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No. of bitstreams: 1 dayselanepimentalopesrezende.pdf: 2005394 bytes, checksum: 680064aadc3bc3261286f3e1eb1492a7 (MD5) Previous issue date: 2017-03-14 / O ensino da Geometria por muitos anos foi deixado em segundo plano e isso trouxe consequências graves que até hoje permeiam as salas de aulas de nossas escolas. Nesse sentido, percebe-se a necessidade da utilização de diferentes metodologias para o ensino da geometria. Diante de tantas inquietações sobre a forma como os conceitos geométricos são abordados em sala de aula, a presente pesquisa tem como foco responder as seguintes indagações: Quais contribuições para o processo de aprendizagem de estudantes do ensino fundamental podem ocorrer a partir do ensino de polígonos com tarefas exploratório-investigativas e com o uso de material didático manipulativo? Quais as contribuições que um trabalho com tarefas exploratório-investigativas com a utilização de material didático manipulável traz para a mudança da prática docente da professora-pesquisadora? Procurando responder essas questões, o estudo tem como objetivo geral ampliar a compreensão acerca de polígonos, trazendo elementos que possam contribuir para a elaboração de atividades que estimulem o desenvolvimento do pensamento crítico, raciocínio lógico e a habilidade argumentativa dos alunos. Para tal, procuramos identificar e analisar de que forma as aulas de cunho exploratório-investigativas, mediadas pelo uso de material didático manipulável, do trabalho em grupo e a intervenção do professor podem favorecer a aquisição do conhecimento geométrico produzido pelos alunos. Também procuramos descrever e refletir sobre as mudanças ocorridas na prática pedagógica da professora-investigadora para a formação e produção do conhecimento. Nesse sentido, a pesquisa foi de cunho qualitativo e realizada com alunos do oitavo ano do Ensino Fundamental de uma escola do interior do Estado do Rio de Janeiro. A coleta e análise de dados foram realizadas a partir do desenvolvimento de uma sequência didática que abordou conceitos relativos a polígonos, utilizando tarefas exploratório-investigativas e materiais didáticos manipuláveis. Os resultados desta pesquisa apontam para a importância das aulas de cunho investigativo para a aprendizagem de polígonos, destacando que esse tipo de tarefa oportuniza a participação individual e coletiva, tornando o aluno mais autônomo e facilitando o desenvolvimento do pensamento geométrico. Também destaca que investigar a própria prática possibilita ao professor refletir e rever seus saberes, propiciando assim, a produção de novos saberes para si e para outros professores de matemática. Por outro lado, o trabalho com investigações matemáticas propiciou a mudança da perspectiva da sala de aula, pois tanto o professor quanto o aluno têm uma alternância de papéis, no qual um novo modelo de comunicação foi estabelecido, permitindo assim, que ambos adquirissem uma postura mais livre e autônoma, permeada por indagações e troca de saberes. / The teaching of geometry for many years was left in the background and it brought serious consequences that pervade classrooms of our schools. In this regard, the need for the use of different methodologies for the teaching of geometry. Faced with so many concerns about how geometric concepts are covered in the classroom, the present research focuses on answering the following questions: What contributions to the learning process of students of elementary school learning of polygons can occur with exploratory-investigative tasks with the use of manipulative courseware? What are the contributions that a job with exploratory-investigative tasks with the use of courseware manipulative brings to the change of the teaching practice of teacherresearcher? Seeking to answer these questions, the study aims to extend the general understanding about polygons, bringing elements that may contribute to the development of activities to stimulate the development of critical thinking, logical reasoning and argumentative ability of students. For this, we seek to identify and analyze how exploratory-oriented classes, mediated by the use of investigative teaching material work group handle and the intervention of the teacher can encourage the acquisition of geometric knowledge produced by the students. Also we seek to describe and reflect on the changes in pedagogical practice of the teacher-researcher for the formation and production of knowledge. In this sense, the research was qualitative measures and held with students in the eighth grade of elementary school to a school in the State of Rio de Janeiro. The data collection and analysis were performed from the development of a didactic sequence which addressed concepts related to polygons, using exploratory-investigative tasks courseware manipulative. The results of this research points to the importance of investigative nature classes for learning of polygons, noting that this type of task it gives individual and collective participation, making the student more and facilitating the development of geometric thinking. Also highlights that investigate the practice allows the teacher to reflect and review their knowledge, thus, the production of new knowledge for themselves and other math teachers. On the other hand, working with mathematical investigations led to the change from the perspective of the classroom as the teacher as student have an alternating roles, in which a new model of communication was established, allowing both to acquire a more free and autonomous, permeated by questions and exchange of knowledge.

In this thesis we consider extensions and generalizations of some classical problems in Combinatorial Geometry. Our work is an offshoot of four classical problems in Combinatorial Geometry. A fundamental assumption in these problems is that the underlying point set is R2. Two fundamental themes entwining the problems considered in this thesis are: “What happens if we assume that the underlying point set is finite?”, “What happens if we assume that the underlying point set has a special structure?”. Let P ⊂ R2 be a finite set of points in general position. It is reasonable to expect that if |P| is large then certain ‘patterns’ in P always occur. One of the first results was the Erd˝os-Szekeres Theorem which showed that there exists a f(n) such that if |P| ≥ f(n) then there exists a convex subset S ⊆ P, |S| = n i.e., a subset which is a convex polygon of size n. A considerable number of such results have been found since. Avis et al. in 2001 posed the following question which we call the n-interior point problem: Is there a finite integer g(n) for every n, such that, every point set P with g(n) interior points has a convex subset S ⊆ P with n interior points. i.e. a subset which is a convex polygon that contains exactly n interior points. They showed that g(1) = 1, g(2) = 4. While it is known that g(3) = 9, it is not known whether g(n) exists for n ≥ 4. In the first part of this thesis, we give a positive solution to the n-interior point problem for point sets with bounded number of convex layers. We say a family of geometric objects C in Rd has the (l, k)-property if every subfamily C′ ⊆ C of cardinality at most l is k-piercable. Danzer and Gr¨unbaum posed the following fundamental question which can be considered as a generalised version of Helly’s theorem: For every positive integer k, does there exist a finite g(k, d) such that if any family of convex objects C in Rd has the (g(k, d), k)-property, then C is k-piercable? Very few results(mostly negative) are known. Inspired by the original question of Danzer and Gr¨unbaum we consider their question in an abstract set theoretic setting. Let U be a set (possibly infinite). Let C be a family of subsets of U with the property that if C1, . . . ,Cp+1 ∈ C are p + 1 distinct subsets, then |C1 ∩ · · · ∩Cp+1| ≤ l. In the second part of this thesis, we show in this setting, the first general positive results for the Danzer Grunbaum problem. As an extension, we show polynomial sized kernels for hitting set and covering problems in our setting. In the third part of this thesis, we broadly look at hitting and covering questions with respect to points and families of geometric objects in Rd. Let P be a subset of points(possibly infinite) in Rd and C be a collection of subsets of P induced by objects of a given family. For the system (P, C), let νh be the packing number and νc the dual packing number. We consider the problem of bounding the transversal number τ h and the dual transversal number τ c in terms of νh and νc respectively. These problems has been well studied in the case when P = R2. We systematically look at the case when P is finite, showing bounds for intervals, halfspaces, orthants, unit squares, skylines, rectangles, halfspaces in R3 and pseudo disks. We show bounds for rectangles when P = R2. Given a point set P ⊆ Rd, a family of objects C and a real number 0 < ǫ < 1, the strong epsilon net problem is to find a minimum sized subset Q ⊆ P such that any object C ∈ C with the property that |P ∩C| ≥ ǫn is hit by Q. It is customary to express the bound on the size of the set Q in terms of ǫ. Let G be a uniform √n × √n grid. It is an intriguing question as to whether we get significantly better bounds for ǫ-nets if we restrict the underlying point set to be the grid G. In the last part of this thesis we consider the strong epsilon net problem for families of geometric objects like lines and generalized parallelograms, when the underlying point set is the grid G. We also introduce the problem of finding ǫ-nets for arithmetic progressions and give some preliminary bounds.

The diploma thesis deals with the topic of concept building process in 2D geometry. The topic is aimed at preschoolers and school-aged children. The theoretical part describes the stages of human development and the cognitive stages of development. Then there are characterized the term concept, the concept building process and the stages of the language development in mathematics. The following part describes two theories about the building knowledge in Maths and the levels of thinking in geometry according to the van Hiele model. The last one chapter of this part describes the geometry curriculum within the primary school education. The method of qualitative research - participated observation - was used for the practical part. This part describes the research that consists of seven experiments. The aim of the experiments was to observe the development of children's and pupils' ideas about 2D geometric shapes. Many activities were prepared for the research. On the basis of the activities reflection the activities were changed or completed. The experiments are described by means of the phenomena that appeared. The phenomena are important for the describing the concept building process of 2D geometric shapes - a square, a circle, a rectangle, a triangle. The information from research is compared...

Title: Educational environment additive polygons and polyhedrons Summary: The main intention of the work is to introduce a new mathematical educational environment that would be especially attractive for pupils in the grades 6. -9., but also in the secondary schools, universities or primary schools The work consists of six parts. In the introduction are mentioned the reasons that led me to choose this topic. The second chapter describes the theoretical basis of the work. The third section describes in detail the environment of additive polygons, both its aspects - mathematical and educational one. Analogously, as it is in the third chapter, is processed the fourth chapter that is dedicated to the environment of additive polyhedrons. The fifth chapter is devoted to the linking of the environment of additive polygons and polyhedrons into the linear algebra. In conclusion are provided further opportunities of work with this environment.

This Master's Project deals with creation of virtual Splint (Orthopedic plate) model and with placement of virtual Splint model onto 3D Bone model. It handles with creating of interpolation curves in 3D (especially Subdivision method) and motion along a curve. This work is only in Czech.

[pt] O objetivo deste trabalho é trazer à tona conceitos importantes da geometria no plano euclidiano sob o título de construções geométricas, cada vez mais esquecidos nos currículos escolares brasileiros. Nossa primeira ideia é mostrar a dificuldade que professores do ensino médio poderão encontrar ao tentar descobrir quais conceitos validam suas práticas já que os argumentos que validam a possibilidade ou a impossibilidade de algumas construções geométricas residem numa álgebra abstrata de difícil compreensão e domínio por parte dos professores, sobretudo aqueles que não cursaram disciplinas mais avançadas em matemática. Vamos comentar sobre os principais problemas da antiguidade que motivaram os matemáticos às descobertas de novas propriedades, apresentar tais construções geométricas e apresentar uma descrição algébrica das construções geométricas. A ideia é que através da álgebra abstrata podemos obter argumentos que validem a possibilidade e impossibilidade de tais construções e assim aumentar a cultura matemática do professor do ensino médio e não transformá-lo num expert no assunto. / [en] The main purpose of this work is to rescue the important concepts in geometric constructions. Concepts that are being progressively forgotten by Brazilian curriculums in schools. First, we want to present the difficulties that high school teachers might face when they will try to formalize concepts like the possibility or not to construct some figures in the Euclidean plane, especially those who have not studied advanced math courses at undergraduation. We comment on the main problems of antiquity that led mathematicians to new discoveries properties, we present geometric constructions as well as an algebraic description of these geometric constructions. The idea is that through abstract algebra we can present arguments about the possibility or impossibility of such constructions. In this work, we will comment that abstract algebra will help teachers to validate some arguments that involves the possibility or not to construct some figures as well as to enlarge high schools teachers culture, not trying to make them experts in the subject.

[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.

No description available.

Étudier l'émergence de savoirs lors de situations didactiques non finalisées par un savoir préfabriqué et pré-pensé nécessite un bouleversement des points de vue, aussi bien épistémologique que didactique. C'est pourquoi, pour l'étude de situations didactiques pour lesquelles le problème est l'essence, nous développons une nouvelle approche historique et repensons des outils pour les analyses didactiques. Nous proposons alors, pour un problème particulier, l'exploration des pavages archimédiens du plan, une enquête historique centrée sur l'activité du savant cherchant et sur l'influence de la relation aux objets dans la recherche. De ce point de vue, l'étude des travaux de Johannes Kepler à la recherche d'une harmonie du monde est particulièrement instructive. Nous proposons également, pour l'analyse des savoirs émergents en situation didactique, une utilisation d'outils liés à la sémiotique qui permet de mettre en évidence la dynamique de l'évolution des objets mathématiques. Nous pouvons finalement conclure quant à la possibilité de construire et mettre en œuvre des ≪ Situations Didactiques de Recherche de Problème ≫ assurant l'engagement du sujet dans la recherche, l'émergence et le développement d'objets mathématiques, la genèse de savoirs. L'étude nous conforte dans la nécessité d'une approche pragmatique des situations et la pertinence d'un regard différent sur les savoirs à l'école / The study of the emergence of knowledges in teaching situations not finalized by a prefabricated and pre-thought knowledge requires an upheaval of point of view, epistemological as well as didactic. For the study of learning situations in which the problem is the essence, we develop a new historical approach and we rethink the tools for didactic analyzes. We propose, then, for a particular problem, exploration of Archimedean tilings of the plane, a historical inquiry centered on the activity of the scientist in the process of research and on the influence of the relationship with objects. From this perspective, the study of Johannes Kepler’s work in search of a world harmony is particularly instructive. We also propose, for the analysis of the emerging knowledge in teaching situations, to use tools related to semiotics, which allows to highlight the dynamic of evolution of mathematical objects. We can finally conclude on the opportunity to build and implement “Didactic Situations of Problem Solving”, which ensure the commitment of the subject in the research, the emergence and development of mathematical objects, the genesis of knowledges. The study reinforces the necessity of a pragmatic approach of situations and the relevance of a different look at the knowledge at school

Dans ce mémoire de thèse, on étudie le morphisme de monoïde $mu$du monoïde libre sur l'alphabet des entiers $nb$,`a valeurs dans le groupe modulaire $PSL_2(zb)$,considéré comme monoïde, défini pour tout entier $a$ par $mu(a)=begin{pmatrix} 0 & -1 1 & a+1 end{pmatrix}.$Les nombres de Catalan apparaissent naturellement dans l'étudede sous-ensembles du noyau de $mu$.Dans un premier temps, on met en évidence deux systèmes de réécriture, l'un sur l'alphabet fini ${0,1}$, l'autresur l'alphabet infini des entiers $nb$ et on montreque ces deux systèmes de réécriture définissent des présentations de monoïde de $PSL_2(zb)$ par générateurs et relations.Par ailleurs, on introduit le morphisme d'indice associé `a l'abélianisé du rev^etement universel de $PSL_2(zb)$,le groupe $B_3$ des tresses `a trois brins. Interprété dans deux contextes différents,le morphisme d'indice est associé au nombre de "demi-tours".Ensuite, dans les quatrième et cinquième parties, on dénombre des sous-ensembles du noyau de $mu_{|{0,1}}$ etdu noyau de $mu$, bigradués par la longueur et l'indice. La suite des nombres de Catalan et d'autres diagonales du triangle de Catalan interviennentsimplement dans les résultats.Enfin, on présente l'origine géométrique de cette étude : on explicite le lien entre l'objectif premier de la thèse qui était l'étudedes polygones convexes entiers d'aire minimale et notre intéret pour le monoïde engendré par ces matrices particulières de $PSL_2(zb)$. / In this thesis, we study a morphism of mono"id $mu$ between the free mono"id on the alphabet of integers $nb$and the modular group $PSL_2(zb)$ considered as a mono"id, defined for all integer $a$by $mu(a)=begin{pmatrix} 0 & -1 1 & a+1 end{pmatrix}.$ The Catalan Numbers arised naturally in the study ofsubsets of the kernel of the morphism $mu$.Firstly, we introduce two rewriting systems, one on the finite alphabet ${0,1}$, and the other on the infinite alphabet of integers $nb$. We proove that bothof these rewriting systems defines a mono"id presentation of $PSL_2(zb)$ by generators and relations.On another note, we introduce the morphism of loop associated to the abelianised of the universal covering group of $PSL_2(zb)$, the group $B_3$ ofbraid group on $3$ strands. In two different contexts, the morphism of loop is associated to the number of "half-turns".Then, in the fourth and the fifth parts, we numerate subsets of the kernel of $mu_{|{0,1}}$ and of the kernel of $mu$,bi-graduated by the morphism of lengthand the morphism of loop. The sequences of Catalan numbers and other diagonals of the Catalan triangle come into the results.Lastly, we present the geometrical origin of this research : we detail the connection between our first aim,which was the study of convex integer polygones ofminimal area, and our interest for the mono"id generated by these particular matrices of $PSL_2(zb)$.