Text in Afrikaans, abstract in Afrikaans and English / Hierdie studie is daarop gemik om te bepaal tot welke mate wiskundeleerlinge op skool en onderwysstudente in wiskunde, onderrig in logika ontvang as agtergrond vir strenge bewysvoering. Die formele aspek van wiskunde op hoerskool en tersiere vlak is besonder belangrik. Leerlinge en studente kom onvermydelik met hipotetiese argumente in aanraking. Hulle leer ook om die kontrapositief te gebruik in bewysvoering. Hulle maak onder andere gebruik van bewyse uit die ongerymde. Verder word nodige en voldoende voorwaardes met stellings en hulle omgekeerdes in verband gebring. Dit is dus duidelik dat 'n studie van logika reeds op hoerskool nodig is om aanvaarbare wiskunde te beoefen. Om seker te maak dat aanvaarbare wiskunde beoefen word, is dit nodig om te let op die gebrek aan beheer in die ontwikkeling van 'n taal, waar woorde meer as een betekenis het. 'n Kunsmatige taal moet gebruik word om interpretasies van uitdrukkings eenduidig te maak. In so 'n kunsmatige taal word die moontlikheid van foutiewe redenering uitgeskakel. Die eersteordepredikaatlogika, is so 'n taal, wat ryk genoeg is om die wiskunde te akkommodeer. Binne die konteks van hierdie kunsmatige taal, kan wiskundige toeriee geformaliseer word. Verskillende bewystegnieke uit die eersteordepredikaatlogika word geidentifiseer, gekategoriseer en op 'n redelik eenvoudige wyse verduidelik. Uit 'n ontleding van die wiskundesillabusse van die Departement van Onderwys, en 'n onderwysersopleidingsinstansie, volg dit dat leerlinge en studente hierdie bewystegnieke moet gebruik. Volgens hierdie sillabusse moet die leerlinge en studente vertroud wees met logiese argumente. Uit die gevolgtrekkings waartoe gekom word, blyk dit dat die leerlinge en studente se agtergrond in logika geheel en al gebrekkig en ontoereikend is. Dit het tot gevolg dat hulle nie 'n volledige begrip oor bewysvoering het nie, en 'n gebrekkige insig ontwikkel oor wat wiskunde presies behels. Die aanbevelings om hierdie ernstige leemtes in die onderrig van wiskunde aan te spreek, asook verdere navorsingsprojekte word in die laaste hoofstuk verwoord. / The aim of this study is to determine to which extent pupils taking Mathematics at school level and student teachers of Mathematics receive instruction in logic as a grounding for rigorous proof. The formal aspect of Mathematics at secondary school and tertiary levels is extremely important. It is inevitable that pupils and students become involved with hypothetical arguments. They also learn to use the contrapositive in proof. They use, among others, proofs by contradiction. Futhermore, necessary and sufficient conditions are related to theorems and their converses. It is therefore apparent that the study of logic is necessary already at secondary school level in order to practice Mathematics satisfactorily. To ensure that acceptable Mathematics is practised, it is necessary to take cognizance of the lack of control over language development, where words can have more than one meaning. For this reason an artificial language must be used so that interpretations can have one meaning. Faulty interpretations are ruled out in such an artificial language. A language which is rich enough to accommodate Mathematics is the first-order predicate logic. Mathematical theories can be formalised within the context of this artificial language. Different techniques of proof from the first-order logic are identified, categorized and explained in fairly simple terms. An analysis of Mathematics syllabuses of the Department of Education and an institution for teacher training has indicated that pupils should use these techniques of proof. According to these syllabuses pupils should be familiar with logical arguments. The conclusion which is reached, gives evidence that pupils' and students' background in logic is completely lacking and inadequate. As a result they cannot cope adequately with argumentation and this causes a poor perception of what Mathematics exactly entails. Recommendations to bridge these serious problems in the instruction of Mathematics, as well as further research projects are discussed in the final chapter. / Curriculum and Institutional Studies / D. Phil. (Wiskundeonderwys)

In this study, senior certificate (standard 10) pupils' attainment of the cognitive and affective aims and objectives of the senior certificate mathematics curriculum was investigated. With regard to the attainment of the cognitive objectives and aims, senior certificate pupils' performance in their mathematics examination, in terms of three broad categories of cognitive abilities (lower level, middle level and higher level mathematical abilities) was analysed and examined. As no norms (criteria) for mathematical attainment in respect of these three categories of cognitive abilities could be identified, these norms had to be firstly developed by the researcher. However, suitable standardised scales were identified and administered to determine senior certificate pupils' attainment of the affective aims and objectives (attitude towards and interest in mathematics). Besides the quantitative analysis, qualitative assessments of senior certificate pupils' attainment of the cognitive and affective aims and objectives were also made using information obtained, by way of a questionnaire, from teachers of senior certificate mathematics classes. The main findings that emerged from this investigation were: * The senior certificate pupils are attaining the desired proficiency levels in the cognitive objectives and aims of the senior certificate mathematics curriculum. However, these pupils are not adequately attaining the affective aims and objectives of the mathematics curriculum. * Qualitative information elicited from senior certificate teachers of mathematics tends to support the above findings which were obtained from the quantitative analysis. * There is a need for curriculum development in certain areas of the senior certificate mathematics curriculum, particularly in Euclidean geometry, for standard grade pupils. In terms of the general findings, certain recommendations were also formulated. In several ways, the present research is a pioneering effort in evaluating the efficacy of the cognitive and affective aims and objectives of the senior certificate mathematics curriculum. It is hoped that this study will serve as a catalyst for future research. / Curriculum and Instructional Studies / D. Ed. (Didactics)

Mathematics anxiety is a personal characteristic which is widespread and continuing. It has a debilitating effect on mathematics performance and contributes to perceptions and attitudes that perpetuate a dislike for mathematics and a lack of confidence when dealing with mathematical problems. An investigation of relevant literature on mathematics anxiety identifies sources and symptoms and emphasises a need for a comprehensive approach to remediation. The historical development of an appropriate measuring instrument is documented and statistical evidence is used to create a mathematics anxiety rating scale suitable for measuring anxiety levels of secondary school pupils and student teachers. The extensive literature interest, research publications and remedial programmes emphasise the problem of mathematics anxiety and thus the need for a comprehensive approach to remediation. Mathematics teaching and curriculum design is expounded to provide the necessary direction to the alleviation of mathematics anxiety. General perspectives on curriculum design are discussed and a cyclical systems approach is recommended. Elements of this approach are detailed and are linked to important personal characteristics to add a humanistic and socio-cultural view of curriculum design in mathematics. The didactic viability of constructivism as an approach to mathematics curriculum design is investigated. Constructivism embodies a philosophy and a methodology which addresses the critical aspects influencing mathematics anxiety. Classroom topics and activities are reviewed in terms of a constructivist approach and the sources of mathematics anxiety are discussed from a constructivist perspective. A longitudinal case study of pupils during their five years at secondary school as well as a study involving student teachers was undertaken. Mathematics performance, perceptions, attitudes and levels of anxiety were investigated by means of tests, questionnaires, and mathematics anxiety rating scales. The statistical results of this research provide evidence to support a comprehensive approach to the remediation of mathematics anxiety. Constructivism is seen as the synthesis of critical aspects of teaching and curriculum development which will stem the perpetuation of mathematics anxiety. Constructivism provides the didactic approach to develop each individual's intellectual autonomy and mathematics power, by instilling a problem solving attitude and a self-confidence when doing mathematics. / Curriculum and Instructional Studies / D. Ed. (Didactics)

This study investigates the possibilities and implications for the teaching of the solving of polynomial equations. It is historically directed and also focusses on the working procedures in algebra which target the cognitive and affective domains. The teaching implications of the development of representational styles of equations and their solving procedures are noted. Since concepts in algebra can be conceived as processes or objects this leads to cognitive obstacles, for example: a limited view of the equal sign, which result in learning and reasoning problems. The roles of sense-making, visual imagery, mental schemata and networks in promoting meaningful understanding are scrutinised. Questions and problems to solve are formulated to promote the processes associated with the solving of polynomial equations, and the solving procedures used by a group of college students are analysed. A teaching model/method, which targets the cognitive and affective domains, is presented. / Mathematics Education / M.A. (Mathematics Education)

In this study, senior certificate (standard 10) pupils' attainment of the cognitive and affective aims and objectives of the senior certificate mathematics curriculum was investigated. With regard to the attainment of the cognitive objectives and aims, senior certificate pupils' performance in their mathematics examination, in terms of three broad categories of cognitive abilities (lower level, middle level and higher level mathematical abilities) was analysed and examined. As no norms (criteria) for mathematical attainment in respect of these three categories of cognitive abilities could be identified, these norms had to be firstly developed by the researcher. However, suitable standardised scales were identified and administered to determine senior certificate pupils' attainment of the affective aims and objectives (attitude towards and interest in mathematics). Besides the quantitative analysis, qualitative assessments of senior certificate pupils' attainment of the cognitive and affective aims and objectives were also made using information obtained, by way of a questionnaire, from teachers of senior certificate mathematics classes. The main findings that emerged from this investigation were: * The senior certificate pupils are attaining the desired proficiency levels in the cognitive objectives and aims of the senior certificate mathematics curriculum. However, these pupils are not adequately attaining the affective aims and objectives of the mathematics curriculum. * Qualitative information elicited from senior certificate teachers of mathematics tends to support the above findings which were obtained from the quantitative analysis. * There is a need for curriculum development in certain areas of the senior certificate mathematics curriculum, particularly in Euclidean geometry, for standard grade pupils. In terms of the general findings, certain recommendations were also formulated. In several ways, the present research is a pioneering effort in evaluating the efficacy of the cognitive and affective aims and objectives of the senior certificate mathematics curriculum. It is hoped that this study will serve as a catalyst for future research. / Curriculum and Instructional Studies / D. Ed. (Didactics)

Mathematics anxiety is a personal characteristic which is widespread and continuing. It has a debilitating effect on mathematics performance and contributes to perceptions and attitudes that perpetuate a dislike for mathematics and a lack of confidence when dealing with mathematical problems. An investigation of relevant literature on mathematics anxiety identifies sources and symptoms and emphasises a need for a comprehensive approach to remediation. The historical development of an appropriate measuring instrument is documented and statistical evidence is used to create a mathematics anxiety rating scale suitable for measuring anxiety levels of secondary school pupils and student teachers. The extensive literature interest, research publications and remedial programmes emphasise the problem of mathematics anxiety and thus the need for a comprehensive approach to remediation. Mathematics teaching and curriculum design is expounded to provide the necessary direction to the alleviation of mathematics anxiety. General perspectives on curriculum design are discussed and a cyclical systems approach is recommended. Elements of this approach are detailed and are linked to important personal characteristics to add a humanistic and socio-cultural view of curriculum design in mathematics. The didactic viability of constructivism as an approach to mathematics curriculum design is investigated. Constructivism embodies a philosophy and a methodology which addresses the critical aspects influencing mathematics anxiety. Classroom topics and activities are reviewed in terms of a constructivist approach and the sources of mathematics anxiety are discussed from a constructivist perspective. A longitudinal case study of pupils during their five years at secondary school as well as a study involving student teachers was undertaken. Mathematics performance, perceptions, attitudes and levels of anxiety were investigated by means of tests, questionnaires, and mathematics anxiety rating scales. The statistical results of this research provide evidence to support a comprehensive approach to the remediation of mathematics anxiety. Constructivism is seen as the synthesis of critical aspects of teaching and curriculum development which will stem the perpetuation of mathematics anxiety. Constructivism provides the didactic approach to develop each individual's intellectual autonomy and mathematics power, by instilling a problem solving attitude and a self-confidence when doing mathematics. / Curriculum and Instructional Studies / D. Ed. (Didactics)

This study investigates the possibilities and implications for the teaching of the solving of polynomial equations. It is historically directed and also focusses on the working procedures in algebra which target the cognitive and affective domains. The teaching implications of the development of representational styles of equations and their solving procedures are noted. Since concepts in algebra can be conceived as processes or objects this leads to cognitive obstacles, for example: a limited view of the equal sign, which result in learning and reasoning problems. The roles of sense-making, visual imagery, mental schemata and networks in promoting meaningful understanding are scrutinised. Questions and problems to solve are formulated to promote the processes associated with the solving of polynomial equations, and the solving procedures used by a group of college students are analysed. A teaching model/method, which targets the cognitive and affective domains, is presented. / Mathematics Education / M.A. (Mathematics Education)