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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.

The narrative permeating higher education institutions globally is the assimilation of advancing information and communications technology into mainstream Mathematics Education. In this paper we report on a mixed mode case study which explored possible mathematical gaps that created barrier/s when engineering students (n=162) worked with the online diagnostic tests. These engineering students were provided with online tasks on the learner management system (LMS) on factorization, fractions and logarithms. The study was carried out at a University of Technology in South Africa. The data collected from the LMS and written responses of students to quizzes were analysed. Findings emanating from the data analysis indicated that 1) all diagnostic tests logged positive increase in group averages, 2) greater repetition by students invoked better procedural proficiency and 3) better understanding of the basic mathematics leads to greater success in higher mathematics.

This study explored the mental constructions displayed by pre-service mathematics teachers (PMTs) when applying Cramer's rule. The aim was to reveal the character of mental constructions made around the nature of the solution set of equations and the role of parameters in the solution of equations with parametric coefficients. These mental constructions occur within the Action-Process-Object-Schema (APOS) theory. In this study, data was generated from 31 first year pre-service teachers by means of an activity sheet and interviews. The interviews were used to clarify pre-service teachers' responses to tasks from the activity sheet. The findings revealed that many PMTs displayed procedural understanding of Cramer's rule. It was also found that they understood what the solution of system of linear equations means. This meant that they were operating at the action stage, in terms of APOS. Additionally, it revealed that the lack of construction of related schemata negatively impacted the PMTs' attempt to construct the necessary mental constructions. Therefore, the researchers provided the genetic decomposition (GD) for the use of Cramer's rule to assist teacher educators to analyse the mental constructions of students. This study explored a new application of APOS theory. Analysing the mental constructions of students by means of research is intended to assist in designing alternative teaching strategies. In this way, this study makes a significant contribution to the solution of system of equations pedagogy.

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 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 qualitative case study in a rural school in Umgungundlovu District in KwaZulu-Natal, South Africa, explored Grade 12 learners' mental constructions of mathematical knowledge during engagement with optimisation problems. Ten Grade 12 learners who do pure Mathemat-ics participated, and data were collected through structured activity sheets and semi-structured interviews. Structured activity sheets with three tasks were given to learners; these tasks were done in groups, and the group leaders were interviewed. It was found that learners tended to do well with routine-type questions, implying that they were functioning at an action level. From the interviews it appeared that learners might have the correct answer, but lacked conceptual understanding. Exploring learners' mental constructions via their responses to activity sheets and interviews enabled common errors and misconceptions to be identified. Themes that emerged were that learners: 1) lacked the understanding of notation dy/dx, 2) had not constructed the derivative and minima/maxima schema, 3) had some difficulty in modelling problems, 4) preferred rules and formulas, and 5) applied algebraic notions incorrectly. Inferences are drawn for curriculum developers and teachers. This study also formulated itemised genetic decompositions for particular tasks, which contribute to APOS theory.

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.)

This paper reports on an exploration of students' learning derived from the implementation of learning materials developed in a previous collaborative project. The purpose of the study was to examine the development of third-year students' understanding of the Reimann Sum. These concepts were taught to undergraduate teacher trainees wishing to specialize in the teaching of mathematics in the FET (further education and training) school curriculum. The learning materials were developed by mathematics teacher educationists from the University of Witwatersrand, the UKZN (University of KwaZulu-Natal) and the Nelson Mandela Metropolitan University. A group of 31 students participated in the project. The study adopted the APOS (action-process-object-schema) approach within constructivism, as a theoretical framework. Data were generated from students' responses to the collaborative tasks. A genetic decomposition of the Reimann Sum is offered and used as a basis for the analysis. This paper, specifically, reports on the investigation of students' responses based on a learning theory within the context of advanced mathematical thinking and makes a contribution to an understanding of how these students constructed the concepts in a collaborative way. Appended is "The Worksheet with Instructional Tasks." (Contains 5 tables.)

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.)

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.

This article employs Kolb's Experiential Learning Theory (ELT) to address weaknesses in South Africa's Mathematics education, focusing on the use of concrete mathematical manipulatives for enhancing conceptual understanding in grade 12 learners. A Mathematical manipulative is defined as any material or object from the real world that children can move around to show a mathematical concept. Kolb's Experiential Theory links learning style to academic achievement. Kolb's ELT involves four learning modes: concrete experience, reflective observation, abstract conceptualization, and active experimentation. The use of manipulatives completes Kolb's learning cycle by providing a concrete experience, interaction and reflection with peers, abstract conceptualisation of triangles in the various planes and finally active experimentation. The study aims at improving the application of the sine rule, cosine rule, and area rule in solving 3-dimensional trigonometric problems. This qualitative study investigated the influence of mathematical models on teaching 3D trigonometric problems based on a sample of sixteen Grade 12 learners. Data was collected through observation, interviews, and written responses. The findings in this paper show that learners using manipulatives displayed heightened interests and improved understanding. [ABSTRACT FROM AUTHOR]

This South African study explored how pre-service teachers understand concepts in analytical geometry. Extending the work of Skemp and Carrillo-Yañez, et al., we captured data and carried out this qualitative study. We noted that the participants displayed complete understanding for some geometrical concepts but lacked relational understanding. Their learning process was interrupted during the covid epidemic, and this had a negative impact on their relational understanding of geometric concepts. [ABSTRACT FROM AUTHOR]

This paper is a report of a qualitative case study conducted in South Africa that sought to identify students' misconceptions and how those misconceptions hinder students' conceptual understanding of mathematical concepts, with a focus on algebraic expressions. The Structure of the Observed Learning Outcomes (SOLO) theory was adopted as the theoretical framework. The media for data collection were: classroom observations, activities, unstructured interviews and standardized tests. Participants were required to explain their solutions to the standardized test via unstructured interviews. This guided the researchers to identify students' misconceptions and how they hindered their conceptual understanding of mathematical concepts. Content analysis was carried out on participants' written and oral responses by applying the SOLO theory. From this study, the researchers established that conceptual misunderstandings are the main students' misconception in algebraic expressions. This misconception serves as barriers to students developing conceptual understanding as it does not invoke students' rational reasoning. [ABSTRACT FROM AUTHOR]

This study is part of ongoing research in undergraduate mathematics education. The study was guided by the belief that understanding the mental constructions the pre-service teachers make when learning matrix algebra concepts leads to improved instructional methods. In this preliminary study the data was collected from 85 pre-service teachers through a structured activity sheet. The aim was to explore their mental constructions of matrix algebra concepts and how they concur with a preliminary genetic decomposition. The study is underpinned by APOS theory (action-process-object-schema). The findings revealed that the mental constructions made by pre-service teachers in most cases concur with the preliminary genetic decomposition. In terms of APOS theory, the responses revealed that most pre-service teachers were operating at action and process stages, with few operating at the object stage. It was observed that the schemata for basic algebra and real number are necessary for the conceptual development of matrix algebra, and that familiarity of the correct use of terminology and notations promotes the earning of matrix algebra. On the basis of the detailed response analysis, modified genetic decompositions are proposed for the matrix order concept, matrix transpose concept and for matrix operations.

Many studies point to the problem of poor mathematics content knowledge of mathematics teachers in South Africa. The purpose of this study was to investigate teachers' knowledge of the mathematics they are themselves teaching. Data was generated from the teachers' (n = 253) written responses to test that was a shortened form of a previous Grade 12 Mathematics Paper One examination. The sample of teachers were studying towards an Advanced Certificate in Education (an upgrading high school mathematics qualification) at the University KwaZulu-Natal in South Africa. The findings revealed that the teachers in this sample obtained an average of 57% in the test. Using an APOS theory analysis it was found that many teachers who were working at an action level of a concept would require help and scaffolding to move to process or object levels of understanding of that concept. Furthermore it was found that on average teachers obtained 29% on questions which were at the problem solving level, raising concerns about how these teachers would mediate tasks that are set at high cognitive levels, with their Grade 12 learners.

There is concern over the learning and teaching of mathematics in South Africa. The aim of this study was to explore high school mathematics teachers' conceptual difficulties in trigonometry and coordinate geometry. The participants were in-service teachers (n = 236), who were enrolled in an in-service programme at a South African university. The research instrument was based on a previous Grade 12 national mathematics paper. The study used Duval's theory on semiotic systems to try to understand the teachers' difficulties experienced with the content. Ethical clearance was obtained from the university research office for the collection of data. All participants provided informed consent to participate in the study. The study used a mixed methods approach where the quantitative data was first analysed. Thereafter a content analysis was conducted on 50 scripts to gain insight into the areas which presented challenges to the teachers. The results showed that the teachers obtained an average of 55% overall in the items. The average for the trigonometry and coordinate geometry was 52% and 61% respectively. The study suggests that teachers need more support in mastering the Grade 12 content and developing representational fluency. [ABSTRACT FROM AUTHOR]

Objective – This paper explores the challenges experienced by mathematics teachers and learners of seven high schools in South Africa. A mixed method approach involving interviews and questionnaires was used. Methodology – The sample included 7 principals or deputy principals, 7 departmental heads who supervised mathematics, 1 mathematics teacher who taught mathematics in grades 10 to 12, and 1 mathematics teacher who taught Mathematics in grades 8 and 9. In addition, 1 learner from each grade 8 to 12 was included in the sample. Each participant was interviewed using a structured interview schedule, followed by participants completing a questionnaire. Findings – Vygotsky's socio-cultural and conceptual development theories in mathematics education formed the basis of this study. The 'Inputs, Processes and Output" model of Howie and Kilpatrick's model of the five strands of achieving 'mathematics proficiency was adopted. We identified many systemic, societal, and pedagogical challenges that teachers and learners experienced, which affected their mathematics teaching and learning. Our findings correlated with research in the literature review. Novelty – The researchers concluded there were always intervention strategies that could assist in minimizing or eradicating the challenges faced by the teachers and learners of mathematics, thus improving their socio-economic standing. [ABSTRACT FROM AUTHOR]