This study investigates the effect of using Van Hiele’s instructional model in the teaching of congruent triangles in grade 10 in Gauteng high schools. Three randomly selected high schools in Gauteng formed the research fields, while intact groups of grade 10 learners in these schools formed the study participants (136 learners) for the study. A mixed method approach which was adopted for the study, using pre-test/post-test matching control group design and classroom observation. The pre-test/post-test was used to collect quantitative data, while classroom observation was used to glean qualitative data. Some of the findings from the quantitative data analysis suggested that the intervention improve the achievement scores in the experimental groups while the qualitative data was revealed that the intervention facilitated the learning of the concepts of congruent. It was recommended that Van Hiele learning and instructional model be adopted and applied in the teaching of other areas of mathematics / Mathematics Education / M. Sc. (Mathematics Education)

South African learners perform poorly in the geometry sections of both national and international assessments. Numerous assessment reports mention multiple errors that keep re-occurring and play a big role in the learners’ poor performance. For this research, the link between the grade 9 learners Van Hiele levels of thought and the typical errors that they made were investigated. In this mixed method study, 194 grade 9 learners in two schools in Port Elizabeth, South Africa were tested using a Van Hiele based test. A test was set up containing multiple-choice and open-ended questions and was used to determine firstly, the predominant level of geometric reasoning of the learners and secondly, to determine their typical errors. Semi-structured interviews were held with six learners to gain more insight into some of the typical errors uncovered in the tests. The quantitative data revealed that the learners’ predominant levels of geometric thought were low. Furthermore, the qualitative data revealed typical error patterns concerning angles and sides, parallel lines, hierarchy of quadrilaterals and incorrect reasons in the proofs. The quantitative and qualitative data was merged to determine if the errors could be linked to the Van Hiele levels. From the findings, it was concluded that most of their typical errors could be linked to the Van Hiele levels of the learners.

The Lesotho Junior Secondary Examination Analysis (2009 and 2010) revealed that students performance in Mathematics in general and Transformation geometry of rotation in particular was generally poor. Only a few number of students that sat for the final Form C Examination passed. This study employed the van Hiele’s levels of learning to investigate and describe the difficulties students have in the learning of rotational transformation geometry. Both a written test and interview were used to solicit information regarding students’ difficulties. This information was collected from 90 students from Qaoling Secondary School in Maseru district in Lesotho. Findings from the study revealed that students had difficulties in identifying and naming transformation of rotation, finding the centre, angle of rotation and locating the exact image of a rotated figure after rotation. Also, they had greater difficulties when using transformation to do proof. The analysis showed that students mostly had difficulties at the level of Abstraction and Deduction. This gave an indication that the vast majority of the students in Form C are reasoning at the lowest two levels of the van Hiele’s model which are Visualization and Description. For these students’ difficulties to be curbed, the analysis demonstrated amongst others that teachers needed to use Manipulative materials and Information Communication Technology (ICT) during the process of teaching and learning. Manipulative materials provide experience in which students can transfer their understanding smoothly from one concept to another. / Mathematics Education / M. Ed. (Mathematics Education)

Submitted by Jean Medeiros (jeanletras@uepb.edu.br) on 2016-03-22T14:06:02Z No. of bitstreams: 1 PDF - Gilmara Gomes Meira.pdf: 7255359 bytes, checksum: 40824b6702d64e230f76026fb78df336 (MD5) / Approved for entry into archive by Secta BC (secta.csu.bc@uepb.edu.br) on 2016-07-22T15:09:09Z (GMT) No. of bitstreams: 1 PDF - Gilmara Gomes Meira.pdf: 7255359 bytes, checksum: 40824b6702d64e230f76026fb78df336 (MD5) / Approved for entry into archive by Secta BC (secta.csu.bc@uepb.edu.br) on 2016-07-22T15:09:19Z (GMT) No. of bitstreams: 1 PDF - Gilmara Gomes Meira.pdf: 7255359 bytes, checksum: 40824b6702d64e230f76026fb78df336 (MD5) / Made available in DSpace on 2016-07-22T15:09:19Z (GMT). No. of bitstreams: 1 PDF - Gilmara Gomes Meira.pdf: 7255359 bytes, checksum: 40824b6702d64e230f76026fb78df336 (MD5) Previous issue date: 2015-04-07 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This research analyzes limits and possibilities from problems solving that consider the level of comprehension of van Hiele Model. Therefore, we want to know how students communicate with each other when they develop activities with geometric problems solving in the referred Model perspective. The target audience for the research development was a third year class of high school from a public school in Cabaceiras city – PB. The theoretical framework emphasizes the Problem Solving, the Geometric teaching and learning relevance, van Hiele Model, the use of manipulable materials and the aspects of social interaction taking into consideration particularly the written and oral students’ communication. This research, developed together with the Program Observatório de Educação/CAPES proposal, from which we are part of, happenned in three steps – with all the class working on Duo; with all the class working individually and; with the Duo selected from its development on van Hiele tests. Such study is of qualitative nature, it happened from the class development on selected activities that after it resulted in three case studies where it is analyzed the respective development on problems solving subsidized by the use of Tangram and the manner in which the double interact and communicate. The data were collected by participant observation, audiorecordings and recordings of oral and written communication from the Duo. Some of the main references used as theoretical support were Boavida et al (2008), Nasser and Sant'Anna (2010), Rego, Rego and Vieira (2012), Van de Walle (2009), Fonseca (2009), Carvalho (2009 ), among others. The results indicate there is fragility in Geometry knowledge from the students who finish High School, reflecting in limitations to solve problems. Also it reveals the potentialities that exist in the work developed from social interaction, raising a progressive communication that leads the students on reflecting by specific development in problems solving. / A presente pesquisa analisa limites e possibilidades a partir da resolução de problemas que levam em consideração o Nível de compreensão segundo o Modelo van Hiele. Dessa forma, queremos saber como os alunos se comunicam quando desenvolvem atividades com resolução de problemas geométricos, na perspectiva do referido Modelo. O público alvo para desenvolvimento da pesquisa foi uma turma do 3º Ano do Ensino Médio de uma escola pública estadual da cidade de Cabaceiras - PB. O quadro teórico enfatiza a Resolução de Problemas, a relevância do ensino e aprendizagem da Geometria, o Modelo van Hiele, o uso de Materiais Manipuláveis e aspectos da interação social tendo em vista, particularmente, a comunicação oral e escrita dos alunos. Essa pesquisa desenvolvida em conjunto com a proposta do Programa Observatório de Educação/CAPES, do qual fazemos parte, aconteceu em três etapas - com a turma toda trabalhando em Díades; com a turma toda trabalhando individualmente e; com as Díades selecionadas a partir do seu desenvolvimento nos testes van Hiele. Esse estudo é de natureza qualitativa, aconteceu a partir do desenvolvimento da turma em atividades selecionadas que, posteriormente, resultou em três estudos de caso nos quais se analisa o respectivo desenvolvimento na resolução dos problemas subsidiados com o uso do Tangram, bem como o modo como as Díades interagem e se comunicam. Os dados foram recolhidos por meio da observação participante, áudio-gravações e registros da comunicação oral e escrita das Díades. Algumas das principais referências que utilizamos como sustentação teórica foram Boavida et al (2008), Nasser e Sant’anna (2010), Rêgo, Rêgo e Vieira (2012), Van de Walle (2009), Fonseca (2009), Carvalho (2009), entre outros. Os resultados analisados apontam para a fragilidade que há no conhecimento de Geometria por parte dos alunos que concluem o Ensino Médio, refletindo em limitações ao resolver problemas. Além disso, revela as potencialidades que há no trabalho desenvolvido a partir da interação social, suscitando em uma comunicação progressiva que leva os alunos a refletirem por meio do desenvolvimento específico na resolução dos problemas.

Submitted by Jean Medeiros (jeanletras@uepb.edu.br) on 2016-05-19T12:19:08Z No. of bitstreams: 1 PDF - Gilberto Beserra da Silva Filho.pdf: 2246546 bytes, checksum: 2ce35dfd1f7a8e15ac073b28740b5161 (MD5) / Approved for entry into archive by Secta BC (secta.csu.bc@uepb.edu.br) on 2016-07-21T20:40:51Z (GMT) No. of bitstreams: 1 PDF - Gilberto Beserra da Silva Filho.pdf: 2246546 bytes, checksum: 2ce35dfd1f7a8e15ac073b28740b5161 (MD5) / Approved for entry into archive by Secta BC (secta.csu.bc@uepb.edu.br) on 2016-07-21T20:41:07Z (GMT) No. of bitstreams: 1 PDF - Gilberto Beserra da Silva Filho.pdf: 2246546 bytes, checksum: 2ce35dfd1f7a8e15ac073b28740b5161 (MD5) / Made available in DSpace on 2016-07-21T20:41:07Z (GMT). No. of bitstreams: 1 PDF - Gilberto Beserra da Silva Filho.pdf: 2246546 bytes, checksum: 2ce35dfd1f7a8e15ac073b28740b5161 (MD5) Previous issue date: 2015-12-04 / Our research work covers the field of geometry in order to investigate how a sequence of activities can contribute to the level advancements under the Van Hiele‟s model, starting from concrete everyday life objects and manipulable materials, relating them with the concepts and properties of geometric solids. To this end, we conducted a qualitative research in a senior year of a state High School in the city of Flores - PE. As our method, a series of activities were apllied, using manipulable materials, the van Hiele model and aspects of the social interaction, in view of the relevance of teaching and learning geometry. It was observed that, among the main implications of this research, the interpersonal relationships, group work and progress on development levels of geometric thinking can be emphasized. This research was conducted in five stages; the first four with the class working in groups, carrying out the presented activities; and the last one was carried out individually, with the students solving a questionnaire to verify whether there has been level progress, based on the van Hiele theory. The data were acquired through analysis of recorded audios, films, oral and written communication records. We rely on van Hiele‟s studies, especially on his levels of learning, which guided the elaboration of the activities sequence. Concerned about an appropriate methodology to more effectively contribute to understanding of the problem, we analyzed some teaching tools, including the Laboratório de Ensino de Geometria and two methodological tendencies of the mathematical teaching, emphasizing those that we considered more suitable for use in spatial geometry teaching. Our goal was to provide objective answers to pertinent solutions in our research and show that it is possible to engage students in the process, in a pleasant way and that they can build their knowledge with meaning. The results show a certain fragility that exists in the student regarding the knowledge of geometry. On the other hand, the results also show the existence of a potentiality for realization of the work through social interaction; therefore, we conclude that the sequence of activities, tied to the way they were experienced by the students, presents itself as a significant contribution so there is advancement in the development of geometrical thinking, based on the van Hiele model. / Nosso trabalho de pesquisa aborda o campo da geometria, com o objetivo de investigar como uma sequência de atividades pode contribuir para o avanço de nível segundo o modelo van Hiele partindo de objetos concretos do cotidiano e de materiais manipuláveis relacionando-os com os conceitos e propriedades de sólidos geométricos. Para tanto, realizamos uma pesquisa de caráter qualitativo, numa turma de 3º Ano do Ensino Médio de uma escola estadual do município de Flores – PE. Utilizamos como método, a realização de uma sequência de atividades, utilizando materiais manipuláveis, o modelo van Hiele e aspectos da interação social, tendo em vista a relevância do ensino e aprendizagem da geometria. Observamos que, entre as principais repercussões desse trabalho, destacamos as relações interpessoais, o trabalho em grupo e o avanço sobre os níveis de desenvolvimento do pensamento geométrico. Essa pesquisa foi desenvolvida em cinco etapas, as quatro primeiras com a turma trabalhando em equipe na realização das atividades propostas, e a última individualmente, na resolução de um questionário para verificarmos se houve avanço de níveis, baseado na teoria de van Hiele. Os dados foram recolhidos por meio da observação de áudios, filmagens, registros da comunicação oral e escrita. Baseamo-nos nos estudos de van Hiele, sobretudo nas fases de aprendizagem, que nos nortearam na elaboração da sequência de atividades. Preocupados com uma metodologia adequada para contribuir de forma mais eficaz para compreensão do problema, fizemos uma análise de algumas ferramentas de ensino, entre elas o Laboratório de Ensino de Geometria e duas tendências metodológicas de ensino de Matemática, dando ênfase àquelas que achamos mais adequadas para utilizar no ensino de geometria espacial. Nossa proposta foi oferecer respostas objetivas para soluções pertinentes na nossa pesquisa e demonstrar que é possível envolver os alunos durante o processo, de forma prazerosa e que eles possam construir seu conhecimento com significado. Os resultados demonstram uma certa fragilidade que há por parte do aluno quanto ao conhecimento de geometria. Por outro lado, evidencia a potencialidade que há na realização do trabalho a partir da interação social, sendo assim concluímos que a sequência de atividades, atrelada à forma como foi vivenciada pelos alunos, contribui de forma significativa para que haja o avanço no desenvolvimento do pensamento geométrico baseado no modelo van Hiele.

The Lesotho Junior Secondary Examination Analysis (2009 and 2010) revealed that students performance in Mathematics in general and Transformation geometry of rotation in particular was generally poor. Only a few number of students that sat for the final Form C Examination passed. This study employed the van Hiele’s levels of learning to investigate and describe the difficulties students have in the learning of rotational transformation geometry. Both a written test and interview were used to solicit information regarding students’ difficulties. This information was collected from 90 students from Qaoling Secondary School in Maseru district in Lesotho. Findings from the study revealed that students had difficulties in identifying and naming transformation of rotation, finding the centre, angle of rotation and locating the exact image of a rotated figure after rotation. Also, they had greater difficulties when using transformation to do proof. The analysis showed that students mostly had difficulties at the level of Abstraction and Deduction. This gave an indication that the vast majority of the students in Form C are reasoning at the lowest two levels of the van Hiele’s model which are Visualization and Description. For these students’ difficulties to be curbed, the analysis demonstrated amongst others that teachers needed to use Manipulative materials and Information Communication Technology (ICT) during the process of teaching and learning. Manipulative materials provide experience in which students can transfer their understanding smoothly from one concept to another. / Mathematics Education / M. Ed. (Mathematics Education)

Made available in DSpace on 2015-05-07T15:08:09Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1832682 bytes, checksum: 130715e1d96b5966792eb1ee98ad3518 (MD5) Previous issue date: 2011-05-06 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This research comprises the result of a process that had beginning as a personal reflection about a specific type of teaching activity that we had developed in the classroom, allowing us to expand our understanding of this topic and work with elements that can be explored in the process of initial and continuing training of teachers of Mathematics. Focusing on the teaching of plane geometry and more specifically exploring relative contents to the study of polygons such as classification, relation between properties, determination of perimeter and area, we take the van Hiele model as the main theoretical reference, which includes establishing different levels of development in student`s geometric thoughts, focusing on the first three (Levels 0, 1 and 2), connecting the action with manipulative materials for teaching geometry and solve problems. As a source of reflection for analysis, we take as the basic proposition a particular type of puzzles with matchsticks in Mathematics textbooks aimed at students from 6 to 9 Years of Elementary Education. We understand that the field of theoretical proposals as highlighted here for the teaching of geometry allows a teacher to do a better plan their actions to the classroom as well as more adequately explore the potential and limitations of various instruments, such textbooks, which are part of everyday school life. We concluded that even activities that may at first seem limited or traditional, can be improved considering references to research in a particular field of research, using them as a starting point for modifying the current reality of our formative student. / O presente trabalho de investigação compreende o resultado de um processo que teve início como uma reflexão pessoal acerca de um tipo específico de atividade didática que desenvolvemos em sala de aula, permitindo-nos ampliar nossa compreensão sobre o tema em tela e colaborar com elementos que podem ser explorados em processos de formação inicial e continuada de professores de Matemática da Educação Básica. Tendo como foco o ensino de Geometria Plana e, mais especificamente, explorando conteúdos relativos ao estudo de polígonos, como a classificação, relação entre propriedades, determinação de perímetro e área, tomamos como principal referencial teórico o Modelo van Hiele, que compreende o estabelecimento de diferentes níveis de desenvolvimento do pensamento geométrico do aluno, centrando-nos nos três primeiros (Níveis 0, 1 e 2), conectando a ação com materiais manipulativos para o ensino de Geometria à Resolução de Problemas. Como fonte de reflexão para análise, tomamos como base a proposição de um tipo particular de quebras-cabeças, com palitos de fósforo, em livros didáticos de Matemática direcionados a alunos do 6º ao 9º Anos do Ensino Fundamental. Compreendemos que o domínio de propostas teóricas como a aqui destacada para o ensino de Geometria, permite ao professor realizar um melhor planejamento de suas ações para a sala de aula, assim como explorar de forma mais adequada as potencialidades e limitações de instrumentos diversos, a exemplo do livro didático, que fazem parte do cotidiano escolar. Concluímos que mesmo atividades que podem, à primeira vista, parecer limitadas ou tradicionais, podem ser melhoradas considerando-se referências de pesquisas realizadas em um determinado campo de investigação, tomando-as como ponto de partida para a modificação da realidade formativa atual de nosso aluno.

Summaries in Afrikaans and English / Outcomes-Based Education forms the foundation of South Africa's revised national education curriculum, known as The Revised National Curriculum Statement Grades R-9. The Van Hiele Model, developed for the teaching and learning of geometry, has already been used successfully in the school curricula of the Netherlands and Russia. Strong support for this theory exists in leading countries like America and Europe. This study has investigated the possible relationship between the Van Hiele theory and the "Revised National Curriculum Statement Grades R-9" and through critical evaluation found a great compatibility between them. Both are giving strong support for the spiral approach to mathematics education. It is recommended that all geometry learning programmes and teaching units be moulded on the Van Hiele Model. / Suid-Afrika se hersiene nasionale onderwyskurrikulum, die "Revised National Curriculum Statement Grades R-9" word deur uitkomsgebaseerde onderwys gefundeer. Die Van Hiele Model vir die onderrig en leer van meetkunde, is suksesvol in skoolkurrikula van Nederland en Rusland toegepas en geniet wye steun in lande soos Amerika en Europa. Hierdie navorsing het die moontlike verband tussen die Van Hiele raamwerk en die nuwe kurrikulum ondersoek en deur kritiese evaluering baie sterk versoenbaarheid tussen die "Revised National Curriculum Statement Grades R-9" en die Van Hiele Model gevind. Verder is ook aangetoon dat beide sterk gebruik maak van die spiraalbenadering in wiskundeonderwys. Aanbevelings in die verband is dat leerprogramme en onderrigeenhede wat die kurrikulum vir meetkunde in skole implementeer op die Van Hiele Model geskoei word. / Mathematical Sciences / M. Sc. (Mathematics Education)

Summaries in Afrikaans and English / Outcomes-Based Education forms the foundation of South Africa's revised national education curriculum, known as The Revised National Curriculum Statement Grades R-9. The Van Hiele Model, developed for the teaching and learning of geometry, has already been used successfully in the school curricula of the Netherlands and Russia. Strong support for this theory exists in leading countries like America and Europe. This study has investigated the possible relationship between the Van Hiele theory and the "Revised National Curriculum Statement Grades R-9" and through critical evaluation found a great compatibility between them. Both are giving strong support for the spiral approach to mathematics education. It is recommended that all geometry learning programmes and teaching units be moulded on the Van Hiele Model. / Suid-Afrika se hersiene nasionale onderwyskurrikulum, die "Revised National Curriculum Statement Grades R-9" word deur uitkomsgebaseerde onderwys gefundeer. Die Van Hiele Model vir die onderrig en leer van meetkunde, is suksesvol in skoolkurrikula van Nederland en Rusland toegepas en geniet wye steun in lande soos Amerika en Europa. Hierdie navorsing het die moontlike verband tussen die Van Hiele raamwerk en die nuwe kurrikulum ondersoek en deur kritiese evaluering baie sterk versoenbaarheid tussen die "Revised National Curriculum Statement Grades R-9" en die Van Hiele Model gevind. Verder is ook aangetoon dat beide sterk gebruik maak van die spiraalbenadering in wiskundeonderwys. Aanbevelings in die verband is dat leerprogramme en onderrigeenhede wat die kurrikulum vir meetkunde in skole implementeer op die Van Hiele Model geskoei word. / Mathematical Sciences / M. Sc. (Mathematics Education)

There are numerous problems associated with the teaching of Euclidean geometry at secondary schools today. Students do not see the necessity of proving results which have been obtained intuitively. They do not comprehend that the validity of a deduction is independent of the 'truth' of the initial assumptions. They do not realise that they cannot reason from diagrams, because these may be misleading or inaccurate. Most importantly, they do not understand that Euclidean geometry is a particular interpretation of physical space and that there are alternative, equally valid interpretations. A possible means of addressing the above problems is tbe introduction of nonEuclidean geometry at school level. It is imperative to identify those students who have the pre-requisite knowledge and skills. A number of interesting teaching strategies, such as debates, discussions, investigations, and oral and written presentations, can be used to introduce and develop the content matter. / Mathematics Education / M. Sc. (Mathematics)