Made available in DSpace on 2015-09-25T12:23:03Z (GMT). No. of bitstreams: 1 PDF - Alexsandro Coelho Alencar.pdf: 6754819 bytes, checksum: c6a8f7156f4dae99004e9377d5a26ce6 (MD5) Previous issue date: 2014-07-14 / This analysis is a qualitative study about the history of mathematics in mathematics textbook. This is an analysis of didactic material according to the methodological framework of discourse analysis. Aims to analyze the discursive practices present in the passages of the history of mathematics in mathematics textbook of high school in three of the seven collections approved by the Programa Nacional do Livro Didático -.PNLD 2012 (National Textbook Program) it was observed that the use of history in science and Mathematics teaching takes an important position in the contexts of teaching and learning currently. So, the history of mathematics is an integral part of the school mathematical knowledge contained in the textbook. It was also observed that the trend toward the use of traditional history is prevalent in the history of mathematics contained in the textbook of high school; that in mathematics textbook is not practiced as part of the mathematical content, but rather as an accessory to it and there are more historical passages informative or motivating character, and few where it uses the story as a teaching resource or as exploration of historical content itself. Therefore concluded that the textbook of mathematics, in regard to the use of mathematical history, reinforces the traditional paradigm, historically constructed and culturally determined in scientific and pedagogical field most commonly accepted in the mathematical community. In this process, the textbook through their discursive practices, plays an important disseminator role, producing meanings and contributing to the production of these because of their many uses and appropriations, sometimes reinforcing, sometimes breaking paradigms from a network of relationships that involves the scientific, pedagogical, marketing and cultural discourses. / O presente trabalho consiste em uma pesquisa de abordagem qualitativa sobre o uso da história da matemática no livro didático de matemática. Trata-se de uma análise de material didático segundo o referencial metodológico da análise do discurso. Tem como objetivo analisar as práticas discursivas presentes nas passagens da história da matemática no livro didático de matemática do Ensino Médio em três das sete coleções aprovadas pelo Programa Nacional do Livro Didático - PNLD 2012. Foi observado que o uso da história no ensino das ciências e da matemática assume uma posição relevante nos contextos de ensino e aprendizagem atualmente. Por isso, a história da matemática é parte integrante do conhecimento matemático escolar constante no livro didático. Foi observado também que a tendência ao uso da história tradicional é predominante na história da matemática contida no livro didático do ensino médio; que a matemática no livro didático não é praticada como parte do conteúdo matemático, mas sim como acessório a ele e que há mais passagens históricas de caráter informativo ou motivador, e poucas onde se usa a história como recurso didático ou como exploração do conteúdo histórico propriamente dito. Concluiu-se, portanto, que o livro didático de matemática, no que se refere ao o uso da história da matemática, reforça o paradigma tradicional, historicamente construído e culturalmente determinado no campo científico e pedagógico mais comumente aceito na comunidade matemática. Nesse processo, o livro didático, através de suas práticas discursivas, desempenha um papel disseminador relevante, produzindo sentidos e contribuindo para a produção destes em virtude dos seus diversos usos e apropriações, ora reforçando, ora quebrando paradigmas a partir de uma rede de relações que envolve os discursos científico, pedagógico, mercadológico e cultural.

O objetivo do presente trabalho é discutir a importância do diálogo para a educação matemática a distância. Parte-se da própria definição do termo, indo para a importância que os diálogos tiveram na história da matemática e chegando por fim ao diálogo em cursos de formação de professores a distância. A primeira parte do trabalho é constituída da análise de algumas definições do termo diálogo principalmente quando pensado para a educação e para a educação matemática. Em seguida, iremos discutir três obras literárias em forma de diálogo, são elas O Menon de Platão (1945), Diálogo sobre os dois máximos sistemas do mundo ptolomaico e copernicano de Galileo Galilei (1632/2004) e por último A lógica do descobrimento matemático: Provas e refutações de Imre Lakatos (1974), todas com importância para a educação matemática, para a história da ciência e que contribuíram para novas pesquisas, ampliando a discussão sobre o que é matemática ou mesmo como fazer matemática. A segunda parte deste trabalho visa analisar o diálogo na educação inserido no contexto da utilização de tecnologias de comunicação. Partimos do questionamento se era possível a ocorrência de diálogo em cursos de formação de professores a distância. Para isso começaremos definindo o que entendemos por educação a distância, analisando alguns dos principais modelos dessa modalidade de ensino. Feito isso partiremos para a discussão do significado do termo diálogo neste contexto, pois, assim como outros autores tais como Pierre Lèvy (1993), Moore & Kearsley (2007), consideramos o diálogo uma forma de interação/interatividade entre os participantes de um curso a distância. Após esses estudos, no último capítulo faremos a analise de momentos especiais de um conjunto de dez videoaulas de matemática do PEC (Programa de Educação Continuada) - Formação Universitária Municípios II. Trata-se de um estudo de caso em que procuramos por indícios de ocorrência de diálogo, como interatividade e suas características principais, durante todo este processo. Ao contrapormos o que foi discutido nos capítulos anteriores com a análise deste conjunto de videoaulas, encontramos indícios fortes da ocorrência de diálogo entre professores e alunos e mesmo entre os próprios alunos. Com este trabalho procuramos contribuir para vincular a prática clássica do diálogo na educação em um contexto de utilização de tecnologias de comunicação em cursos a distância, e constatamos que esta prática tende a aproximar os participantes, fazem com que eles estejam juntos mesmo que de maneira virtual, estando separados por grandes distâncias. / The aim of the present work is to discuss the important of the dialogue in mathematics distance education. From the very definition of the term \'dialogue\', to the importance of dialogues in Mathematics history and, at last, to the use of dialogue in teacher distance education programs. The first part of this work offers an analysis of some of the most significant definitions of the term \'dialogue\', mainly when used in Education and in Mathematics Education. Following that, we discuss three works based on dialogues: Plato\'s Menon (1945); Dialogue Concerning the Two Chief World Systems, by Galileo Galilei (1632/2004); and Proofs and Refutations: The Logic of Mathematical Discovery, by Imre Lakatos (1974). All these works have had impact on Mathematics Education and on Science history, and also contributed to new researches by broadening up the discussion about what Mathematics is about or even how to do Mathematics. The second part of this study aims to analyze the use and meaning of dialogue on Education, taking into account its use within communication technologies. We star by questioning whether such dialogue would be possible in teacher distance education programs. In order to do so, we define what we understand as distance learning, and we analyze some of the models for this approach. After that we discuss the meaning of dialogue in such context. As other researchers - Pierre Lèvy (1993) and Moore & Kearsley (2007), among others -, we conceive dialogue as a form of interaction/interactivity among pupils of a distance learning course. The last chapter brings the analysis of special moments of a set of ten video classes on Mathematics produced to a continuing education program named PEC (Programa de Educação Continuada) - \"Formação Universitária Municípios II\" (Municipal University Degree II). In such case study research, we looked for evidences of ocurrences of dialogue of interactivity and the main features observed along the process. The comparison between what was discussed on previous chapters with the analysis of this set of video classes gives us strong evidences of occurrences of dialog between teachers and students, and also among students. With this study, we want to contribute to create a link between the classical practices of dialogue in Education and its use within the communication technologies in distance learning. Our findings are that this practices tend to approximate the participants and collaborate to keep them virtually together, although sometimes separated by long distances.

O objetivo do presente trabalho é discutir a importância do diálogo para a educação matemática a distância. Parte-se da própria definição do termo, indo para a importância que os diálogos tiveram na história da matemática e chegando por fim ao diálogo em cursos de formação de professores a distância. A primeira parte do trabalho é constituída da análise de algumas definições do termo diálogo principalmente quando pensado para a educação e para a educação matemática. Em seguida, iremos discutir três obras literárias em forma de diálogo, são elas O Menon de Platão (1945), Diálogo sobre os dois máximos sistemas do mundo ptolomaico e copernicano de Galileo Galilei (1632/2004) e por último A lógica do descobrimento matemático: Provas e refutações de Imre Lakatos (1974), todas com importância para a educação matemática, para a história da ciência e que contribuíram para novas pesquisas, ampliando a discussão sobre o que é matemática ou mesmo como fazer matemática. A segunda parte deste trabalho visa analisar o diálogo na educação inserido no contexto da utilização de tecnologias de comunicação. Partimos do questionamento se era possível a ocorrência de diálogo em cursos de formação de professores a distância. Para isso começaremos definindo o que entendemos por educação a distância, analisando alguns dos principais modelos dessa modalidade de ensino. Feito isso partiremos para a discussão do significado do termo diálogo neste contexto, pois, assim como outros autores tais como Pierre Lèvy (1993), Moore & Kearsley (2007), consideramos o diálogo uma forma de interação/interatividade entre os participantes de um curso a distância. Após esses estudos, no último capítulo faremos a analise de momentos especiais de um conjunto de dez videoaulas de matemática do PEC (Programa de Educação Continuada) - Formação Universitária Municípios II. Trata-se de um estudo de caso em que procuramos por indícios de ocorrência de diálogo, como interatividade e suas características principais, durante todo este processo. Ao contrapormos o que foi discutido nos capítulos anteriores com a análise deste conjunto de videoaulas, encontramos indícios fortes da ocorrência de diálogo entre professores e alunos e mesmo entre os próprios alunos. Com este trabalho procuramos contribuir para vincular a prática clássica do diálogo na educação em um contexto de utilização de tecnologias de comunicação em cursos a distância, e constatamos que esta prática tende a aproximar os participantes, fazem com que eles estejam juntos mesmo que de maneira virtual, estando separados por grandes distâncias. / The aim of the present work is to discuss the important of the dialogue in mathematics distance education. From the very definition of the term \'dialogue\', to the importance of dialogues in Mathematics history and, at last, to the use of dialogue in teacher distance education programs. The first part of this work offers an analysis of some of the most significant definitions of the term \'dialogue\', mainly when used in Education and in Mathematics Education. Following that, we discuss three works based on dialogues: Plato\'s Menon (1945); Dialogue Concerning the Two Chief World Systems, by Galileo Galilei (1632/2004); and Proofs and Refutations: The Logic of Mathematical Discovery, by Imre Lakatos (1974). All these works have had impact on Mathematics Education and on Science history, and also contributed to new researches by broadening up the discussion about what Mathematics is about or even how to do Mathematics. The second part of this study aims to analyze the use and meaning of dialogue on Education, taking into account its use within communication technologies. We star by questioning whether such dialogue would be possible in teacher distance education programs. In order to do so, we define what we understand as distance learning, and we analyze some of the models for this approach. After that we discuss the meaning of dialogue in such context. As other researchers - Pierre Lèvy (1993) and Moore & Kearsley (2007), among others -, we conceive dialogue as a form of interaction/interactivity among pupils of a distance learning course. The last chapter brings the analysis of special moments of a set of ten video classes on Mathematics produced to a continuing education program named PEC (Programa de Educação Continuada) - \"Formação Universitária Municípios II\" (Municipal University Degree II). In such case study research, we looked for evidences of ocurrences of dialogue of interactivity and the main features observed along the process. The comparison between what was discussed on previous chapters with the analysis of this set of video classes gives us strong evidences of occurrences of dialog between teachers and students, and also among students. With this study, we want to contribute to create a link between the classical practices of dialogue in Education and its use within the communication technologies in distance learning. Our findings are that this practices tend to approximate the participants and collaborate to keep them virtually together, although sometimes separated by long distances.

Ce travail se concentre sur le cours d'analyse enseigné à l'École polytechnique de 1795 à 1809. En devenant professeurs, plusieurs mathématiciens au tournant du 19ème siècle y ont contribué par des ouvrages importants d’Analyse. Parmi eux, Joseph-Louis Lagrange (1736-1813) joua un rôle central, en y devenant le premier Institutor d'analyse. Les trois premiers chapitres de cette thèse se focalisent sur les leçons que Lagrange donna de 1795 à 1799. En insistant sur le fait que Lagrange enseignait l'arithmétique à l’École Polytechnique avant son cours d'analyse, la première partie de cette thèse clarifie les raisons pour lesquelles de Lagrange incorporait ces éléments d’arithmétique et leur relation avec le cours d’analyse. Cette étude fournit une discussion détaillée des concepts fondamentaux des mathématiques dans les cours de Lagrange. Ainsi, on y montre que l'intention de Lagrange est de lier des branches différentes de l'analyse à l'algèbre à l'arithmétique. Ce travail montre de quelles façons et en quels termes Lagrange unifie ces branches. De plus, cette thèse met l'accent sur les valeurs épistémologiques que Lagrange poursuit et défend dans ses travaux mathématiques, sur la base desquelles Lagrange a choisi la méthode des développements des fonctions en séries pour présenter les principes du calcul différentiel. La but de la deuxième partie de cette thèse est de montrer à quel point le cours de Lagrange à l'Ecole Polytechnique a influencé l'enseignement de trois autres professeurs: Joseph Fourier (1768-1830), Jean-Guillaume Garnier (1766-1840) et Sylvestre-François Lacroix (1765-1843). Fourier inventa une nouvelle méthode en croisant la méthode de Lagrange et la méthode des limites. Garnier et Lacroix suivent essentiellement la méthode de Fourier, mais avec quelques modifications. En comparant les deux traités du calcul différentiel de Lacroix, cette étude montre que la pratique de l’enseignement, ainsi que la destination des élèves de l’École Polytechnique ont constitué des facteurs importants dans l’évolution des principes du calcul différentiel et de leur présentation / This work studies the courses of analysis taught at the Ecole Polytechnique (EP) from 1795 until 1809. Several mathematicians of the eighteenth century contributed important works as they practiced the teaching of analysis at this school. Joseph-Louis Lagrange (1736-1813) was the central figure, who had been the first professor of the course of analysis at the EP and had great impact on his successors. In order to show in which way and to what degree the lectures that Lagrange gave exerted influence on the teaching of analysis at the EP, this dissertation gives a detailed discussion on Lagrange’s publications and courses of analysis, as well as those by other teachers, i.e. Joseph Fourier(1768-1830), Jean-GuillaumeGarnier(1766-1840)andSylvestre-FrançoisLacroix (1765-1843). It achieves the following conclusions. First, Lagrange, taking into account the utility for students, chose to found analysis on the method of the developments of functions in series, so that analysis could be united with algebra, and arithmetic as well. Second, Lagrange’s approach to differential calculus, as well as the epistemic values he pursued in his mathematical works, provided influential source for the teaching of analysis by other professors. The thesis is that the three professors who taught beside or after Lagrange followed Lagrange’s ideas, although each made some modifications on his own course

Hourya Sinaceur présente dans son ouvrage Corps et Modèles la théorie des caractéristiques de Kronecker comme la fin d’« un cycle d’idées sturmiennes », la situant ainsi dans une histoire de l’algébrisation du théorème de Sturm. Pourtant, cette théorie est souvent aussi présentée comme le point de départ de certains des premiers travaux de topologie de la toute fin du dix-neuvième siècle. Nous souhaitons dans cette thèse rendre compte de la façon dont ces deux histoires se rencontrent au sein de la théorie de Kronecker. Pour cela, nous disposons d’un matériel particulièrement éclairant et jusqu’alors très peu exploité : les manuscrits des cours que Kronecker a donnés à l’Université de Berlin entre 1872 et 1891 sur la Théorie des équations algébriques. Nous commencerons par présenter ces manuscrits, leur contenu et leur contexte de rédaction. Nous nous intéresserons ensuite plus précisément à la reprise du théorème de Sturm par Kronecker, et nous montrerons que la théorie des caractéristiques n’est pas seulement un prolongement algébrique de ce théorème, mais qu’elle se transforme pour fournir certains des outils analytiques que Poincaré utilisera lors de la construction de son Analysis Situs. L’exposition de la théorie des caractéristiques dans ses cours est l’occasion pour Kronecker de reprendre trois des quatre démonstrations que Gauss a données du théorème fondamental de l’algèbre, et nous montrerons comment, dans la pratique de Kronecker, la notion même de racine est interrogée / In her book Corps et Modèles, Hourya Sinaceur presented Kronecker’s theory of characteristics as the end of a “cycle of Sturmian ideas”, making it a step in history of the algebraization of Sturm’s theorem. However, this theory is often also introduced as the starting point of some of the early works of topology of the very end of the nineteenth century. In this PhD thesis, I will describe how these two stories are connected in Kronecker’s theory. To achieve this, I used material which has seldom been discussed before : the manuscripts of the courses that Kronecker gave at the University of Berlin between 1872 and 1891 on the Theory of algebraic equations. I begin with the presentation of these manuscripts, their contents and their writing contexts. I then look more closely at Kronecker’s rework of the theorem of Sturm and show that the theory of characteristics is not only an algebraic extension of this theorem but also that it is transformed in order to provide some of the analytic tools that Poincaré will use for the construction of his Analysis Situs. The exposition of the theory of characteristics in his courses is an opportunity for Kronecker to take up three of the four demonstrations from Gauss of the fundamental theorem of algebra, and I will show how, in Kronecker’s practice, the very notion of root is questioned

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.

<p>The aim of this thesis is to examine how the history of mathematics can be used in teaching and whether pupils’ attitudes to mathematics can be affected by this. In order to do this a school project about the history of mathematics is planned, implemented and analysed.</p><p>The following questions are addressed:</p><p>▪ Why use the history of mathematics in teaching?</p><p>▪ How can the history of mathematics be used in teaching?</p><p>▪ Can a historical perspective in teaching affect pupils’ views on mathematics?</p><p>The school curriculum emphasizes the importance of history in teaching. Literature, scientific articles and government inquiries call attention to the positive effects of including a historical perspective in teaching.</p><p>The school project, which underlies this thesis, is put to practice by 25 pupils in a mathematics science profile class. The outcome is one written and one oral report. A first questionnaire is completed by the pupils before the history project is introduced and a second one after the project work is entirely finished. The result shows positive effects of using the history of mathematics in teaching.</p>

The aim of this thesis is to examine how the history of mathematics can be used in teaching and whether pupils’ attitudes to mathematics can be affected by this. In order to do this a school project about the history of mathematics is planned, implemented and analysed. The following questions are addressed: ▪ Why use the history of mathematics in teaching? ▪ How can the history of mathematics be used in teaching? ▪ Can a historical perspective in teaching affect pupils’ views on mathematics? The school curriculum emphasizes the importance of history in teaching. Literature, scientific articles and government inquiries call attention to the positive effects of including a historical perspective in teaching. The school project, which underlies this thesis, is put to practice by 25 pupils in a mathematics science profile class. The outcome is one written and one oral report. A first questionnaire is completed by the pupils before the history project is introduced and a second one after the project work is entirely finished. The result shows positive effects of using the history of mathematics in teaching.

This dissertation deals with aspects of axiomatization, intuition and visualization in the history of mathematics. Particular focus is put on the end of the 19th century, before David Hilbert's (1862–1943) work on the axiomatization of Euclidean geometry. The thesis consists of three papers. In the first paper the Swedish mathematician Torsten Brodén (1857–1931) and his work on the foundations of Euclidean geometry from 1890 and 1912, is studied. A thorough analysis of his foundational work is made as well as an investigation into his general view on science and mathematics. Furthermore, his thoughts on geometry and its nature and what consequences his view has for how he proceeds in developing the axiomatic system, is studied. In the second paper different aspects of visualizations in mathematics are investigated. In particular, it is argued that the meaning of a visualization is not revealed by the visualization and that a visualization can be problematic to a person if this person, due to a limited knowledge or limited experience, has a simplified view of what the picture represents. A historical study considers the discussion on the role of intuition in mathematics which followed in the wake of Karl Weierstrass' (1815–1897) construction of a nowhere differentiable function in 1872. In the third paper certain aspects of the thinking of the two scientists Felix Klein (1849–1925) and Heinrich Hertz (1857–1894) are studied. It is investigated how Klein and Hertz related to the idea of naïve images and visual thinking shortly before the development of modern axiomatics. Klein in several of his writings emphasized his belief that intuition plays an important part in mathematics. Hertz argued that we form images in our mind when we experience the world, but these images may contain elements that do not exist in nature.

<p>This dissertation explores the ideational underpinnings of the rise of the west through a comparison of ancient Greek geometry, medieval Arabic algebra, and early modern European calculus. Blending insights from Thomas Kuhn, Michel Foucault, and William H. Sewell, I assert that there is an underlying logic, however clouded, to the unfolding of a given civilization, governed by a cultural episteme that delineates the boundaries of rational thought and the accepted domain of human endeavor. Amid a certain conceptual configuration, the rise of the west happens; under other circumstances, it does not. Mathematics, as an explicit exhibition of logic premised on culturally determined axioms, presents an outward manifestation of the lens through which a civilization surveys the world, and as such offers a window on the fundamental assumptions from which a civilization's trajectory proceeds. To identify the epistemological conditions favorable to the rise of the west, I focus specifically on three mathematical divergences that were integral to the development of calculus, namely analytic geometry, trigonometry, and the fundamental theorem of calculus. Through a comparative/historical analysis of original source documents in mathematics, I demonstrate that the logic in the earlier cases is fundamentally different from that of calculus, and furthermore, incompatible with the key developments that constitute the rise of the west. I then examine the conceptual similarities between calculus and several features of the rise of the west to articulate a description of the early modern episteme.</p> / Dissertation