Este trabalho apresenta uma proposta de abordagem de modelação matemática voltada principalmente aos últimos anos da educação básica. Com base nos pressupostos da resolução de problemas, explora-se diversos conteúdos com significação. O fenômeno social a ser modelado consiste no formato e variações nas dimensões de embalagens a serem despachadas pelos Correios. As representações geométrica, gráfica e algébrica serão valorizadas no decorrer do desenvolvimento das atividades. A título de aprofundamento dos estudos, serão sugeridas algumas articulações possíveis entre conteúdos da educação básica e da graduação, como por exemplo, entre a função quadrática e estudo de máximos e mínimos, limites e derivadas. / This work presents a proposal for a mathematical modeling approach focused mainly on the last years of basic education. Based on the assumptions of solving problem, it explores several meaningful content. The social problem to be modeled consists of the format and variations in the dimensions of packages to be despatched by the Correios. The geometric, graphic and algebraic representations will be valued during the development of activities. In order to deepen the studies, it will be suggested some possible articulations between contents of basic and undergraduate education, for example, between the quadratic function and study of maximum and minimum, limits and derivatives.

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)

Capes / Na busca por uma melhor representação da realidade tridimensional, as Geometrias não- Euclidianas oferecem uma alternativa ao euclidianismo clássico e um dos destaques e a Geometria Projetiva. Assim, o objetivo deste trabalho e, através de ilustrações, contribuir para a assimilação de definições como perspectiva, projeção e o principio da dualidade. E, a partir de resultados importantes como o Teorema de Desargues, o Teorema de Pappus e o Teorema de Pascal, queremos facilitar a compreensão e a visualização de algumas das técnicas de perspectiva que podem ser adaptadas para o uso na sala de aula pelos professores da Educação B ́ sica. A aplicação de uma oficina de Geometria Projetiva em uma turma do 6o ano do Ensino Fundamental e a avaliação dos resultados revelaram que o tema pode ser desenvolvido de maneira promissora com os estudantes na Educação B ́ sica, obtendo uma melhor compreensão do objeto real e associando-o ao conteúdo matemático envolvido. / In search for a better representation of three-dimensional reality, non-Euclidean Geometries offer an alternative to the classic euclidianism and the Projective Geometry is one of the highlights. The purpose of this word is contribute to the assimilation of definitions such as perspective, projection, and the principle of duality, through illustrations. And, from important results as Desargues’ Theorem, Pappus’ Theorem and Pascal’s Theorem, we want to facilitate understanding and viewing some of the perspective techniques that can be adapted for use in classroom by Basic Education teachers. The application of a workshop of Projective Geometry in a class of 6th grade of elementary school and the evaluation of the results revealed that the theme can be developed in a promising way with students in basic education, getting a better comprehension of the real object and associating it to the mathematical content involved.

Acompanha: Ângulos: o ensino de geometria / Esta pesquisa, de abordagem qualitativa, apresenta como tema o ensino de Geometria e a inclusão de alunos com Múltiplas Deficiências em classes de ensino regular. Busca-se, por meio dessa, evidenciar as contribuições que as Tecnologias Assistivas trazem para o ensino e aprendizagem do conteúdo matemático de Geometria para esses alunos, de forma a favorecer sua autonomia na realização de atividades comuns nas turmas de ensino regular. A revisão de literatura, nesta investigação, considerou estudos relacionados ao tema ensino de geometria e a inclusão escolar, utilizando Tecnologias Assistivas. Na intervenção pedagógica foram abordados conceitos matemáticos em Geometria, retas, semirretas, construção e medição de ângulos, em uma turma do quinto ano do Ensino Fundamental de uma escola pública, do interior do Paraná, que contava com uma aluna com Múltipla Deficiência inclusa. Para o desenvolvimento das atividades, elaborou-se o material didático “Transferidor Adaptado”, para auxiliar alunos com Múltiplas Deficiências e/ou com alunos com Deficiência Física Neuromotora. Após a aplicação das atividades, foi possível observar a contribuição do uso da Tecnologia Assistiva para o ensino de geometria, aos alunos com Múltiplas Deficiências, permitindo a eles a compreensão e aprendizagem de conceitos matemáticos. / This paper, in a quality approach, aims to present the teaching of Geometry and the inclusion of students with Multiple Disabilities in classes of regular education. It also aims to elucidate the contributions of Assistive Technologies in teaching and learning mathematical contents of Geometry encouraging students’ autonomy when carrying out regular activities in regular education classes. The revision of literature in this investigation contemplated studies on teaching Geometry and scholar inclusion, using Assistive Technologies. In the pedagogical intervention mathematical concepts in Geometry were addressed, straight lines, semistraight lines, construction and measurement of angles, in a fifth grade class of the Primary Education in a public school, in the countryside in the state of Paraná, which has included a girl student with Multiple Disabilities. In order to develop those activities, a didactic material “Adapted Protractor” was elaborated, to help the students with Multiple Disabilities and/or students with Neuromotor Physical Disabilities. After the application of activities, it was possible to observe the contribution of the use of Assistive Technology to the teaching of Geometry for students with Multiple Disabilities, allowing them to understand and learn mathematical concepts.

Capes / Na busca por uma melhor representação da realidade tridimensional, as Geometrias não- Euclidianas oferecem uma alternativa ao euclidianismo clássico e um dos destaques e a Geometria Projetiva. Assim, o objetivo deste trabalho e, através de ilustrações, contribuir para a assimilação de definições como perspectiva, projeção e o principio da dualidade. E, a partir de resultados importantes como o Teorema de Desargues, o Teorema de Pappus e o Teorema de Pascal, queremos facilitar a compreensão e a visualização de algumas das técnicas de perspectiva que podem ser adaptadas para o uso na sala de aula pelos professores da Educação B ́ sica. A aplicação de uma oficina de Geometria Projetiva em uma turma do 6o ano do Ensino Fundamental e a avaliação dos resultados revelaram que o tema pode ser desenvolvido de maneira promissora com os estudantes na Educação B ́ sica, obtendo uma melhor compreensão do objeto real e associando-o ao conteúdo matemático envolvido. / In search for a better representation of three-dimensional reality, non-Euclidean Geometries offer an alternative to the classic euclidianism and the Projective Geometry is one of the highlights. The purpose of this word is contribute to the assimilation of definitions such as perspective, projection, and the principle of duality, through illustrations. And, from important results as Desargues’ Theorem, Pappus’ Theorem and Pascal’s Theorem, we want to facilitate understanding and viewing some of the perspective techniques that can be adapted for use in classroom by Basic Education teachers. The application of a workshop of Projective Geometry in a class of 6th grade of elementary school and the evaluation of the results revealed that the theme can be developed in a promising way with students in basic education, getting a better comprehension of the real object and associating it to the mathematical content involved.

Acompanha: Ângulos: o ensino de geometria / Esta pesquisa, de abordagem qualitativa, apresenta como tema o ensino de Geometria e a inclusão de alunos com Múltiplas Deficiências em classes de ensino regular. Busca-se, por meio dessa, evidenciar as contribuições que as Tecnologias Assistivas trazem para o ensino e aprendizagem do conteúdo matemático de Geometria para esses alunos, de forma a favorecer sua autonomia na realização de atividades comuns nas turmas de ensino regular. A revisão de literatura, nesta investigação, considerou estudos relacionados ao tema ensino de geometria e a inclusão escolar, utilizando Tecnologias Assistivas. Na intervenção pedagógica foram abordados conceitos matemáticos em Geometria, retas, semirretas, construção e medição de ângulos, em uma turma do quinto ano do Ensino Fundamental de uma escola pública, do interior do Paraná, que contava com uma aluna com Múltipla Deficiência inclusa. Para o desenvolvimento das atividades, elaborou-se o material didático “Transferidor Adaptado”, para auxiliar alunos com Múltiplas Deficiências e/ou com alunos com Deficiência Física Neuromotora. Após a aplicação das atividades, foi possível observar a contribuição do uso da Tecnologia Assistiva para o ensino de geometria, aos alunos com Múltiplas Deficiências, permitindo a eles a compreensão e aprendizagem de conceitos matemáticos. / This paper, in a quality approach, aims to present the teaching of Geometry and the inclusion of students with Multiple Disabilities in classes of regular education. It also aims to elucidate the contributions of Assistive Technologies in teaching and learning mathematical contents of Geometry encouraging students’ autonomy when carrying out regular activities in regular education classes. The revision of literature in this investigation contemplated studies on teaching Geometry and scholar inclusion, using Assistive Technologies. In the pedagogical intervention mathematical concepts in Geometry were addressed, straight lines, semistraight lines, construction and measurement of angles, in a fifth grade class of the Primary Education in a public school, in the countryside in the state of Paraná, which has included a girl student with Multiple Disabilities. In order to develop those activities, a didactic material “Adapted Protractor” was elaborated, to help the students with Multiple Disabilities and/or students with Neuromotor Physical Disabilities. After the application of activities, it was possible to observe the contribution of the use of Assistive Technology to the teaching of Geometry for students with Multiple Disabilities, allowing them to understand and learn mathematical concepts.

MEd (Curriculum Studies) / Department of Curriculum Studies / Learning Geometry has been identified as an area of Mathematics that poses various problems for many secondary school learners’. Many learners’ fail to develop an adequate understanding of geometrical concepts and to demonstrate good reasoning and problem solving skills. Poor performance of learners’ in Euclidian Geometry has been a concern to Mathematics teachers, parents and government. The purpose of this study was to enhance learners’ performance in Euclidian Geometry as a branch of Mathematics in the FET band of secondary schools. The research question was: What factors facilitate the enhancement of learners’ performance in Euclidian Geometry in the FET band of secondary schools? The study adopted the mixed methods design. The population were all the FET band learners who chose Mathematics as a field of study at secondary schools in Nzhelele East Circuit. Purposive sampling was used to select participants who contributed to qualitative data. For quantitative data, clusters and random sampling methods were used to select learners who participated in the study. The sample consisted of 6 school principals, 6 heads of departments and 6 teachers. These groups of people participated in the interviews. Nine schools were sampled from the population and questionnaires were administered to 405 learners of which 45 were randomly selected from each school and 15 learners were randomly selected from each grade level. Thematic analysis was used to analyze qualitative data and the SPSS Version 22 was employed to analyze quantitative data. The data collected revealed the causes of poor performance in Geometry at Grades 10-12. The results of this study described how learners in Grades 10-12 could improve their performance in Geometry.

This action research study summarizes the investigation of journal writing and cooperative grouping with 8th grade geometry students in the construction of proof. Students written responses to prompts in journals were analyzed over the course of twelve weeks. Case studies of four students were developed from the researchers’ three geometry classes. All four students in the study participated in an academic program called Pre International Baccalaureate Preparation. Standardized test scores and an attitude scale categorized the four students. The ATMAT survey (Appendix A) measured each student’s attitude toward mathematics. Writing prompts focused on the students’ perceptions of the group process in constructing proofs and the development of geometric proof. The results suggested the students were engaged in learning within their cooperative groups but they also desired individual time before coming to a group setting. In addition, students’ written responses to journal prompts may provide an informal assessment and help students convey their own understanding of proof before any formal assessments.

The subject of teacher competencies or knowledge has been a key issue in mathematics education reform. This study attempts to identify and analyze teacher competencies necessary in the orchestration of a problem-centred approach to dynamic geometry teaching and learning. The advent of dynamic geometry environments into classrooms has placed new demands and expectations on mathematics teachers. In this study the Teacher Development Experiment was used as the main method of investigation. Twenty third-year mathematics major teachers participated in workshop and microteaching sessions involving the use of the Geometer's Sketchpad dynamic geometry software in the teaching and learning of the geometry of triangles and quadrilaterals. Five intersecting categories of teacher competencies were identified: mathematical/geometrical competencies. pedagogical competencies. computer and software competences, language and assessment competencies. / Mathematical Sciences / M. Ed. (Mathematical Education)

Considering that geometry is taught according to certain principles that do not encourage creativity, I have decided to employ the mixed methods philosophical framework applying the concurrent transformative design in the form of an exploratory case study. The case study to (i) explore and design a model that influences learning using polygon pieces and mathematics dictionary in the teaching and learning of geometry to grade 8 learners; (ii) investigate if the measurement of angles and sides of polygons using polygon pieces assisted by mathematics dictionary promote learners’ comprehension of geometry and (iii) investigate how mathematics teachers should use polygon pieces along with mathematics dictionary to teach properties of triangles in order to promote learners’ conceptual understanding. Drawing from my research findings a model has been developed from the use of polygon pieces and mathematics dictionary. The model use of mathematics dictionary in teaching and learning geometry is to develop learners’ mathematics vocabulary and terminology proficiency. Polygon pieces are to enhance the comprehension of geometric concepts. The quantitative data emerged from marked scripts of the diagnostic and post-intervention tests, the daily reflective tests and intervention activities were analysed as percentages and presented in line and bar graphs. Qualitative data obtained from observation notes and transcribed interviews were analysed in three forms: thematically, constant comparison and keywords in context. These findings support other research regarding the importance of using physical manipulatives with mathematics dictionary in teaching and learning geometry. They align with other findings that stress that manipulatives are critical facilitating tools for the development of mathematics concepts. The investigations led into the designing of a teaching model for the topic under study for the benefit of the mathematics community in the teaching and learning of geometry, focusing on properties of triangles. The model developed during this study adds to the relatively sparse teaching models but growing theoretical foundation of the field of mathematics. / Mathematics Education / Ph. D. (Mathematics Education)