https://ir.unisa.ac.za/handle/10500/3016 2026-08-07T12:14:38Z https://ir.unisa.ac.za/handle/10500/32749 Exploring misconceptions and errors that manifest when grade 10 learners simplify tasks involving algebraic fractions: a case of selected high school in Mopani West District, Limpopo Province Mkhari, Cynthia Poor mathematics performance is a global issue, including in South Africa, with a reported decline in skills worldwide Berisha et al. (2024). Learners struggle with understanding algebraic fractions, which are essential in mathematics. Inadequate algebraic knowledge can impact learners' confidence and success in Further Education and Training, limiting their career prospects in Science and Engineering. The purpose of this study was therefore to explore the misconceptions and errors that manifest when Grade 10 learners simplify tasks involving algebraic fractions. Thirty-six (36) learners from Mopani West District in Limpopo studying mathematics were purposively sampled to participate in the study. The qualitative case study method used included data collection techniques such as a written test and semi-structured interviews. This dissertation draws upon Newman’s theory to provide a framework for understanding and explaining the phenomenon. Findings reveal that learners manipulate rules they learned in whole numbers and inappropriately apply them in algebraic fractions. This misapplication is evident in errors related to comprehension, procedural skills, transformation, and encoding. Based on the findings, the study recommends that learners should work on problems that foster connections in mathematics. A further recommendation is that learners need to have a deep understanding of the underlying principles governing the operations with fractions. Further research is necessary to explore alternative learner-centered strategies, such as inquiry-based learning and active notetaking, to assess their potential impact on enhancing learner performance in algebraic fractions. Investigating the efficacy of these approaches will provide valuable insights into their effectiveness in fostering deeper understanding and improving learner performance in mathematics education. 2025-02-01T00:00:00Z https://ir.unisa.ac.za/handle/10500/32707 Topics in point-free rings of continuous functions Wamugunda, Sharon Nyokabi Let L be a completely regular frame and RL be the ring of real-valued continuous functions on L. In this dissertation, we will study the maximal ideals and prime ideals of the ring RL which were described by Dube in [20] and [25]. We will also study the z-ideals and d-ideals of point-free function rings and how they are related. In the class of f-rings, sufficient conditions for the sum of z-ideals (resp. d-ideals) to be a z-ideal (resp. d-ideal) will be given. In [29], Dube and Ighedo show that, as in C(X), the sum of two z-ideals in RL is a z-ideal. In [64], the authors study algebraic properties of quotient rings in the ring C(X) of real-valued continuous functions on a Tychonoff space X. We show that this result is actually purely ring theoretic and thus deduce its extensions to the f-rings RL of real-valued continuous functions on a frame L. We will show that the maximal essential ideals of RL are precisely the M-ideals indexed with the nowhere dense sublocales of βL (the Stone-ˇCech compactification of L). We will also show that the nowhere dense sublocales of βL are perfect if and only if every prime ideal in RL/MA is essential, where A is a closed sublocale of βL. 2026-04-01T00:00:00Z https://ir.unisa.ac.za/handle/10500/32652 Investigating grade eleven learners’ mathematical modelling competencies in algebraic problem solving Dlamini, Reuben The National Curriculum Statement (NCS), implemented in 2003, marked the formal introduction of mathematical modelling into the South African curriculum. This represented a significant shift, as mathematical modelling had previously been largely absent from the curriculum. This study investigated the mathematical modelling proficiency of Grade 11 learners from three high schools in the Pongola Circuit, KwaZulu-Natal Province, with a specific focus on determining their competency in solving non-routine problems. The study was motivated by the observation that many teachers did not receive adequate training in mathematical modelling during their pre-service education. Existing research indicates that inadequately trained teachers can negatively affect learners’ academic performance. Furthermore, the teacher is widely regarded as the most critical agent in the successful implementation of instructional reforms at the classroom level (Shepherd, 2019; Theophile et al., 2020). It was therefore necessary to examine learners’ competencies in mathematical modelling. Learners’ competencies were assessed using the five-step modelling process proposed by Kaiser and Stender (2013), which includes: (1) understanding the problem, (2) formulating a mathematical model, (3) solving the model, (4) interpreting the results, and (5) validating the results. For a learner to be considered competent in mathematical modelling, all five stages of the process had to be correctly executed. A total of 75 Grade 11 learners from three purposively selected schools participated in the study. Qualitative data were collected through document analysis. The data analysis process involved familiarisation with learners’ written responses, followed by thematic analysis to interpret meaning, identify patterns, and generate insights related to the research questions. The findings revealed that all learners demonstrated incomplete competency in mathematical modelling. Specifically: • None of the learners successfully completed all five stages of the modelling process in any of the four test questions. • Learners did not make or attempt to make assumptions, which are essential in solving real-life problems. - 6 - • Variables were used without clear definitions, and final solutions were often expressed in terms of unidentified variables. This indicated a lack of interpretation of results within the context of the real-world problems. Interpretation involves translating mathematical outcomes back into meaningful real-life conclusions. • Learners did not verify or validate their solutions. Validation is critical for assessing the accuracy and appropriateness of both the mathematical model and its results in relation to the real-world context. The findings suggest that Mathematics teachers should be encouraged to adopt a mathematical modelling approach in teaching and learning. It can be inferred that learners had limited or no exposure to mathematical modelling, as key processes—such as defining variables, making assumptions, interpreting results, and validating solutions—were consistently omitted. According to the Curriculum and Assessment Policy Statement (CAPS), mathematical modelling should serve as a central focus of the Mathematics curriculum (Department of Basic Education [DBE], 2011). Therefore, the Department of Education has a responsibility to ensure the effective implementation of mathematical modelling, emphasising the integration of real-life contexts across all aspects of the curriculum. 2026-01-01T00:00:00Z https://ir.unisa.ac.za/handle/10500/32304 Compartmental model for the spread of infectious disease with hererogenous population: a case study of COVID-19 and Lassa fever Oluwagunwa, Abiodun Peter This study investigates the transmission dynamics of COVID-19 and Lassa fever, with particular attention to the risks and implications of co-infection. By dividing the human population into epidemiological compartments, the models capture the real course of disease spread in a structured and realistic way. The mathematical correctness of the models was validated by proving positivity, boundedness, and reliability of solutions for public health interpretation. For COVID-19, the basic SYR framework was analysed to obtain the reproduction number RY , and the model was further extended to an SEAIHR structure to include exposed, asymptomatic, infectious, and hospitalised individuals. For Lassa fever, a deterministic compartmental model was developed and subjected to stability analysis. Numerical simulations were carried out for both diseases to assess intervention strategies and transmission behaviour. The COVID–19 analysis revealed that the disease can be eliminated when RY < 1, but once RY > 1, infection becomes persistent. The extended SEAIHR model also produced the overall reproduction number R0, showing how reductions in contact rates, timely detection, effective hospital care, and faster recovery can substantially suppress transmission. For Lassa fever, the disease-free equilibrium remained stable only when the reproduction number was kept below one. Simulations highlighted the significant influence of asymptomatic carriers and showed that no single intervention— whether treatment, health education, or rodent control—can fully control the disease on its own. Instead, the most meaningful reduction in cases occurred when human-focused measures were combined with strong rodent control, leading to the elimination of infectious rodents by the 35th day. Together, these results emphasise the urgency of studying co-infections in regions such as West Africa, where both diseases circulate at the same time. Incorporating optimal control theory provides a systematic and cost-effective framework for coordinating interventions across multiple pathways of transmission. This study therefore deepens our understanding of the dynamics of both COVID– 19 and Lassa fever and offers practical insights for improving interventions, epivdemic preparedness, and public health responses. The centre-manifold analysis further shows that both models experience a forward transcritical bifurcation at R0 = 1: once transmission exceeds this threshold, a stable endemic state emerges. This reinforces a critical message—maintaining transmission below the threshold is essential for preventing long-term persistence of either disease. 2015-10-01T00:00:00Z