Neste trabalho estudamos o Modelo de van Hiele, os níveis de desenvolvimento do pensamento geométrico e fases de aprendizagem. Utilizando esse conhecimento elaboramos um Instrumento de Pesquisa a fim de identificar o Nível de Desenvolvimento do Pensamento Geométrico (Níveis de van Hiele) dos alunos do Ensino Fundamental II em relação ao conteúdo de Polígonos. Aplicamos este Instrumento de Pesquisa a 237 alunos de um colégio público (estadual) em Curitiba e realizamos uma análise dos dados obtidos. Aperfeiçoamos as questões do Instrumento de modo que possa ser utilizado pelo professor em sala de aula, auxiliando no diagnóstico do nível que o aluno de encontra em relação ao conteúdo proposto. / This work studies the van Hiele model, the levels of development of geometric thinking and its learning phases. Using this knowledge, we prepared a Research Instrument to identify the Level of Development in Geometric Thinking (Levels of van Hiele) of Middle School students, related to contents of Polygons. We have applied this Research Instrument to 237 students from a public school (state) in Curitiba, and we made an analysis of the acquired data. We have improved the Instrument’s questions so that it can be used by teachers during the class. Helping to identify to which level content the student belongs, related to the proposed.

CAPES / Nesse trabalho identificamos os objetivos e as orientações nos Parâmetros Curriculares Nacionais, do terceiro e quarto ciclos, sobre o estudo da Geometria. Pesquisamos sobre a Biografia de Tales de Mileto, onde fazemos um relato, da região e história, da época em que ele viveu, contamos alguns de seus feitos, e enumeramos os teoremas cujas demonstrações lhe são atribuídas. Analisamos seis livros didáticos do 9º ano do ensino fundamental, que integram o Plano Nacional do Livro Didático 2014, observamos a forma como a Geometria é trabalhada, e quais as demonstrações e atividades apresentadas em relação ao Teorema de Tales. Usamos e recomendamos o software Geogebra para o estudo da Geometria. Propomos atividades diversificadas, para serem utilizadas em sala de aula, quando o Teorema de Tales for trabalhado. Sugerimos uma demonstração para o Teorema de Tales, onde utilizamos a definição de área do triângulo, e as propriedades do paralelogramo. / In this work we identify the goals and guidelines the National Curriculum Guidelines, the third and fourth cycles, on the study of geometry. We searched on the Biography of Thales of Miletus, where we do a story, and history of the region, the era in which he lived, we count some of their deeds, and enumerate the theorems whose statements are allocated. We analyzed six textbooks in 9th grade of elementary school, comprising the National Plan of Didactic Book 2014, observed how the geometry is crafted, and what activities and statements made with respect to the Thales’ Theorem. We use and recommend the Geogebra software for the study of geometry. We propose diversified activities for use in the classroom when the Thales’ Theorem is working. We suggest a demonstration of Thales’ Theorem, where we use the definition of the triangle area, and properties of the parallelogram.

Neste trabalho estudamos o Modelo de van Hiele, os níveis de desenvolvimento do pensamento geométrico e fases de aprendizagem. Utilizando esse conhecimento elaboramos um Instrumento de Pesquisa a fim de identificar o Nível de Desenvolvimento do Pensamento Geométrico (Níveis de van Hiele) dos alunos do Ensino Fundamental II em relação ao conteúdo de Polígonos. Aplicamos este Instrumento de Pesquisa a 237 alunos de um colégio público (estadual) em Curitiba e realizamos uma análise dos dados obtidos. Aperfeiçoamos as questões do Instrumento de modo que possa ser utilizado pelo professor em sala de aula, auxiliando no diagnóstico do nível que o aluno de encontra em relação ao conteúdo proposto. / This work studies the van Hiele model, the levels of development of geometric thinking and its learning phases. Using this knowledge, we prepared a Research Instrument to identify the Level of Development in Geometric Thinking (Levels of van Hiele) of Middle School students, related to contents of Polygons. We have applied this Research Instrument to 237 students from a public school (state) in Curitiba, and we made an analysis of the acquired data. We have improved the Instrument’s questions so that it can be used by teachers during the class. Helping to identify to which level content the student belongs, related to the proposed.

CAPES / Nesse trabalho identificamos os objetivos e as orientações nos Parâmetros Curriculares Nacionais, do terceiro e quarto ciclos, sobre o estudo da Geometria. Pesquisamos sobre a Biografia de Tales de Mileto, onde fazemos um relato, da região e história, da época em que ele viveu, contamos alguns de seus feitos, e enumeramos os teoremas cujas demonstrações lhe são atribuídas. Analisamos seis livros didáticos do 9º ano do ensino fundamental, que integram o Plano Nacional do Livro Didático 2014, observamos a forma como a Geometria é trabalhada, e quais as demonstrações e atividades apresentadas em relação ao Teorema de Tales. Usamos e recomendamos o software Geogebra para o estudo da Geometria. Propomos atividades diversificadas, para serem utilizadas em sala de aula, quando o Teorema de Tales for trabalhado. Sugerimos uma demonstração para o Teorema de Tales, onde utilizamos a definição de área do triângulo, e as propriedades do paralelogramo. / In this work we identify the goals and guidelines the National Curriculum Guidelines, the third and fourth cycles, on the study of geometry. We searched on the Biography of Thales of Miletus, where we do a story, and history of the region, the era in which he lived, we count some of their deeds, and enumerate the theorems whose statements are allocated. We analyzed six textbooks in 9th grade of elementary school, comprising the National Plan of Didactic Book 2014, observed how the geometry is crafted, and what activities and statements made with respect to the Thales’ Theorem. We use and recommend the Geogebra software for the study of geometry. We propose diversified activities for use in the classroom when the Thales’ Theorem is working. We suggest a demonstration of Thales’ Theorem, where we use the definition of the triangle area, and properties of the parallelogram.

Orientador: Edson Agustini / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Cientifica / Made available in DSpace on 2018-08-17T04:14:16Z (GMT). No. of bitstreams: 1 Staib_Armando_M.pdf: 3998901 bytes, checksum: 260b0ccc34231b7e4c21f0bddc18fdcd (MD5) Previous issue date: 2010 / Resumo: Este trabalho trata do ensino das Geometrias Hiperbólica e Euclidiana utilizando softwares de Geometria Dinâmica, em especial o software NonEuclid. O objetivo deste trabalho é ser uma proposta de atividades em Geometria Hiperbólica com o uso do software. O computador introduz uma diversidade dinâmica ao estudo, proporcionando ao aluno, verificar, conjecturar e investigar. As figuras planas podem ser manipuladas e transformadas de diferentes maneiras mantendo as suas propriedades geométricas. Elaboramos algumas atividades de Geometria Hiperbólica utilizando o software NonEuclid para alunos da graduação em matemática e fizemos também atividades que relacionam ambas as geometrias. Os futuros professores precisam saber mais do que irão lecionar e, em geometria, a utilização dos softwares de Geometria Dinâmica contribuem na evolução gradual da aprendizagem de ambas Geometrias: Hiperbólica e Euclidiana, potencializando as habilidades dos alunos pela visualização, experimentação e compreensão das propriedades geométricas / Abstract: This work deals with the teaching of Euclidian and Hyperbolic Geometry using software in the Dynamic Geometry area, especially the software by the name of NonEuclid". The objective of this work is to be a proposal for activities in Hyperbolic Geometry using this software. The computer introduces a dynamic diversity to the study, allowing students to examine, investigate and conjecture in this area. The plane figures can be manipulated and processed in different ways while maintaining their geometric properties. We can prepare some activities in Hyperbolic Geometry using the software NonEuclid for graduate students in mathematics and related activities that we also both geometries. Future teachers need to know more than material they present to their students, the use of Dynamic Geometry software contributes to the gradual evolution of learning of geometry, both Euclidean and Hyperbolic. This increases the students' abilities to visualize and experiment and therefore their understanding of geometric properties / Mestrado / Mestre em Matemática

Made available in DSpace on 2016-01-26T18:49:48Z (GMT). No. of bitstreams: 1 .htaccess: 82 bytes, checksum: d923c631439b8b61d73aee22e866c9a7 (MD5) Previous issue date: 2012-12-13 / This research entitled "Geometry in Early Childhood Education: training and pedagogical knowledge necessary to practice" is linked to the research line 2-Training and Professional Practice Pedagogical Faculty of the Master s Program in Education Unoeste of Presidente Prudente - SP. Aimed to analyze the knowledge of teachers in order to identify the theoriesthat guidethe teaching of Geometry in Early Childhood Education. A descriptive qualitative research approach involving action research-type intervention. In this investigative process intervention walked together to investigative practice, practice and reflective educational practice. The data obtained was analyzed by content analysis. The study was conducted at fortnightly meetings with kindergarten teachers of a private school in the city of Presidente Prudente. During the activities the teachers were able to identify the concept of early childhood education, content and methodology worked as well as the relationship of these data with the pedagogic approaches to teaching in kindergarten, with documents such as the National Curriculum Reference to Education Children and the National Curriculum. Teacher training was analyzed as well as the knowledge of geometry that make your practice and starting points that served as support, rose data that could determine the basis on which rests the work of teachers with geometry. These bases indicated that the actions of the teachers are not directly linked to documents, initial training, or educational guidelines they receive, but are built from the readings that teachers make their classroom practice, embodied in the reference training basic, and that according to the theory of van Hiele are still at the basic level in tending to level 1. / Esta pesquisa intitulada Geometria na Educação Infantil: formação e saberes necessários à prática pedagógica está vinculada à linha de pesquisa 2 Formação e Prática Pedagógica do Profissional Docente, do Programa de Mestrado em Educação da Unoeste de Presidente Prudente - SP. Teve como objetivo analisar os saberes dos docentes, a fim de identificar as teorias que orientam o ensino de Geometria na Educação Infantil. A pesquisa de enfoque qualitativo descritivo envolveu pesquisa-ação do tipo intervenção. Neste processo investigativo de intervenção, caminharam juntas a prática investigativa, a reflexiva e a educativa. Os dados obtidos foram trabalhados pela análise de conteúdo. A intervenção foi desenvolvida em encontros quinzenais com professores de Educação Infantil de um colégio particular do município de Presidente Prudente. No decorrer das atividades desenvolvidas com os professores, foi possível identificar a concepção de Educação Infantil, os conteúdos e a metodologia trabalhada, bem como a relação desses dados com as orientações pedagógicas para o ensino na Educação Infantil, com base em documentos como o Referencial Curricular Nacional para a Educação Infantil e as Diretrizes Curriculares Nacionais. A formação dos professores foi analisada, bem como os saberes de Geometria que fazem da sua prática e valendo-se do referencial que serviu de suporte, foram levantados dados que puderam determinar as bases em que se assenta o trabalho dos professores com a Geometria. Essas bases indicaram que as ações dos docentes não estão diretamente ligadas aos documentos, à formação inicial, ou às orientações pedagógicas que recebem, mas são construídas por meio das leituras que os professores fazem da sua prática em sala de aula, consubstanciados nas referências da formação básica e que, segundo a teoria de van Hiele, se encontram ainda no nível básico, tendendo para o nível 1.

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.

Made available in DSpace on 2016-07-18T17:54:18Z (GMT). No. of bitstreams: 1 .htaccess: 82 bytes, checksum: d923c631439b8b61d73aee22e866c9a7 (MD5) Previous issue date: 2012-12-13 / This research entitled "Geometry in Early Childhood Education: training and pedagogical knowledge necessary to practice" is linked to the research line 2-Training and Professional Practice Pedagogical Faculty of the Master s Program in Education Unoeste of Presidente Prudente - SP. Aimed to analyze the knowledge of teachers in order to identify the theoriesthat guidethe teaching of Geometry in Early Childhood Education. A descriptive qualitative research approach involving action research-type intervention. In this investigative process intervention walked together to investigative practice, practice and reflective educational practice. The data obtained was analyzed by content analysis. The study was conducted at fortnightly meetings with kindergarten teachers of a private school in the city of Presidente Prudente. During the activities the teachers were able to identify the concept of early childhood education, content and methodology worked as well as the relationship of these data with the pedagogic approaches to teaching in kindergarten, with documents such as the National Curriculum Reference to Education Children and the National Curriculum. Teacher training was analyzed as well as the knowledge of geometry that make your practice and starting points that served as support, rose data that could determine the basis on which rests the work of teachers with geometry. These bases indicated that the actions of the teachers are not directly linked to documents, initial training, or educational guidelines they receive, but are built from the readings that teachers make their classroom practice, embodied in the reference training basic, and that according to the theory of van Hiele are still at the basic level in tending to level 1. / Esta pesquisa intitulada Geometria na Educação Infantil: formação e saberes necessários à prática pedagógica está vinculada à linha de pesquisa 2 Formação e Prática Pedagógica do Profissional Docente, do Programa de Mestrado em Educação da Unoeste de Presidente Prudente - SP. Teve como objetivo analisar os saberes dos docentes, a fim de identificar as teorias que orientam o ensino de Geometria na Educação Infantil. A pesquisa de enfoque qualitativo descritivo envolveu pesquisa-ação do tipo intervenção. Neste processo investigativo de intervenção, caminharam juntas a prática investigativa, a reflexiva e a educativa. Os dados obtidos foram trabalhados pela análise de conteúdo. A intervenção foi desenvolvida em encontros quinzenais com professores de Educação Infantil de um colégio particular do município de Presidente Prudente. No decorrer das atividades desenvolvidas com os professores, foi possível identificar a concepção de Educação Infantil, os conteúdos e a metodologia trabalhada, bem como a relação desses dados com as orientações pedagógicas para o ensino na Educação Infantil, com base em documentos como o Referencial Curricular Nacional para a Educação Infantil e as Diretrizes Curriculares Nacionais. A formação dos professores foi analisada, bem como os saberes de Geometria que fazem da sua prática e valendo-se do referencial que serviu de suporte, foram levantados dados que puderam determinar as bases em que se assenta o trabalho dos professores com a Geometria. Essas bases indicaram que as ações dos docentes não estão diretamente ligadas aos documentos, à formação inicial, ou às orientações pedagógicas que recebem, mas são construídas por meio das leituras que os professores fazem da sua prática em sala de aula, consubstanciados nas referências da formação básica e que, segundo a teoria de van Hiele, se encontram ainda no nível básico, tendendo para o nível 1.

South African learners continue to perform poorly in high school Circle Geometry. Lack of learner confidence and motivation in Mathematics may be the contributing factors to the low success rate in Circle Geometry. These factors, coupled with Conventional Teaching Instructions (CTI), may have contributed to the significant reduction in the number of learners enrolled for Mathematics, and provided a rationale for the study that explored a possible alternative teaching pedagogy to motivate and eventually improve learner performance in Circle Geometry. The study aimed to investigate the comparative effects of Computer Assisted Instruction (CAI) and CTI on the performance and motivation of Grade 11 learners in the topic of Circle Geometry. The population of the study consisted of Grade 11 Mathematics learners from 65 secondary schools in the Ekurhuleni North District of Gauteng province in South Africa. Using convenience sampling techniques two schools with n=136 Grade 11 Mathematics learners and two teachers participated in the study. One school formed the experimental group (n=71) and the other school formed the control group (n=65). The study followed a quasi-experimental design with a non-equivalent control group approach consisting of pre-and post-test measures. Intact classes participated in the study as it was not possible to randomly select participants for the study. Both groups wrote a standardised achievement pre-test to ascertain their performance status at the beginning of the study. CAI was implemented in the experimental school while CTI was implemented in the control school. A similar post-test was administered on both groups to measure the comparative effects of each teaching method on the performance of learners. A questionnaire was also administered to both groups to measure the motivation of learners. A purposive sample (n=12) was selected from both groups to participate in semi-structured interviews to account for the results of the test and the questionnaire. A socio-constructivist theory framed the study. Analysis of Covariance (ANCOVA) was used to analyse data. The results of this investigation indicated that the use of the computer software, GeoGebra, in the teaching and learning of Circle Geometry improved the performance and motivation of Grade 11 learners. / Mathematics Education / M. Ed. (Mathematics Education)

Submitted in fulfillment of the requirements of the Degree of Master of Technology: Management, Durban University of Technology, 2012. / The emphasis of geometry is of such significance that the current National Curriculum Statement has included the learning of two-and three-dimensional shape from grade R. However it is observed that teachers rely on textbooks for their knowledge of a dynamic topic such as nets of solids. Learners do not have an opportunity to explore the different orientations of solid shapes. Not many teachers use technology to assist their pedagogy. Quality processes in mathematics education emphasise the use of technology in teaching and learning. A study was performed using “Poly”, which is free open-source software for mathematics teaching and learning. This software was chosen because it was an easy to use application. It was able to show the different orientations of the solid shapes. Three-dimensional geometry can be explored using this software. The research is based on a social constructivist view of learning and the methodology used is a case study. The Piagetian and van Hiele stages of development will be the basis of the researcher’s investigation. Piaget’s theory is based on age development whilst van Hiele alludes to the different stages of geometrical development. A control group was compared to the experimental group consisting of 20 learners each. The research was conducted in KwaZulu-Natal and involved a teacher with a class of 40 learners. Qualitative and quantitative data were collected and were analysed. The data consisted of classroom observations and learner questionnaires and interviews. The findings of this study affirm that the use of technology in the teaching of geometry can enhance conceptual understanding. Classroom management breaks from routine while using Poly. Poly has the potential to improve learners' educational experiences; it can enable the effective application of constructive, cognitive and collaborative models of learning. Poly is not just a mathematical tool but also a tool for thinking and helping to enhance learners’ learning. It can serve as a vehicle for helping learners to foster fundamental geometrical concepts. The assessment of the use of computers in mathematics by the learners of the experimental classes indicates that application of computers enables increasing the interest of the learners in mathematics and introduces more variety to the studies, making them more enjoyable and interesting. / National Research Foundation