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)

While professional developers have been encouraging teachers to plan for discourse around problem solving tasks as a way to orchestrate mathematically productive discourse (Stein, Engle, Smith, & Hughes, 2008; Stein, Smith, Henningsen, & Silver, 2009) no research has been conducted explicitly examining the relationship between the plans that teachers make for orchestrating discourse around problem solving tasks and the outcomes of implementation of those plans. This research study is intended to open the door to research on planning for discourse around problem solving tasks. This research study analyzes how 12 middle school mathematics teachers participating in the Mathematics Problem Solving Model professional development research program implemented lesson plans that they wrote in preparation for whole-class discussions around cognitively demanding problem solving tasks. The lesson plans consisted of the selection and sequencing of student solutions to be presented to the class along with identification of the mathematical ideas to be highlighted in the student solutions and questions that would help to make the mathematics salient. The data used for this study were teachers' lesson plans and the audio-recordings of the whole-class discussions implemented by the teachers. My research question for this study was: How do teachers' written plans for orchestrating mathematical discourse around problem solving tasks influence the opportunities teachers create for students to reason mathematically? To address this research question, I analyzed the data in three different ways. First, I measured fidelity to the literal lesson by comparing what was planned in the ISAs to what was actually took place in the implemented debriefs. That is, I analyzed the extent to which the teachers were implementing the basic steps in their lesson (i.e. sharing the student work they identified, addressing the ideas to highlight and the planned questions). Second, I analyzed the teachers' fidelity to the intended lesson by comparing the number of high-press questions in the lesson plans (that is, questions that create opportunities for the students to reason mathematically) to the number of high-press questions in the implemented discussion. I compared these two sets of data using a linear regression analysis and t-tests. Finally, I conducted a qualitative analysis, using grounded theory, of a subset of four teachers from the study. I examined the improvisational moves of the teachers as they addressed the questions they had planned, building a theory of how the different ways that teachers implemented their planned questions affected the opportunities for their students to reason mathematically around those planned questions. My findings showed that it was typical for the teachers to implement most of the steps of their lesson plans faithfully, but that there was not a statistically significant correlation between the number of high-press questions they planned and the number of high-press questions they asked during the whole-class discussions, indicating that there were other factors that were influencing the frequency with which the teachers were asked these questions that prompted their students to reason mathematically. I hypothesize that these factors include, but are not limited to, the norms in the classrooms, teachers' knowledge about teaching mathematics, and teachers' beliefs about mathematics. Nevertheless, my findings did show that in the portions of the whole-class discussions where the teachers had planned at least one high-press question, they, on average, asked more high-press questions than when they did not plan to ask any. Finally, I identified four different ways that teachers address their planned questions which impacted the opportunities for students to reason mathematically. Teachers addressed their questions as drop-in (they asked the question and then moved on as soon as a response was elicited), embedded (the ideas in the question were addressed by a student without being prompted), telling (the teacher told the students the `response' to the question without providing an opportunity for the students to attempt to answer the question themselves) and sustained focus (the teacher sustained the focus on the question by asking the students follow-up questions).

This dissertation focused on the study of the Spanish academic language proficiency of novice bilingual teachers during the act of teaching mathematics in elementary grades. Four first year teachers in a large urban school district in central Texas participated in the study. At the time of the study two participants were fully certified and had attended four-year teacher preparation programs. The additional participants had completed all certification requirements including content examinations and the Texas Oral Proficiency Test (TOPT); they were completing their certification requirements through alternative certification programs. The study sought to identify the moments in their teaching of mathematics in Spanish when their instruction broke down, i.e. when they appeared unable to communicate ideas to students, and the reasons for these breakdowns. Findings revealed that the teachers in the study demonstrated linguistic and/or pedagogical breakdowns and that certain factors influenced their knowledge and language competencies. Linguistic breakdowns were manifested when teachers switched to English, used repetitive language when teaching, or provided limited academic language. Pedagogical breakdowns were identified as a lack of: student talk or discussion, effective teacher questioning, or diverse presentation of content. Overall, the teachers struggled with limited language in Spanish and limited pedagogical reasoning skills while teaching mathematical concepts to their students. These limitations were exacerbated by the pressures of high stakes testing and countered by the fact that all four teachers shared linguistic and cultural affiliation with their students. This collective case study was conducted from within a constructivist theoretical framework focusing on theories of academic language, communicative competence, and Vygotsky’s sociocultural perspective of learning. Recommendations for future training and practice of bilingual teachers are provided specifically on the importance of Spanish language proficiency of this group of educators. / text

It is widely recognized that educational research and theory should be motivated by the desire to continually improve the practice of teaching. However, bridging the divide between theoretical research outcomes and the practical constraints of classroom-based teaching has proved somewhat challenging. The involvement of teachers as the 'bridge-builders' between theory and practice could provide an effective mechanism for achieving this integration. The purpose of this study is thus to investigate whether the involvement of teachers in developing and implementing a theory-based teaching module would improve teaching practice in the classroom. A teaching module was collaboratively developed by a group of teachers for Grade 9 linear functions using: the principles of mathematical proficiency postulated by Kilpatrick, Swafford and Findell, (2001); the teaching phases formulated by van Hiele (1986); and the cognitive classification of classroom activities developed by Stein and Smith (1998). This module was then taught to six Grade 9 classes by four teachers in one school in the Eastern Cape, South Africa over a period of 5 weeks. The effectiveness of the module, and its application in the classroom, was assessed in terms of: (i) the extent to which theory could be used to inform the design and development of teaching materials; (ii) the efficacy of this teaching material in promoting teaching for mathematical proficiency; and (iii) the effects of extraneous influences on the usefulness of the module in teaching for mathematical proficiency. While the theoretical framework provided a sound basis for developing the teaching module, it was found that collaboratively transforming this theory into a teaching module for practical use in the classroom is certainly possible, but it requires considerable time and effort that practising teachers do not have. Developing the depth of understanding required for mathematical proficiency also takes time - a commodity often in short supply as teachers grapple with the demands of the curriculum. Teaching for mathematical proficiency is a layered process. It starts with thinking about an idea (like a graph) that is developed out of a related concept that then has a set of characteristic algorithms and actions which are learnt and performed in sequence. Building understanding in this way ends with a student being able to visualize and conceive the graph as a structure that can be described as if it were an object (encapsulating all the previous concepts belonging to similar graphs in one idea). This development of understanding is important for mathematical proficiency but is not necessarily easy. When teaching with the module, it was necessary to create an extra opportunity for students to use procedural knowledge and repetition in order to provide enough examples to help them see the link: between linear number patterns and linear graphs. Extraneous influences on teaching for mathematical proficiency were grouped into two categories - endogenous and exogenous influences. Endogenous influences were teacher related and included the attitudes, decisions and disposition of the teacher. Exogenous influences were more contextual (and in effect out of the control of the teacher) and included teaching time available, curriculum, external assessments etc. Both of these influences were seen to affect teaching for mathematical proficiency, either promoting or inhibiting it. This research affirmed the central role that teachers play in teaching for mathematical proficiency. It is considered critical that research actively involve teachers in the evolution of mathematical theory. The development of an enabling environment (including institutional support, time, capacity, resources, skills and tools) for teachers will further enhance their capacity to teach for mathematical proficiency.

The purpose of the study was to explore the relationship between mathematics teachers’ subject matter knowledge and their teaching effectiveness. A convenient sample of 19 grade 11 mathematics teachers and 418 students were initially selected for the study and took part in some stages of the study. Of this lot, only 11 teachers and 246 students participated in all the stages of the study. Explanatory Mixed methods research design which entails the use of a co-relational study and a descriptive survey design were employed in the study. Data was collected from the teachers using a self report questionnaire, Teacher Subject Matter Knowledge of Trigonometric Functions Scale (TSMKTFS) and peer evaluation questionnaire, and from students using teacher evaluation questionnaire and Student Trigonometric Functions Performance Scale (STFPS). All the instruments had their validity and reliability accordingly determined. Quantitative data gathered was analysed using descriptive and inferential statistics while qualitative data gathered from teachers’ and students’ tests were analysed using task performance analysis. It was found that a positive, statistically significant relationship existed between teachers’ subject matter knowledge and the composite measure of their teaching effectiveness. The relationships between teachers’ subject matter knowledge and students’ achievement and also between teachers’ subject matter knowledge and students’ rating of the teachers’ teaching effectiveness were found to be positive and statistically significant. However, the relationships between teachers’ subject matter knowledge and teachers’ self rating as well as teachers’ subject matter knowledge and peers’ rating of teachers’ teaching effectiveness were not found to be statistically significant though they were positive. Further data analysis showed that there was a difference between the subject matter knowledge of effective and ineffective teachers and also between the students taught by effective teachers and the students taught by the ineffective teachers. / Institute of Science and Technology Education / PhD (Mathematics Education)

Constructivist teaching strategies are undeniably accepted as effective in achieving the desired educational goals of constructing knowledge through active and creative inquiry. Inasmuch as teachers would love to adopt these strategies in their teaching, mathematics teachers find themselves in a situation where they are forced not to use them. This study investigated the factors that impacted on the selection and adoption of constructivist teaching strategies in selected Gauteng’s urban schools. Four (4) public schools and sixteen (16) mathematics teachers participated in the study. The parallel mixed methods design was employed in the study to produce both quantitative and qualitative data. The data were therefore analysed both quantitatively and qualitatively. It was found that the participating mathematics teachers had an understanding of constructivist theories of teaching and that they perceived their classroom environments to be constructivist in character. The study also found that the adoption of constructivist teaching strategies was hindered by teachers’ lack of skills and competencies to handle a curriculum that they felt was handed down to them without their full involvement at all the stages of its development. Learners’ family backgrounds were also identified as a major social factor that impacted negatively against selection of constructivist strategies. Based on these findings, recommendations were made on how constructivist views can be realised in the teaching of mathematics in South African schools. / Mathematics Education / M. Ed. (Mathematics Education)

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)

CAPES / Esta dissertação está dividida em duas partes. A primeira parte é uma introdução da teoria básica de grupos necessária para o desenvolvimento do teorema da órbita-estabilizador, que permite fazer as contagens das simetrias dos poliedros regulares. A segunda parte é a descrição de uma atividade aplicada em sala de aula. / This dissertation is divided in two parts. The first part is an introduction to basic group theory required for the development of the orbit-stabilizer theorem, that allows the counts of symmetries of the regular polyhedra. The second part is the description of an activity applied in classroom.

CAPES / O assunto Equações Algébricas é estudado na 3ª Série do Ensino Médio. Basicamente, os estudantes aprendem que, se a equação possui solução racional então esta pertencerá a um conjunto que pode ser estabelecido. No entanto, caso a equação possua alguma raiz irracional, nenhum método é abordado. Além disso, pelo fato de os exemplos praticados serem todos de equações com raízes racionais, muitos estudantes concluem o ciclo básico acreditando que se pode resolver qualquer equação utilizando o algoritmo de Briot-Ruffini. Com objetivo de completar esta lacuna deixada na formação do estudante, o presente trabalho propõe um recurso mais amplo para o estudo de Equações Algébricas, que combina três ingredientes: visão histórica do tema, implementação de um método numérico simples e eficaz (o da Bisseção), e utilização da Tecnologia Educacional proporcionada pelo aplicativo GeoGebra. / The Algebraic Equations subject are studied in the High School. Basically, the students learn that, if the equation has rational solution then this will belong to a set that can be established. However, if the equation has some irrational root, no method is approached. Furthermore, because the examples are all practiced with equation with rational roots, many students complete the basic cycle believing that we can solve any equation using the algorithm of Briot-Ruffini. In order to complete this gap left in the student’s formation, this work proposes a broader approach to the study of Algebraic Equations, which combines three ingredients: historical overview of the issue, implementing a simple and effective numerical method (of the Bisection) and use of Educational Technology provided by GeoGebra.

CAPES / Este trabalho traz um relato a respeito da Hipótese de Riemann, com o objetivo de tornar os conceitos referentes a esse problema acessíveis ao professor da educação básica, que pretenda abordá-los em sala de aula quando tratar de conteúdos a ele relacionados. A pesquisa foi inteira bibliográfica, apoiada em sua grande parte em textos de História da Matemática, tornando este trabalho divulgador dos problemas que ocupam parte das pesquisas matemáticas deste século, em especial da Hipótese de Riemann. / This study presents a report about the Riemann Hypothesis, leaving the underlying concepts behind this problem more accessible to a high school teacher. The literature review was based mainly on History of Mathematics texts. This research aims to study significant topics of mathematical research throughout this century, particularly to popularize the Riemann Hypothesis.