Die tesis is voorgelê ter verwerwing van die graad: Magister in Opvoedkunde in die Fakulteit Onderwys en Sosiale Wetenskappe aan die Kaapse Skiereiland Universiteit van Tegnologie 2014 / This study attempted to address the extremely poor results for Mathematics and total lack of relevant teaching and learning material in multi-grade schools. In the case of multi-grade classes, the teacher has to cope with a large variety of development levels, grades and subjects in one classroom. Challenges that arise as a result of multi-grade teaching require instructional materials with unique characteristics and learners and teachers who develop the skills to put this material to use successfully. During the study of the relevant literature, evidence was found that proved that the learner–material approach offers the best outcomes within the unique circumstances of multi-grade teaching. This approach shifts the focus to the teaching materials and the learners while the teacher facilitates the process. In both multi-grade teaching and Mathematics teaching, this method holds benefits for learners and teachers. It teaches the learners to accept responsibility for their own learning and to know at all times what to do because the material provides clear guidance, giving the teacher more time in class to help those who need it. After a literature study and a context analysis, design principles were established with which the above material should comply. By this means, scientific criteria can be developed against which future materials would be measured. The Centre for Multigrade Education (CMGE) has undertaken to develop the necessary prototype Mathematics material. Using this material, the researcher could put the validity of the design principles and the expected usefulness of the material to the test. This brings together the theory and practice in a remarkable manner. The primary objective of the study was to improve quality, not only of the prototype, but also of future teaching and learning material.

D.Ed. (Curriculum Studies) / This study investigated the impact of mathematics interventions on learner participation and performance in mathematics within Gauteng 47 high schools in the Johannesburg area over a five year period. Motivating the study was the perpetual implementation of mathematics interventions against the backdrop of persistent mediocrity in learner matriculation achievement in this subject. The essence of the research problem was the scarcity of knowledge relating to the effectiveness and impact of mathematics interventions. The strategy of inquiry employed was an explanatory research design which entailed a sequential collection and analysis of quantitative and qualitative data. Quantitative findings revealed that only 10% of learners who participated in interventions were enabled to enroll in mathematics essential to pursue mathematics related professions at tertiary level. Thus, the intention to redress the prevalent limited access to professional careers requiring a strong basis in mathematics has not been dented. Regarding learner performance in mathematics, Friedman tests for nonparametric hypothesis testing revealed that there was no significant evidence that the interventions had an impact on learner performance over the five year observation period. Qualitative data analysis explained the quantitative findings and further uncovered the positive impact of mathematics interventions. Furthermore, strengths and weaknesses of the mathematics interventions were identified as opposed to the quantitative findings which seemed to negate the existence of the impact of mathematics interventions. Through the findings of this study, factors that may be limiting the effectiveness of mathematics interventions were uncovered. These were interwoven into a base knowledge that can influence positive practices and future research related to optimizing the impact of mathematics interventions in high schools.

M.Ed. / Our world is becoming more mathematical. We are constantly surrounded by mathematical situations and are regularly required to make mathematical decisions. These decisions require number sense, estimation skills, ability to analyse data intelligently, knowledge of two and three-dimensional geometry and many other abilities not often taught in school. Halpern (1992:1) states: "...as a nation we not only need competent scientists and engineers, we also need a citizenry that is literate in mathematics issues." Reyes and Stannic (1998:26) support this view as they state: "Knowledge of mathematics is essential for all members of our society. To participate in our democratic processes and to be unrestricted in career choice and advancement, people must be able to apply mathematical ideas." Learners leaving school need to be able to use available technology and to reason mathematically, be confident of their abilities, be able to communicate mathematically and be problem solvers. The introduction of technology into the lives of many people has called upon their background of mathematics to cope with mathematical problems and manipulating technological instruments. The relationship between mathematics and technology is emphasised by Dowling and Noss (1990:24) when they say: "New technology is a powerful tool which opens up new areas of mathematics and changes the way in which society makes use of mathematics in the factory, office and home". The Cockcroft Committee (1982), in their report on an inquiry into the teaching of Mathematics in schools, sees the main task of mathematics teachers as: "Enabling each learner to develop within his own capabilities the mathematical understanding and skills required for adult life, for employment and further study". The most important contribution for the purpose of this study, was the emphasis placed on cognitive aims to equip learners with numerical understanding and skills. Other contributions were to develop logical thinking and to expand their ability to look for patterns and explain them and to develop an awareness of the link between mathematics and everyday situations. This report (1982) motivated the researcher to review how teachers are teaching. This means that there will be an investigation into how children learn mathematics and not just what should be taught. Naidoo, Smit and Van Heerden (1995:7) also emphasise this by stating that "...the advance in technology at this time changed educational thinking more, making it even more important to further investigate how children actually learn." Yildirim (1994:28) argues that "... improving student's thinking ability is accepted as an important goal of education, and schools are considered places where thinking skills can be promoted. However, ways in which this should be done is a matter of controversy among educators." He further postulates that the best way to improve student thinking involves deep and thoughtful subject matter instruction in which students are encouraged to think reflectively rather than to merely cite the facts. The central characteristics of mathematical thinking are the determination of relationship and their application. It includes the ability to analyse a given situation or experience; to distinguish between applicable data and those which are not applicable; to classify and arrange these data; to abstract relationships from them, and finally to symbolize them for future use and application in new situations. Therefore it is important and essential for students to be taught how to approach the problems. This is one way of encouraging inquisitive and creative mathematics. Well-developed thinking skills are useful in almost every life situation, therefore there is a need to design instructional programmes that focus on the acquisition and uses thinking skills that are needed to find and solve mathematical problems.

Dissertation submitted in compliance with the requirements for the Masters Diploma in Technology: (Post School Education), Technikon Natal, 1993. / Die voorbereiding van hierdie tesis behels die inagneming van algemene en spesifieke faktore (Hoofstuk 1) wat gelei het tot navorsing in self-onderrig in 'n komponent van technikon wiskunde II asook 'n ondersoek van vier algemene doelwitte, naamlik / M

M.Ed. / The research conducted for the purpose of this study clearly showed that outcomes-based education (OBE) potentially offers a remedy for this country's ineffective education system which often fails to equip learners adequately to face the future with confidence and achieve success in the adult world. OBE enables each learner to achieve optimally within the limitations of his/her personal ability and prepares learners for life after school. Since teachers play a vital role in the successful implementation of OBE, it is essential that they should truly understand this paradigm shift and should wholeheartedly support the accompanying change in focus. In teaching mathematics, the aim is that the learner should understand it well enough to actually apply mathematic principles in everyday life. To start with, a special effort should be made to make the syllabus content more meaningful by showing learners how their knowledge of mathematics can be applied in everyday situations and how it can be of value once they have left school. Once the OBE approach has been implemented, Mathematics pupils will be much more skilled when they leave school. In addition to being able to use their knowledge of mathematics confidently, they will have achieved a higher level of emotional maturity, which is an important asset, regardless of whether they continue their studies at a tertiary institution or enter the world of work. Teachers who consider OBE in this light will find that the change-over is much easier. The best way to prepare for the new system of education is undoubtedly to read as much as possible about it and to access any other available sources of information. Once a person has accepted and embraced the idea of OBE, he/she will find that the many new related terms an phrases are much easier to understand. This can then be followed by practical implementation, which will demand that teachers design their own strategies for the effective achievement of specific outcomes by their pupils. Much more than ever before teachers will have to develop their entrepreneurial skills, and this should be an exciting experience! The urgent need for a more effective approach to teaching in South Africa is spelled out most clearly by the poor results obtained in Mathematics at school level, and by the anxiety generally associated with the study of this subject. As a result relatively few learners are interested in continuing their studies in the fields of mathematics and science, which will inevitably lead to a critical shortage of skilled engineers, economists, etc.

M.A. (Anthropology) / This study explores the role of four teachers’ communicative styles in a multilingual and multicultural classroom focusing on the role of gesture when teaching. To compare their gestural behaviour under similar conditions, I filmed four grade one teachers (two Setswana mother tongue and two Afrikaans mother tongue) teaching the mathematical concept of halving. I classified the gestures and their sematic relation to speech on ELAN using an adapted version of Colletta et al.’s (2009) coding scheme. I found gestures formed by the teachers functioned according to Tellier’s (2006) three roles of ‘teaching gestures.’ They functioned to explain, evaluate and manage. The teachers used mainly representational, performative and deictic gestures when teaching these lessons. They used similar types of gestures, but the manner in which they used their gesture as semiotic resources varied. It appeared that conceptually accurate gestures used at key stages in the learning process aided learning. When used in a conceptually accurate way gesture functioned to mediate the transition from objects to mathematical signs when testing the children’s halving skills using semiotic resources. Overall gestures functioned to integrate, supplement and complement teachers’ speech. The teachers varied cultural and linguistic backgrounds did not appear to influence their gesturing styles, but differences can be noted due to the subject matter, context and content, idiosyncratic styles (pedagogic styles) and the manner in which they structured their lesson in relation to the teaching aids used.

M.Ed. (Subject Didactics) / Please refer to full text to view abstract

In the past the emphasis was given to intelligence as the 'sole major determining factor where academic success was concerned and Warburton (1969) wrote: The ignorance of educationists in the field of personality makes a striking contrast to their ability to estimate educational and intellectual capacity, imperfect as that may be. The understanding of children's personality and the teacher's ability to assess these traits is perhaps the most important topic in the whole field of education'. Where would we be today if we assumed that all children are of equal intelligence? We cannot therefore assume they are equally anxious, introverted and have the same persistence in the performing of a task, just as we cannot assume they have equal intellectual capabilities. " ... Any attempt to understand the complete causal chain associated with school attainment must include the effect of personality on the child's work in the school." (Eysenck , 1972). Intro. p. 1.

This study reports on the findings of my research, which was based on an intervention focused on recovery of early arithmetic strategies with one Grade 4 class of learners in a township school in Port Elizabeth in the Eastern Cape. Learners came from poor socio-economic backgrounds and initial evaluations showed that the majority of learners still relied on concrete methods, like tally counting, to perform addition and subtraction calculations even with numbers less than 10. This is not uncommon in the South African context especially with learners in low Socio-economic Status (SES) schools. The results of numerous assessments including the Department of Education’s Annual National Assessments point to a crisis in primary mathematics education where intermediate phase learners are generally operating several grade levels below the grade they are in. A large drop in mathematics performance is seen in the ANA results in grade 4 learners (the first grade of the transition from foundation phase to intermediate phase). Within this context, and my background in learning support for students, my research aimed to understand the possibilities and constraints of the implementation of a recovery program adapted from the widely implemented work of Wright et al. (2006, 2012). The primary adaptation made to the MR program involved administering the assessments and intervention with groups of (rather than individual) learners. Within the context of the many low SES under-resourced schools in SA, individualised interview based assessments and recovery is not seen as a possible remediation strategy. Drawing on a socio- constructivist perspective, my study used action research with one class of 23 learners and found that adaptation of the MR program for a group, based on eight recovery sessions, was useful for enabling some progress for all learners in terms of their early arithmetic strategies and conceptual place value. Although the need for a longer recovery period is acknowledged, the adapted program enabled some progress in levels and stages of conceptual knowledge (as conceptualized by Wright et al.’s (2006) Learning Framework in Number) for these two domains. The study concludes with some reflections and recommendations for the future.

This study investigates the possible effects that access to selected multiple representations (words, arrows and colours) have in terms of cognitive load and learner achievement when presented with algebraic problems at grade nine level. The presentation of multiple representations (the intervention) was intended to decrease extraneous cognitive load, manage the intrinsic cognitive load (algebraic problems) and optimise germane cognition (schema acquisition and automation). An explanatory sequential mixed-method design was employed with six hundred and seventy three learners in four secondary schools. Quantitative data were generated via pre-, intervention and post-tests/questionnaires, while qualitative data were obtained from open-ended questions in the pre-, intervention, and post-tests/questionnaires, eight learner focus group interviews (n = 32), and four semi-structured, open-ended teacher interviews. Statistically and practically significant improvement in mean test scores from the pre- to intervention test scores in all schools was noted. No statistically and practically significant improvement was noted in further post-tests except for post-test 2 which employed more challenging problems (statistically significant decrease with a small practical effect). Learners expressed their preference for arrows, followed by colours and then words as effective representations. Teacher generated qualitative data suggests that they realise the importance of using multiple representations as an instructional strategy and implicitly understand the notion of cognitive load. The findings, when considered in the light of literature on cognitive load, suggest that a reduction in extraneous cognitive load by using a more effective instructional design (multiple representations) frees working memory capacity which can then be devoted to the intrinsic cognitive load (algebraic problems) and thereby increase germane cognition (schema acquisition and automation).