The purpose of this study was to determine the relationship between High School learners‟ mathematics-related belief systems and their approaches to mathematics non-routine problem-solving. A mixed methods approach was employed in the study. Survey questionnaires, mathematics problem solving test and interview schedules were the basic instruments used for data collection. The data was presented in form of tables, diagrams, figures, direct and indirect quotes of participants‟ responses and descriptions of learners‟ mathematics related belief systems and their approaches to mathematics problem solving. The basic methods used to analyze the data were thematic analysis (coding, organizing data into descriptive themes, and noting relations between variables), cluster analysis, factor analysis, regression analysis and methodological triangulation. Learners‟ mathematics-related beliefs were grouped into three Learners‟ mathematics-related beliefs were grouped into three categories, according to Daskalogianni and Simpson (2001a)‟s macro-belief systems: utilitarian, systematic and exploratory. A number of learners‟ problem solving strategies were identified, that include unsystematic guess, check and revise; systematic guess, check and revise; trial-and-error; logical reasoning; non-logical reasoning; systematic listing; looking for a pattern; making a model; considering a simple case; using a formula; numeric approach; piece-wise and holistic approaches. A weak positive linear relationship between learners‟ mathematics-related belief systems and their approaches to non-routine problem solving was discovered. It was, also, discovered that learners‟ mathematics-related belief systems could explain their approach to non-routine mathematics problem solving (and vice versa). / Mathematics Education / D.Phil. (Mathematics Education)

The aim of this study was to investigate the comparative effects of Constructivist-Based Teaching Method (CBTM) and the Traditional Teaching Method (TTM) on Grade 11 Mathematics learners’ errors in algebra. The constructivist learning theory (CLT) was used to frame this study. Mainly, CLT was used to influence the design of CBTI to hone participants’ errors in algebra that militate against their performance in Mathematics. The study was conducted in the Mpumalanga Province of South Africa with a four-week intervention programme in each of the two participating secondary schools. Participants consisted of n=78 Grade 11 Mathematics learners and one Grade 11 Mathematics teacher. A non-equivalent control group design consisting of a pre-test and post-test measure was employed. The Grade 11 teacher in the control school employed the TTM while the researcher implemented CBTM in the experimental school. The main aspects of CBTM entailed participants’ construction of their own knowledge from the base of prior knowledge and through group learning approach and exploratory talk in which discussions included argumentation, verbalising explanations, justifications and reflections. Participants in experimental school became familiar with the basic principles of CBTI such as group work, which enhanced the construction of conceptual understanding of algebraic concepts. This reduced most of the errors they commit in algebra and elevated their performance in Mathematics. The principal instruments for data collection consisted of a standardised Algebra Concept Achievement Test and lesson observations. The pre-test was used to determine participants’ initial errors in algebra before the intervention. A post-test was given at the end of intervention to ascertain change in participants’ errors in algebra over a four-week intervention period. Using descriptive and inferential statistical techniques, the study found that participants in experimental school significantly reduced their errors in algebra than those in control school. The study showed that CBTM was a more effective pedagogy that improved the errors Grade 11 learners commit in algebra than the TTM. / College of Education / M. Ed. (Mathematics Education)

The study intends to investigate the nature and cognitive demands of contextual word-problems posed in the FET mathematics examinations of IEB and NSC. The analysis of the mathematization of real-life situations to form contextual word-problems is based on the theory of authentic task situations. The theoretical basis for analyzing mathematics teaching and learning is the Realistic Mathematics Education (RME) theory. Data was obtained using the schedule of mathematization of real-life situations and the schedule of total marks of contextual word-problems and national performance. All contextual word-problems included in the 2008-2013 question papers of IEB and NSC mathematics examinations were analysed. The research revealed that 509 marks out of 1800 marks were allocated to contextual word-problems in IEB examinations; whereas 473 marks out of 1800 marks were allocated to contextual word-problems in NSC examinations. / Mathematics Education / M. Sc. (Mathematical Education)

The aim of this study was to identify types of behaviour manifested by learners during mathematics instruction, and the impact that this behaviour might have on the mathematics performance of learners. The study was conducted in Far East cluster of Johannesburg East District, in the province of Gauteng, South Africa. At the time of the study the Far East cluster of Johannesburg East District consisted of a population of seven public high schools, of which two schools were randomly sampled to participate in the study. Participants consisted of (n=10) Grade 10 mathematics learners, 2 mathematics teachers and 2 heads of mathematics departments (HODs). Data from learners were collected using a set of their assessment scores accumulated over a period of six months, that is, from January 2014 to June 2014 (Section 1.3.3). Also, semi-structured interviews were carried out with learners to determine types of classroom behaviour they perceived to influence their mathematical performance. The aim of documenting learners‟ assessment scores (document analysis) was to determine their average performance in Grade 10 mathematics over a stipulated period. Teachers and HODs completed questionnaires to identify types of classroom behaviour that learners manifested during mathematics instruction. The study followed a qualitative approach with phenomenology research design (Section 3.2). The study identified several types of classroom behaviour that characterized mathematics instruction in both schools, such as making noise and not doing classwork and homework activities. In addition, the study established that forms of behavioural patterns that are manifested by learners during a mathematics instruction influenced their performance in the subject. Huitt‟s (1997) model was used to conceptualize and interpret the results. / Mathematics Education / M. Ed. (Mathematics Education)

This mixed methods research design, which was modelled on the constructivist view of schooling, sets out to investigate the effect of developing mathematical problem solving skills of grade 8 learners on their performance and achievement in mathematics. To develop the mathematical problem solving skills of the experimental group, a problem-centred teaching and learning environment was created in which problem posing and solving were the key didactic mathematical activity. The effect of the intervention programme on the experimental group was compared with the control group by assessing learners’ problem solving processes, mathematical problem solving skills, reasoning and cognitive processes, performance and achievement in mathematics. Data were obtained through questionnaires, a mathematical problem solving skills inventory, direct participant observation and questioning, semi-structured interviews, learner journals, mathematical tasks, written work, pre- and post- multiple-choice and word-problem tests. Data analysis was largely done through descriptive analysis and the findings assisted the researcher to make recommendations and suggest areas that could require possible further research. / Mathematics Education / M. Ed. (Mathematical Education)

The aim of this study was to investigate the effect of context-based problem solving instruction (CBPSI) on the problem solving performance of Grade 10 learners, who performed poorly in mathematics. A cognitive load theory (CLT) was used to frame the study. In addition, CLT was used to: 1) facilitate the interpretation and explanation of participants‟ problem solving performance; and, 2) influence the design of CBPSI to hone participants‟ problem solving skills. The study was conducted in the Gauteng province of South Africa and involved a two-week intervention program in each of the nine participating high schools. Participants consisted of 783 learners and four Grade 10 mathematics teachers. A non-equivalent control group design was employed, consisting of a pre- and post- measure. In addition, classroom observations and semi-structured interviews were conducted with teachers and learners. Teachers employed conventional problem solving instructions in four control schools while the researcher implemented CBPSI in five experimental schools. Instruction in experimental schools entailed several worked-out context-based problem solving examples given to participants in worksheets. The main aspects of CBPSI embraced elements of the effects of self-explanation and split-attention, as advocated by CLT. Due to the design of CBPSI participants in experimental schools became familiar with the basic context-based problem solving tasks that were presented to them through the worked-out example samples. In turn, the associated cognitive load of problem solving tasks was gradually reduced. The principal instrument for data collection was a standardized Functional Mathematics Achievement Test. The pre-test determined participants‟ initial problem solving status before intervention. A post-test was given at the end of intervention to benchmark change in the functionality of CBPSI over a two-week period. Using one-way analysis of covariance (ANCOVA), Analysis of Variance (ANOVA), and other statistical techniques the study found that participants in experimental schools performed significantly better than participants in control schools on certain aspects of problem solving performance. In addition, semi-structured interviews and classroom observations revealed that participants rated CBPSI highly. On the whole, the study showed that CBPSI is an effective instructional tool to enhance the problem solving performance of Grade 10 mathematics learners. / Mathematics Education / D. Phil. (Mathematics, Science and Technology Education)

Smartphone technology can be integrated in mathematics education, since most learners immerse in this technology. Teachers should be facilitators of confined learning in traditional classrooms, including digital facilitators. The aim of the study was to determine the effect of integrating smartphone technology on Grade 10 learners’ achievement in algebra. The study focused on two Grade 10 classes from different township schools in the Gauteng Province. The study was conducted during the first term of 2016. A quasi-quantitative experimental research design was used. Smartphones were integrated in algebra teaching and learning, using the Microsoft Math online application: https://www.math.microsoft.com. Despite challenges encountered during the study, the results indicated a positive impact on learners’ achievements in algebra, amongst experimental group learners. A need exists for future research in integrating smartphone technology in mathematics education. / Mathematics Education / M. Ed. (Mathematics Education)

University of Macau / Faculty of Education

The Eastern Cape Department of Education has a constitutional and legislative mandate to provide quality public education in all public schools within the province. The commitment to improve also involves ensuring effective educator personnel provisioning in public secondary xii schools. However, despite various legislative and policy measures relating to educator personnel provisioning, serious concerns have been raised. There have been concerns over the shortage of educators. There are concerns over the lack of suitably qualified educators, especially in Mathematics and Science subjects. Disparities have been noted between legislative and policy directives with regard to educator personnel provisioning and the expectations of educators based on the realities in public secondary schools. The complications and challenges thereof have called upon for the appraisal of educator provisioning. The objective of the study is to describe and explain the nature and place of personnel provisioning in public secondary schools. Thereafter, evaluate its application at selected public secondary schools in the Eastern Cape Province. The basic intention is to prove that the educator personnel provision policy implementation in public schools needs to be examined and possibly changed for effective results. Basically, personnel provisioning is the first step in the personnel process. It is classified into: Human resources determination; and the Filling of posts The process of filling public personnel posts involves recruitment, selection, appointment, placement, transfer and promotion. In evaluating personnel provisioning policy in public secondary schools, a research study has to be conducted. As an integral part of scope of study, the survey area includes public secondary schools, provincial department offices and educator union/association offices. The questionnaire is used as the appropriate data collection instrument for this survey. The total population for the study are provincial department officials, public secondary school principals, school governing body chairpersons and educator unions/associations. xiii When data was analysed and interpreted, there were various significant findings. The respondents’ demographic details provided a significant insight into the study and its findings. They ensured divergence of opinions and understanding of the personnel provisioning. Regarding human resources determination and the filling of posts in public secondary schools, it was found that there are challenges in terms of the employment of suitably qualified educators. These include among other issues educator dissatisfaction and lack of effective procedures in the filling of posts. This also results in educator shortage in public secondary schools.

The study attempted to determine the causes of poor performance among Grade 12 learners in tasks involving calculus, especially in cubic graphs and the application of differential calculus. The study was conducted in three schools of the Msukaligwa 1 Circuit in the Gert Sibande District, Mpumalanga Province in South Africa. Differential calculus is a branch of mathematics that is concerned mainly with the study of the rate of change of functions with respect to their variables especially through the use derivatives and differentials. Students have difficulties in learning and mastering this section of calculus as is revealed by examiners’ and moderators’ reports year after year. The purpose of this study was to investigate the possible reasons for the poor performance by Grade 12 learners in calculus-based tasks, especially in cubic graphs and the application in optimisation. The study sought to investigate the causes of the poor performance by Grade 12 learners in tasks based on these two subtopics of calculus. Three schools were selected by means of purposive sampling: one former model C, one Mathematics, Science and Technology Academy (MSTA) and one other school that does not fall in either of these two categories. This enabled the study to have participants from diverse backgrounds. A qualitative research design was used. Data was collected using learners’ scripts for the three formal tasks: May common test, June (midyear) and Trial (preparatory) examinations. Only the questions involving cubic graphs and the application of calculus were part of the study. Analysis was done in order to determine learners’ challenges, common mistakes, and misconceptions, but also of good responses given by learners. / Mathematics Education / M. Ed. (Mathematics Education)