published_or_final_version / Education / Master / Master of Education

The study reported here utilised a theory of levels of geometric thinking. This theory was proposed and developed by two Dutch mathematics educators, Pierre van Hiele and his wife, Dina van Hiele-Geldof. The van Hiele theory enables investigations into why many students experience difficulties in learning geometry. In many nations, such as the UK, the USA, Netherlands, the USSR and to a certain extent, Nigeria and South Africa, research evidence has indicated that the overall students’ mathematical competencies are linked to their geometric thinking levels. This study is the first of its kind to apply the van Hiele theory of geometric thinking in the Namibian context to analyse geometrical conceptualisation in Grade 12 mathematics students. In all, 50 Grade 12 students (20 from School A and 30 from School B) were involved in this study. These students wrote a van Hiele Geometry Test adapted from the Cognitive Development and Achievement in Secondary School Geometry test items. Thereafter, a clinical interview with the aid of manipulatives was conducted. The results from this study indicated that many of the School A and School B students who participated in the research have a weak conceptual understanding of geometric concepts: 35% of the School A and 40% of the School B subsamples were at the prerecognition level. 25% and 30% of the School A, and 20% and 23.3% of the School B students were at van Hiele levels 1 and 2 respectively. An equal number of students but different in percentages, 2 (10%) in School A and 2 (6.7%) in School B, were at van Hiele level 3. Only one student from School B attained van Hiele level 4. These results were found to be consistent with those of previous similar studies in UK, USA, Nigeria and South Africa. The findings of this study also highlight issues of how the Namibian Grade 12 geometry syllabus should be aligned with the van Hiele levels of geometric thinking as well as the use of appropriate and correct language in geometrical thinking and problem solving.

This study reports on the extent to which selected mathematics teachers facilitate the teaching and learning of geometry at the van Hiele levels 1 and 2 at a Grade 10 level in selected schools in Namibia. It also addresses and explores the teaching strategies teachers employ in their classrooms. Kilpatrick et al.’s model for proficient teaching and the van Hiele model of geometric thinking were used to explore the type of teaching strategies employed by selected mathematics teachers. These two models served as guidelines from which interview and classroom observation protocols were developed. Given the continuing debate across the world about the learning and teaching of geometry, my thesis aims to contribute to a wider understanding of the teaching of geometry with regard to the van Hiele levels 1 and 2. There are no similar studies on the teaching of geometry in Namibia. My study concentrates on selected Grade 10 mathematics teachers and how they teach geometry using the van Hiele theory and the five Kilpatrick components of proficient teaching. As my research looks at teaching practice it was important to deconstruct teaching proficiency with a view to understanding what makes good teachers effective. The results from this study indicated that the selected Grade 10 mathematics teachers have a good conceptual understanding of geometry as all of them involved in this study were able to facilitate the learning and teaching of geometry that is consistent with the van Hiele levels 1 and 2.

The research presented in this thesis is a case study located in the interpretive paradigm of qualitative research. The focus is on the use of van Hiele's theory to explore problems encountered in circle geometry by grade 11 learners and making some policy recommendations concerning the curriculum structure and teaching of the geometry at all grades. The interpretation is based to the learners' background in geometry i.e. their prior knowledge and experience of learning geometry. The study was carried out over a period of three years. The data collection process took a period of two months (April and May 2003) with a group of 21 grade 11 mathematics learners in a rural senior secondary school in the Eastern Cape. The researcher used document analysis, worksheets, participants' observation, van Hiele tests, a questionnaire and semi-structured interviews to collect data. The study showed that the structure of the South African geometry syllabus consists of a some what disorganized mixture of concepts. It is not sequential and hierarchical and it sequences concepts in a seemingly unrelated manner. The study revealed that the South African high school geometry curriculum is presented at a higher van Hiele level than what the learners can attain. The findings of the study showed that many of the grade 11 learners were under-prepared for the study of more sophisticated geometry concepts and proofs. Three categories of reasons could be ascribed to this: Firstly, there was insufficient preparation of learners during the primary and senior phases. Secondly the study indicated that there is overload of geometry at the high school level in the South African mathematics curriculum. Thirdly, the over-reliance on the traditional approach to teaching geometry, poor presentation of mathematical technical concepts and language problems, were identified as possible additional reasons for the poor learner understanding of geometry in general and circle geometry in particular. The study recommends that the structure of the South African geometry curriculum should be revisited and redesigned. Teachers should be empowered and developed to be more effective in teaching geometry through further studies in mathematics and in-service workshops. They should also be engaged in the process of implementing the van Hiele's theory in the teaching of geometry in their classrooms.

This study investigated the effect of integration of GeoGebra into the teaching of circle geometry on Grade 11 students’ achievement. The study used a quasiexperimental, non-equivalent control group design to compare achievement, Van Hiele levels, and motivation of students receiving instruction using GeoGebra and those instructed with the traditional ‘talk-and-chalk’ method. Two samples of sizes n = 22 (experimental) and n = 25 (control) drawn from two secondary schools in one circuit of the Vhembe district, Limpopo Province in South Africa were used. A pilot study sample of size n = 15, was carried out at different schools in the same circuit, in order to check the reliability and validity of the research instruments, and statistical viability. The results of the pilot study were shown to be reliable, valid and statistically viable. The study was informed by the action, process, object, schema (APOS) and Van Hiele theories, as the joint theoretical framework, and the literature search concentrated on technology integration, especially GeoGebra, in the teaching and learning of mathematics. The literature was also reviewed on the integration of computer technology (ICT) into mathematics teaching and learning, ICT and mathematical achievement, and ICT and motivation. The study sought to answer three research questions which were hypothetically tested for significance. The findings of this study revealed that there was a significant difference in the achievement of students instructed with GeoGebra compared to those instructed with the traditional teaching method (teacher ‘talk-andchalk’). The average achievement of the experimental group was higher than that of the control group. Significant differences were also established on the Van Hiele levels of students instructed with GeoGebra and those instructed without this software at Levels 1 and 2, while there were no significant differences at Levels 3, 4 and 5. The experimental group achieved a higher group average at the visualisation and analysis Van Hiele levels. It was also statistically inferred from questionnaires through chi-square testing, that students instructed with GeoGebra were more motivated to learn circle geometry than those instructed without the software / Mathematics Education / M. Sc. (Mathematics Education)

Orientador: Évelin Meneguesso Barbaresco / Banca: Ermínia de Lourdes Campello Fanti / Banca: Francielle Rodrigues de Castro Coelho / Resumo: Este trabalho tem como objetivo principal estudar as isometrias - no plano e no espaço - dentro da sala de aula. Inicialmente é realizado um breve resgate histórico a respeito da relação entre os conceitos de isometria e beleza, através dos tempos. Algumas das isometrias foram observadas na natureza em animais e plantas, e em obras construídas pelo Homem, como objetos, obras arquitetônicas famosas e obras de arte (quadros e esculturas). Buscamos conceituar e demonstrar os tipos de isometria no plano, bem como suas composições, além de conceituar os tipos de isometrias no espaço. Também apresentamos o Teorema Fundamental das Isometrias que caracteriza todos os tipos de isometrias no plano. Após a fundamentação teórica, descrevemos a realização de atividades em sala de aula utilizando diferentes tipos de malhas e imagens recortadas de revistas e jornais, com o objetivo de levar o aluno a identificar algum tipo de isometria. No final, apresentamos uma análise de como o ensino-aprendizagem de isometrias são abordados em alguns documentos oficiais de ensino no Brasil (PCNs, Currículo do Estado de São Paulo e matrizes de referências do SARESP) / Abstract: This work has as main objective to study the isometries - in the plane and in the space - inside the classroom. Initially is performed a brief historic rescue about the relation between the concepts of isometry and beauty through the ages. Some of isometries have been observed in nature in animals and plants, and works constructed by mens, such as objects, famous architectural works and artworks (paintings and sculptures). We seek to conceptualize and demonstrate the types of isometries in the plane, as well as their compositions, and conceptualize the types of isometries in space. After the theoretical foundation, we describe the realization of activities in the classroom using different types of grids and pictures cut from magazines and newspapers, with the goal of bringing students to identify some sort of isometry. In the end, we present an analysis of how the teaching and learning of isometries are dealt in some official documents of education in Brazil (PCNs, Curriculum of São Paulo and arrays of references SARESP) / Mestre

Orientador: José Carlos Rodrigues / Coorientador: Aylton Pagamisse / Banca: Vera Lucia Carbone / Banca: Marco Antonio Piteri / Resumo: Com o objetivo de expandir as experiências com a utilização da Metodologia de Resolução de Problemas, visando o aprimoramento da prática do professor e a motivação ao uso desta estratégia metodológica em suas dinâmicas de aula, propõe-se a discussão de como os discentes podem aprender por meio da utilização de situações problema, afim de promover a superação de suas dificuldades e enganos, contribuir no aperfeiçoamento de seus argumentos e na validação de suas respostas. No desenvolvimento deste projeto abordam-se alguns aspectos que se julgam pertinentes ao ensino da Geometria, mais especificamente aos cálculos de perímetro e área de figuras planas. Ao relatar experiências coletadas dos trabalhos realizados com alunos da 1ª Série A do Ensino Médio da Escola Estadual Profª Maria Evanilda Gomes, situada em Itororó do Paranapanema, distrito da cidade paulista de Pirapozinho, acompanhados a partir de agosto de 2014. Para promover uma visão geral do rendimento escolar desta sala, busca-se coletar os dados apresentados pela Secretaria da Educação do Estado de São Paulo, no Relatório Pedagógico sobre o SARESP (Sistema de Avaliação de Rendimento Escolar do Estado de São Paulo) 2013, IDESP (Índice de Desenvolvimento da Educação do Estado de São Paulo) 2013 e resultados obtidos das provas diagnósticas aplicadas referentes à turma em questão. Para tanto, utilizam-se indicadores importantes que retratam a realidade da escola e que serviram de base para o desenvolvimento desse trabalho / Abstract: With the objective expand the experiences with the update of Methodology of Resolution of Problems, angled for best the practical of teacher and the motivation of the use this strategy methodological in their dynamics of class, propose the discussion of how the instructors can learn through of use of problem situation, in order to promote the overcoming of their difficulties and mistakes, contribute to the perfecting of their arguments and in validation of their answers. In development this project approach some aspects that think relevant to geometry teaching, more specific of calculations of perimeters and area plane figures. To report experiences collected the work performed with first year of high school students in Profª Maria Evanilda Gomes school, located in Itororó do Paranapanema, district of Pirapozinho city, accompanied from august of 2014. To promote a vision general of school performance this class, search collect the informations submitted by Secretaria da Educação do Estado de São Paulo, in Relatório Pedagógico about the SARESP (Sistema de Avaliação de Rendimento Escolar do Estado de São Paulo) 2013, IDESP (Índice de Desenvolvimento da Educação do Estado de São Paulo) 2013 and results obtained in diagnostics tests referents the class in question. Therefore utilized important indicators that depict the reality of the school and that formed the basis for the development of this Project / Mestre

Transformation geometry has been neglected in our schools because teachers are often not proficient enough to teach it, as it was not part of the syllabus during their training. The study investigates effective ways of teaching transformation geometry in grade 9, applying van Hiele’s theory (1986) of geometry teaching and learning and Kilpatrick’s model of mathematical proficiency. The teaching programme activities require consistent use of physical manipulatives by the teacher for effective teaching, learning and understanding of geometric concepts. The type of study is a case study. Data collection tools are: - baseline evaluation, teacher & learner interviews (pre & post programme intervention) and observation (pre & post) during the implementation of the teaching programme. Results were analysed both qualitatively and quantitatively. My research findings show some improvement of learner performance after the application of the programme. Baseline evaluation shows that some learners attained below and above 30%. Interviews showed that some learners had problems before the implementation of the programme and some problems were eliminated by the use of the programme activities and learning progression was evident. Learner performance showed that learners had acquired some knowledge and critical thinking and reasoning skills, reflection skills, communication through LOLT improved, commitment to activities of the programme and teaching practice had improved. Learner performance showed that a learner can be in two different levels at the same time. Consistent use of manipulatives resulted in effective teaching and learning of geometry in grade 9. The results of this research support other researchers’ views of teaching geometry using van Hiele’s theory (1986) and Kilpatrick et al. (2001). Shaw (2002) argues that teaching geometry with manipulatives enhances conceptual understanding by the learner. In my opinion, it also promotes immediate intervention by the teacher as soon as the learner picks an incorrect object. The project enhanced and improved levels of communication between the learner, teacher and others in the classroom.

Submitted by Filipe dos Santos (fsantos@pucsp.br) on 2018-11-28T09:08:19Z No. of bitstreams: 1 Jéssica Barbosa dos Passos Sanchez.pdf: 1368421 bytes, checksum: e97575f8dc010d81f8e95b68791a2861 (MD5) / Made available in DSpace on 2018-11-28T09:08:19Z (GMT). No. of bitstreams: 1 Jéssica Barbosa dos Passos Sanchez.pdf: 1368421 bytes, checksum: e97575f8dc010d81f8e95b68791a2861 (MD5) Previous issue date: 2018-05-21 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The objective of this research was to map, describe, analyze, discuss thematic trends and theoretical-methodologies in the field of Spatial Geometry. In this way, we made systematically cut-outs in Brazilian researches, produced in the period from 2007 to 2017, in strict sense graduate programs in Mathematics Education. We use a state-of-the-art methodology step called mapping. The data were selected from the search of academic productions in the Bank of Thesis and Dissertations of CAPES. We identified 11 academic masters and 3 doctoral theses related to our mathematical object: Spatial Geometry. The topics developed in the selected researches were: metric geometry, regular polyhedral, prisms (triangular and quadrangular bases), pyramids, cylinders, cones and spheres, including elements of these solids, surface area, surface planning and volume measurement, we conclude that the themes did not change during the studied period, because we did not find any research reflected in the curricular structure. To do so, we created three axes of analysis: The first axis refers to the analysis of the works by which they used games, manipulative materials and non-digital resources; in the second axis we analyze the researches, whose objective was the resources of information technology and communication (ICT) and, in the last axis, since they deal with strategies for the teaching of spatial geometry. We conclude, among the three axes - the predominance of investigations in high school. Although, we obtained the highest concentration of works in the second axis, to the detriment of the first and third, evidencing a tendency of change in the teaching strategies, whose consequence was the use of digital technologies / O objetivo desta pesquisa foi mapear, descrever, analisar, discutir tendências temáticas e teórico-metodologias no campo da Geometria Espacial. Desse modo, fizemos um recorte de forma sistematizada em pesquisas brasileiras, produzidas no período de 2007 a 2017, em programas de pós-graduação strictu sensu na área da Educação Matemática. Utilizamos uma etapa da metodologia estado da arte, intitulada de mapeamento. Os dados foram selecionados a partir da busca das produções acadêmicas no Banco de Teses e Dissertações da CAPES. Identificamos 11 mestrados acadêmicos e 3 teses de doutorado, relacionados ao nosso objeto matemático: Geometria Espacial. Os temas desenvolvidos nas pesquisas selecionadas foram: geometria métrica, poliedros regulares, prismas (base triangular e quadrangular), pirâmides, cilindros, cones e esferas, incluindo os elementos destes sólidos, área de superfície, planificação de superfícies e o cálculo da medida de volume, concluímos que os temas não mudaram ao longo do período estudado, até porque não encontramos nenhuma pesquisa questionando o que está posto no currículo. Criamos três eixos de análise: O primeiro eixo refere-se à análise dos trabalhos que utilizaram jogos, materiais manipulativos e recursos não digitais; no segundo eixo analisamos as pesquisas que utilizaram recursos de tecnologia da informação e comunicação (TIC) e, no último eixo, os que tratam de estratégia para o ensino de Geometria Espacial. Concluímos que entre os três eixos temos a predominância de investigações no ensino médio. Ainda como resultado de pesquisa, obtivemos a maior concentração de trabalhos no segundo eixo, em detrimento do primeiro e terceiro, mostrando uma tendência de mudança nas estratégias de ensino decorrente do uso de tecnologias digitais

Submitted by Filipe dos Santos (fsantos@pucsp.br) on 2018-12-11T11:59:03Z No. of bitstreams: 1 Suzana Nobre.pdf: 29189043 bytes, checksum: da5a6e67fdc9a55355dfee73aa58ad19 (MD5) / Made available in DSpace on 2018-12-11T11:59:03Z (GMT). No. of bitstreams: 1 Suzana Nobre.pdf: 29189043 bytes, checksum: da5a6e67fdc9a55355dfee73aa58ad19 (MD5) Previous issue date: 2018-09-17 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This research aims to identify dimensions of the affective domain that emerge from the interactions among students during a didactic sequence involving geometry. The study considered as descriptors of the affective domain the beliefs, the emotions and the attitudes, with its subcategories. The hypotheses formulated presuppose that cognition and affection cannot be investigated separately as affection cannot be investigated separately from the context in which the student is inserted. The subjects of the research were the students of the 5th year of Elementary School in a public school in the city of São Paulo. The focus of the observations fell particularly on the students indicated for the parallel recovery of Mathematics. The data collection was done with the video recordings of the groups and the classroom during the application of the didactic sequence, which was elaborated based on the Theory of Didactic Situations and took into account the cognitive conditions of learning of the geometry. The qualitative analysis of the emotional episodes – extracts of events in the classroom recorded in fragments of video recording was made. The analysis of the data allowed to identify the emergence of beliefs, attitudes and emotions, intrinsically related to learning during the group activities of the didactic sequence. It was also possible to identify the cognitive, affective and intentional components of attitudes that influence behavior. In addition, it showed the interdependence of the dimensions of the affective domain, that is, the influence that one dimension exerts on the emergence of another dimension. Thus, the hypotheses were verified, indicating that it is possible and necessary to definitively break with the dichotomy cognition and affection / Esta pesquisa tem o objetivo de identificar dimensões do domínio afetivo que emergem das interações entre alunos durante uma sequência didática envolvendo a geometria. O estudo considerou como descritores do domínio afetivo as crenças, as emoções e as atitudes, com suas subcategorias. As hipóteses formuladas pressupõem que cognição e afeto não podem ser investigados separadamente, assim como o afeto também não pode ser investigado separadamente do contexto no qual o aluno está inserido. Os sujeitos da pesquisa foram os alunos do 5º ano do Ensino Fundamental de uma escola pública do município de São Paulo. O foco das observações recaiu particularmente nos alunos indicados para a recuperação paralela de Matemática. A coleta dos dados foi feita com as videogravações dos grupos e da sala de aula durante a aplicação da sequência didática, que foi elaborada com base na Teoria das Situações Didáticas e levou em conta as condições cognitivas de aprendizagem da geometria. Foi feita a análise qualitativa dos episódios emocionais – extratos de acontecimentos na sala de aula registrados em fragmentos de videogravação. A análise dos dados permitiu identificar a emergência de crenças, atitudes e emoções, intrinsecamente relacionadas à aprendizagem durante as atividades em grupo da sequência didática. Foi possível também identificar as componentes cognitiva, afetiva e intencional das atitudes que influenciam o comportamento. Além disso, evidenciou a interdependência das dimensões do domínio afetivo, isto é, a influência que uma dimensão exerce no aparecimento ou emergência de outra. Dessa forma, as hipóteses foram verificadas, indicando que é possível e necessário romper definitivamente com a dicotomia cognição e afeto