The way learners define mathematics shapes their perceptions of its nature as a subject and its relevance to their lives. In South Africa, many learners have limited exposure to meaningful mathematical experiences, which contributes to low achievement levels. This study investigates how 17 Grade 11 learners understood and solved trigonometric equations. Grounded in constructivist learning theory, the research explores how learners build mathematical knowledge through problem-solving. A qualitative approach was used, involving purposive sampling of learners from an afternoon mathematics programme. Data were collected through worksheets and semi-structured interviews. According to the Curriculum and Assessment Policy Statement (CAPS) curriculum, Grade 11 learners are expected to deepen their understanding of trigonometric equations based on foundational knowledge from Grade 10. However, findings revealed that while learners attempted to construct knowledge they encountered significant challenges, particularly in applying algebraic rules. Contribution: The study concludes that reinforcing basic algebraic concepts and using varied teaching strategies can enhance learners’ comprehension of trigonometric equations.
This Table of Contents reflects the print compilation of peer-reviewed articles published in the journal. Each article listed was originally published online under the journal’s open access model and remains individually accessible and citable. This compilation has been created solely for print distribution, reference, and archival purposes. No new research content is introduced. The publisher affirms that all articles included in this compilation have undergone the journal’s standard editorial and peer-review processes.
Based on the first author’s doctoral research, this article explores the connection between Sfard’s commognition theory and Polya’s four steps of problem-solving. The article highlights how commognition, through its focus on language and discourse, helps explain learners’ problem-solving approaches as either ritualistic or explorative. The study emphasises that for multilingual learners, understanding mathematical problems, especially in geometry, depends on grasping terminology and visual elements. By combining commognition with Polya’s framework, the analysis suggests that learners’ problem-solving and visualisation skills can be improved, though language remains a key challenge. Contribution: This article establishes a theoretical connection between Sfard’s commognition and Polya’s problem-solving steps, highlighting their complementary roles in mathematics education. The article validates how commognitive elements, particularly word use and visual mediators, support multilingual learners in understanding and solving geometry problems. The study offers insights into how discourse influences learners’ problem-solving routines, with implications for teaching in linguistically diverse classrooms.
South African schools face stark inequalities in infrastructure, connectivity, language, and teacher preparedness. These contextual factors profoundly shape what artificial intelligence (AI) can and cannot do for mathematics teaching and learning. This article synthesizes recent peer-reviewed scholarship, policy texts, and book chapters to argue that AI adoption must be context-responsive: aligned to local resource constraints, multilingual realities, professional development ecosystems, and regulatory frameworks (notably POPIA). This article emphasizes that without attention to connectivity, electricity, devices, teacher TPACK, multilingual pedagogy, and data protection, AI may amplify—rather than reduce—existing inequities. In mitigation, this article provides practical, evidence-based principles for context-aware AI implementation in South African mathematics education.
The development of computational thinking has been declared a necessity for everyone in the current digital era. Most studies dedicated to the development of computational thinking have focused on using plugged activities. However, in developing countries like South Africa, most schools struggle with resource provision which can inhibit the integration of computational thinking in all learners. This article considers an alternative approach to computational thinking in mathematics education using pen-and-paper problem-solving. Through the adoption of exploratory qualitative research with a purposeful sample of five preservice teachers, the manifestation of computational thinking in pen-and-paper problem-solving is explored. Adopting the combination of the metacognitive framework and Schoenfeld’s framework for problem-solving, this study explains from the perspectives of preservice teachers how computational thinking practices manifest in pen-and-paper problem-solving. The thematic analysis conducted in this study revealed that computational thinking practices such as abstraction, decomposition and algorithmic thinking, and evaluation were observed in preservice teachers’ pen-and-paper problem-solving. Two interesting findings emerged from this study: (1) there was no discernible difference between decomposition and algorithmic thinking in preservice teachers’ pen-and-paper problem-solving, and (2) there was no clear evidence of pattern recognition as an important computational thinking practice in preservice teachers’ pen-and-paper problem-solving. Contribution: The first finding indicates that during pen-and-paper problem-solving, it might be difficult to differentiate between decomposition and algorithmic thinking. The second finding indicates that more studies need to be conducted, directed towards how pattern recognition manifest in pen-and-paper problem-solving. Conclusions and recommendations are made for this study.
The global concern over low mathematical achievement consistently challenges educational ministries. Poor development of learners’ algebraic representations contributes to poor mathematical achievement from the Foundation Phase classes (Grades 1–3). In short, in teaching algebraic representation in Grade 3, success is measured by the extent to which the learners can reason, handle abstractions, manipulate symbols and finally be able to communicate mathematical ideas. This article explores the views and experiences of Grade 3 teachers in developing learners’ algebraic representations. The concrete-representational-abstract (CRA) approaches underpin this study as they guide teachers in using concrete objects, drawings, and symbols to build learners’ algebraic representations. A single case study research design using a qualitative approach was employed to explore the phenomenon under study. A homogenous purposive sampling was used to select six Grade 3 teachers from three schools in Limpopo province, South Africa. Semi-structured interviews, interpretivist document analysis and non-participant observations were used to collect data. Thematic narrative analysis was used to interpret the data to explore the views and experiences of Grade 3 teachers in developing learners’ algebraic representations. The findings indicate that teachers use concrete manipulatives and indigenous games to develop Grade 3 learners’ algebraic representations. However, the lesson plans and observations revealed that Grade 3 teachers face challenges teaching abstract number symbols, as learners rely more on concrete objects. Contribution: Considering these findings, the novelty of this article contributes to teachers’ knowledge on expanding the CRA approach with scaffolding approaches to teach abstract number symbols to develop Grade 3 learners’ algebraic representations. As a result, this research addresses Africanisation, decolonisation, and pedagogical transformation in the alarmingly poor results of mathematics.
This article contributes to the body of research on equivalence relations by delineating the levels of sophistication and efficiency in the intuitive strategies employed by pre-service teachers (PSTs). Using an inductive approach, we analysed 1102 responses to two open equivalence relations problems, drawn from a 27-item baseline assessment, completed by 551 first-year PSTs across three cohorts (2022–2024). From the analysis, we began teasing out a conceptual framework that could explain the reasoning behind the responses. To authenticate the conceptual framework, we needed to unearth the strategies employed by PSTs, so we conducted follow-up task-based interviews with nine selected participants. These interviews provided insights into the reasoning behind the PSTs’ written responses and helped illuminate the developmental progression in their understanding of reasoning when solving equivalence relations problem. In particular, our analysis of the interviews revealed three distinct stages of sophistication (quantitative, additive, and multiplicative) and four levels of efficiency in PSTs’ reasoning when solving equivalence relations problems. The findings show that PSTs encountered significant difficulties with solving open equivalence relations problems, with 46% correct responses: 14% of which involved quantity-based reasoning, while 17% and 15% were additive and multiplicative-based reasoning. Interestingly, in 2023, multiplicative-based reasoning (which is the most sophisticated of the three types of reasoning), had the highest responses at 21.1%, compared to 11.4% in 2022 and 11.2% in 2024. Moreover, the strategy that involves working with a large equivalence difference (which strategy is the least efficient) was found to be the predominant strategy in the PSTs’ workings within the additive-based reasoning. Thus, these interview findings provided deeper insights into the patterns observed in the written responses regarding PSTs’ intuitive strategies. Contribution: These findings provide insights into a possible teaching framework that delineates the stages of sophistication that students go through in the development of their knowledge of open equivalence relations problems. By introducing and classifying open equivalence relations problems as belonging to the Comparative Relational category of equivalence tasks, this study elaborates on the highest category of equivalence tasks. Also, this study offers a possible analytical framework for analysing levels of efficiency in students’ reasoning when solving open equivalence relations problems. Consequently, the findings have implications for teacher training programmes and intervention studies with an interest in building sophistication and efficiency into PSTs’ strategies for solving open equivalence relations problems.
As an academic discipline, mathematics is more than a set of computing mechanisms; it is argued by scholars that the quality of mathematics education is in decline today in human society. Historically, mathematics and philosophy had a close interrelationship enabling the understanding of various natural phenomena. According to Platonism of pure mathematics, the dynamics of natural phenomena and human societies follow invariant and absolute mathematical principles. This article presents a socio-philosophical argument that the concept of society within the natural space can be classified into natural society and synthetic society based on the concept of mathematical purifications, and mathematics education has a role in it. The existence of logical inversions of various forms in the synthetic societies are analysed and corrective roles of mathematics education are explained. The Pythagorean philosophy of mathematics in building an improved as well as just society is an appropriate solution that calls for a relook into mathematics education in order to reduce utilitarian distortion in mathematics education today and to promote intellectualism as well as harmony while reforming mathematics education. Contribution: This article presents detailed analysis of the existing critical issues related to mathematics education today in human society and it compares various social forms in nature, including human, in light of the concepts of socio-philosophy. It is illustrated that these two aspects are interrelated in view of developments in education and evolution of human society. The historical perspectives of mathematics education forming a free and harmonious society following the Pythagorean philosophy of mathematics and society are presented, which can be a solution to the multidimensional problems in human society and in mathematics education today.
This article reports on the effectiveness of GeoGebra as a modelling tool to mitigate undergraduate engineering mathematics students’ misconceptions and errors associated with complex numbers. GeoGebra is a transformative open-source mathematical software that allows students to visualise and manipulate mathematical objects on different types of digital devices. Despite the centrality of complex numbers in studying vital mathematical concepts such as vectors, eigenvalues, and eigenvectors, studies revealed a prevalence of misconceptions and errors associated with complex numbers. Some students confuse the complex number’s representations; others view the representations as autonomous and unrelated. The study adopted a methodological pragmatism research design. It involved volunteering first-year first-semester engineering mathematics students from purposefully selected specialisation groups that included mechanical, industrial, and electrical engineering, at a South African university. The empirical intervention was underpinned by the Realistic Mathematics Education (RME) framework; the data for students’ misconceptions and errors were collected from their pre-test and post-test scripts and analysed qualitatively using Donaldson and Orton’s errors categories as a lens and quantised or quantitised using a chi-square test. The total frequencies of misconceptions and errors yielded a chi-square statistic of 7.9584 and a p-value of 0.004787, which was statistically significant at p 0.05. Contribution: The study’s key findings strongly suggest that GeoGebra-facilitated intervention effectively mitigates undergraduate engineering mathematics students’ total misconceptions and errors associated with complex numbers more than the traditional intervention. This indicates teachers can harness GeoGebra, reducing students’ misconceptions and errors associated with complex numbers and improving the quality of teaching and learning complex numbers and tertiary engineering mathematics education.
This study analysed first-year preservice teachers’ understanding of trigonometric equations at a South African university in the Eastern Cape province. We employed the Action-Process-Object-Schema (APOS) framework to analyse the mental constructions made by preservice teachers in solving trigonometric equations. A qualitative case study design was employed to analyse test scripts from 223 preservice teachers, complemented by follow-up interviews with eight of these participants. Findings show that the success rate in the two analysed items was low. Students who had not developed specific mental structures could not solve the given problems. Only 15.5% of the participants reached the Object level, while 76% remained at the Action or Process stages. Conversely, 8.5% of the participants were at the pre-Action stage, having not shown evidence of action mental structures conjectured in the genetic decomposition. Challenges encountered include difficulties with algebraic manipulations, reference angles, angle relationships across quadrants, and conversions between degrees and radians. The analysis further revealed a lack of understanding of the periodic nature of trigonometric functions and the general solution derivation. Contribution: These findings reflect global trends in mathematical struggles across various educational levels, particularly in solving trigonometric equations. The study highlights the importance of assessing preservice teachers’ mathematical knowledge both at the entry and exit points of their training programmes. Such dual assessments could improve their content mastery and teaching effectiveness. This suggests that adjusting educational strategies to address these identified gaps could foster significant growth.
The study examined the impact of concept-based instruction on the teaching and learning of mathematics. Functions is the topic that was implemented in order to identify the improvements in learners’ understanding. A total of 35 Grade 11 learners from a South African township school in Limpopo province participated in this study. Data were collected using tests, questionnaires, and semi-structured interviews. The constructivist learning theory underpinned the exploration. All the 35 participants wrote a test twice, completed a questionnaire and only 6 of them were interviewed. The study employed the sequential explanatory research design because of its richness in data findings as it incorporates both quantitative and qualitative research methods. The initial stages of data analysis started with emerging themes from data being coded and categorised. To identify changes in learners’ performance in the administered pre and post assessment tests, a dependent t-test was carried out, while questionnaires and interviews were used to assess learners’ attitudes and perceptions. The findings from the inquiry indicated positive gains in deep understanding, critical thinking, long-term retention of information, transferable skills, engagement and integration, which are all directly linked to conceptual understanding, resulting in enhanced performance. Both numerical and descriptive analyses confirmed that concept-based instruction enables learners to construct their own knowledge and enhance their conceptual understanding which in the end improves their performance in mathematics. The findings confirm that learners responded positively to concept-based instruction and suggest its broad adoption in mathematics education to address issues of conceptual gaps. Contribution: The study provides teachers with an opportunity to implement innovative teaching and learning approaches such as the concept-based instruction to improve learners’ conceptual understanding.
