Thesis (M.Ed. (Education)) - Rhodes University, 2009. / Submitted in partial fulfilment of the requirements for the degree of Master in Education (Mathematics Education)

Orientador: Marcia Regina Ferreira de Brito / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Educação / Made available in DSpace on 2018-07-28T07:03:29Z (GMT). No. of bitstreams: 1 Rezi_Viviane_M.pdf: 10245249 bytes, checksum: c00579782d968a91333fdcf06adbaa0d (MD5) Previous issue date: 2001 / Resumo: O presente estudo teve por objetivo contribuir para o desenvolvimento da compreensão sobre alguns componentes da habilidade matemática intrínsecos às atividades que envolviam conceitos geométricos, através da abordagem de solução de problemas, procurando investigar quais as relações existentes entre o nível de desenvolvimento do pensamento em geometria, e componentes das habilidades matemáticas, como a percepção geométrica e a habilidade para conceitos espaciais. Para isso, foram sujeitos da pesquisa 201 alunos concluintes do ensino médio de duas escolas, uma pública e outra particular, submetidos a cinco instrumentos do tipo lápis e papel, durante o período de aula. Foi identificada uma relação linear significativa entre esses constructos, sendo que quanto maior o nível de desenvolvimento em Geometria do sujeito, melhor era o seu desempenho em provas que avaliavam a percepção geométrica, as habilidades para trabalhar com conceitos espaciais e o raciocínio espacial. Os dados foram analisados também através de análise fatorial, sendo que as provas se agruparam em três fatores de avaliação: problemas com enunciado verbal, problemas que requerem processamento visual e problemas que requerem representação e manipulação mental de objetos / Abstract: The main objective of this investigation was to isolate some mathematics abilities components that appear in activities involving geometric concepts, during problems solving situation. The relations between geometric thinking development level and mathematics abilities components, as the geometric perception and the presence of spatial concepts, were investigated. Subjects were 201 High school students from private and public schools. Instruments were five test and students were submitted to their at class period. A significance linear relationship between these constructs was identified, being that the higher the development level in Geometry presented by the subject, the better was his/her performance on the exams evaluating geometric perception, and also the presence of spatial concepts and spatial reasoning. Data analysis underwent a factorial analysis approach, where exams were gathered upon three evaluation criteria: word problems, visual processing and mental representation / Mestrado / Educação Matematica / Mestre em Educação

Orientador: Lucila Diehl Tolaine Fini / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Educação / Made available in DSpace on 2018-07-23T07:22:58Z (GMT). No. of bitstreams: 1 Lujan_MariaLuciaS_M.pdf: 10073307 bytes, checksum: bae5592f3cbc3594c472c171d14176fb (MD5) Previous issue date: 1997 / Resumo: Esta pesquisa teve como fundamento o modelo de desenvolvimento do pensamento geométrico de van Hiele e apoiou-se na Psicologia Genética de Piaget. Foram investigadas algumas atividades geométricas em alunos em início de escolarização. Utilizou-se, para isso, materiais não industrializados e confeccionados especialmente para este trabalho. Foram sujeitos desta pesquisa 44 crianças da 12 série do 1 Q grau de uma escola da rede oficial do Estado de São Paulo, que constituíram o grupo experimental e o grupo de controle. Esses sujeitos foram submetidos ao pré-teste, intervenção pedagógica e pós-teste. Os resultados indicaram que os sujeitos do grupo experimental, e que foram submetidos a um trabalho de intervenção pedagógica, apresentaram um desempenho significativamente melhor no pós-teste que o grupo de controle que não participou do trabalho de intervenção pedagógica. A pesquisa mostrou, também, que alunos de 1º série, podem adquirir conceitos geométricos, como os estudados nesta pesquisa, se a proposta de trabalho pedagógico for condizente ao nível cognitivo dos educandos / Abstract: This research had as basis the geometric thought development model of van Hiele and was assisted by the Piaget's genetic psychology. A few geometrical activities of students in the elementary school were investigated. For this purpose non-industrialized materials were used and made specially for this work. For this research 44 children first graders (elementary school) from an official school of the São Paulo State formed the experimental and the control groups. Those students were submitted to an initial test, pedagogical intervention and a post test. The results indicated that the students from the experimental group, who were submitted to pedagogical intervention, presented a significantly better performance in the post test than the control group, who did not participate of the pedagogical intervention. This research has also shown that students of the first grade of the elementary school can acquire geometrical concepts, as those used in this work, only if the proposed pedagogical work was adequate for their cognitive level / Mestrado / Educação Matematica / Mestre em Educação

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.

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.