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This article reports on a study that explored how a group of in-service mathematics teachers integrated ethnomathematics approaches into the teaching and learning of geometry. The study used a convergent parallel mixed-methods design, which combined both quantitative and qualitative methods, to provide a deeper understanding of how the participants integrate ethnomathematics approaches into the teaching and learning of geometry. The data for the study were gathered from 40 in-service mathematics teachers through the use of questionnaires and focus group discussions. Results showed that the in-service teachers integrate ethnomathematics approaches into the teaching and learning of geometry as learning materials, resources, and the learning context. Based on the study, it is suggested that teachers should consider the incorporation of ethnomathematics approaches into the teaching and learning of geometry using technological means such as the internet, TV, and films. The findings have implications for continuous teacher professional development in the forms of workshops for the teachers involving the use of ethnomathematics approaches.

This article uses action-process-object-schema theory to study the mental constructions about the limit of series during a Calculus 1 course at a university. The researchers also used the theory's teaching methodology to teach the topic. A plethora of research on the limit concept is available and suggests that the concept is on record as being difficult for students to learn and comprehend. The study proposed a genetic decomposition on how undergraduate students might demonstrate their mental constructions in learning the limit of a series. Students were taken through the activities-classroom discussions-exercises cycle. Thirty students participated in answering questions based on the convergence of a series. The students' written responses together with the interviews were analysed and based on the findings a revision of the preliminary genetic decomposition was done. We found that there were students who did not display the predicted mental construction indicated by the preliminary decomposition in the application of the definition for the convergence of a series, but displayed the predicted mental construction for the application of the series convergence tests. We also found that certain schema were necessary for the achievement of a complete understanding of the convergence of a series concept. The mental constructions within the missing schema were included in the modified genetic decomposition. This empirically enriched model is now expected to inform pedagogy on the convergence of a series concept.

This qualitative research study explored barriers that impinged on teachers' use of their knowledge of students' learning styles in teaching mathematics. It was a case study of three secondary schools in the Makoni District of Manicaland Province in Zimbabwe. Participants were fifteen mathematics teachers. Results revealed that the barriers were related to the teachers, the students, the curriculum and the socio-economic status of the schools. Collaborative teaching, formation of mathematics clubs, mobilisation of resources, use of excursions and in-servicing of teachers were recommended as ways of reducing the effects of the barriers.

The study intended to find out how in-service teachers were trained to teach geometry in Zimbabwe. Mixed method was employed in this study, in which qualitative and quantitative methods are concurrently used. While the questionnaire with closed ended questions constituted the quantitative aspect of the study, the open ended questionnaire questions and focus-group discussions were used for the qualitative aspect. The study participants consisted of 40 in-service teachers who completed the questionnaires and then participated in focus-group discussions. The findings reveal that 52.5 per cent of the teachers were adequately prepared to teach geometry and both teacher-centred and learner-centred approaches were used during their training. It emerged that teachers are likely to teach using the approaches that they experienced during their training. It is recommended that initial teacher training programs should adequately train teachers to teach geometry for them to be able to teach it effectively in schools.

Geometry and culture are interrelated, making school geometry closely connected to the environment as well as culture in which it is taught. With regard to this connectedness, the Zimbabwean mathematics syllabus indicates that geometry should be connected to the learners' environment and culture. This article explores teacher-related challenges to the integration of ethnomathematics approaches into the teaching of geometry. Findings are based on feedback received from questionnaires and focus-group discussions in which 40 in-service mathematics teachers expressed their views on the challenges that affect the integration of ethnomathematics approaches into the teaching of geometry. Major challenges included lack of knowledge on ethnomathematics approaches and how to integrate these approaches into the teaching of geometry; teachers' lack of geometry content knowledge, teachers' views of geometry taught in schools, teachers' competence in teaching geometry, teaching and professional experience as well as resistance to change by teachers. The study recommends that teacher training institutions need to redesign their curricula to include ethnomathematics approaches and that there is need for in-service training on ethnomathematics approaches.

A qualitative research design comprising of open-ended questionnaires and focus group discussions was used to investigate learner-related variables that impede the integration of ethnomathematics approaches into the teaching and learning of geometry. Proportional stratified sampling and recruitment based sampling were used to select the 40 in-service teachers who took part in this study. The data was analysed inductively. The findings revealed that learners' attitudes towards ethnomathematics approaches, learners' lack of knowledge on cultural examples, cultural diversity, resistance to change by learners, lack of listening skills by learners and learners' attitudes towards geometry impede the integration of ethnomathematics approaches into the teaching and learning of geometry. It is recommended that teachers should find ways to use cultural examples that accommodate all the learners from the varied cultural backgrounds. Teachers should also find ways of motivating the learners to learn geometry.

This study reports on in-service mathematics teachers' views of geometry in the context of ethnomathematics. A convergent parallel mixed method design which includes both quantitative and qualitative methods was used to provide a deeper understanding of how the participants view geometry from an ethnomathematics perspective. Questionnaires and focus group discussions were used to gather data from 40 in-service mathematics teachers. The teachers' views of geometry were classified under the cultural, social and practical dimensions of ethnomathematics. The views held by the teachers influence the way they teach geometry. The study suggested that knowledge construction and conceptual understanding in geometry are essential for its application in everyday life.

The purpose of this study, conducted with 44 participants, was to explore their experiences of learning linear algebra using the online content and materials during COVID-19. The conceptual framework was guided by the notion of social presence and connectivisim in the context of remote learning. The study used a questionnaire comprising closed and open-ended items where students were asked about their experiences using the online platforms. Our study found that most students preferred traditional classroom learning to online learning. Students reported that unreliable internet connections as well as home settings that were not conducive to studying made their studies challenging. It was found that the provision of recorded videos of lectures and pdfs of those lectures were useful to the students. However, the students did not find it helpful to participate in tutorials and other supplementary instruction during remote learning. Our study also found that those students preferred online assessments (tests and quizzes) compared with written examinations. We recommend, among other things, that adequate internet facilities be provided for students, especially those living in the rural or disadvantaged areas who cannot access the online contents and the medium through which they were uploaded. Further, it is recommended that the module website should be revised to include more explicit instructions about accessing the support structures that are available.

This convergent mixed method study aimed at gaining insight into teachers' awareness of ethnomathematics approaches and the ethnomathematical practices that could be used in the teaching and learning of geometry. The data were gathered through questionnaires and focus group discussions. The findings showed that the in-service teachers had varying definitions of ethnomathematics approaches and they were aware of the ethnogeometrical practices found in their cultural practices and experiences that could be integrated into the teaching and learning of geometry.

For teachers to integrate ethnomathematics approaches, it is essential to understand their views related to the implementation challenges of such approaches. In schools, teachers held previous views and experiences that influenced their integration of ethnomathematics approaches. This study investigates school-related factors that impede the integration of ethnomathematics approaches into the teaching and learning of geometry as well as suggesting possible solutions into mathematics instruction in Zimbabwe. A convergent mixed methods design, using questionnaires, focus group discussions and interviews, was used. Stratified random sampling, research-based recruitment and purposive sampling were used to select 42 participants. Inductive analysis was used to analyse qualitative data and descriptive statistics was employed to analyse quantitative data. The study revealed that curriculum-related challenges, insufficient resources such as textbooks and very few cultural examples and activities affect the integration of ethnomathematics approaches into the teaching and learning of geometry. There is a need for curriculum developers to engage more teachers during the curriculum design in order to develop material that suits all the learners in Zimbabwe.

In this paper we focused on university mathematics students' mental constructions on the concept limit of a sequence. The aim was to explore the nature of the mental constructions and to contribute to Action-Process-Object-Schema (APOS) theory in terms of instructional strategy for teaching limit of sequences. The belief that understanding students' mental constructions in learning mathematics leads to improved instructional strategy guided this study. Maple software was applied in the learning of limit of sequences to foster students' understanding of the topic. Thirty university students took part in this study. The data was collected through the test questions and interviews. The results revealed that in most cases, the preliminary genetic decomposition could be used to appropriately describe students' constructions of their responses. Furthermore, many students operated at an action/process/object level with the exception of a few students who were not able to operate at these levels. It was also revealed that few students carried out procedures effectively without the conceptual understanding of the concept limit of a sequence. It is recommended that the preliminary genetic decomposition be extended to include aspects necessary for conceptual understanding of the limit of sequences in modified genetic decomposition, such as prerequisites, concepts which include rules of inequalities and properties of the modulus function. These concepts are required for the understanding of proving or disproving the given limits of a sequence.

This article reports on a study which used the APOS (action-process-object-schema) Theory framework and a classification of errors to investigate university students' understanding of the integration concept and its applications. Research was done at the Westville Campus of the University of KwaZulu-Natal in South Africa. The relevant rules for finding antiderivatives, the link between derivatives and antiderivatives, interpreting a definite integral as area under the relevant curve and their context-based applications were taught to undergraduate science students. This paper reports on the analysis of two students' responses to questions on integrals and their applications. The findings of this study suggest that those students had difficulty in applying the rules for integrals and their applications, and this was possibly the result of them not having appropriate mental structures at the process, object and schema levels.

This article reports on a study which investigated first year university engineering students' construction of the definition of the concept of the chain rule in differential calculus at a University of Technology in South Africa. An APOS (Action-Process-Objects-Schema) approach was used to explore conceptual understanding displayed by students in learning the chain rule in calculus. Structured worksheets based on instruction designed to induce construction of conceptual understanding of the chain rule were used. A number of students used the straight form technique in differentiating complicated tasks while very few used either the link and Leibniz form techniques. In this manner differentiation of each function within the composite function was accomplished. Students either operated in the Inter- or Trans stages of the Triad. It was found that even students who had inadequate understanding of composition of functions, performed well in the application of the chain rule.

This article reports on a study which used the APOS (action-process-object-schema) theoretical framework to investigate university students' understanding of derivatives and their applications. Research was done at the Westville Campus of the University of KwaZulu-Natal in South Africa. The relevant rules for finding derivatives and their applications were taught to undergraduate science students. This paper reports on the analysis of students' responses to six types of questions on derivatives and their applications. The findings of this study suggest that those students had difficulty in applying the rules for derivatives and this was possibly the result of many students not having appropriate mental structures at the process, object and schema levels.

Students have experienced difficulty in understanding and using the chain rule. This study aims at assisting the students to understand and apply the chain rule and thus inform the author's teaching for future learning of students. A questionnaire will be designed to explore the conceptual understanding of the concept of the chain rule by first year university of technology students using APOS (action-processes-objects-schema) which proposes in the form of the genetic decomposition a set of mental constructions that the students might make in order to learn the concept of the chain rule in calculus and accessing it when needed. This instrument will be used to collect data on how students learn derivatives of trigonometric functions in calculus, using the chain rule. This will be with a view to clarify their understanding of the composition of functions, derivative and the chain rule. The study consists of two phases, both using a qualitative approach. A structured way to describe an individual student's understanding of the chain rule is developed and applied to analyzing the evolution of the understanding for each of 30 first year students. Other ways to collect data include tests, written exercises and classroom observations. The purpose of the questionnaire will be to establish the correlation between the students' ability to deal with composition of functions and using the chain rule successfully. Students (n = 10) will then be interviewed based on their written responses to elicit their thinking involved when answering. The analysis of written responses and interviews should establish whether the instrument provided substantial information for identification of certain mental constructions that the researches proposed to consider.

Continuity of functions appears throughout the grades in South African high school (FET (further education and training)) topics as prescribed by the final draft of the Curriculum and Assessment Policy Statement. This article reports on the use of a combined framework of APOS (action-process-object-schema) and DCT (dual coding theories) to analyze data captured in a study which investigated second-year teacher trainees' understanding of the concept of continuity. The study is qualitative in that it reports on these teacher trainees' mental constructions of the concept of continuity of single-valued functions, obtained from analysis of their responses to structured activity sheets. The 12 students in this study specialize in the teaching of mathematics for the FET high school curriculum at a South African education faculty. A two-tiered concurrent approach was attempted, one through student-collaborations and the other through an instructional design worksheet, to develop mathematical understandings of the concept of continuity. (Contains 2 tables and 6 figures.)

The authors explored the teaching and learning of fractions by reflecting on teachers and learners' views on practical work and their classroom practices. The teachers and learners were from two primary schools in a rural area in KwaZulu-Natal (South Africa). Questionnaires, in which teachers and learners expressed their views on practical work and fraction teaching and learning, were administered to teachers and their learners. Lessons on the division of fractions were observed to determine teachers' practices in relation to the researchers' assumptions and claims by literature. Data yielded by these research instruments confirmed assumptions and literature claims. Although it was a small scale qualitative research, interesting observations were made that could have pedagogical implications. The paper is presented in two parts. The first considers the teaching of fractions and the second the learning of fractions. (Contains 2 tables and 7 figures.)

I report on the findings from research and literature on (a) use of symbols in mathematics, (b) algebraic/trigonometric expressions, (c) solving equations, and (d) functions and calculus. From these, some insights and implications for teaching and learning are derived.

Teachers' views on practical work and their classroom practices were investigated to confirm or refute existing assumptions and literature claims. The teachers were from two primary schools in a rural area of the Hammarsdale Circuit in KwaZulu-Natal, South Africa. Questionnaires in which teachers expressed their views on practical work and fraction teaching were administered to teachers. Lessons on the division of fractions were observed to determine teachers' practices in relation to the researcher's assumptions and claims by literature. Data yielded by these research instruments confirmed assumptions and literature claims. Although this was a small-scale, qualitative study, interesting observations were made that could have pedagogical implications.