Submitted by (lucia.rodrigues@ufrpe.br) on 2016-10-21T12:13:53Z No. of bitstreams: 1 Alceu Domingues Alves.pdf: 2706251 bytes, checksum: b08b282eb347f2211c929e290db2c050 (MD5) / Made available in DSpace on 2016-10-21T12:13:53Z (GMT). No. of bitstreams: 1 Alceu Domingues Alves.pdf: 2706251 bytes, checksum: b08b282eb347f2211c929e290db2c050 (MD5) Previous issue date: 2008-08-29 / The present work of research proposes to teach the fractal geometry in high school classroom, with approach in the forms the objects natural and build by the man. Despite of the utility of the fractal geometry for description of the natural objects, this geometry is a subject that has been taught poor in the last series of the high school. The objective of the work is: i. to identify as the students conceive the geometric forms of objects and processes of the nature, without previous knowledge of fractal geometry; the procedure methodological is to carry the students for to apply the Euclidian and fractal in the description of the different shape natural an build by the man. Educational software of dynamic geometry will be used to work with the Euclidean and fractal geometry. The object used will be some students the last year of the high school from a public school of the state of Pernambuco. The theory of the Kelly personal constructs were be used in the analysis of the data. / O presente trabalho propõe introduzir o conceito e propriedades da Geometria Fractal no Ensino Médio, com enfoque numa abordagem baseada nas descrições das formas dos objetos construídos pelo homem e pela natureza. A Geometria Fractal é um tema que tem sido explorado de maneira bastante superficial nas séries finais do ensino médio, apesar da sua extrema utilidade na descrição das formas construídas pela natureza. O principal objetivo do trabalho é investigar como os alunos concebem as formas geométricas dos objetos e processos da natureza. A proposta metodológica para a realização da pesquisa consistiu em utilizar objetos construídos pela natureza e pelo homem e levar os alunos a descreverem suas formas a partir da geometria euclidiana (estudada previamente) e da geometria fractal (discutida numa oficina realizada durante a pesquisa). Softwares educacionais de geometria dinâmica foram usados para trabalhar com os alunos as duas geometrias. A amostra trabalhada foi constituída de alunos de uma turma de terceiro ano do ensino médio de uma escola pública da rede oficial de ensino do Estado de Pernambuco. A teoria dos construtos pessoais de George Kelly foi usada para analisar os dados.

Orientador: Edson Agustini / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Cientifica / Made available in DSpace on 2018-08-17T04:14:16Z (GMT). No. of bitstreams: 1 Staib_Armando_M.pdf: 3998901 bytes, checksum: 260b0ccc34231b7e4c21f0bddc18fdcd (MD5) Previous issue date: 2010 / Resumo: Este trabalho trata do ensino das Geometrias Hiperbólica e Euclidiana utilizando softwares de Geometria Dinâmica, em especial o software NonEuclid. O objetivo deste trabalho é ser uma proposta de atividades em Geometria Hiperbólica com o uso do software. O computador introduz uma diversidade dinâmica ao estudo, proporcionando ao aluno, verificar, conjecturar e investigar. As figuras planas podem ser manipuladas e transformadas de diferentes maneiras mantendo as suas propriedades geométricas. Elaboramos algumas atividades de Geometria Hiperbólica utilizando o software NonEuclid para alunos da graduação em matemática e fizemos também atividades que relacionam ambas as geometrias. Os futuros professores precisam saber mais do que irão lecionar e, em geometria, a utilização dos softwares de Geometria Dinâmica contribuem na evolução gradual da aprendizagem de ambas Geometrias: Hiperbólica e Euclidiana, potencializando as habilidades dos alunos pela visualização, experimentação e compreensão das propriedades geométricas / Abstract: This work deals with the teaching of Euclidian and Hyperbolic Geometry using software in the Dynamic Geometry area, especially the software by the name of NonEuclid". The objective of this work is to be a proposal for activities in Hyperbolic Geometry using this software. The computer introduces a dynamic diversity to the study, allowing students to examine, investigate and conjecture in this area. The plane figures can be manipulated and processed in different ways while maintaining their geometric properties. We can prepare some activities in Hyperbolic Geometry using the software NonEuclid for graduate students in mathematics and related activities that we also both geometries. Future teachers need to know more than material they present to their students, the use of Dynamic Geometry software contributes to the gradual evolution of learning of geometry, both Euclidean and Hyperbolic. This increases the students' abilities to visualize and experiment and therefore their understanding of geometric properties / Mestrado / Mestre em Matemática

Submitted by Jaqueline Silva (jtas29@gmail.com) on 2016-08-31T19:46:48Z No. of bitstreams: 2 Dissertação - Ricardo Anselmo Zanotto - 2015.pdf: 7706833 bytes, checksum: 26c6e884d0e3a03a3daebaa4ab5764a4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Jaqueline Silva (jtas29@gmail.com) on 2016-08-31T19:47:01Z (GMT) No. of bitstreams: 2 Dissertação - Ricardo Anselmo Zanotto - 2015.pdf: 7706833 bytes, checksum: 26c6e884d0e3a03a3daebaa4ab5764a4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2016-08-31T19:47:01Z (GMT). No. of bitstreams: 2 Dissertação - Ricardo Anselmo Zanotto - 2015.pdf: 7706833 bytes, checksum: 26c6e884d0e3a03a3daebaa4ab5764a4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2015-12-12 / Outro / This is a research about a part of the non-Euclidean geometry that has recently been very studied. It was addressed initial themes of the non-Euclidean geometry and it was exposed the studies abut fractals, its history, buildings and main fractals (known as classic fractals). It was also addressed the relation among the school years contents and how to use fractals; as well as some of its applications that have helped a lot of researches to spread and show better results. / Este trabalho é uma pesquisa sobre parte da geometria não euclidiana que há pouco vem sendo muito estudada, os fractais. Abordamos temas iniciais da geometria nãoeuclidiana e no decorrer do trabalho expomos nosso estudo sobre fractais, seu histórico, construções, principais fractais (conhecidos como fractais clássicos). Também abordamos relações entre conteúdos dos anos escolares e como usar fractais nos mesmos; como também algumas de suas aplicações que vem ajudando muitas pesquisas a se difundirem e apresentarem melhores resultados.

The central question of this thesis is: what is mathematics about? The answer arrived at by the thesis is an unsettling and unsatisfying one. By examining two of the most promising contemporary accounts of the nature of mathematics, I conclude that neither is as yet capable of giving us a conclusive answer to our question. The conclusion is arrived at by a combination of historical and conceptual analysis. It begins with the historical fact that, since the middle of the nineteenth century, mathematics has undergone a radical transformation. This transformation occurred in most branches of mathematics, but was perhaps most apparent in geometry. Earlier images of geometry understood it as the science of space. In the wake of the emergence of multiple distinct geometries and the realization that non-Euclidean geometries might lay claim to the description of physical space, the old picture of Euclidean geometry as the sole correct description of physical space was no longer tenable. The first chapter of the dissertation provides an historical account of some of the forces which led to the destabilization of the traditional picture of geometry. The second chapter examines the debate between Gottlob Frege and David Hilbert regarding the nature of geometry and axiomatics, ending with an argument suggesting that Hilbert’s views are ultimately unsatisfying. The third chapter continues to probe the work of Frege and, again, finds his explanations of the nature of mathematics troublingly unsatisfying. The end result of the first three chapters is that the Frege-Hilbert debate leaves us with an impasse: the traditional understanding of mathematics cannot hold, but neither can the two most promising modern accounts. The fourth and final chapter of the thesis investigates mathematical structuralism—a more recent development in the philosophy of mathematics—in order to see whether it can move us beyond the impasse of the Frege-Hilbert debate. Ultimately, it is argued that the contemporary debate between ‘assertoric’ structuralists and ‘algebraic’ structuralists recapitulates a form of the Frege-Hilbert impasse. The ultimate claim of the thesis, then, is that neither of the two most promising contemporary accounts can offer us a satisfying philosophical answer to the question ‘what is mathematics about?’.

The aim of this work is to derive mathematical descriptions of gravitation. Postulating gravitation as a force field, Newton's law of gravitation is heuristically derived by considering linear differential operators invariant under euclidean isometries and by finding the fundamental solution to Helmholtz equation in three dimensions. Thereafter, the theory of differential geometry is introduced, providing a framework for the subsequent review of gravitation as curvature. Lastly, in the light of Einstein's postulates and equivalence principle, Lovelock's proof of uniqueness of Einstein's field equations is presented.

[pt] Consideramos duas formas de complexidade em geometria euclidiana plana.Na primeira, problemas são descritos algebricamente, e a complexidade é cotada essencialmente pelo grau de um polinômio. Como consequência, mostramos que vários resultados gerais e familiares em geometria podem ser demonstrados a partir da simples verificação de dois ou três casos particulares. A segunda forma faz uso da descrição sintática dos teoremas, que permite uma quantificação da complexidade em termos lógicos (número de quantificadores e átomos de uma fórmula). Inspirados por esta última abordagem, são descritos alguns procedimentos de demonstração automática. Alguns grupos habituais de operções em geometria são apresentados com a intenção de simplificar as duas abordagens.Através do estudo de técnicas mais avançadas em matemática trazemos novos pontos de vista a assuntos estudados no ensino médio. / [en] Two forms of complexity in Euclidean plane geometry are considered. In the first one, problems are described algebraically, and the complexity level is measured essentially by the degree of a polynomial. As a consequence, many familiar and general results in geometry can be proved by inspecting two or three special cases. The second form uses the syntactic description of a theorem allowing for a quanti.cation of the complexity in logic terms (number of quantifiers and atoms in the formula). Inspired by this approach, some procedures in mechanized proofs are described. We also present some traditional groups of operations in geometry which simplify the two approaches. The study of more advanced techniques in mathematics sheds new light on standard high school topics.

Made available in DSpace on 2016-04-27T16:57:55Z (GMT). No. of bitstreams: 1 dissertacao_eliane_cabariti.pdf: 4199946 bytes, checksum: 7b4a1cc8c562d90ec0b98ff672a71a8b (MD5) Previous issue date: 2004-09-07 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The main aim of this work is to contribute to the process of teaching and learning of geometry, in particular the non-Euclidean geometries, seeking to support the implementation of proposals associated with the introduction of a hyperbolic model, with the help of a computational tool, in mathematics teacher education courses. To this end, we conducted an experimental study to investigate the possible relations that teacher educators of Euclidean geometry establish when asked to solve situations involving notions of hyperbolic geometry, using the software Cabri-géomètre. The activities developed for the experimental study were inspired by the principals for the development of thought-revealing tasks, described by Lesh et al. (2000). Our analyses were based on two aspects: the dynamics behind movements between the geometrical domains Euclidean geometry and hyperbolic geometry as well as interactions between the spatio-graphical and theoretical fields (Laborde, 1999) and the role of Cabri as a tool for construction, exploration and validation, especially with respect to its dynamic aspects and the different possible drag modes (Olivero, 2002). Through our analysis of teachers' interactions with these situations, we confirmed the importance of the use of the hyperbolic menu of Cabri, fundamental for access to representations of hyperbolic objects favouring the understanding of concepts, properties and relations involved in this domain. The results of this study enabled us to reconsider some choices, leading to the re-design of the activities included in our initial proposal, in particular with reference to the makeup and use of the tools available in Cabri-géomètre. As a consequence, we were able to present a new pedagogic proposal consistent with the original aims / Este trabalho tem como objetivo principal contribuir para o processo de ensino e aprendizagem de Geometria, em particular das Geometrias não Euclidianas, procurando subsidiar a implementação de propostas que visam a introdução de um modelo hiperbólico, com o auxílio de uma ferramenta computacional, em cursos de formação de professores de Matemática. Para nos auxiliar no delineamento dessa proposta, realizamos um estudo experimental que teve como intuito investigar as possíveis relações que professores-formadores de Geometria Euclidiana, estabelecem quando solicitados a resolver situações envolvendo noções de Geometria Hiperbólica, com o auxílio do software Cabri-géomètre. As atividades desenvolvidas para o estudo experimental foram inspiradas nos princípios para o desenvolvimento de tarefas thought revealing descritos por Lesh et al. (2000). Nossas análises foram baseadas em dois aspectos: a dinâmica das trocas entre os domínios geométricos geometria Euclidiana e Hiperbólica além das interações entre os campos espaço-gráfico e teórico (Laborde, 1999) e o papel do Cabri como ferramenta de construção, exploração e verificação, especialmente relacionadas ao seu aspecto dinâmico, nos diferentes modos de arrastar (Olivero, 2002). Por meio das interações dos professores nessas situações, confirmamos a importância do uso da barra do menu hiperbólico do Cabri, fundamental para o acesso às representações de objetos hiperbólicos favorecendo a compreensão de conceitos, propriedades e relações envolvidos nesse domínio. Os resultados desse estudo permitiram-nos reconsiderar algumas escolhas, levando-nos à reelaboração das atividades de nossa proposta inicial, em particular no que se refere à constituição e utilização das ferramentas disponibilizadas no Cabri-géomètre. Consolidamos assim, uma nova proposta pedagógica com os mesmos objetivos iniciais

Tento projekt se věnuje problematice zobrazení velkých hierarchických struktur, zejména možnostem vizualizace stromových grafů. Cílem je implementace hyperbolického prohlížeče ve webovém prostředí, který využívá potenciálu neeukleidovské geometrie k promítnutí stromu na hyperbolickou rovinu. Velký důraz je kladen na uživatelsky přívětivou manipulaci se zobrazovaným modelem a snadnou orientaci.

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)

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)