of this study was to explore the learning of trigonometric identities in grade 11 mathematics. The study sought to discover and understand the learning of trigonometric identities by grade 11 mathematics learners. This study was guided by qualitative methods, framed within the interpretive paradigm and employed case study methods. The study was underpinned by the principles of Kilpatrick’s strands of mathematical proficiency. In the study, ten grade 11 mathematics learners were purposively selected at a high school in the province of KwaZulu-Natal, South Africa. The study generated primary data through semi-structured interviews and an activity worksheet. In interpreting data that was collected, the researcher applied both inductive and deductive approaches to data analysis. Thematic data analyses methods were employed for analysing data. The findings in this case study indicated that learners had an understanding of the fundamental trigonometric identities, and were able to prove these identities making use of the unit circle. In addition, the results showed that learners experience challenges when they have to apply algebraic manipulations in the learning of trigonometric identities. The findings of this study were similar to other research studies regarding the learning of trigonometry and how learners experience difficulty in this topic. Hence, making use of the unit circle when trigonometric identities are introduced was seen as a tool to develop conceptual and procedural understanding of mathematical concepts. Also, the use of knowledge from other topics when teaching trigonometric identities was seen as crucial.
Exploring the factors influencing Grade 11 learner performance in Geometry in two high schools in the Umlazi District.
(2019) Zamisa, Paulos Musawenkosi.; Mudaly, Vimolan.
Performance of learners in geometry has persistently been poor compared to other sections of mathematics. Learning geometry has been identified as an area of mathematics that poses various problems for many secondary school learners. Many learners fail to develop an adequate understanding of geometrical concepts and demonstrate good reasoning and problem-solving skills. This has a negative impact on he overall mathematics results of learners. My study was aimed at exploring factors that influence learner performance in geometry and establish the strategies that can be adopted to improve performance in geometry by learners in public secondary schools of Umlazi District. The study explored(school-based, home-based, Department of Education based and learner-based) contributory factors that influence learners’ performance in geometry in secondary schools. The theory guiding the study, was social constructivism which was derived from the work of Vygotsky. Vygotsky’s social constructivism was used as a theoretical framework to understand the ways that learners acquire knowledge about groups.His theoretical framework states that social constructivist teaching, knowledge construction is achieved through learners' collective social interaction.The study used a qualitative methodology to provide rich data and compare and incorporate findings and get a holistic view of the phenomenon. The sample size constituted of16 participants.The study used focus group discussions with grade 11learnersandface-to-face semi-structured interviews with educators, Heads of Department and Principals of the two secondary schools under the Umlazi District.The qualitative data were collected by means of interviews with the participants. Thematic analysis was used to analyse qualitative data. The findings of the study led to the conclusion that the significant factors contributing to poor performance included lack of prior knowledge by learners,negative attitude toward geometry, lack of motivation,anxiety and fear of geometry,parental educational background, parental involvement, poor teaching strategies, geometry educators’ knowledge, lack of professional development of educators, class size,lack of resources and facilities.
Exploring learners’ understanding of mathematical concepts necessary in the learning of grade 11 algebraic functions: the case of three schools in uMgungundlovu District.
(2019) Ndlovu, Nkosinathi Emmanuel.; Goba, Barbara Busisiwe.
The purpose of this study was to explore learners’ understanding of mathematical concepts in the learning of grade 11 algebraic functions in uMgungundlovu district, KwaZulu-Natal. In order to gain insights into learners’ understanding of mathematical concepts in learning grade 11 algebraic functions, APOS theory was used as a theoretical lens to explore learners’ level of understanding of functions. This study describes the mathematical concepts that are important in the learning of grade 11 algebraic functions. The CAPS document was used to analyse the mathematical concepts for functions to be learnt in grade 11. The data was gathered through written tasks and interviews of grade 11 learners in three schools in one district in KwaZulu-Natal. The research approach used for this study was the mixed method. Sixty grade 11 learners (twenty in each school) were purposively selected; however, this sample selection was conveniently done since learners were able to participate in the study after school. This study employed the interpretive paradigm and nine learners (three from each school) were interviewed during data collection. Multiple methods were employed for data collection in this study. Qualitative data was organised using interview transcripts and quantitative data was organised using the APOS analytical framework. The findings of this study confirm that learners’ level of understanding of algebraic functions at an object level is extremely poor.

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