Submitted by isabela.moljf@hotmail.com (isabela.moljf@hotmail.com) on 2016-08-15T14:10:30Z No. of bitstreams: 1 denisemansoldosalazar.pdf: 6189821 bytes, checksum: 46c6fab6e5197499e4198d05e4cb4214 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-02-02T11:08:56Z (GMT) No. of bitstreams: 1 denisemansoldosalazar.pdf: 6189821 bytes, checksum: 46c6fab6e5197499e4198d05e4cb4214 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-02-07T11:21:23Z (GMT) No. of bitstreams: 1 denisemansoldosalazar.pdf: 6189821 bytes, checksum: 46c6fab6e5197499e4198d05e4cb4214 (MD5) / Made available in DSpace on 2017-02-07T11:21:23Z (GMT). No. of bitstreams: 1 denisemansoldosalazar.pdf: 6189821 bytes, checksum: 46c6fab6e5197499e4198d05e4cb4214 (MD5) Previous issue date: 2015-12-17 / Apresentamos uma pesquisa que investiga as potencialidades do software GeoGebra como instrumento tecnológico favorável à aprendizagem das funções trigonométricas no Ensino Médio, a partir das representações gráficas destas funções. Trata-se uma pesquisa caracterizada por uma abordagem qualitativa em que foi usada uma sequência didática de atividades, com tarefas em um nível de exigência crescente. Cada etapa de execução desta pesquisa está respaldada pelos pressupostos teóricos da Engenharia Didática, metodologia adotada em pesquisas que envolvem uma parte experimental. Os argumentos advindos das análises dos dados estão fundamentados na Teoria Antropológica do Didático. Trabalhamos com a elaboração de recursos educacionais digitais para o ensino da Trigonometria utilizando o software GeoGebra, que serão aplicados a um grupo de alunos do Ensino Médio. Relacionada à pesquisa está a elaboração de um produto educacional que reunirá as tarefas aplicadas com uma orientação para a sua utilização em sala de aula. / We present a research which investigates the GeoGebra software capabilities as technological instrument conducive to learning of the trigonometric functions in high school, from the graphical representations of these functions. It is a survey characterized by a qualitative approach in which a didactic sequence of activities was used, with tasks at a level of growing demand. Each of this research execution phase is supported by the theoretical assumptions of the Didactic Engineering methodology used in research involving an experimental part. The arguments arising from the analysis are based on the Anthropological Theory of Didactic. For the development of digital educational resources for teaching trigonometry, which will apply to a group of high school students, used the GeoGebra software. Related to research, educational product offered here brings together the applied tasks and with guidance for using it in the classroom.

Abstracts in English, Afrikaans and Zulu / The purpose of this study was to explore the learners’ understanding of trigonometric functions using GeoGebra software. A qualitative case study approach was used with six Grade 11 learners at a school in Tshwane South District. The data were collected during a seven-day period using multiple methods: a diagnostic test, worksheets, a smart recorder, a trigonometric functions test, one-on-one interviews and focus-group discussions. The findings showed clearly that the use of GeoGebra enhanced Grade 11 learners’ understanding of trigonometric functions. The findings also showed that the use of GeoGebra helped the learners to understand the basic trigonometric functions graphs. This enabled them to sketch freely without using the point-by-point method. Based on this study, it is recommended that GeoGebra should be made available to all Grade 11 learners. This will encourage them use the software out of regular school hours. / Die doel van hierdie studiegids was om die leerder se kennis van die gebruik van GeoGebra sagteware, ten opsigte van trigonometriese funksies te ondersoek. ‘n Kwalitatiewe benadering is gevolg met ses Graad 11 leerders by ‘n skool in die Tshwane Suid Distrik. Die data is ingewin oor ‘n periode van sewe dae, deur gebruik te maak van verskeie metodes: ‘n diagnostiese toets, werkkaarte, ‘n “smart” opname, ‘n trigonometriese funksies toets, een-tot-een onderhoude en fokusgroepe waar besprekings plaasgevind het. Die data-analise wys duidelik dat die gebruik van GeoGebra, die Graad 11 leerders gehelp het om trigonometriese funksies beter te verstaan. Die uitkomste wys ook duidelik dat GeoGebra die leerders gehelp het met ‘n beter begrip van die basiese trigonometriese funksies. Dit het hulle gehelp om vryhand sketse te doen en nie noodwendig die punt-tot-punt metode nie. Gebaseer op hierdie studie beveel ons aan dat GeoGebra beskikbaar gemaak moet word aan aale Graad 11 leerders. Dit sal leerders motiveer om ook die sagteware op hulle eie buite skoolure te gebruik. / Injongo yocwaningo lolu bekuyikuhlola ukuzwisisa kwabantwana amagrafu e-trigonometry ma bewafunda nge softhiwe ye GeoGebra. Abantwana abayisithupha abakubanga letshumi lanye abaphuma kusigodi se Tshwane South babambiqhaza kulolucwaningo lwendlela ye ‘qualitative case study’. Imininingo iqoqwe kumalanga ayisikhombisa kusetshenziswa indlela ezilandelayo: ukuhlolwa kwe-diagnostic, amaphepha okusebenzela, isingxoxo zamunye ngamunye lezingxoxo leqembu. Iziphumo ezinkulu zikhombe ngokucacile ukuthi ukusetshenziswa kwesofthiwe yeGeoGebra kukhulise ukuzwisisa kwamagrafu e-trigonometry ngabafundi bebanga letshumi lanye.Iziphumo njalo zibonise ukuthi abafundi bazwisisa izinto eziyisiseko ngala magrafu ma bewafunda ngesofthiwe ye GeoGebra. Lokhu kunike abafundi amandla okudwebadweba lamagrafu ngokushesha ngokukhululeka. Kusekelwa ngalezi ziphumo, kunconyiwe ukuthi abafundi bafumane isofthiwe yeGeoGebra ngaso sonke isikhathi. Lokhu kuzabakhuthaza ukuthi basebenzise lesofthiwe noma bengaphandle kwesikolo. / Mathematics Education / M. Ed. (Mathematics Education)

The purpose of the study was to explore the relationship between mathematics teachers’ subject matter knowledge and their teaching effectiveness. A convenient sample of 19 grade 11 mathematics teachers and 418 students were initially selected for the study and took part in some stages of the study. Of this lot, only 11 teachers and 246 students participated in all the stages of the study. Explanatory Mixed methods research design which entails the use of a co-relational study and a descriptive survey design were employed in the study. Data was collected from the teachers using a self report questionnaire, Teacher Subject Matter Knowledge of Trigonometric Functions Scale (TSMKTFS) and peer evaluation questionnaire, and from students using teacher evaluation questionnaire and Student Trigonometric Functions Performance Scale (STFPS). All the instruments had their validity and reliability accordingly determined. Quantitative data gathered was analysed using descriptive and inferential statistics while qualitative data gathered from teachers’ and students’ tests were analysed using task performance analysis. It was found that a positive, statistically significant relationship existed between teachers’ subject matter knowledge and the composite measure of their teaching effectiveness. The relationships between teachers’ subject matter knowledge and students’ achievement and also between teachers’ subject matter knowledge and students’ rating of the teachers’ teaching effectiveness were found to be positive and statistically significant. However, the relationships between teachers’ subject matter knowledge and teachers’ self rating as well as teachers’ subject matter knowledge and peers’ rating of teachers’ teaching effectiveness were not found to be statistically significant though they were positive. Further data analysis showed that there was a difference between the subject matter knowledge of effective and ineffective teachers and also between the students taught by effective teachers and the students taught by the ineffective teachers. / Institute of Science and Technology Education / PhD (Mathematics Education)

Made available in DSpace on 2016-04-27T16:57:00Z (GMT). No. of bitstreams: 1 Luciane Santos Rosenbaum.pdf: 6907932 bytes, checksum: 73fa79dd753c44164be51ff0aab23315 (MD5) Previous issue date: 2010-10-04 / Secretaria da Educação do Estado de São Paulo / This present work aims to verify: as compatible constructivist perspectives of learning with the planning of Trigonometric Functions; teaching the as researches in mathematics education field , which brings important results on the learning process may contribute to the organization of the Trigonometric Functions teaching that leverage best learning situations for students, as the performance of teachers of mathematics is revealed, with regard to planning activities in the teaching of Trigonometric Functions, consistent with a constructivist view of learning. We developed a qualitative study with two teachers and 70 students from the 2nd Grade of high school in a public school in the State of São Paulo. Its theoretical work of Simon (1995) on the use of HLT in teaching mathematics to formulate models of teaching based on constructivism. As a component of the Mathematics Teaching Cycle developed by Simon, the elaborate HLT was made use of the research findings for the development of Trigonometric Functions through activities and solve problems involving: constructions with ruler and compass, manipulative material, scientific calculator, construct graphs using software GeoGebra and paper and pencil. The results led us to conclude that the use of research contributes to the education organization of Trigonometric Functions; however you must provide access to such teachers to such research. Although the HLT are potentially rich, complex is the task of developing activities to accomplish a constructivist learning perspective. We note that participation in tasks involving the use of technology and material handling enhances the learning of Trigonometric Functions. However, the HLT is not prepared enough for learning to occur, because the teacher performance has a decisive role in mediating the construction of knowledge of the students. In the same way we experience the interaction and participation between students and teacher which is essential for learning / O presente trabalho tem como objetivo verificar: como compatibilizar perspectivas construtivistas de aprendizagem com o planejamento do ensino de Funções Trigonométricas; como as pesquisas na área de Educação Matemática, que trazem resultados importantes sobre a aprendizagem, podem contribuir para a organização do ensino de Funções Trigonométricas que potencialize boas situações de aprendizagem aos alunos; como a atuação do professor de Matemática se revela, no que se refere às atividades de planejamento do ensino de Funções Trigonométricas, de forma compatível com uma perspectiva construtivista de aprendizagem. Desenvolvemos um estudo de natureza qualitativa com 2 professores e 70 alunos da 2.ª série do Ensino Médio de uma escola da rede pública do Estado de São Paulo. Este trabalho, tem como fundamentação teórica os trabalhos de Simon (1995) sobre o uso de THA no ensino de Matemática para formular modelos de ensino baseados no construtivismo. Como componente do Ciclo de Ensino de Matemática desenvolvido por Simon, a THA elaborada fez uso de resultados de pesquisas para o desenvolvimento de Funções Trigonométricas por meio de atividades e resolução de problemas que envolveram: construções com régua e compasso, material manipulativo, calculadora científica, construção de gráficos usando o software Geogebra e papel e lápis. Os resultados obtidos nos levaram a concluir que o uso de pesquisas contribui para a organização do ensino de Funções Trigonométricas, no entanto é necessário possibilitar o acesso dos professores a tais pesquisas. Verificou-se que embora as THAs sejam potencialmente ricas, é complexa a tarefa de elaboração de atividades para que se efetive uma aprendizagem numa perspectiva construtivista. Constatamos que a participação em tarefas que envolvem o uso de tecnologia e manipulação de materiais potencializa o aprendizado de Funções Trigonométricas. Porém, a THA elaborada não é suficiente para que a aprendizagem ocorra, pois a atuação do professor tem papel decisivo na mediação da construção do conhecimento dos seus alunos. Da mesma forma vimos que a interação entre alunos, e estes com o professor são essenciais para uma aprendizagem significativa

This paper investigates four important methods of solving summation and product problems in mathematics competitions. Chapter 1 presents the basic concepts of sequence and series, including arithmetic sequence (series), geometric sequence (series) and infinite geometric sequence (series). Chapter 2 handles the binomial coefficients and binomial theorem and show they how can be applied to compute series sum. Chapter 3 deals with power series, including interchanging summation and differentiation; interchanging summation and integration; and generating function which expresses a sequence as coefficients arising from a power series in variables. Chapter 4 provides four methods of telescoping sum, including antidifference, partial fractions, trigonometric functions, and factorial functions. Chapter 5 discusses the telescoping product which the main ideas and techniques are analogous to telescoping sum. Two types of telescoping product including difference of two squares and trigonometric functions are investigated.

We report a class of the mathematics in which an animation technology (calculating and plotting capabilities) of the software Mathematica is utilized. This class is taught for university students in a computer laboratory during a second semester. It is our purpose to make a student realize the usefulness and the importance of mathematics easily through visualization. In addition, we hope that students will acquire a new power of mathematics needed in the 21st century. For several years, we have continued this kind of class, and have continued to investigate the effectiveness that our teaching method (especially visualization) brings in the understanding of the mathematics. In this paper, we present some of this teaching method, which is performed in our class. From the questionnaire survey, it is found that our teaching method not only convinces students that the mathematics is useful or important but also deepens the mathematic understanding of students more.

The purpose of the study was to explore the relationship between mathematics teachers’ subject matter knowledge and their teaching effectiveness. A convenient sample of 19 grade 11 mathematics teachers and 418 students were initially selected for the study and took part in some stages of the study. Of this lot, only 11 teachers and 246 students participated in all the stages of the study. Explanatory Mixed methods research design which entails the use of a co-relational study and a descriptive survey design were employed in the study. Data was collected from the teachers using a self report questionnaire, Teacher Subject Matter Knowledge of Trigonometric Functions Scale (TSMKTFS) and peer evaluation questionnaire, and from students using teacher evaluation questionnaire and Student Trigonometric Functions Performance Scale (STFPS). All the instruments had their validity and reliability accordingly determined. Quantitative data gathered was analysed using descriptive and inferential statistics while qualitative data gathered from teachers’ and students’ tests were analysed using task performance analysis. It was found that a positive, statistically significant relationship existed between teachers’ subject matter knowledge and the composite measure of their teaching effectiveness. The relationships between teachers’ subject matter knowledge and students’ achievement and also between teachers’ subject matter knowledge and students’ rating of the teachers’ teaching effectiveness were found to be positive and statistically significant. However, the relationships between teachers’ subject matter knowledge and teachers’ self rating as well as teachers’ subject matter knowledge and peers’ rating of teachers’ teaching effectiveness were not found to be statistically significant though they were positive. Further data analysis showed that there was a difference between the subject matter knowledge of effective and ineffective teachers and also between the students taught by effective teachers and the students taught by the ineffective teachers. / Institute of Science and Technology Education / PhD (Mathematics Education)

Dans la première partie de la thèse, nous présentons un nouveau système permettant de produire des mouvements de caméra réalistes pour l’animation stopmotion. Le système permettra d’enrichir les logiciels d’animation 3D classiques (comme par exemple Maya et 3D Studio Max) afin de leur faire contrôler des mouvements de caméra pour la stop motion, grâce à l’utilisation d’une interface haptique. Nous décrivons le fonctionnement global du système. La première étapeconsiste à récupérer et enregistrer les données envoyées par le périphérique haptique de motion capture. Dans la seconde étape, nous réélaborons ces données par un procédé mathématique, puis les exportons vers un logiciel de 3D pour prévisualiser les mouvements de la caméra. Finalement la séquence est exécutée avec un robot de contrôle de mouvement et un appareil photo. Le système est évalué par un groupe d’étudiants du Master "Art plastiques et Création numérique" de l’Université de Valenciennes. Dans la deuxième partie, nous définissons une nouvelle classe de courbes à partir des courbes polynomiales paramétriques à hodographe pythagorien (PH) construite sur un espace algébrique-trigonométrique. Nous montrons leurs propriétés fondamentaleset leurs avantages importants par rapport à leur équivalent polynomial, grâce à l’utilisation d’un paramètre de forme. Nous introduisons une formulation complexe et nous résolvons le problème d’interpolation de Hermite. / In the first part, we present a new system that allows to create realistic cameramovements for a stop motion animation. The system improves traditional 3D software animation programs (for example Maya and 3D Studio Max) for creatingstop motion camera movements by using an haptic interface. After describing thewhole system, we explain in detail the mathematical processing to obtain differentcamera movements by using an haptic interface for motion capture. The recordedhaptic positions, once elaborated, are exported, frame by frame, to the motioncontrol software, which allows to calibrate the motion control robot, to control thecamera settings and, finally, to execute the sequences. A class of students of the"Art plastiques et Création numérique" Master of the University of Valenciennesevaluated the system. In the second part, we define a new class of Pythagorean Hodograph curves built upon a five dimensional mixed algebraic trigonometric space, and show their fundamental properties and important advantages over their well known polynomial counterpart. A complex representation for these curves is introduced and constructive approaches are provided to solve the first order Hermite interpolation problem.

We report a class of the mathematics in which an animation technology (calculating and plotting capabilities) of the software Mathematica is utilized. This class is taught for university students in a computer laboratory during a second semester. It is our purpose to make a student realize the usefulness and the importance of mathematics easily through visualization. In addition, we hope that students will acquire a new power of mathematics needed in the 21st century. For several years, we have continued this kind of class, and have continued to investigate the effectiveness that our teaching method (especially visualization) brings in the understanding of the mathematics. In this paper, we present some of this teaching method, which is performed in our class. From the questionnaire survey, it is found that our teaching method not only convinces students that the mathematics is useful or important but also deepens the mathematic understanding of students more.

Chapter 1 presents the definitions and basic properties of trigonometric functions including: Sum Identities, Difference Identities, Product-Sum Identities and Sum-Product Identities. These formulas provide effective tools to solve the problems in trigonometry. Chapter 2 handles the most important two theorems in trigonometry: The laws of sines and cosines and show how they can be applied to derive many well known theorems including: Ptolemy¡¦s theorem, Euler Triangle Formula, Ceva¡¦s theorem, Menelaus¡¦s Theorem, Parallelogram Law, Stewart¡¦s theorem and Brahmagupta¡¦s Formula. Moreover, the formulas of computing a triangle area like Heron¡¦s formula and Pick¡¦s theorem are also discussed. Chapter 3 deals with the method of superposition, inverse trigonometric functions, polar forms and De Moivre¡¦s Theorem.