This study employed a qualitative approach to investigate the use of cooperative learning to enhance conceptual understanding of trigonometry in a Grade 11 mathematics classroom, conducted at a high school in Moletlane Circuit, Capricorn District in Limpopo Province, South Africa. A single case study was used as a research design to get an in-depth analysis and collect detailed data using semi-interviews and lesson observation of the cooperative learning of trigonometry in Grade 11 from the learners and the teacher. Participants were purposely chosen and consisted of (n=30) Grade 11 mathematics learners and their mathematics teacher. Data from the participants were collected through semi-structured interviews and observation, with the aid of observation guide (Appendix C) for three weeks. The salient findings from the study showed how cooperative learning was used, research questions, the approaches, the teacher did not adequately highlight the importance of trigonometry to students without integrating the topic to real-life situations. Some students said that the teacher did not teach trigonometry in a manner that they understood, which made trigonometry challenging for them. Concerning cooperative learning, the study found that many learners were passively engaged, listened to or watched the teacher. Mainly, the study recommends teacher-training institutions to host practical workshops to help teachers integrate theoretical training and practical cooperative learning experience. While this study was qualitative in nature, future researchers could conduct quantitative data collection. This would allow for the collection of numerical findings through survey questionnaires. / Mathematics Education / M. Ed. (Mathematics Education)

Os caleidociclos têm sido utilizados como forma artística de apresentação de imagens, pinturas ou como parte de trabalhos artísticos, principalmente de imagens com simetrias; talvez os mais conhecidos sejam os trabalhos de M. C. Escher. As poucas publicações encontradas da teoria matemática envolvida nos caleidociclos dão base para imaginar e criar aplicações no desenvolvimento de habilidades e competências trabalhadas na escola. Para aumentar as possibilidades de aplicações de conceitos, teoremas e relações matemáticas estudadas no ensino básico, o presente trabalho apresenta algumas propostas de atividades utilizando os caleidociclos. As propostas foram elaboradas de acordo com o nível de ensino, ou seja, simetrias para o 7o ano, teorema de Pitágoras para os 8o e 9o anos do Ensino Fundamental, lei dos cossenos e relação fundamental da trigonometria para a 1a série e volume e área de superfície de sólidos geométricos para 2a série do Ensino Médio; algumas das propostas apresentam variações para se adequar ao nível de desenvolvimento em que a turma se encontra. Todos os moldes utilizados e outras possibilidades de caleidociclos, incluindo sólidos encaixantes aos caleidociclos, foram organizados ao final deste trabalho em um dos apêndices. Há também um apêndice com outros tipos de sólidos geométricos com movimentos, que podem ser usados no mesmo intuito de aplicação diferenciada da geometria espacial. / Kaleidocycles have been used asan artistic formof presentation of pictures, paintings or a part of artworks, especially images with symmetries; perhaps the best known works are M. C. Eschers. The few finded publications of the mathematical theory related to these three-dimensional rings give rise to imagine and create applications for developing skills to be worked in classroom. In order to increase the possibility of applications of concepts, theorems and mathematical relations, the present work proposes some activities dealing with kaleidocycles. The proposals were prepared in accordance with the students level of education, i.e., symmetries for the7th grade, the Pythagorean theorem for the 8th and 9th grades, law of cosines and the fundamental relation of trigonometry, volume and surface area of geometric solids for high school students; some of the proposals have variations to suit the level of development in which the class is at. All the molds used and other possibilities of kaleidocycles, including solids which fit into kaleidocycles, were organized at the end of this dissertation in one of the appendices. There is also an appendix with other types of mobile geometric solids that can be used in the same purpose in different applications of spatial geometry.

Made available in DSpace on 2016-04-27T17:13:02Z (GMT). No. of bitstreams: 1 Alessandra Nascimento.pdf: 1945223 bytes, checksum: f6950936e78897c5cfc27568b8cd6159 (MD5) Previous issue date: 2005-07-17 / Made available in DSpace on 2016-08-25T17:25:34Z (GMT). No. of bitstreams: 2 Alessandra Nascimento.pdf.jpg: 2104 bytes, checksum: c4715912a635b5fbde63d2a9b070733f (MD5) Alessandra Nascimento.pdf: 1945223 bytes, checksum: f6950936e78897c5cfc27568b8cd6159 (MD5) Previous issue date: 2005-07-17 / The objective of this study is to construct a trigonometrical table, on the basis of historical surveys of the works of Ptolomeu and other mathematicians of Old Greece, to investigate the appropriation of the meaning of the concepts of the trigonometrical reasons: sine, cosine and tangent, in the rectangular triangle, for students of first year Average education. We look for to answer to the research question: How to teach trigonometry in the rectangular triangle in significant way? E also decurrent questions: Which factors influence the acquisition of such knowledge? How to distanciar the use of Trigonometry in Average Education of mechanization? For in such a way we use the estimated theoreticians of Vygotsky in that if relates to the importance attributed to the social interaction, the language and the simbolizaction in the gradual domain of a conceptual field for the pupils, of the estimated theoreticians of Vergnaud, when dealing with the operations invariants: concept-in-action and theorem-in-action, of its conception, of conceptual field and concept, e also in the model presented for Parzysz for a theoretical picture of the education of geometry, where it detaches four stages of the development of the geometric thought. The results of the experimentation point with respect to an imbalance in Geometry and Algebra. The experimentation showed that despite this, an education of the Trigonometry of the generating rectangular triangle of motivations, including diversified activities, with problems situations, that stimulate thinking, the inquiry and carrying through, contributes so that the pupils construct the meaning of the trigonometrical reasons, besides favoring the argument and modifying some wrong conceptions / O objetivo deste estudo é construir uma tabela trigonométrica, com base em levantamentos históricos dos trabalhos de Ptolomeu e outros matemáticos da Grécia Antiga, para investigar a apropriação do significado dos conceitos das razões trigonométricas: seno, cosseno e tangente, no triângulo retângulo, por estudantes do 1o ano do Ensino Médio. Procuramos responder à questão de pesquisa: Como ensinar trigonometria no triângulo retângulo de maneira significativa? E também questões decorrentes: Quais fatores influenciam a aquisição de tal conhecimento? Como distanciar a utilização da Trigonometria no Ensino Médio da mecanização? Para tanto utilizamos os pressupostos teóricos de Vygotsky no que se refere à importância atribuída à interação social, à linguagem e à simbolização no progressivo domínio de um campo conceitual pelos alunos, dos pressupostos teóricos de Vergnaud, ao tratar dos invariantes operatórios: conceito-em-ação e teorema-em-ação, de sua concepção de campo conceitual e de conceito, e também no modelo apresentado por Parzysz para um quadro teórico do ensino da geometria, onde ele destaca quatro etapas do desenvolvimento do pensamento geométrico. Os resultados da experimentação apontam para uma defasagem em Geometria e em Álgebra. A experimentação mostrou que apesar disso, um ensino da Trigonometria do triângulo retângulo gerador de motivações, incluindo atividades diversificadas, com situações problematizadoras, que estimulem o pensar, a investigação e o realizar, contribui para que os alunos construam o significado das razões trigonométricas, além de favorecer a argumentação e modificar várias concepções errôneas

Made available in DSpace on 2016-04-28T14:16:13Z (GMT). No. of bitstreams: 1 Armando Pereira Loreto Junior.pdf: 1989486 bytes, checksum: b270828c0f687f19dacbba6800156d40 (MD5) Previous issue date: 2001-06-26 / This research investigates which were the mathematical contents, sequence and manner of exposition taught in the sphere class of Colégio de Santo Antão-OVelho, in Lisbon, during the second half of the 18th century. The current work describes, within the context of that time, the most relevant features of the compendium Trigonometria Plana e Esferica com o Canon Trigonometrico, Linear e Logarithmico, published in 1737. Its author is the Jesuit priest Manoel de Campos, who was one of the teachers of that class. This study reveals interesting approaches made by him, which led to important results, like the correction of the Canon Trigonometrico, and the generation of the logarithmic curve. It was verified that the trigonometry presented by Campos had as its first objective the application to the Nautical Science and Astronomy. However, the author did not present these applications in his book, but promised to show them in another publication, which existence we did not find register of. Because of that, some examples of those applications, included in Arte de Navegar of Manoel Pimentel, were selected and presented in this work / Esta pesquisa investiga quais eram os conteúdos de matemática ensinados, sua ordem seqüencial e a forma de exposição, na aula da esfera do Colégio de Santo Antão-O-Velho da cidade de Lisboa, na primeira metade do século XVIII. O presente trabalho descreve, dentro do contexto científico daquela época, os aspectos mais relevantes do compêndio Trigonometria Plana e Esferica com o Canon Trigonometrico, Linear e Logarithmico, publicado em 1737, cujo autor é o padre jesuíta Manoel de Campos, que foi um dos professores daquela aula. Este estudo revela abordagens interessantes por ele efetuadas, que conduziram a resultados importantes, como a correção do Canon Trigonometrico e a geração da linha logarítmica. Ficou constatado que a trigonometria apresentada por Campos tinha como primeiro objetivo a sua aplicação à náutica e à astronomia. Contudo, o autor não apresentou tais aplicações no seu compêndio, prometendo mostrá-las oportunamente em outra publicação, de cuja existência não encontramos registro. Em virtude disso, alguns exemplos dessas aplicações, contidos na Arte de Navegar de Manoel Pimentel, contemporâneo de Campos, foram selecionados e apresentados neste trabalho

Made available in DSpace on 2016-04-27T16:59:01Z (GMT). No. of bitstreams: 1 Americo Augusto Barbosa.pdf: 1871211 bytes, checksum: b1d025c51844683af5341ce09c48231b (MD5) Previous issue date: 2009-12-08 / Secretaria da Educação do Estado de São Paulo / This dissertation is part of the research project developed by researchers at the Pontifícia Universidade Católica of Sao Paulo, which seeks to develop support materials and curricular innovations to school, taking reference to the notion of Hypothetical Learning Trajectories (THA), a concept used by Dr Simon (1995) as part of the 'model' Cycle of Teaching and Learning of Mathematics. The objective of this research is to analyze the possibility of reconciling perspectives of learning by teaching plans related to the reasons and trigonometric functions and verify the performance of a mathematics teacher in front of a proposal aimed at teaching a constructivist perspective. To answer our questions we developed a qualitative study, collecting data through contents the observation of lessons with teachers and students from three different groups of 2nd year high school public school Education of the State of Sao Paulo. The results indicate that: it is possible to reconcile learning perspective with the planning of education and how important the work of mathematics teacher for learning to occur. We conclude that not just one good result of education, interaction and participation among students and teachers are the main instruments to be made effective a significant learning in a constructivist perspective / Esta Dissertação faz parte do projeto de pesquisa desenvolvido por pesquisadores da Pontifícia Universidade Católica de São Paulo, que procura desenvolver materiais de apoio e inovações curriculares para o Ensino Médio, tomando com referência a noção de Trajetórias Hipotéticas de Aprendizagem (THA), conceito utilizado pelo pesquisador Simon (1995) como parte do modelo de Ciclo de Ensino e de Aprendizagem de Matemática. O objetivo da pesquisa é analisar a possibilidade de compatibilizar perspectivas de aprendizagem com a planificação de ensino relacionada às razões e às funções trigonométricas e verificar a atuação do professor de matemática diante de uma proposta de ensino visando uma perspectiva construtivista. Para responder às nossas questões desenvolvemos uma pesquisa de caráter qualitativo, coletando os dados por meio de relatórios de observação das aulas com professores e alunos de três diferentes turmas de 2º ano do Ensino Médio da rede pública do Ensino Oficial do Estado de São Paulo. Os resultados obtidos indicam que: é possível compatibilizar perspectiva de aprendizagem com a planificação de ensino, e o quanto é importante a atuação do professor de matemática para que ocorra aprendizagem. Concluímos que não basta uma boa sequência de ensino, a interação e a participação entre alunos e professores são os principais instrumentos para que se efetive uma aprendizagem significativa numa perspectiva construtivista

CAPES / Utilizando-se da relação entre a Matemática e a Física e tendo a música como fonte de recursos associados a conceitos matemáticos, este trabalho objetiva contribuir com o desenvolvimento de habilidades nos estudantes na aprendizagem de conteúdos matemáticos como as funções trigonométricas. Conceitos importantes relacionados à acústica são apresentados, mostrando a importância da Série de Fourier Contínua no desenvolvimento de sintetizadores analógicos aditivos e seu uso no ensino de trigonometria. / Using the relationship between Mathematics and Physics and having music as a source of features associated with mathematical concepts, this work aims to contribute to the development of skills in students who are learning mathematical subjects such as trigonometric functions. Important concepts related to acoustics are presented, showing the importance of Continuous Fourier Series (CFS) in the development of Additive Synthesizer and its use in trigonometry.

Made available in DSpace on 2016-06-02T20:02:50Z (GMT). No. of bitstreams: 1 4002.pdf: 7792183 bytes, checksum: f1d845672bf0e825f73ed21d4b2c4b4e (MD5) Previous issue date: 2011-08-30 / This work is the development, implementation and evaluation of a Virtual Learning Environment to support teaching trigonometry Curriculum Proposal according to the State of São Paulo, with strong use of dynamic geometry viewers. That proposal, implemented since 2008 with the name São Paulo Faz Escola, brings a series of actions to improve of education in face of national assessments of São Paulo, and the international quality indicators established by the Ministry of Education. The teaching material from these actions required a recycling of the practice, focusing on learning skills and abilities. Within this overall picture we develop teaching sequences with the use of ICT (Information and Communication Technologies) to show the student as the protagonist of the construction of knowledge. The implementation of the environment occurred in the first grade of Public Hight School, Newton Prado, during the 4th school grade of 2010 and had part of its five weekly classes held in the computer lab. We believe that these results will undoubtedly contribute to the improvement and development of basic education, especially math. / Este trabalho expõe o desenvolvimento, aplicação e avaliação de um Ambiente Virtual de Aprendizagem de apoio ao ensino de trigonometria segundo a Proposta Curricular do Estado de São Paulo, cujo uso de visualizadores de geometria dinâmica é forte e pedagogicamente promissor. Tal Proposta, implantada a partir de 2008 com o nome São Paulo Faz Escola, traz uma série de ações que visam melhorar a situação da educação paulista frente às avaliações nacionais, internacionais e aos indicadores de qualidade estabelecidos pelo Ministério da Educação. O material didático proveniente destas ações exigiu uma reciclagem das práticas exercidas, focando a aprendizagem nas competências e habilidades. Dentro deste panorama, procuramos desenvolver sequências didáticas com o uso de Tecnologias de Informação e Comunicação (TIC) que evidenciassem o aluno como protagonista do processo de construção de conhecimento. A aplicação do ambiente se deu nas primeiras séries do ensino médio da Escola Estadual Newton Prado que, durante o 4° bimestre letivo de 2 010, tiveram parte de suas cinco aulas semanais realizadas no laboratório de informática. Consideramos que os resultados apresentados poderão contribuir de forma indubitável à melhoria e aperfeiçoamento da educação básica, especialmente, de matemática.

Os caleidociclos têm sido utilizados como forma artística de apresentação de imagens, pinturas ou como parte de trabalhos artísticos, principalmente de imagens com simetrias; talvez os mais conhecidos sejam os trabalhos de M. C. Escher. As poucas publicações encontradas da teoria matemática envolvida nos caleidociclos dão base para imaginar e criar aplicações no desenvolvimento de habilidades e competências trabalhadas na escola. Para aumentar as possibilidades de aplicações de conceitos, teoremas e relações matemáticas estudadas no ensino básico, o presente trabalho apresenta algumas propostas de atividades utilizando os caleidociclos. As propostas foram elaboradas de acordo com o nível de ensino, ou seja, simetrias para o 7o ano, teorema de Pitágoras para os 8o e 9o anos do Ensino Fundamental, lei dos cossenos e relação fundamental da trigonometria para a 1a série e volume e área de superfície de sólidos geométricos para 2a série do Ensino Médio; algumas das propostas apresentam variações para se adequar ao nível de desenvolvimento em que a turma se encontra. Todos os moldes utilizados e outras possibilidades de caleidociclos, incluindo sólidos encaixantes aos caleidociclos, foram organizados ao final deste trabalho em um dos apêndices. Há também um apêndice com outros tipos de sólidos geométricos com movimentos, que podem ser usados no mesmo intuito de aplicação diferenciada da geometria espacial. / Kaleidocycles have been used asan artistic formof presentation of pictures, paintings or a part of artworks, especially images with symmetries; perhaps the best known works are M. C. Eschers. The few finded publications of the mathematical theory related to these three-dimensional rings give rise to imagine and create applications for developing skills to be worked in classroom. In order to increase the possibility of applications of concepts, theorems and mathematical relations, the present work proposes some activities dealing with kaleidocycles. The proposals were prepared in accordance with the students level of education, i.e., symmetries for the7th grade, the Pythagorean theorem for the 8th and 9th grades, law of cosines and the fundamental relation of trigonometry, volume and surface area of geometric solids for high school students; some of the proposals have variations to suit the level of development in which the class is at. All the molds used and other possibilities of kaleidocycles, including solids which fit into kaleidocycles, were organized at the end of this dissertation in one of the appendices. There is also an appendix with other types of mobile geometric solids that can be used in the same purpose in different applications of spatial geometry.

Dieses Manuskript entstand aus Vorlesungen über Geodätische Berechnungen an der Hochschule für Technik und Wirtschaft Dresden. Da diese Lehrveranstaltung im ersten oder zweiten Semester stattfindet, werden noch keine Methoden der höheren Mathematik benutzt. Das Themenspektrum beschränkt sich deshalb weitgehend auf elementare Berechnungen in der Ebene.:0 Vorwort 1 Ebene Trigonometrie 1.1 Winkelfunktionen 1.2 Berechnung schiefwinkliger ebener Dreiecke 1.3 Berechnung schiefwinkliger ebener Vierecke 2 Ebene Koordinatenrechnung 2.1 Kartesische und Polarkoordinaten 2.2 Erste Geodätische Grundaufgabe 2.3 Zweite Geodätische Grundaufgabe 3 Flächenberechnung und Flächenteilung 3.1 Flächenberechnung aus Maßzahlen. 3.2 Flächenberechnung aus Koordinaten 3.3 Absteckung und Teilung gegebener Dreiecksflächen 3.4 Absteckung und Teilung gegebener Vierecksflächen 4 Kreis und Ellipse 4.1 Kreisbogen und Kreissegment 4.2 Näherungsformeln für flache Kreisbögen 4.3 Sehnen-Tangenten-Verfahren 4.4 Grundlegendes über Ellipsen 4.5 Abplattung und Exzentrizitäten 4.6 Die Meridianellipse der Erde 4.7 Flächeninhalt und Bogenlängen 5 Ebene Einschneideverfahren 5.1 Bogenschnitt 5.2 Vorwärtsschnitt 5.3 Anwendung: Geradenschnitt 5.4 Anwendung: Kreis durch drei Punkte 5.5 Schnitt Gerade ⎼ Kreis oder Strahl ⎼ Kreis 5.6 Rückwärtsschnitt 5.7 Anwendung: Rechteck durch fünf Punkte 6 Ebene Koordinatentransformationen 6.1 Elementare Transformationsschritte 6.2 Rotation und Translation. 6.3 Rotation, Skalierung und Translation 6.4 Ähnlichkeitstransformation mit zwei identischen Punkten 6.5 Anwendung: Hansensche Aufgabe 6.6 Anwendung: Kleinpunktberechnung 6.7 Anwendung: Rechteck durch fünf Punkte 6.8 Ebene Helmert-Transformation 6.9 Bestimmung der Parameter bei Rotation und Translation 6.10 Ebene Affintransformation 7 Lösungen / This manuscript evolved from lectures on Geodetic Computations at the University of Applied Sciences Dresden (Germany). Since this lecture is given in the first or second semester, no advanced mathematical methods are used. The range of topics is limited to elementary computations in the plane.:0 Vorwort 1 Ebene Trigonometrie 1.1 Winkelfunktionen 1.2 Berechnung schiefwinkliger ebener Dreiecke 1.3 Berechnung schiefwinkliger ebener Vierecke 2 Ebene Koordinatenrechnung 2.1 Kartesische und Polarkoordinaten 2.2 Erste Geodätische Grundaufgabe 2.3 Zweite Geodätische Grundaufgabe 3 Flächenberechnung und Flächenteilung 3.1 Flächenberechnung aus Maßzahlen. 3.2 Flächenberechnung aus Koordinaten 3.3 Absteckung und Teilung gegebener Dreiecksflächen 3.4 Absteckung und Teilung gegebener Vierecksflächen 4 Kreis und Ellipse 4.1 Kreisbogen und Kreissegment 4.2 Näherungsformeln für flache Kreisbögen 4.3 Sehnen-Tangenten-Verfahren 4.4 Grundlegendes über Ellipsen 4.5 Abplattung und Exzentrizitäten 4.6 Die Meridianellipse der Erde 4.7 Flächeninhalt und Bogenlängen 5 Ebene Einschneideverfahren 5.1 Bogenschnitt 5.2 Vorwärtsschnitt 5.3 Anwendung: Geradenschnitt 5.4 Anwendung: Kreis durch drei Punkte 5.5 Schnitt Gerade ⎼ Kreis oder Strahl ⎼ Kreis 5.6 Rückwärtsschnitt 5.7 Anwendung: Rechteck durch fünf Punkte 6 Ebene Koordinatentransformationen 6.1 Elementare Transformationsschritte 6.2 Rotation und Translation. 6.3 Rotation, Skalierung und Translation 6.4 Ähnlichkeitstransformation mit zwei identischen Punkten 6.5 Anwendung: Hansensche Aufgabe 6.6 Anwendung: Kleinpunktberechnung 6.7 Anwendung: Rechteck durch fünf Punkte 6.8 Ebene Helmert-Transformation 6.9 Bestimmung der Parameter bei Rotation und Translation 6.10 Ebene Affintransformation 7 Lösungen

Metacognition is commonly understood in the context of the learners and not their teachers. Extant literature focusing on how Mathematics teachers apply their metacognitive skills in the classroom, clearly distinguishes between teaching with metacognition (TwM) referring to teachers thinking about their own thinking and teaching for metacognition (TfM) which refers to teachers creating opportunities for learners to reflect on their thinking. However, in both of these cases, thinking requires a language, in particular appropriate mathematical language to communicate the thinking by both teacher and learners in the Mathematics classroom. In this qualitative study, which forms part of a bigger project within SANPAD (South Africa Netherlands Research Programme on Alternatives in Development), the metacognitive skills and mathematical language used by Mathematics teachers who teach at two township schools were interrogated using the design-based research approach with lesson study. Data collection instruments included individual interviews and a trigonometric assessment task. Lessons were also observed and video-taped to be viewed and discussed during focus group discussions in which the teachers, together with five Mathematics lecturers, participated. The merging of the design-based research approach with lesson study brought about teacher-lecturer collaboration, referred to in this study as the Mathematics Educators’ Reflective Inquiry (ME’RI) group, and enabled the design of a hypothetical teaching and learning trajectory (HTLT) for the teaching of trigonometric functions. A metacognitive performance profile for the two grade 10 teachers was also developed. The Framework for Analysing Mathematics Teaching for the Advancement of Metacognition (FAMTAM) from Ader (2013) and the Teacher Metacognitive Framework (TMF) from Artzt and Armour-Thomas (2002) were adjusted and merged to develop a new framework, the Metacognitive Teaching for Metacognition Framework (MTMF) to analyse the metacognitive skills used by mathematics teachers TwM as well as TfM. Without oversimplifying the magnitude of these concepts, the findings suggest a simple mathematical equation: metacognitive skills + enhanced mathematical language = conceptualization skills. The findings also suggest that both TwM and TfM are required for effective mathematics instruction. Lastly the findings suggest that the ME’RI group holds promise to enhance the use of the metacognitive skills and mathematical language of Mathematics teachers in Mathematics classrooms. / PhD (Mathematics Education), North-West University, Potchefstroom Campus, 2014