The purpose of this study was to extend our knowledge about mathematical literacy learning with the focus on a foundational preservice teacher education module required for prospective teachers. The construct of mathematical proficiency provided a framework for understanding how successful learning depends on a multiplicity of competences, and in particular to highlight the pivotal role of a productive disposition towards mathematics in becoming mathematically literate. The main questions that guided the study were as follows: What is the nature and strength of the productive disposition strand of mathematical proficiency evident in preservice teachers entering a Mathematical Literacy module and how does this productive disposition change over the course of the module? and What pedagogical practices and learning behaviours best enable preservice teachers to achieve mathematical literacy? The study was undertaken as two overlapping case studies, the first describing the preservice teachers at the onset of their studies in the Mathematical Literacy for Educators module, and in the second, a three part story-telling case study of the unfolding of the module over three years from 2003 to 2005. The mathematics autobiographies of 254 preservice teachers and the data obtained from a premodule questionnaire and introductory class activities contributed to the first case study which was summarised in the form of three fictional letters. Written reflections, final module evaluations and the insights of my co-workers contributed to the second case study which documented the successes and struggles of the preservice teachers as the module unfolded each year. Complementary mixed methods techniques were used to analyse the multiple sources of data and to weave strong ropes of evidence to support the findings. Statistical analysis pointed to themes which were supported or tempered by qualitative evidence reported in the voices of the preservice teachers themselves. The analysis revealed that many of the preservice teachers entering the Mathematical Literacy for Educators module had found their school experience of mathematics to be dispiriting and consequently had developed negative dispositions towards the subject. The change in this disposition depended on their success in the module and the empathy shown by the lecturer. Helpful pedagogical practices were found to be those that supported language difficulties in learning mathematics, assisted in organising learning, remediated for poor schooling background in mathematics and took account of the diversity amongst the students. I argue that many of the lessons learned and insights gained from teaching the Mathematical Literacy for Educators module are relevant to the expanding number of mathematics courses required as part of humanities programmes. In addition, they can inform practices at school level and in both in mathematics and mathematical literacy teacher education. / Thesis (PhD.)-University of KwaZulu-Natal, 2007.

This study had two components: 1) Investigating the conceptual understanding of teachers teaching elementary mathematics at primary schools in the province of KwaZulu-Natal, who had been successful in their mathematics modules in the National Professional Diploma in Education (NPDE) teacher upgrading program, and 2) Investigating the influence of their mathematical lifehistories on their understanding and personal philosophies about mathematics. It firstly required the NPDE students from the University of KwaZulu-Natal to complete a questionnaire adapted from the TELT interview schedule used by Liping Ma (1999). This questionnaire was to assess whether these high scoring teachers had an understanding of basic mathematical concepts that could have been regarded as being profound. The second part of the study was designed in order to get these teachers to examine their mathematical life histories and then to look at how their life histories could have influenced their level of understanding. It was found that these teachers were procedurally capable and were aware of the algorithms that could be used to solve the problems posed, but they lacked deep understanding of the concepts and were thus conceptually weak. None of the teachers demonstrated an understanding of the fundamental mathematics concepts that were assessed, that could be regarded as been ‘profound’. The mathematical life history portion of this study revealed that these teachers, having experienced mathematics education very differently due to their Apartheid influenced education, mentioned that there were definite influences that had a marked effect on their outlook on the subject and thus their belief in their ability to do basic/ fundamental mathematics. / Thesis (M.Ed. (School of Education and Development)) - University of KwaZulu-Natal, Pietermaritzburg, 2007.

Preservice teachers have had a twelve year "apprenticeship of observation" in the practice of teaching and as a consequence have internalised a set of beliefs about Mathematics teaching and learning. These beliefs are usually implicit but nevertheless influence the teaching practice of the preservice teachers to the extent that they "teach as they were taught." A preservice Mathematics Education course, falling as it does between the prospective teachers' experiences as scholars, and their future teaching experience provides an ideal opportunity for preservice teachers to review their personal beliefs prior to carrying them over to teaching practice. In order to facilitate this review, a series of activities was designed as part of a Mathematics Education course for preservice secondary phase Mathematics teachers. These activities provided opportunities for student teachers to examine their beliefs, to discuss and write about these beliefs, to read about the beliefs of others, and finally to decide whether they wished to retain or modify their personal beliefs. Data on the personal theories of the preservice teachers was obtained from the written responses to various critical incidents, from metaphors for the teaching and learning of Mathematics drawn and described by the preservice teachers, and from interviews with selected participants. These theories were classified into qualitatively different categories. After completing several developmental activities and a five week period of classroom teaching, the preservice teachers were invited to reconsider their personal theories and amend their metaphors. This provided evidence of reflection and development in their thinking. It is contended that the personal theories of preservice teachers are not only reflected in their classroom practice but also function as barriers to impede acceptance of novel ideas and innovations. The findings of this study contribute to the understanding of the thinking of preservice teachers and inform the development of a curriculum for the Mathematics Education component of a Professional Studies course. / Thesis (M.Ed.)-University of Natal, 1999.

The aim of this study was to explore the concept of pedagogical content knowledge, or PCK, and its use in the practice of teaching. Teacher knowledge is a significant factor in determining learner gains in all school subjects. However, little is known about the role of the different types of knowledge that teachers are supposed to possess in particular in a developing world context. PCK was introduced by Lee Shulman in 1986 and has since been the subject of much research in teacher education. Pedagogical content knowledge is thought to be a highly specialised form of teacher knowledge that intertwines subject matter (content) knowledge and general pedagogic knowledge. In this study, I examined the levels of PCK of 39 mathematics teachers; I tried to determine how they used PCK in their teaching of mathematics; what determined their PCK; and to what extent PCK influenced the mathematical achievement of their learners. The methodology that I used was lesson observation of 42 video-recorded grade 6 mathematics lessons from various schools in the greater Umgungundlovu district of Pietermaritzburg in KwaZulu-Natal. These schools were selected through random stratified sampling to participate in a larger regional achievement study, designed to investigate the factors which influence learning in schools. I was part of a research team that analysed the videos of the mathematics lessons, with the intention of getting the ‘big picture’ of mathematics teaching and learning in South Africa. Using the data from my observations, I developed a PCK instrument and attempted to measure the teachers’ PCK. I then tried to link these PCK scores to other variables in my study, which included a teacher’s test and learner tests. I tested the consistency of my instrument and the teachers’ PCK scores appeared fairly consistent across lessons, but that more research is needed to interrogate that. My initial findings suggested that all teachers possess PCK in some form, though their observed PCK levels were limited. The opportunity to develop proficiency, the use of examples and some engagement with learners’ prior knowledge though mostly in the form of checking homework were the areas most prevalent. The focus was mostly on procedural aspects. Only a minority of the teachers used representations, showed more than one method, displayed longitudinal coherence or engaged in more substantial ways with learner thinking (misconceptions and errors). Crucially, it emerged that a sound teachers’ knowledge of mathematical content was necessary for a high PCK rating, but there was no significant relationship between teachers’ PCK and learner gains in mathematics. It is likely that there are other factors which have a greater impact on learners’ learning than effective teachers, factors such as the socio-economic backgrounds of the learners. Given the random sampling of the schools in the study, and various attempts to ensure consistency in my coding and analysis, I hoped that these results would be valid for the greater KwaZulu-Natal area. However, because I used mainly the video analysis of lessons, and only a part of the teachers’ test, to determine the teachers’ PCK, it is possible that I may not have been able to get the full picture of the teachers’ PCK as I would have if I had also interviewed them.

I embarked upon this study to investigate mathematics teachers' practices that have led to the persistence of traditional teaching methods in grade 9 mathematics classrooms in Rundu. The study was conducted from a learner-centred (L-C) perspective or reform approach in the Namibian context. It is a qualitative study oriented in the interpretive paradigm- a paradigm that seeks to understand the meanings attached to human actions. The participants involved in this study were purposively selected and they are composed of two mathematics teachers and their grade 9 learners. This study was conducted at two schools in Rundu. One is an urban school while the other one is a rural school on the outskirts of Rundu. The research tools employed in this study are questionnaires, interviews and observations. The questionnaires were used to identify and select my participants while the observations were used to investigate the participants teaching strategies. The purpose of the interviews was mainly to investigate teachers' understanding, interpretation and implementation of learner-centre education (LCE). Amongst other findings, this study reveals that inadequate teacher-training, controversial educational policies and challenges such as overcrowdedness in mathematics classrooms, lack of teaching and learning materials, lack of cooperation among mathematics teachers and learners' negative attitude towards mathematics are some of the contributing factors to the persistence of traditional teaching methods in mathematics classrooms. Moreover, the study reveals that the persistence of traditional teaching methods in mathematics classes can no longer be attributed to the lack of understanding of LCE. The teachers in this study seem to be well acquainted with the theoretical aspect of the LCE framework while the implementation aspect seems to be a concern.

The South African education is undergoing transformation. The introduction of Curriculum 2005 and Outcome Based Education (OBE) are important aspects of this transformation process. The implementation of the new curriculum however, has not been smooth. A lack of adequately qualified and trained teachers and effective learning support materials have been identified as some of the major problems facing the implementation of OBE at school level. Even though the colleges of education in the country were not brought on board in the planning and implementation levels of the new curriculum, they could have played an important role in training of educators and developing learning materials for the successful take off of the new curriculum. In the first research project I look into the preparedness of the Mathematics Department of a college of education in the Eastern Cape towards the implementation of Outcome Based Education. The first part of the research project consists of a literature review on Curriculum 2005, Outcome Based Education and the expectations of teachers in the new curriculum and the research methodologies used. The findings revealed that the department, as a whole, was not adequately prepared for the implementation of outcome-based education even though there were some indications that the department effected some modifications in its curriculum and practice teaching. As an OBE facilitator and a college lecturer, I developed some learning materials in Linear Programming. These activity-oriented materials were based on constructivist principles and were used by my first year Secondary Teachers Diploma students. In the second research project, I reflect on the results of using these learning materials by my students. In the first chapter of the project, the context and background of the research and the reasons for selecting Linear Programming as the topic for preparing the learning material are described. This is followed by a brief overview of constructivism together with a brief explanation of the reasons for considering the material to be constructivist. The research paradigm followed in the project, the research techniques employed in evaluating the learning material and the strengths and weaknesses of the evaluation techniques are given in the next chapter. In the following chapter, the findings from the various data gathering methods and the results of the implementation of the material are described. The concluding chapter presents a critical reflection on the whole process involved in the material development. The post 1994 government in South Africa seems to attach much importance to mathematics, science and technology education. The majority of the population who were previously denied access to these subjects is given more opportunities to learn them. The international Mathematics Union declared 2000 as the World Mathematical Year. One of the aims of the activities organized as part of the celebrations was improving the public image of mathematics to realize the vision of "Mathematics For All". The South African government shows keen interest to improve mathematics education in the country in an attempt to realize the vision of Mathematics For All. Even though the accessibility rate to mathematics has increased, the success rate has not yet increased as anticipated. In this context I, as a post-graduate student in Mathematics Education, thought of /reviewing the concept of Mathematics For All in the South African context. In the third research project, which is a literature review, initially an attempt is made to unpack the concept of Mathematics for ALL. In analyzing the concept, answers are sought for questions like: What is mathematics and why should it be taught? It is followed by a brief review of some goals of mathematics education. Then the current situation of mathematics education in South Africa and the efforts to improve it are also looked at. This is followed by an analysis pf the reasons for the general unpopularity of mathematics. In the concluding part some suggestions for improving mathematics education in the country are given.

M.Ed. / Students who are being trained as teachers for the primary school must undergo subject-didactical training in mathematics. The attitude of these students towards mathematics is apparently not always positive. A survey of the literature indicates a universal concern of teacher-trainers, namely that it is precisely these future teachers who have to teach mathematics to the pupil in his formative years, who demonstrate a negative attitude towards the subject. It is imperative that those who develop a subject-didactics curriculum for mathematics at a college of education should determine in a scientifically responsible manner what the nature and extent of the students' attitude towards mathematics is. Attitude is a determining factor in each of the principles in curriculum design. An empirical investigation was conducted with first year students at the Goudstad College of Education as respondents. Their attitude towards mathematics in general and towards certain dimensions of mathematics in particular was measured by means of Visser's attitudinal scales. It was found that there is no meaningful difference in attitude towards mathematics between man and woman students, but a meaningful difference was found between students who had a matriculation qualification in mathematics and those who did not. The former group indicated a more positive attitude towards mathematics than the latter. The attitude of students who are prepared to teach mathematics as a school subject in the future is significantly more positive than that of those who are not prepared to do so. Negative attitudes towards mathematics appear particularly in relation to a person's self confidence and motivation to be involved with mathematics. The most important implication of this investigation for the design of a subject-didactics curriculum is that it should be borne in mind that certain groups of students reveal clear negative attitudes towards mathematics. The total curriculum should be directed to developing positive attitudes where necessary and to improving and strengthening existing positive attitudes.

D.Phil. / The research problematised the learning of mathematics in South African high schools in a Pedagogical Content Knowledge context. The researcher established that while at best, teachers may command mathematics content knowledge, or pedagogic knowledge, that command proves insufficient in leveraging the learning of mathematics and differentiation. Teachers' awareness of their learners' errors and misconceptions on a mathematics topic is critical in developing appropriate pedagogical content knowledge. The researcher argues that the study of learner errors in mathematics affords educators critical knowledge of the learners' Zones of Proximal Development. The space where learners experience misconceptions as they attempt to assign meaning to new mathematical ideas to which they may or may not have obtained semiotic mediation. In their Zones of Proximal Development learners may harbour concept images that are incompetition with established mathematical knowledge.Educators need to study and understand those concept images (amateur or alternative conceptions), and how learners come to have them, if they are to help learners learn mathematics better. Besides the socio-cultural v1ew, the study presumed that the misconceptions formed by learners in mathematicsmay also beexplained within a constructivist perspective of learning. The constructivist perspective of learning assumes that learners interpret new knowledge on the basis of the knowledge they already have. However, some of the knowledge that learners construct though meaningful to them may be full of misconceptions. This may occur through overgeneralisation of prior knowledge to new situations. The researcher presumed that the ideas that learners have of particular mathematical concepts were concept images they construct. Though some of the concept images may be deficient or defective from a mathematics expert's point of view, they are still used by the learners to learn new mathematics concepts and to solve mathematics problems. The lack of success in mathematics that results in the application of erratic concept images ultimately leads to unsuccessful learning of mathematics with the danger of snowballing if there are no practicable interventions. Differentiation is a new topic in the South African mathematics curriculum and most teachers and learners have registered problems in teaching and learning it. Hence it was imperative to do research on this topic from an angle of learner errors on that topic. The significance of the study is that this research isolated the differentiation learner errors and misconceptions that teachers can focus on for the improvement of learning and achievement in the topic of introductory differentiation. The research focused on the nature of errors and misconceptions learners have on introductory differentiation as exhibited in their 2008 examination scripts. It sought to identify, categorise (form a database) and discuss the errors and their conceptual links. A typology of errors and misconceptions in introductory calculus was constructed. The study mainly used qualitative methods to collect and analyse data. Content analysis techniques were used to analyse the data on the basis of a conceptual framework of mathematics and calculus errors obtained from literature. One thousand Grade 12, Mathematics Paper 1 examination scripts from learners of both sexes emanating from diverse social backgrounds provided data for the study. The unit of analysis was students' errors in written responses to differentiation examination items.

This qualitative study reports on critical reflective teaching by three mathematics teachers and how it shapes their classroom practice. The study was carried out in three secondary schools in Rundu in northern Namibia. The study employed a case study method. The selection of teachers was based on their rich practical professional knowledge and exemplary teaching practices. Data collection and analysis was done through an interpretive approach. Interviews and document analyses were the two research tools used, not only for the collection of data but for triangulation also. Interpretations of the findings were validated through member checking. Critical reflective teaching involves thought and action, and it raises teachers’ consciousness of what they do. Through critical reflective practice, teachers scrutinize their beliefs and knowledge of the subject and their practice. Furthermore critical reflective practice may get teachers into a disposition to find alternatives to improve their teaching. In this study, the findings are that participants reflect extensively on their classroom practice. The teachers pointed out that reflection on practice enables them to analyse and evaluate their teaching in line with effective mathematics teaching. They emphasised that critical reflection leads to the identification of weaknesses in teachers’ classroom practice. This culminates in better planning whereby alternative approaches to teaching are exercised. Because of its potential to improve teaching and enhance professional development it is therefore recommended that mathematics teachers be exposed to skills that enhance critical reflective teaching practice. Teachers need to familiarise themselves with the concept of critical reflective teaching in mathematics to meet the demands of superior quality teaching.

D.Ed. / Please refer to full text to view abstract