It's almost like a dinner party and a buffet. A dinner party you get dished up stuff and you eat it and a buffet you can choose what you want out of a range of stuff. I mean think about it when you go to a dinner party- they always dish up something you don't like and you don't want to eat it anyway. It's the same if you have a lecturer in the front that just dishes out what you're going to learn, ... you have to learn it. But in terms of the curriculum negotiation process, I've got to choose what I wanted to leam. (Melissa II 27/11/01: 3) This thesis focuses on the process of negotiating the curriculum with twelve pre-service teachers registered for the Bachelor of Education (Primary) Degree during their Mathematics Education Course in their third year of study. The research is presented methodologically as an action research located within two paradigmatic positions, interpretive and critical. The research attempted to understand, interpret and critique the process of curriculum negotiation within the context of teacher education in South Africa In order to understand the negotiated curriculum process, experiences of the participants are presented through the use of their voices within the thesis. The interpretation is based on the construction and reconstruction of meaning during the enactment of the negotiated curriculum process and during the writing and reviewing of this thesis. The critique is rooted in the historical, cultural and social contexts of both the students and the author. The main contention of this thesis is that curriculum negotiation is not necessarily a suitable vehicle for developing a critical pedagogy in pre-service teacher education when all the participants form a homogenous group in this case, white middle-class women. The democratic values promoted within the context of our curriculum negotiation were fraught with dilemmas and entrenched the values of western liberalism. At most, the curriculum negotiation process and the development of a democratic learning environment, promoted a conscientisation at an individual level, namely a ' transformation of consciousness' . The democratic values promoted in our pedagogy were not sufficient in bringing about social change, a 'transformation for social action'.

This study involves an investigatioh into the nature and effectiveness of the pre-service training, where provided, of secondary school Mathematics teachers at the 11 colleges of education administered by the House of Representatives in the Republic of South Africa. The purpose of this study is to determine whether the HDE (Secondary) course offered at some of the aforementioned colleges of education is justified, In spite of the fact that it contravenes the requirements of the Education Act, No. 73 of 1969. Three different questionnaires were designed and during 1986 were sent to heads of Mathematics departments at the aforementioned colleges of education , to Mathematics teacher educators at these colleges and to beginning secondary school Mathematics teachers, teaching at schools administered by the House of Representatives in the Eastern Cape and Natal. The findings of this study show that the secondary course for Mathematics teachers is only offered at 5 of the 11 colleges of education and only a small percentage of students take this course. During 1986 most of the Mathematics teacher educators at the colleges of education were not suitably qualified. The study also shows that the Mathematics curriculum for the training of secondary school teachers is inadequate and a revision thereof is thus recommended. An obvious conclusion drawn from the findings is that the training of secondary school Mathematics teachers at these colleges of education is still in an early stage of development. One of the main recommendations of this study is that the training of secondary school Mathematics teachers at colleges of education administered by the House of Representatives should be terminated , in view of its ineffectiveness and In accordance with Education Act , No . 73 of 1969. On the other hand, in case this is not possible, suggestions are also made for the improvement of the pre-service training of secondary school mathematics teachers at these colleges of education.

Submitted by LAHIS BRAGA SOUZA null (lahisbraga@yahoo.com.br) on 2017-01-17T19:02:32Z No. of bitstreams: 1 BRAGA_LS_ MESTRADO.pdf: 2372806 bytes, checksum: 23846e7d98f294cacd2e594586c6b7f1 (MD5) / Approved for entry into archive by LUIZA DE MENEZES ROMANETTO (luizamenezes@reitoria.unesp.br) on 2017-01-20T15:45:00Z (GMT) No. of bitstreams: 1 souza_lb_me_rcla.pdf: 2372806 bytes, checksum: 23846e7d98f294cacd2e594586c6b7f1 (MD5) / Made available in DSpace on 2017-01-20T15:45:00Z (GMT). No. of bitstreams: 1 souza_lb_me_rcla.pdf: 2372806 bytes, checksum: 23846e7d98f294cacd2e594586c6b7f1 (MD5) Previous issue date: 2016-12-16 / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Fundamentada na metodologia qualitativa, a presente pesquisa investigou as concepções dos professores de Matemática da Educação Básica em relação às Tecnologias Digitais, nos anos finais do Ensino Fundamental. Esta tem como pergunta diretriz "O que enunciaram e apresentaram os professores participantes de um curso de formação continuada sobre o uso das Tecnologias Digitais em aulas de Educação Básica?". Os sujeitos da pesquisa foram professores da rede pública estadual pertencente à Diretoria de Ensino de São José do Rio Preto, que participaram de um curso de formação continuada voltado para uso de tecnologias no ensino de Geometria. Os resultados apresentados mostram que os professores veem como importante o uso das Tecnologias Digitais durante suas aulas de Geometria, principalmente devido às potencialidades que os softwares de Geometria Dinâmica proporcionam, como a visualização, investigação e experimentação. No entanto destaca-se a necessidade de uma formação específica para o uso das tecnologias, principalmente direcionada para a elaboração das atividades e aponta alguns fatores que podem influenciar na maneira como o professor concebe e utiliza as tecnologias em suas aulas, tais como, a falta de apoio da gestão escolar, a insegurança gerada devido a falta de formação e a falta da infraestrutura das salas de informáticas. / Grounded in the qualitative methodology approach, this study investigated the conceptions of Basic Education Mathematics teachers regarding to Digital Technologies (DT), in Middle School. This research has as leading question “What did the teachers participating in an in-service teacher training course enunciate and present about the use of Digital Technologies in Basic Education classes?”. The investigation subjects were state public school teachers belonging to the Board of Teaching of São José do Rio Preto, who participated in an in-service teacher training course focused on the use of technologies in Geometry teaching. The presented results show that the teachers see as important the use of Digital Technologies in his Geometry classes, mainly because of the potentialities that the Dynamic Geometry softwares provides, as visualization, investigation and experimentation. However, it stands out the necessity of a specific training to the use of technologies, specially directed at the elaboration of tasks and it suggests some factors that can influence on the way that the teacher conceives and employs the technologies in his classes, such as, the lack of support of the school management, the insecurity caused by the lack of training, and, the lack of infrastructure in the computer labs.

CAPES / Este trabalho tem como objetivo aprofundar o conhecimento dos professores do ensino médio fundamental a respeito de congruência e polinômios. Apesar de congruência não ser abordado nas escolas, este assunto justifica alguns conceitos repassados aos alunos, como por exemplo a divisibilidade de um número por outro. A congruência ainda pode auxiliar na verificação de raízes de polinômios. Aqui, os polinômios são tratados como elementos de um anel, o anel dos polinômios, e vários resultados utilizados em sala de aula são justificados a partir da estrutura desse anel. Com esses dois conceitos, ainda e feito um breve estudo de congruência polinomial. / The aim of this work is to deepen the knowledge of elementary and high school teachers about congruence and polynomials. Although congruence is not studied in schools, this subject justifies some concepts passed to the students, such as the divisibility of one number by another. The congruence can also help to verify roots of polynomials. Here, polynomials are treated as elements of a ring, the ring of polynomials, and several results used in the classroom are justified from the structure of this ring. These concepts are used for a brief study of polynomial congruence.

CAPES / O presente trabalho tem dois objetivos: o primeiro está relacionado ao ensino da análise combinatória nas séries do ensino fundamental 2 e ensino médio. O segundo objetivo e buscar aprofundar meus conhecimentos relacionados aos conceitos combinatoriais. Com relação ao primeiro objetivo, o ensino da análise combinatória, na minha trajetória como docente, tem sido uma das tarefas mais árduas que o professor de matemática da educação básica enfrenta. Diante disso, surgem algumas perguntas. Por que um assunto totalmente aplicável ao cotidiano tem gerado tanta dificuldade de compreensão? Um dos objetivos desse trabalho e buscar respostas para essa pergunta e propor sugestões que possam melhorar o entendimento desse conceito. Como segundo objetivo proposto, busquei compreender conceitos que até então, por mim, não dominados, aprofundando meu conhecimento combinatorial. Para tanto, esse trabalho possui uma parte dedicada ao estudo de conceitos combinatorias mais complexos, não o abordados junto aos alunos de ensino médio mas que permitem compreender situações combinatoriais mais complexas. / The present work has two major goals. The first one is related to the teaching of combinatorics in elementary school and high school. The second one is to seek further knowledge related to combinatorial concepts. Regarding the first goal, the teaching of combinatorics, in my trajectory as a teacher, has been one of the most arduous tasks that the math teacher of basic education faces. Therefore, some questions arise. Why a subject fully applicable to everyday, has generated so much trouble understanding? One of the goals of this work is to seek answers to this question and propose suggestions that can improve the understanding of this concept. As a second proposed goal, I sought to understand concepts that hitherto were not dominated, deepening my combinatorial knowledge. Therefore, this work has section devoted to the study of more complex combinatory concepts, not addressed to the students of high school but they allow us to understand more complex combinatorial situations.

CAPES / Este trabalho tem como objetivo aprofundar o conhecimento dos professores do ensino médio fundamental a respeito de congruência e polinômios. Apesar de congruência não ser abordado nas escolas, este assunto justifica alguns conceitos repassados aos alunos, como por exemplo a divisibilidade de um número por outro. A congruência ainda pode auxiliar na verificação de raízes de polinômios. Aqui, os polinômios são tratados como elementos de um anel, o anel dos polinômios, e vários resultados utilizados em sala de aula são justificados a partir da estrutura desse anel. Com esses dois conceitos, ainda e feito um breve estudo de congruência polinomial. / The aim of this work is to deepen the knowledge of elementary and high school teachers about congruence and polynomials. Although congruence is not studied in schools, this subject justifies some concepts passed to the students, such as the divisibility of one number by another. The congruence can also help to verify roots of polynomials. Here, polynomials are treated as elements of a ring, the ring of polynomials, and several results used in the classroom are justified from the structure of this ring. These concepts are used for a brief study of polynomial congruence.

Quality mathematics education relies on effective pedagogy which offers students appropriate and rich opportunities to develop their mathematical proficiency (MP) and intellectual autonomy in learning mathematics. This qualitative case study aimed to explore and analyse selected effective mathematics teachers' proficiency in the area of geometry in five secondary schools in five different Namibia educational regions. The sample was purposefully selected and comprised five mathematics teachers, identified locally as being effective practitioners by their peers, Education Ministry officials and the staff of the University of Namibia (UNAM). The schools where the selected teachers taught were all high performing Namibian schools in terms of students' mathematics performance in the annual national examinations. The general picture of students' poor performance in mathematics in Namibia is no different to other sub-Saharan countries and it is the teachers who unfortunately bear the brunt of the criticism. There are, however, beacons of excellence in Namibia and these often go unnoticed and are seldom written about. It is the purpose of this study to focus on these high achievers and analyse the practices of these teachers so that the rest of Namibia can learn from their practices and experience what is possible in the Namibian context. The mathematical content and context focus of this study was geometry. This qualitative study adopted a multiple case study approach and was framed within an interpretive paradigm. The data were collected through individual questionnaires, classroom lesson observations and in-depth open-ended and semi-structured interviews with the participating teachers. These interviews took the form of post lesson reflective and stimulated recall analysis sessions. An adapted framework based on the Kilpatrick, Swafford and Findell's (2001) five strands of teaching for MP was developed as a conceptual and analytical lens to analyse the selected teachers' practice. The developed coding and the descriptive narrative vignettes of their teaching enabled a qualitative analysis of what teachers said contributed to their effectiveness and how they developed MP in students. An enactivist theoretical lens was used to complement the Kilpatrick et al.'s (2001) analytical framework. This enabled a deeper analysis of teacher teaching practice in terms of their embodied mathematical knowledge, actions and interactions with students. procedural fluency (PF) and productive disposition (PD), were addressed regularly by all five participating teachers. Evidence of addressing either the development of students' strategic competence (SC) or adaptive reasoning (AR) appeared rarely. Of particular interest in this study was that the strand of PD was the glue that held the other four strands of MP together. PD was manifested in many different ways in varying degrees. PD was characterised by a high level of content knowledge, rich personal experience, sustained commitment, effective and careful preparation for lessons, high expectations of themselves and learners, collegiality, passion for mathematics and an excellent work ethic. In addition, the teachers' geometry teaching practices were characterised by making use of real-world connections, manipulatives and representations, encouraging a collaborative approach and working together to show that geometry constituted a bridge between the concrete and abstract. The findings of the study have led me, the author, to suggest a ten (10) principles framework and seven (7) key interrelated factors for effective teaching, as a practical guide for teachers. This study argues that the instructional practices enacted by the participating teachers, who were perceived to be effective, aligned well with practices informed by the five strands of the Kilpatrick et al.'s (2001) model and the four concepts of autopoesis, co-emergence, structural determinism and embodiment of the enactivist approach. The study concludes with recommendations for effective pedagogical practices in the teaching of geometry, and opportunities for further research.

The purpose of this project was to determine if there was a correlation between teachers' scores on fractions items on project assessments and the percentage of participation time in professional development activities.

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)