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.

Visualization is widely recognized as a powerful support for problem-solving across disciplines such as mathematics and science. However, research on visualization has often treated it as a language-neutral cognitive tool, paying limited attention to the linguistic and cultural resources of multilingual learners (MLs). This review synthesizes empirical research on the role of visualization in problem-solving for MLs, clarifying its conceptualization, its study, and the evidence regarding its effectiveness for this population. Drawing on cognitive, cognitive-linguistic, and sociocultural perspectives, the review examines how visual representations interact with language demands, disciplinary practices, and learners’ meaning-making processes. The analysis reveals several tensions in the literature, including limited consideration of learners’ language proficiency and a predominance of monolingual research designs. The review also identifies methodological and theoretical gaps, particularly the need for integrated frameworks that account for both visual and linguistic dimensions of problem-solving. Implications for future research and instructional practice are discussed, highlighting directions for more inclusive, theoretically grounded investigations of visualization in multilingual learning contexts.

Definition of geometric concepts forms a critical part of teachers' content knowledge for the teaching of geometry. The purpose of the study reported on here was to analyse pre-service mathematics teachers' geometric thinking with regards to definitions and classification of quadrilaterals. Data were generated from 8 pre-service teachers based on their performance in a test administered to those who voluntarily accepted to take part in the study. The 8 participants were divided into 2 groups of 4 each-4 in Group A who performed well in the test, and 4 in Group B whose performance was not so good. The results show that even though the pre-service teachers (PSTs) had difficulties with definitions, those in Group A demonstrated moderate thinking on the use of literate words in definitions of quadrilaterals compared to their counterparts in Group B who showed deficiencies and used colloquial words in their definitions. We also found that a few of the participants in Group A showed an explorative way of thinking in the classification of quadrilaterals. Thus, generally, the PSTs' geometric thinking on inclusion relations of quadrilaterals was more ritualised in nature.

Learners in general struggle when working with word problems. South African learners in particular have an added barrier owing to the many official languages that have been legislated. The South African Department of Basic Education, through its curriculum statements, envisages that problem solving will play a fundamental part in mathematics teaching and learning. Whilst this has been embraced by teachers of mathematics, learners still experience difficulties when solving chosen word problems. This empirical, qualitative study explored 28 Grade 9 learners’ use of visualisation when solving word problems. After all the due ethical clearance protocols were observed, one class of learners completed a task-based exercise and five learners were interviewed. The results showed that when learners drew a diagram that represented the problem, they were more likely to solve it. Those learners who drew no diagram, or drew an incorrect one, failed to arrive at a suitable solution. An important finding is that the visualisation technique is a powerful tool when working with the solving of problems in mathematics.

This study explores how pre-service mathematics teachers (PMTs) in South Africa use visualization and self-discourse to solve mathematical problems. Visualization is known to enhance mathematics learning, while effective communication skills are critical for teaching and learning mathematics, especially in contexts, where the language of instruction may not be the first language of students or teachers. By understanding the visualization techniques and discursive properties employed by PMTs, insights can be gained into how to improve mathematics learning and teaching. The study is informed by the commognitive framework and uses qualitative data from a purposive sample of 10 PMTs who participated in a performance test and semi-structured interviews. The study found that PMTs rely on mental visualization for simpler mathematical problems but use both symbolic and iconic visual mediators for more complicated problems. The use of language to engage in self-discursive activity during problem-solving was found to be key for successful visualization.

This article is an advanced theoretical study as a result of a chapter from the first author’s PhD study. The aim of the article is to discuss the relationship between commognition and the Van Hiele theory for studying discourse during Euclidean geometry problem-solving. Commognition is a theoretical framework that can be used in mathematics education to explain mathematical thinking through one’s discourse during problem-solving. Commognition uses four elements that characterise mathematical discourse and the difference between ritualistic and explorative discourses to explain how one displays mastery of mathematical problem-solving. On the other hand, the Van Hiele theory characterises five levels of geometrical thinking during one’s geometry learning and development. These five levels are fixed and mastery of one level leads to the next, and there is no success in the next level without mastering the previous level. However, for the purpose of the Curriculum and Assessment Policy Statement (CAPS) we only focused on the first four Van Hiele levels. Findings from this theoretical review revealed that progress in the Van Hiele levels of geometrical thinking depends mainly on the discourse participation of the preservice teachers when solving geometry problems. In particular, an explorative discourse is required for the development in these four levels of geometrical thinking as compared to a ritualistic discourse participation.

Visualization has been a subject of much research, and recently, technology in terms of the Fourth Industrial Revolution movement has also been in vogue. While visualization serves as a strong tool for problem-solving, technology offers learners the possibility of experiencing mathematics and science in a dynamic environment, with diagrams changing by simply dragging or implementing a code. If these changes are visible and understandable, then they offer opportunities for an increased conviction that something is either true or not. This interpretivist qualitative study combined these areas of study to explore the possibility of engaging learners using technology from a visual perspective. Thirteen in-service teachers were asked to design lessons that incorporated visuals and learners were allowed to engage in these lessons actively. These participants then became co-researchers of the study. The research sites varied, and therefore the lessons planned and delivered were not the same for all participants. The participants reported an increase in learner confidence and a subsequent improvement in understanding of concepts. The framework that was used as a lens to look at the data was the Iterative Visualization Cycle, which was an adaptation of Kolb’s Experiential Learning Theory. Much of what is written is from the participants’ perspectives because it was their voices that needed to be highlighted.

For several years we, a group of mid-to-late-career academics in science and mathematics education, experienced increasing discomfort in our classrooms.Our professional anxiety intensified as we found ourselves trapped in the liminal space between the demands of the academic institutions, and the call of students to decolonise the curriculum.This was inextricably linked to our complicity in perpetuating the tradition of importing 'good' science and mathematics, grounded in Euro-western paradigms, and transplanting it into African classrooms, with its practices of epistemological exclusion.We were, and still are, uneasy about continuing to use science and mathematics education, which is informed by a monolithic canon, as a vehicle of fragmentation and conquest instead of leading towards social cohesion.Within this complex space we adopted narrative inquiry to become more thoughtful, retrospective, and introspective of our practice.We reflected on our past and present identities and our active agency in becoming

In mathematics, problem-solving can be considered to be one of the most important skills students need to develop, because it allows them to deal with increasingly intricate mathematical and real-life issues. Often, teachers attempt to try to link a problem with a drawn diagram or picture. Despite these diagrams, whether given or constructed, the student still individually engages in a private discourse about the problem and its solution. These discourses are strongly influenced by their a priori knowledge and the given information in the problem itself. This article explores first-year pre-service teachers’ mental problem-solving skills. The emphasis was not on whether they solved the problems, but rather on their natural instincts during the problem-solving process. The research shows that some students were naturally drawn to construct mental images during the problem-solving process while others were content to simply leave the question blank. The data were collected from 35 first-year volunteer students attending a second semester geometry module. The data were collected using task sheets on Google Forms and interviews, which were based on responses to the questions. An interpretive qualitative analysis was conducted in order to produce deeper meaning (insight). The findings point to the fact that teachers could try to influence how students think during the problem-solving process by encouraging them to engage with mental images.

It is widely recognized that students encounter difficulties with proof across all grades and beyond, yet standardized instruments related specifically to students’ perceived self-efficacy for mathematical proof have not been readily available. The purpose of this study was to develop and investigate preliminary validity evidence for a new instrument for measuring self-efficacy for mathematical proof that can be of importance to the field. The new Perceived Self-Efficacy for Proof (PSEP) questionnaire is a self-administered, 8-item questionnaire that quantifies experimentation, conjecturing, inductive reasoning, justification, and validation. To validate the PSEP, two studies with 260 eleventh grade students—recruited from three Dinaledi schools in EThekwini metropolitan area, South Africa—were conducted. In Study 1 (n=128), face and content validity were evaluated, and an exploratory factor analysis (EFA) was performed. In Study 2 (n=132), a confirmatory factor analysis (CFA) was conducted and external validity was investigated. In both samples, the PSEP was found to possess good internal consistency reliability with relatively high factor loadings on a single component. Although the findings in this report represent preliminary validation evidence, it can be concluded that the PSEP is a valid, reliable and sensitive measure of 11th grade students’ perceptions of their ability to construct a proof and may serve as a meaningful outcome in mathematical proof research and classroom proof education.

During the COVID-19 pandemic, many surveys in education were conducted. These revealed alarming statistics about learners losing half of the academic year, parents' anxiety about sending children to school, and a minority of education institutions being able to offer online teaching. In response to a cacophony from teachers' and students' unions, school governing body representatives, scientists and education experts, the government decided to close education institutions as part of what was known as the hard lockdown. Against this background, we used critical policy analysis (CPA) to explore decision-making by education departments and the enactment of these decisions at schools. This qualitative study revealed iniquity and inequity as departments of education made decisions to close and reopen institutions. The findings revealed a tension between expectations of producers of policy and enactors of policy within unequal school settings. We recommend a repositioning from the perspective of the dispossessed to inform future policy.

This paper reports on a study conducted with Grade 11 learners as participants, in four public high schools in a rural district of the Eastern Cape Province of South Africa. The central objective was to investigate the mathematical discourses of Grade 11 learners related to the word, algebraic and graphic representations of asymptotes of the hyperbola and exponential functions. Commognition and social learning theories were referred to, with particular emphasis on the characteristics of the mathematical discourse, that is, word use, visual mediators, endorsed narratives and routines. The study adopted a qualitative research design, with exploratory, descriptive and interpretive elements complementing both its data collection and analysis processes. A purposive sampling strategy was opted for in the study. Data was collected by means of a test administered to a total of 112 Grade 11 participants from the selected four high schools. Data was thematically analysed by means of a developed Discourse Profile of the Hyperbola and Exponential Function adapted from the Arithmetic Discourse Profile. The study's findings revealed that, while there was evidence of learning functions, for these leareners, the process of de-ritualisation was not yet complete. There were also challenges in linking different representations of a function. Learners could work efficiently on procedure tasks, but struggled on action-oriented tasks.

There is concern about mathematics success and its related pedagogy. Society has seen rapid changes in the economy and technology with a call for these changes to be reflected within the classroom. New methods of teaching mathematics are being sought with the purpose to improve teaching and learning while making mathematics relatable to the new generation of learners. The incorporation of technology within the classroom has been seen an option to make this change. The purpose of this research was to determine how effective the use of the GeoGebra app is in allowing learners to successfully discover the properties of straight line graphs. Furthermore, the research looked at learner responses to using the app. A qualitative research design was used with data generated through a task-based investigation, as well as individual interviews. Results of the research showed that the use of GeoGebra aided learners successfully in discovering the properties of straight-line graphs with the majority of learners understanding both concepts. The results also showed that learners had a positive outlook to the use of the app and enjoyed the experience. Keywords: GeoGebra, iPad technology, mathematics teaching, linear functions, software manipulation.

Learners in South African schools often respond poorly in questions related to the asymptote. Despite the fact that there are only a few functions in the South African curriculum that actually explore the asymptote, learners still show some deficiency in their understanding of the concept. This research examined Grade 11 learners’ mathematical discourses about the asymptotes of the hyperbola and exponential functions. Data were analysed using the Realisation Tree of a Function, an adaptation of the Realisation Tree Assessment tool from Weingarden, Heyd-Metzuyanim and Nachlieli. While the Realisation Tree Assessment tool focused on teacher talk, the Realisation Tree of a Function focused on learner expression and responses. A qualitative research design was essentially adopted, with exploratory, descriptive and interpretive elements complementing both its data collection and analysis. A purposive sampling strategy was implemented. Data were collected by means of a test administered to a total of 112 Grade 11 participants from four selected secondary schools. Focus group interviews were conducted with 24 of the best-performing participants by using their responses from the written mathematical tests. The results revealed that the learners’ mathematical discourse is not coherent. While learners’ work on each representation was often mathematical there seemed to be a struggle when the task had an unusual orientation. Different expressions of the same mathematical object elicited different responses. The challenge is that learners exhibited a fragmented relationship between the mathematical objects of the function. Keywords: commognition, realization tree, ritualised learning, visual mediators.

Ponte Academic JournalSep 2018, Volume 74, Issue 9 LANGUAGE: A BARRIER WHEN TEACHING AND LEARNING BUSINESS STUDIESAuthor(s): Vimolan Mudaly ,Kajal SinghJ. Ponte - Sep 2018 - Volume 74 - Issue 9 doi: 10.21506/j.ponte.2018.9.4 Abstract:This study set out to explore how a Critically Reflective approach to teaching and learning assists Business Studies English Second Language Learners’ in understanding terms exclusive to the subject. The objective of this study was to explore how an action research intervention can assist these learners in understanding Business Studies. This study was informed by the critical paradigm which sought to bring about change both professionally and social. The integration of the critical reflection theory and Kolb’s experiential learning theory were used as theoretical frameworks. Both theories were used for the purpose of improving teaching and learning by questioning the individual’s presuppositions. This study reveals how the teacher and learners’ gradually change their existing frame of reference by becoming questioning minds. Various data collection methods were employed to retrieve rich detailed information. Some of these methods included introspection, observation and feedback from participants. The findings reveal how English Second Language Learners (ESLLs) construct new knowledge and make meaning of concepts when exposed to visual aids. When ESLLs are exposed to resources that they can relate to, meaningful learning occurs. Download full text:Check if you have access through your login credentials or your institution Username Password

Ponte Academic JournalNov 2018, Volume 74, Issue 11 THE EFFECTS OF THE GEOBOARD ON LEARNER UNDERSTANDING OF GEOMETRY THEOREMSAuthor(s): Mandlenkosi Richard Sibiya ,Vimolan MudalyJ. Ponte - Nov 2018 - Volume 74 - Issue 11 doi: 10.21506/j.ponte.2018.11.8 Abstract:This study investigated the effects of the Geoboard on learners understanding of geometric theorems. The study employed a qualitative research design. The study was conducted in two secondary schools. Twenty five participants were selected randomly from each of two conveniently selected secondary schools. Four focus groups were eventually established, two from each school. All 50 participants were taught geometric theorems using the Geoboard for two weeks. The data were analysed using thematic analysis. The results revealed that Geoboards improved learners understanding of geometric theorems, especially understanding the geometric terminology and reasoning. Download full text:Check if you have access through your login credentials or your institution Username Password

Ponte Academic JournalSep 2018, Volume 74, Issue 9 ALGEBRA IN TEXTBOOKS: A COMPARISON OF SOUTH AFRICAN\r\nAND ANGOLAN TEXTBOOKSAuthor(s): Vimolan Mudaly ,Isabel PedroJ. Ponte - Sep 2018 - Volume 74 - Issue 9 doi: 10.21506/j.ponte.2018.9.5 Abstract:This study compares the algebra sections in three textbooks, one South African grade 9, one South African grade 10, and one Angolan grade 10 textbook. The comparison is fourfold. First, I compare the textbooks in terms of topics. Second, I compare the distribution of the text on explanations, examples and exercises. Third, I use the levels of understanding of algebraic expressions suggested by Sfard and Linchevski (1994) to interrogate progression and consistency of the texts. And finally, I look at the number of steps required to move from task to solution in the examples and exercises, providing a different measure of progression. The data shows that the Angolan textbook has more explanations than the South African textbooks, and far fewer exercises. However, the progression both in terms of levels of understanding of algebraic expressions and in terms of the number of steps required to solve tasks is swifter in the Angolan textbook. Also, there is less focus on symbol manipulation in the Angolan textbook. Though much depends on the ways in which textbooks are used in the classroom, this suggests that the Angolan textbook offers the learners more opportunities to learn. Download full text:Check if you have access through your login credentials or your institution Username Password