Made available in DSpace on 2017-07-21T20:56:28Z (GMT). No. of bitstreams: 1 Fabiana Reis.pdf: 2755681 bytes, checksum: e3dc0c10579da17a989cb45f6078d987 (MD5) Previous issue date: 2016-02-26 / The trigonometry has many applications not only in mathematics, but in different areas. Some problems can only be resolved with the use of its concepts. This paper presents a historical survey of the emergence of trigonometry, first, as a part of astronomy, after opening as a part of mathematics. Therefore, we engage the leading mathematicians and their contributions to obtain trigonometry as it currently is. We include the main settings, their properties, some demonstrations and also the trigonometric functions as a way to deepen their knowledge on the subject. Is out finally, some applications of trigonometric concepts in various areas in order to show that trigonometry goes far beyond simple repetitions of exercises in the classroom. / A trigonometria tem muitas aplicações não apenas em Matemática, mas em diversas áreas. Alguns problemas só podem ser resolvidos com o uso de seus conceitos. Neste trabalho apresenta-se um levantamento histórico sobre o surgimento da trigonometria, primeiramente, como uma parte da astronomia, depois se abrindo como uma parte da matemática. Para tanto, envolvemos os principais matemáticos e suas contribuições até obtermos a trigonometria como ela é atualmente. Incluímos as principais definições, suas propriedades, algumas demonstrações e também as funções trigonométricas como forma de aprofundar o conhecimento sobre o tema. Destacam-se, por último, algumas aplicações dos conceitos trigonométricos nas diversas áreas com o objetivo de mostrar que a trigonometria vai muito além de simples repetições de exercícios em sala de aula.

Orientador: João Peres Vieira / Banca: Maria Gorete Carreira Andrade / Banca: Suzinei Aparecida Siqueira Marconato / Resumo: O objetivo deste trabalho é propor uma abordagem no ensino de trigonometria desde o 9 ano do ensino fundamental até o final do ensino médio, respeitando o currículo básico da matemática e o nível de aprofundamento do conteúdo de acordo com a faixa etária dos estudantes. Para isso, são apresentadas atividades para serem aplicadas em sala de aula de modo que os alunos participem da formação e construção do conteúdo com ênfase nas aplicações e nos contextos históricos, contando com o auxílio de softwares matemáticos / Abstract: This study aims at proposing a new approach to the teaching trigonometry starting at 9th grade in elementary school to senior year in high school, taking into account the basic Mathematics syllabus and the degree of difficulty of the topics studied concerning the age group the students belong to. In order to achieve that goal, this study presents activities to be used in classroom so that the students are active in the building and construction of content, emphasizing its real use and its historical context, with the support of mathematical software / Mestre

A sociedade necessita cada vez mais de pessoas aptas a interagir com o conhecimento de maneira ativa e participativa. Nesse sentido, aprender é adquirir vivências e estabelecer uma dialética entre conhecimento e sujeito. Para que se formem cidadãos capazes dessa tarefa, é necessário um sistema de ensino pautado na construção de habilidades e competências, não apenas específicas às disciplinas, mas também no trabalho em equipe. Porém os resultados de avaliações externas, como SARESP e PISA, mostram que é necessária uma mudança no sistema educacional Brasileiro, uma vez que nossos alunos não têm demonstrado bons resultados. Nesse sentido, o presente trabalho utiliza como metodologia de ensino dois jogos pedagógicos, sendo um de estratégia e outro de conhecimento, respectivamente: \"Baralho Trigonométrico\" criado pelo professor-pesquisador e \"Trigonometrilha\" proposto por Smole (2008). O projeto foi aplicado, nos anos de 2015, 2016 e 2017, em uma escola pública do estado de São Paulo a alunos do 2° ano do ensino médio. Em todos os anos foi realizada a avaliação diagnósticas (préteste e pós-teste). As atividades dos jogos foram realizadas em grupo e a metodologia foi aplicada em duas sequências diferentes, seguida de análise de conteúdo. Foi possível verificar, a partir dos resultados da avaliação diagnostica que mesmo com as turmas não avançando de forma proporcional, seus índices de acertos no pós-teste aumentaram principalmente em questões relacionadas diretamente aos conteúdos utilizados nos jogos. Sendo que a média de acertos passou de 29% para 63% em 2015, de 30% para 49% em 2016 e de 20% para 31% em 2017, reforçando o fato que os jogos são uma importante ferramenta para o ensino. / From an active and participatory means of communication. In this sense, learning is to acquire experiences and establish a dialectic between knowledge and subject. In order for people to become familiar with this task, a system of teaching in the area of skills and competences is needed, not only in the disciplines, but there is also no teamwork. The results of the second, as well as the SARESP and the PISA, were presented as an educational program, since our students do not have good results. In this sense, the present work was based on two pedagogical games, one of strategy and another of knowledge, mainly: \"Trigonometric Deck\" created by the teacher-researcher and \"Trigonometrilha\" by Smole (2008). The project was applied, in the years 2015, 2016 and 2017, in a public school in the state of São Paulo, a second year of high school. In all years a diagnostic evaluation (pre-test and posttest) was performed. The game rules were applied in a group and the methodology was applied in two different sequences, followed by a content analysis. It was possible to verify, from the results of the evaluation of the same level of groups of proportional form, their indices of performance in the post-test increased substantially in relation to the expenses of the games. The average number of hits increased from 29% to 63% in 2015, from 30% to 49% in 2016 and from 20% to 31% in 2017, reinforcing what games are an important teaching tool.

Let Ω ⊂ Rn be a bounded Lipschitz domain and consider the energy functional F[u, Ω] := ∫ Ω F(∇u(x)) dx, over the space Ap(Ω) := {u ∈ W 1,p(Ω, Rn): u|∂Ω = x, det ∇u> 0 a.e. in Ω}, where the integrand F : Mn×n → R is quasiconvex, sufficiently regular and satisfies a p-coercivity and p-growth for some exponent p ∈ [1, ∞[. A motivation for the study of above energy functional comes from nonlinear elasticity where F represents the elastic energy of a homogeneous hyperelastic material and Ap(Ω) represents the space of orientation preserving deformations of Ω fixing the boundary pointwise. The aim of this thesis is to discuss the question of multiplicity versus uniqueness for extremals and strong local minimizers of F and the relation it bares to the domain topology. Our work, building upon previous works of others, explicitly and quantitatively confirms the significant role of domain topology, and provides explicit and new examples as well as methods for constructing such maps. Our approach for constructing strong local minimizers is topological in nature and is based on defining suitable homotopy classes in Ap(Ω) (for p ≥ n), whereby minimizing F on each class results in, modulo technicalities, a strong local minimizer. Here we work on a prototypical example of a topologically non-trivial domain, namely, a generalised annulus, Ω= {x ∈ Rn : a< |x| <b}, with 0 <a<b< ∞. Then the associated homotopy classes of Ap(Ω) are infinitely many when n =2 and two when n ≥ 3. In contrast, for constructing explicitly and directly solutions to the system of Euler-Lagrange equations associated to F we introduce a topological class of maps referred to as generalised twists and relate the problem to extremising an associated energy on the compact Lie group SO(n). The main result is a surprising discrepancy between even and odd dimensions. In even dimensions the latter system of equations admits infinitely many smooth solutions, modulo isometries, amongst such maps whereas in odd dimensions this number reduces to one. Even more surprising is the fact that in odd dimensions the functional F admits strong local minimizers yet no solution of the Euler-Lagrange equations can be in the form of a generalised twist. Thus the strong local minimizers here do not have the symmetry one intuitively expects!.

Submitted by Maria Aparecida (cidazen@gmail.com) on 2017-04-17T13:52:33Z No. of bitstreams: 1 Gleyciane Araujo.pdf: 1343747 bytes, checksum: e6395dbd7922b8c1b320e094320e26df (MD5) / Made available in DSpace on 2017-04-17T13:52:33Z (GMT). No. of bitstreams: 1 Gleyciane Araujo.pdf: 1343747 bytes, checksum: e6395dbd7922b8c1b320e094320e26df (MD5) Previous issue date: 2017-02-20 / The objective of this work is to spread the discussion on the subject in the academic environment. Thereunto, the work is based on a literature search that is based on a critical eye on the traditional mathematics teaching, with a focus on trigonometry. Thus, the main concepts will be presented on Mathematical Modeling and the historical context and theoretical foundation of Trigonometry, and present two practical proposals for modeling for the classroom. / O objetivo deste trabalho é contribuir no âmbito acadêmico com o tema, para tanto, o trabalho é fundamentado em uma pesquisa bibliográfica que toma por base um olhar crítico acerca do ensino tradicional da Matemática, com enfoque na Trigonometria. Assim, serão apresentadas as principais concepções sobre a Modelagem Matemática, bem como o contexto histórico e fundamentação teórica da Trigonometria, além de apresentadas duas propostas práticas de modelagem para sala de aula, sob perspectiva dessa metodologia de ensino.

Este trabalho tem por objetivo oferecer uma alternativa para um aprendizado mais significativo, especialmente na introdução à trigonometria. Queremos mostrar aos estudantes que a Matemática é uma ciência em movimento e que vem sendo construída há milênios conforme a necessidade e curiosidade humana. Para alcançar tal objetivo estamos sugerindo uma atividade baseada na metodologia de resolução de problemas e investigação matemática. Acreditamos que apresentando problemas da antiguidade que foram importantes motivadores do desenvolvimento deste ramo da matemática, podemos ao mesmo tempo despertar interesse e atribuir significado à construção dos conceitos a partir do contexto histórico. Para fechar a sequência de atividades, estamos propondo um experimento em que os estudantes tenham que aplicar os conhecimentos adquiridos. Desta forma esperamos mostrar que essas ferramentas podem ser poderosas aliadas no processo de ensino e aprendizagem mostrando ao estudante que ele também pode fazer parte desta história e ajudar a continuar construindo a Matemática. / The aim of this study is to provide an alternative for a more meaningful learning, specially in regard to introduction to trigonometry. We intend to show students that mathematics is a live science, one that is being built over the centuries, according to humans curiosity and needs. In order to achieve such goal, we suggest an activity based on problem solving and mathematics investigation theory. We believe that by introducing ancient problems which were key motivators to the development of this field of mathematics, we may increase students interest as well as help convey meaning to the building of concepts through the historical context. As a wrap up activity, we propose an experiment in which the students have to put their knowledge to practice. By doing so, we hope to demonstrate that these tools can be powerful allies in the learning process, showing students that they can be part of this history and help continue building mathematics.

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Nowadays, technology plays a fundamental role in education, in general, and in mathematics education in particular. The graphing calculator has been an important technological tool in mathematics classrooms since its invention and introduction in 1985 by Casio. As graphing calculators provided so many uses, their contribution to the teaching and learning process has been investigated by many researchers who have shown the use of such technology can have a significant effect on improving mathematics teaching and learning. Investigating the impact of graphing calculators on student learning is important. It is also essential to research teachers’ perspectives on how using graphing calculators in mathematics determines how such use affects their teaching and learning. However, there are few studies on this issue. Therefore, this dissertation study may fill the gap in the literature in terms of examining high school mathematics teachers’ perspectives when they teach a precalculus course with technology integrated in the curriculum materials. In this study, I analyzed eleven teachers’ perspectives about using graphing calculator technology in a precalculus course, titled Functions, Statistics, and Trigonometry (FST). This study was a descriptive intrinsic case study in which I analyzed teachers’ perspectives about how they use graphing calculators in the FST course, specifically about their teaching and students’ learning with available graphing calculator technology. Additionally, I explored teachers' perspectives about the issues they face when using the available technology and for what topics teachers frequently used it. I used mixed methods to examine eleven mathematics teachers’ perspectives about their teaching, students’ learning, and issues that arise when they use graphing calculator technology. In the quantitative part of the study, I created an Index of Teachers’ Initial Perceived Attitude and Experience Level and an Index of Teachers’ Use of Graphing Calculators to measure teachers’ perspectives on technology use at the beginning and end of the school year, respectively. In the qualitative inquiry, I analyzed teachers’ responses to semi-structured interview questions by using thematic analysis. The results of this study showed eight of the eleven mathematics teachers’ students used graphing calculators with Computer Algebra System (CAS) capability loaned by The University of Chicago School Mathematics Project (UCSMP). Five teachers had a high initial perceived attitude and experience level and the other six teachers had a medium level. All teachers reported that helping students learn to use a symbolic manipulator was equally or less important than to use a graphing calculator. The themes (1) Teachers’ use of graphing calculators, (2) Teachers’ opinions about students’ use of graphing calculators, and (3) Teachers’ issues with graphing calculator technology were created to explain teachers’ responses to interview questions related to their graphing calculator perspectives throughout the year. Teachers typically used graphing calculators almost every day for such purposes as exploring mathematics, solving problems, and checking work. Some teachers reported the benefits of using graphing calculators in terms of instruction were focusing on the concepts and showing additional solution approaches. Teachers who wanted their students to be able to do some work without graphing calculators used no calculator tests or questions on which graphing calculators were not allowed as part of their assessment process. Teachers mentioned the need for a manual showing the steps for using graphing calculators with CAS. Teachers’ opinions about students’ use of graphing calculators included that students generally liked them. Teachers reported graphing calculators positively affected students’ learning because students were able to find the answers for problems and have better visualization opportunities. However, teachers reported some meaning was missing and students’ arithmetic skills were negatively affected because of the presence of graphing calculators. Additionally, five teachers indicated their students relied on the graphing calculators too much. The most common issue teachers had relative to graphing calculator technology was the liability issue of the graphing calculators sent by UCSMP for students to loan. Teachers were responsible for those loaned graphing calculators. Additionally, cheating, using features that minimized the mathematics, and not being familiar with the type of graphing calculators loaned from UCSMP were other issues teachers reported. Teachers’ graphing calculator use was demonstrated based on the index of teachers’ use of graphing calculators. Seven teachers were high in terms of their use of graphing calculators at the end of the school year and four teachers had a medium use of graphing calculators. For implications of this study, mathematics teacher educators can use the results to improve professional development programs for teachers. They might create workshops based on teachers’ perspectives and their initial perceived attitude and experience level. Additionally, textbook developers can create more exploration activities with graphing calculators, especially with CAS.

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. Nur im Kapitel 7 kommen einige Methoden der Vektorrechnung zum Einsatz.

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