Orientador: Marcelo de Carvalho Borba / Banca: Siobhan Victoria Healy / Banca: Marcus Vinicius Maltempi / Resumo: Neste trabalho investigo como se dá a produção matemática de alunos-professores em um curso de extensão universitária à distância sobre "Tendências em Educação Matemática". As interações entre os participantes aconteceram, em geral, por meio de encontros semanais síncronos e a distância, nos quais eram discutidas questões relacionadas a algumas das tendências em Educação Matemática e sobre o desenvolvimento de atividades de geometria euclidiana espacial, sendo que este último tema consiste no foco de estudo desta pesquisa. Para as construções geométricas sugeri o uso do software gratuito Wingeom, contudo, outros recursos como materiais manipulativos, bem como diferentes estratégias de resolução foram observadas. Essa dinâmica evidenciou a coordenação de diferentes mídias durante o processo investigativo, que exigiu dos participantes grande envolvimento e empatia para melhor compreender a explicação apresentada durante a discussão no chat. A sala de batepapo do TelEduc, ambiente utilizado, apresentou algumas limitações com relação à troca do fazer matemática, contudo, isso não impediu que a discussão acontecesse e que a produção matemática se consolidasse de um modo muito particular. Analisei os dados baseando-me no construto teórico seres-humanos-com-mídias de Borba e Villarreal (2005) e nas idéias de Lévy (1993, 1999, 2003) no que se refere ao pensamento coletivo e à inteligência coletiva. Os resultados obtidos indicaram que as mídias (lápis e papel, materiais manipulativos, Wingeom, Internet e suas diferentes interfaces) em um ambiente virtual de aprendizagem, condicionaram a forma que os participantes discutiram as conjecturas formuladas durante as construções geométricas e transformaram a produção matemática. / Abstract: In this study, I investigate how teacher-students produce mathematics in a university extension distance course entitled “Trends in Mathematics Education”. The interactions between participants generally occurred in weekly synchronous on-line sessions in which issues were discussed related to some of the current trends in mathematics as well as development of spatial Euclidean geometry, the latter being the focus of this study. I suggested the use of the free software Wingeom for the geometrical constructions, but other resources, such as manipulatives, as well as different strategies for problem solving were observed. This dynamic showed evidence of the coordination of different media during the inquiry process, which demanded considerable involvement and empathy on the part of the participants to better understand the explanation presented during the on-line chat discussions. The chat room of TelEuc, the environment used, presented some limitations with respect to the exchange of mathematical activity; nevertheless, this did not impede the discussion nor prevent the mathematical production from consolidating in a very specific way. I based the data analysis on Borba and Villarreal’s (2005) theoretical construct humans-with-media and the ideas of Lévy (1993, 1999, 2003) regarding collective thinking and collective intelligence. The results suggest that the different media (paper-and-pencil, manipulatives, Wingeom, and the Internet with its various interfaces) in a virtual learning environment conditioned the way the participants discussed the conjectures formulated during the geometric constructions and transformed the production of mathematics. / Mestre

Submitted by Sandra Pereira (srpereira@ufrrj.br) on 2017-01-25T10:55:06Z No. of bitstreams: 1 2015 - Cassio Marins Ferreira.pdf: 2336646 bytes, checksum: ec3b1734c73a206999579144e063ab1f (MD5) / Made available in DSpace on 2017-01-25T10:55:06Z (GMT). No. of bitstreams: 1 2015 - Cassio Marins Ferreira.pdf: 2336646 bytes, checksum: ec3b1734c73a206999579144e063ab1f (MD5) Previous issue date: 2015-08-28 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior - CAPES / This work intends to present the Dynamic Geometry software GeoGebra to elementary school students. It uses the GeoGebra as a tool for building a step by step procedure to convince students of the veracity of results that have simple statements, but not trivial proofs. The following theorems were worked: Morley's Theorem, Hipparchus Theorem, Stewart's Theorem, Menelaus Theorem and 9-Point Circle Theorem. Demonstrations are held strictly in the traditional way, and in parallel it is used the GeoGebra software, thus giving a playful view of the statements. It is understood that this approach will be more attractive, enabling a better understanding of the theorems by students. The work culminates with the application of a motivational test to a class of basic education after a class in which they worked the 9-Point Circle using GeoGebra / Este trabalho tem a inten??o de apresentar o softwarede Geometria Din?micaGeoGebra ao aluno do Ensino B?sico. Utiliza-se o GeoGebra como ferramenta para a constru??o din?mica de um passo-a-passo para convencer os alunos da veracidade de resultados que possuem enunciados simples, mas demonstra??es n?o triviais. Foram trabalhados os seguintes teoremas: Teorema de Morley, Teorema de Hiparco, Teorema de Stewart, Teorema de Menelau e Teorema do C?rculo de 9 Pontos. As demonstra??es s?o realizadas rigorosamente, da forma tradicional, e em paralelo ? feito o uso do softwareGeoGebra, dando assim uma vis?o l?dica das demonstra??es. Entende-se que esta abordagem ser? mais atrativa, possibilitando uma melhor compreens?o dos teoremas pelos alunos. O trabalho culmina com a aplica??o de um teste motivacional a uma turma do Ensino B?sico ap?s uma aula em que se trabalhou o Teorema do C?rculo dos 9 Pontos utilizando-se o GeoGebra

Made available in DSpace on 2016-06-02T20:29:23Z (GMT). No. of bitstreams: 1 5405.pdf: 3455845 bytes, checksum: 749a1a7ca8b7d3b9399b8de99a655175 (MD5) Previous issue date: 2013-08-16 / Financiadora de Estudos e Projetos / This project's main goal is propose the application of a Teaching Sequence about how finding the notable points of a triangle, in particular, the circumcenter. Introducing in the sequence geometric objects primaries of Euclidean geometry with the axioms and the theorems necessary to construct the circumcenter. The geometric objects will be built through educational software Geogebra. The teaching of geometry, often present to the end of the textbook, is faced with the difficulty of the students to manipulate instruments as ruler, protractor and compass. On the other hand, the fascination of students by computers facilitates the use of geometric softwares. The sum of these factors, have inspired this project. Based on the methodology of the didactic engineering I intend to show the construction steps of mathematical knowing from the student's research experimenting, visualizing, conjecturing, generalizing even demonstrating the mathematical basement in this context. / O presente trabalho objetiva relatar a aplicação de uma Sequência Didática sobre como encontrar os pontos notáveis de um triângulo, em particular o circuncentro, apresento na sequência entes geométricos primários da geometria euclidiana plana junto aos axiomas e teoremas necessários para a construção do circuncentro. Os objetos geométricos serão construídos através do software educacional Geogebra. O ensino da geometria, muitas vezes relegado ao final do livro didático, se depara com a dificuldade que os alunos têm em manipular instrumentos como régua, transferidor e compasso. Por outro lado, o fascínio dos estudantes por computadores facilita a utilização de softwares da Geometria dinâmica. A soma desses fatores, inspirou esse trabalho. Baseado na metodologia da engenharia didática, pretendo mostrar etapas da construção do conhecimento matemático a partir da investigação do aluno, experimentando, visualizando, conjecturando, generalizando e até mesmo demonstrando todo embasamento matemático envolvido nesse contexto.

abstract: Continuous monitoring of sensor data from smart phones to identify human activities and gestures, puts a heavy load on the smart phone's power consumption. In this research study, the non-Euclidean geometry of the rich sensor data obtained from the user's smart phone is utilized to perform compressive analysis and efficient classification of human activities by employing machine learning techniques. We are interested in the generalization of classical tools for signal approximation to newer spaces, such as rotation data, which is best studied in a non-Euclidean setting, and its application to activity analysis. Attributing to the non-linear nature of the rotation data space, which involve a heavy overload on the smart phone's processor and memory as opposed to feature extraction on the Euclidean space, indexing and compaction of the acquired sensor data is performed prior to feature extraction, to reduce CPU overhead and thereby increase the lifetime of the battery with a little loss in recognition accuracy of the activities. The sensor data represented as unit quaternions, is a more intrinsic representation of the orientation of smart phone compared to Euler angles (which suffers from Gimbal lock problem) or the computationally intensive rotation matrices. Classification algorithms are employed to classify these manifold sequences in the non-Euclidean space. By performing customized indexing (using K-means algorithm) of the evolved manifold sequences before feature extraction, considerable energy savings is achieved in terms of smart phone's battery life. / Dissertation/Thesis / M.S. Electrical Engineering 2014

Este trabalho investiga as possibilidades de ensino de ideias fundamentais das geometrias não-euclidianas sob a perspectiva da Divulgação Científica e identifica as principais características presentes nas pesquisas que relatam experiências de ensino destas geometrias. nas pesquisas que relatam experiências de ensino destas geometrias. Bibliográfica, a pesquisa fundamenta teoricamente a Divulgação Científica, a educação nãoformal e o ensino das geometrias não-euclidianas. Em relação às geometrias, enfatizou-se o processo histórico de seu surgimento, em especial as tentativas de prova do quinto postulado de Euclides, pois esse processo evidencia uma quebra de paradigma no conhecimento matemático, incluindo a concepção de verdade matemática. Sobre o ensino das geometrias, debateu-se sua inserção no currículo da educação básica e o crescente número de menções ao tema em orientações educacionais oficiais, tanto no Brasil como no exterior. Procurou-se compreender os objetivos dos educadores que se propõem a ensinar as geometrias nãoeuclidianas e percebeu-se que tais objetivos não se vinculam unicamente ao pressuposto de que a aprendizagem da geometria euclidiana se torna significativa quando se proporciona o contato com as não-euclidianas. Foi feito um mapeamento de algumas pesquisas que apresentam experiências de ensino das geometrias e elencaram-se seus êxitos. A análise das pesquisas que relatam estudos de caso esteve focada nos recursos normalmente utilizados, nos principais pressupostos e nos públicos escolhidos. Nessas pesquisas, percebeu-se forte presença da geometria esférica e da geometria hiperbólica, abordadas principalmente por intermédio de materiais concretos e de software de geometria dinâmica, respectivamente. Ficou evidenciada a possibilidade de ensino de ideias fundamentais das geometrias nãoeuclidianas para diferentes públicos. / This paper investigates the teaching possibilities of fundamental ideas of the non- Euclidean geometries under the perspective of scientific popularization and identifies the main characteristics in the teaching of these geometries. This bibliographical research justifies scientific theory, non-formal education, and the teaching of non-Euclidean geometries. In relation to various geometries, we have emphasized the historical process of its emergence, in particular attempts to prove Euclid\'s fifth postulate since this process shows a paradigm in mathematical knowledge, including the concept of mathematical truth. On the teaching of geometry, we have discussed its inclusion in the curriculum of basic education and the growing number of references to the subject in official educational guidelines, both in Brazil and abroad. We have tried to understand the goals of educators who purport to teach non- Euclidean geometries and realized that these goals do not connect solely to the assumption that learning of Euclidean geometry becomes significant when it provides the contact with non-Euclidean geometries. We have mapped some of the studies with teaching experiences of geometries and listed their successes. The analysis of the research that reports case studies focused on the resources normally used, on the main assumptions, and on chosen audiences. In the analyzed research, we have encountered a strong presence of Spherical Geometry and Hyperbolic Geometry, mainly approached via concrete materials and dynamic geometry pieces of software, respectively. The teaching possibility of basic principles of the non- Euclidean geometries for different publics became evident.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-02-22T15:29:17Z No. of bitstreams: 1 marcelocunhafigueiredo.pdf: 1448320 bytes, checksum: d5d065ce34898025ffe848fe7561fe67 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-02-26T14:09:07Z (GMT) No. of bitstreams: 1 marcelocunhafigueiredo.pdf: 1448320 bytes, checksum: d5d065ce34898025ffe848fe7561fe67 (MD5) / Made available in DSpace on 2016-02-26T14:09:07Z (GMT). No. of bitstreams: 1 marcelocunhafigueiredo.pdf: 1448320 bytes, checksum: d5d065ce34898025ffe848fe7561fe67 (MD5) Previous issue date: 2014-02-27 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O presente trabalho tem por finalidade mostrar uma metodologia de ensino dos números reais com base em fundamentos da Geometria Euclidiana. A régua e o compasso serão instrumentos de grande importância na construção dos conjuntos numéricos. Partindo das imagens geométricas dos números naturais e das operações entre seus elementos, iremos, gradativamente, construindo o conjunto dos números inteiros e dos racionais. Provaremos a existência de números que não são racionais e uma característica desses números que os livros didáticos, em sua maioria, não abordam: a questão da densidade dos conjuntos dos números racionais e irracionais no conjunto dos reais. A geometria euclidiana como suporte nos números reais facilita o entendimento do aluno e traz dinâmica nas operações entre esses números. Apresentamos também uma possibilidade de continuação da proposta de trabalho. / This paper aims to show a teaching methodology of real numbers on the grounds of Euclidean geometry. The ruler and compass are instruments of great importance in the construction of numerical sets. Based on the geometric images of the natural numbers and operations between its elements, we will gradually building the set of integers and rational numbers. We prove the existence of numbers that are not rational and a propertie of those numbers that textbooks mostly do not address: the question of density of the sets of rational and irrational in the set of real numbers. Euclidean geometry as real numbers in support facilitates student understanding and produces dynamic operations between these numbers. We also present a possible continuation of the proposed work.

The aim of this thesis are problematics about LDPC codes. There are described metods to create parity check matrix, where are important structured metods using finite geometry: Euclidean geometry and projectice geometry. Next area in this thesis is decoding LDPC codes. There are presented four metods: Hard-Decision algorithm, Bit-Flipping algorithm, The Sum-Product algorithm and Log Likelihood algorithm, where is mainly focused on iterative decoding methods. Practical output of this work is program LDPC codes created in environment Matlab. The program is divided to two parts -- Practise LDPC codes and Simulation LDPC codes. The result reached by program Simulation LDPC codes is used to create a comparison of creating and decoding methods LDPC codes. For comparison of decoding methods LDPC codes were used BER characteristics and time dependence each method on various parameters LDPC code (number of iteration or size of parity matrix).

This study sought to (a) investigate the effect of Van Hiele theory-based instruction on Grade 11 students’ geometric proofs learning achievement, (b) explore students’ views on their geometry learning experiences, and (c) develop a framework for better teaching and learning of Grade 11 Euclidean geometry theorems and non-routine geometric proofs. The study is based on Van Hiele’s theory of geometric thinking. The research involved a convenience sample of 186 Grade 11 students from four matched secondary schools in the Capricorn district of Limpopo province, South Africa. The study employed a sequential explanatory mixed-methods design, which combined quantitative and qualitative data collection methods. In the quantitative phase, a non-equivalent groups quasi-experiment was conducted. A Geometry Proof Test was used to assess students’ geometric proof construction abilities before and after the teaching experiment. Data analysis using non-parametric analysis of covariance revealed that students from the experimental group of schools performed significantly better than their counterparts from control group schools. In the qualitative phase, data were collected using focus group discussions and students’ diary records. Results revealed that the experimental group students had positive views on their geometry learning experiences, whereas students from the control group of schools expressed negative views towards the teaching of Euclidean geometry and geometric proofs in their mathematics classes. Based on the quantitative and qualitative data findings, it was concluded that in addition to organizing instruction according to the Van Hiele theory, teachers should listen to students’ voices and adjust their pedagogical practices to meet the expectations of a diverse group of students in the mathematics class. A framework for better teaching and learning of Grade 11 Euclidean geometry theorems and non-routine geometric proofs was thus developed, integrating students’ views and Van Hiele theory-based instruction. The study recommends that teachers should adopt the modified Van Hiele theory-based framework to enhance students’ mastery of non-routine geometric proofs in secondary schools.

En recherchant la loi de force centripète inscrite dans les Principes Mathématiques de la Philosophie Naturelle, Newton donna au temps un statut de grandeur privilégiée de la philosophie naturelle. Cependant, celui-ci apparaît de façon ambiguë, tantôt grandeur discrète, tantôt grandeur continue. Sa manipulation mathématique, qui repose essentiellement sur la Méthode des premières et dernières raison et sur la loi des aires, laisse, en outre, apparaître un temps de nature géométrique. Confronté, dans la proposition X du livre II, à la résolution du mouvement d'un mobile qui éprouve une résistance en raison du carré de sa vitesse, Newton ne parvient pas à résoudre cette proposition au moyen de la géométrie. Il est contraint de reprendre son raisonnement et de recourir à une méthode algébrique pour énoncer de manière juste, dans l'édition de 1713, la solution de cette proposition, dans laquelle le temps apparaît alors sous une forme algébrisée, représenté par une lettre. Ainsi, d'un temps géométrisé, figuré par un élément d'espace dans l'édition de 1687, Newton en fit un être per se représenté par une lettre dans la proposition X de l'édition de 1713. Cependant, c'est à Varignon, qui aborda les propositions des Principia de Newton à l'aide du calcul différentiel, que l'on doit la fin de la mathématisation et la finalisation du concept de temps mathématique / By looking for the law of centripetal force registered in the Mathematical Principles of the Natural Philosophy, Newton gave to time a status of privileged magnitude of natural philosophy. However, this one appears in a ambiguous way, sometimes discrete magnitude, sometimes continuous magnitude. Its mathematical manipulation, which rests essentially on the Method of first and last ratios and on the law of areas, lets appear a time of geometrical nature. Confronted, in the proposal x of the book II, with the resolution of the movement of a mobile which tests a resistance which is proportional in the square of its speed, Newton does not succeed in solving this proposal by means of the geometry. It is forced to resume its reasoning and to resort to an algebraic method in order to express in a just way the solution of this proposal, in which the time appears then under an algébraic shape, represented by a letter. So, from a geometrical time, represented by an element of space in the edition of 1687, Newton made an entity per se represented by a letter in proposal x of the 1713 edition. But it is to Varignon, who approached the proposals of the Principia by means of the differential calculus, that we owe the end of the "mathematization" and the finalization of the concept of mathematical time

PORTELA, Antonio Edilson Cardoso. Noções de geometria projetiva. 2017. 58 f. Dissertação (Mestrado Profissional em Matemática em Rede Nacional) - Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017. / Submitted by Jessyca Silva (jessyca@mat.ufc.br) on 2017-09-06T17:17:00Z No. of bitstreams: 1 2017_dis_aecportela.pdf: 1065928 bytes, checksum: 468c05aa35745f3fd2761f13aa26eff1 (MD5) / Rejected by Rocilda Sales (rocilda@ufc.br), reason: Boa tarde, Estou devolvendo a Dissertação de ANTONIO EDILSON CARDOSO PORTELA, para que o mesmo realize algumas correções na formatação do trabalho. 1- SUMÁRIO ( A formatação do sumário está incorreta, primeiro, retire o último ponto final que aparece após a numeração dos capítulos e seções (Ex.: 3.1. Axioma....; deve ser corrigido para: 3.1 Axioma.....), o alinhamento dos títulos deve seguir o modelo abaixo 1 INTRODUÇÃO.....................00 2 O ESPAÇO...........................00 3 GEOMETRIA........................00 3.1 Axiomas...............................00 REFERÊNCIAS...................00 (OBS.: não altere a formatação do negrito, pois já estava correta) 2- TITULO DOS CAPÍTULOS E SEÇÕES ( retire o ponto final que aparece após o último dígito da numeração dos capítulos e seções, seguindo o modelo do sumário. Retire o recuo de parágrafo dos títulos das seções. Ex.: 3.1 Axioma.......) 3- REFERÊNCIAS ( substitua o termo REFERÊNCIAS BIBLIOGRÁFICAS apenas por REFERÊNCIAS, com fonte n 12, negrito e centralizado. Retire a numeração progressiva que aparece nos itens da referência. Atenciosamente, on 2017-09-06T17:56:50Z (GMT) / Submitted by Jessyca Silva (jessyca@mat.ufc.br) on 2017-09-11T14:48:40Z No. of bitstreams: 1 2017_dis_aecportela.pdf: 944228 bytes, checksum: 3ab4691817df04ba5d7818fd02e5095f (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2017-09-11T15:30:32Z (GMT) No. of bitstreams: 1 2017_dis_aecportela.pdf: 944228 bytes, checksum: 3ab4691817df04ba5d7818fd02e5095f (MD5) / Made available in DSpace on 2017-09-11T15:30:32Z (GMT). No. of bitstreams: 1 2017_dis_aecportela.pdf: 944228 bytes, checksum: 3ab4691817df04ba5d7818fd02e5095f (MD5) Previous issue date: 2017 / In this work, initially, some results of Linear Algebra are presented, in particular the study of the Vector Space R^n, which becomes, together with Analytical Geometry, the language used in the chapters that follow. We present a study from an axiomatic point of view, from the perspectives of Hilbert's axioms and we elaborate models of planes for the Euclidean, Elliptic and Projective Geometries. The validity of the Incidence and Order axioms for Euclidean Geometry is verified. In R^3, an approach is made to the study of the plane and the unitary sphere, highlighting the elliptical line obtained by the intersection of these sets, thus making an approach to the Elliptic Geometry. With the concepts and definitions studied in the Vector Space R^n, Three-dimensional Space and in the Euclidean and Elliptic Geometries we will approach the study of Projective Geometry, demonstrating propositions and verifying its axioms. / Neste trabalho, inicialmente, apresenta-se alguns resultados da Álgebra Linear, em especial o estudo do Espaço Vetorial R^n, que passa a ser, juntamente com a Geometria Analítica, a linguagem empregada nos capítulos que se seguem. Apresentamos um estudo de um ponto de vista axiomático, sob a ótica dos axiomas de Hilbert e elaboramos modelos de planos para as Geometrias Euclidiana, Elíptica e Projetiva. É verificada a validade dos axiomas de Incidência e Ordem para a Geometria Euclidiana. No R^3, é feita uma abordagem do estudo de plano e da esfera unitária, destacando a reta elíptica obtida pela interseção destes conjuntos, passando assim a fazer uma abordagem da Geometria Elíptica. Com os conceitos e definições estudadas no Espaço Vetorial R^n, Espaço tridimensional e nas Geometrias Euclidiana e Elíptica, abordaremos o estudo da Geometria Projetiva, demonstrando proposições e verificando os seus axiomas.