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.

The central position in this study is that the professional identities, and consequently the classroom practices, of mathematics teachers are continuously being shaped by their narratives of past and present experiences. The primary research question explores the narratives that shape the professional identities of seven mathematics teachers; and the secondary research question, how their narratives shape their professional identities. Furthermore, the potential implications of this study for the design and implementation of pre-service teacher education programmes and in-service teacher development initiatives are considered. This study is framed by Socioculturalism; a theoretical perspective of human thinking as social in origin and of learning as participation in social practices. Futhermore, in line with Situated Learning Theory, the key theoretical notions are: identity (or learning as becoming); community (or learning as belonging); practice (or learning as doing); and meaning (or learning as experience). Identity is construed here as a conceptual bridge between learning and its cultural settings; and also between the individual and the social. In this study, the identity-shaping narratives of seven mathematics teachers, all purposively sampled from schools in the Port Elizabeth and Grahamstown education districts of the Eastern Cape Province of South Africa, were accessed via a narrative inquiry; followed by a horizontal analysis to identify common patterns or recurring themes in the narratives of all seven participants; and, a vertical analysis of the narratives of four of the participants to determine how their narratives shape their professional identities. Recurring themes that emerged during the horizontal analysis include the influence of: family support; role models; changing work environments; continuous professional development; professional recognition; religion; and, micro-politics. The vertical analysis demonstrated how, through a process of interpreting the narratives and restorying them into a meaningful core narrative; it is possible to gain insights into how personal narratives shape a professional identity. This study highlights the importance of listening to the narratives of mathematics teachers; because their professional identities, and consequently their teaching practices, are continuously being shaped by their narratives. It is anticipated that this research will be of interest and benefit to researchers, policy-makers, and teachers; especially in the area of Mathematics Education, where both narrative inquiry as a research method and research into teachers’ professional identities are relatively new.

The impetus for this study came from the findings of an evaluation of a Maths and Science through Arts and Culture (MStAC) Curriculum Intervention undertaken with Grade R teachers registered for a BEd(in-service) qualification at Rhodes University, South Africa. The intervention aimed to enrich Grade R teachers’ teaching of mathematics. Post-intervention classroom observations showed that, in spite of the intervention, teachers’ classroom practices did not change, and they were not using visual arts to teach mathematical concepts. This, together with the lack of research in the field of mathematics in early childhood, particularly in South Africa, motivated this research, a case study, which investigates how visual arts can be used to teach space and shape conceptualization in Grade R. I designed a research intervention underpinned by a constructivist model of teacher professional development located in reflective practice (Borko & Putman, 1995; Zeichner & Liston, 1996; Wilmot, 2005). Guided by Stacey’s (2009) notion of an emergent curriculum, I designed a three phase research intervention which involved selected Grade R teachers undertaking classroom-based research. Phase I built awareness around the notion of creativity; Phase II focused on making meaning of children’s behaviour and interests; and Phase III applied the knowledge and ideas from the Phases I and II to the teaching of space and shape. As an interpretive research study, it closely examines the participating teachers’ perceptions, experiences and reflections which were articulated in reflective reports and assignments. Following action research processes, the participant teachers engaged in the process of an emergent curriculum. They observed the behaviour interests of Grade R children, interpreted and made meaning of the evident behaviours, made decisions regarding extension activities, and planned accordingly. The findings of the study illuminate a model of teacher professional development that can support and enhance teachers’ practice. Understanding the notion of creativity and the ability to create a classroom conducive to creativity, are necessary components for teaching space and shape through visual arts activities. An emergent curriculum approach is proposed as an appropriate pedagogy for teaching children about space and shape through visual arts activities.

This dissertation examines elementary teacher candidates' perceptions of hip-hop culture and utilizing hip-hop pedagogy in a mathematics classroom. This study demonstrates how elements of hip-hop may be integrated into an elementary mathematics methods course to develop pedagogical knowledge that challenges teacher candidates to explore the benefits of utilizing hip-hop as a tool in the classroom. This study contributes to the growing body of research that investigates the use of hip-hop pedagogy in educator preparatory programs. Participants in this study were teacher candidates at a large university in Texas enrolled in the final year of their educator preparatory program. This research shows that as a result of integrating hip-hop pedagogy in the mathematics methods course, teacher candidates had increased knowledge and more positive perceptions of hip-hop culture, and they demonstrated a greater willingness to integrate hip-hop pedagogy in their future classrooms.

Continuing Professional Development (CPD) of teachers is increasingly becoming a priority in most countries throughout the world. It is widely viewed as the most effective approach to prepare teachers adequately, and to improve their instructional and intervention practices, for when they enter the workforce (Fraser et al 2007). Despite the general acceptance of CPD programmes as essential to the improvement of education, reviews of professional development research constantly point out the ineffectiveness of most of these programmes (see Cohen and Hill, 1998 and 2000). Furthermore, many teachers express dissatisfaction with the professional development opportunities made available to them in schools and insist that the most effective development programmes they have experienced have been self-initiated (National Research Council, 2007). There is a consensus that many CPD programmes have yet to understand professional development from a teacher‘s perspective. This perspective acknowledges what drives teachers to enlist in these programmes and how such programmes can make a difference to them and their classrooms. Therefore, this study seeks to return the emphasis of professional development to the teachers. The study explores the teachers‘ perspectives of CPD in general, the personal meaning of CPD, and its meaning in the context of their work. By interviewing the teachers who were part of the Mpumalanga Secondary Science Initiative (MSSI) project (a seven year science and mathematics professional development intervention), I explore: the teachers‘ opinions of the intervention; its meaning to them and their work; and its impact on their classroom practices and students for the duration of the intervention and beyond. In this study, I explore data from an extensive and longitudinal study of teachers who were part of the CPD programme in greater detail. In discussing my data, I propose that CPD, however well intentioned and executed, is received differently by each teacher as a result of their personal circumstances and investment in the programme. I argue that the greater the unity between the personal circumstances and motivations of the teachers and those of the CPD intervention, the more likely the outcome will be meaningful for the participating teachers. In turn, the ability to sustain the benefits of the intervention will be enhanced. / Further Teacher Education / D.Ed. (Curriculum Studies)

This study attempted to design and implement an intervention programme to improve the self-concepts and attitudes of prospective primary school teachers. The possible effects of the programme were researched by means of a combined quantitative and qualitative approach. A questionnaire to determine the effectiveness of the programme was also designed. Important findings of this study are that: (a) the Cronbach Alpha Coefficient indicated a 0,9 reliability of the designed questionnaire; (b) the average self-concepts of prospective teachers improved after the intervention programme was implemented, though not significantly and (c) the self-concepts of prospective teachers not majoring in Mathematics also improved after the intervention, though not significantly. A focus group interview was conducted to improve the intervention programme. On the basis of the responses of the participants in the focus group interview as well as of the quantitative study and of the literature reviewed, an improved version of the intervention programme was designed. According to the research, the programme should include: cooperative learning; everyday life contexts; manipulatives; hands-on activities, encouraging the asking of questions; a positive relationship between lecturer and student; writing assignments before doing a new topic; content relevance; a positive communication of assessment; games; various approaches to solving problems; understandable language; gender sensitivity and positive role modeling. The intervention programme implemented in this study was tested over a period of three months; hence the recommendation in this research for the replication thereof over a longer period. The study further recommends that since the study was conducted in one college of education only, there is a need for it to be conducted in a number of colleges, in order to elicit a broader and more representative picture of the impact of the intervention programme. The findings of this study advocate the need to combine both quantitative and qualitative research in studies: the results of the quantitative research did not conclusively indicate a positive impact of the intervention programme. However, the qualitative research saw respondents expressing improved confidence and attitudes. / Psychology of Education / D. Ed. (Psychology of Education)

University of Macau / Faculty of Education

Continuing Professional Development (CPD) of teachers is increasingly becoming a priority in most countries throughout the world. It is widely viewed as the most effective approach to prepare teachers adequately, and to improve their instructional and intervention practices, for when they enter the workforce (Fraser et al 2007). Despite the general acceptance of CPD programmes as essential to the improvement of education, reviews of professional development research constantly point out the ineffectiveness of most of these programmes (see Cohen and Hill, 1998 and 2000). Furthermore, many teachers express dissatisfaction with the professional development opportunities made available to them in schools and insist that the most effective development programmes they have experienced have been self-initiated (National Research Council, 2007). There is a consensus that many CPD programmes have yet to understand professional development from a teacher‘s perspective. This perspective acknowledges what drives teachers to enlist in these programmes and how such programmes can make a difference to them and their classrooms. Therefore, this study seeks to return the emphasis of professional development to the teachers. The study explores the teachers‘ perspectives of CPD in general, the personal meaning of CPD, and its meaning in the context of their work. By interviewing the teachers who were part of the Mpumalanga Secondary Science Initiative (MSSI) project (a seven year science and mathematics professional development intervention), I explore: the teachers‘ opinions of the intervention; its meaning to them and their work; and its impact on their classroom practices and students for the duration of the intervention and beyond. In this study, I explore data from an extensive and longitudinal study of teachers who were part of the CPD programme in greater detail. In discussing my data, I propose that CPD, however well intentioned and executed, is received differently by each teacher as a result of their personal circumstances and investment in the programme. I argue that the greater the unity between the personal circumstances and motivations of the teachers and those of the CPD intervention, the more likely the outcome will be meaningful for the participating teachers. In turn, the ability to sustain the benefits of the intervention will be enhanced. / Further Teacher Education / D.Ed. (Curriculum Studies)

This study attempted to design and implement an intervention programme to improve the self-concepts and attitudes of prospective primary school teachers. The possible effects of the programme were researched by means of a combined quantitative and qualitative approach. A questionnaire to determine the effectiveness of the programme was also designed. Important findings of this study are that: (a) the Cronbach Alpha Coefficient indicated a 0,9 reliability of the designed questionnaire; (b) the average self-concepts of prospective teachers improved after the intervention programme was implemented, though not significantly and (c) the self-concepts of prospective teachers not majoring in Mathematics also improved after the intervention, though not significantly. A focus group interview was conducted to improve the intervention programme. On the basis of the responses of the participants in the focus group interview as well as of the quantitative study and of the literature reviewed, an improved version of the intervention programme was designed. According to the research, the programme should include: cooperative learning; everyday life contexts; manipulatives; hands-on activities, encouraging the asking of questions; a positive relationship between lecturer and student; writing assignments before doing a new topic; content relevance; a positive communication of assessment; games; various approaches to solving problems; understandable language; gender sensitivity and positive role modeling. The intervention programme implemented in this study was tested over a period of three months; hence the recommendation in this research for the replication thereof over a longer period. The study further recommends that since the study was conducted in one college of education only, there is a need for it to be conducted in a number of colleges, in order to elicit a broader and more representative picture of the impact of the intervention programme. The findings of this study advocate the need to combine both quantitative and qualitative research in studies: the results of the quantitative research did not conclusively indicate a positive impact of the intervention programme. However, the qualitative research saw respondents expressing improved confidence and attitudes. / Psychology of Education / D. Ed. (Psychology of Education)