Correspondências científicas como uma relação didática entre história e ensino de matemática: o exemplo das cartas de Euler a uma princesa da Alemanha

Lettres àune Princesse d'Allemagne sur divers sujets de physique et de philosophie (Letters to a Princess of Germany on various topics of physics and philosophy) is the work taken as an object of study of this thesis. It is a literary success written in the eighteenth century by the Swiss mathematician and physicist Leonhard Paul Euler (1707-1783) in order to meet a request from the Prussian king, Frederick II, the Great (1712-1786) to accept to guide the intellectual education of his niece, the young princess Anhalt-Dessau (1745-1808). The method of teaching and learning through letters elected to the education of the German monarch resulted in a collection of 234 matches in which Euler theory is about music, Philosophy, Mechanics, Optics, Astronomy, Theology and Ethics among others. The research seeks to point out mathematical content contained in this reference work based on the exploitation and adaptation of original historical works as an articulator of development activities for teaching mathematics in basic education and in accordance with the National Curriculum Parameters of Mathematics (NCP) work. The general objective point out the limits and didactic potential of Lettres à une Princesse d'Allemagne sur divers sujets de physique et de philosophie as a source of support for teachers of basic education in developing activities for teaching mathematics. The discussions raised point to concrete possibilities of entanglement between the extracted mathematical content of the bulge of the work with current teaching methodologies from resizing the use of letters according to Freire's pedagogical perspective of the correspondence, and especially the use of new communication channels in the century XXI, both aimed at dialogue and approximation between those who write and those who read.

Formação docente para a inserção da história e filosofia da ciência no ensino: textos histórico-pedagógicos em discussão

The present dissertation focuses on specific problems in the educational context: challenges in the construction of historical narratives for pedagogical use as well as the difficult task of using them in the classroom. In this context, we seek to work in teacher training for insertion of History and Philosophy of Science (HPS) in classroom, and historical narratives become mediation elements to advance the dialogue with this specific audience. This initiative is in line with a recurring concern: one of the main challenges related to the didactic transposition of HFC would be the lack of teacher preparation. Historical contents and Nature of Science are still absent in classrooms. Insecurity and lack of knowledge by teachers are often mentioned as factors that contribute to this situation. It is important, therefore, that teachers (active and in training) take part in discussions concerning the inclusion of HPS in classroom. It is relevant that they know examples of historic-philosophical didactic proposals to address science and contents on science, develop skills to adapt them to their specific contexts and to develop their own proposals. It is believed that these issues are significant to undertake conscious initiatives to insert HPS in classrooms. It is considered that adapting educational proposals to particular educational contexts depends on understanding what these proposals indeed mean and how flexible they can be. In order to address these objectives, we elaborated an educational product, a didactic material focused on teacher training, which was used in an extension course at UFRN. The didactic material discusses the role of HPS in Education, Nature of Science and historiographical issues. It presents a series of dialogical activities on aspects of didactic transposition of HPS, especially those regarding historical narratives. A set of historicpedagogical texts on the History of Vacuum and Atmospheric Pressure is used as a mediation element in discussions. We address potential, possibilities and limitations historical narratives. To carry out the course, it was taken into account methodological concerns of so-called action research. There have been expected changes, modifications and effective actions in the own teacher training material in face of the experience of the researcher-lecturer in interactions with the participants of the course as well as in face of impressions reported by the participants. Developments in this direction have been incorporated into the teacher training material.

Modelagem matemática para o transporte de partículas sujeitas a múltiplos mecanismos de retenção

Discrepancies between classical model predictions and experimental data for deep bed filtration have been reported by various authors. In order to understand these discrepancies, an analytic continuum model for deep bed filtration is proposed. In this model, a filter coefficient is attributed to each distinct retention mechanism (straining, diffusion, gravity interception, etc.). It was shown that these coefficients generally cannot be merged into an effective filter coefficient, as considered in the classical model. Furthermore, the derived analytic solutions for the proposed model were applied for fitting experimental data, and a very good agreement between experimental data and proposed model predictions were obtained. Comparison of the obtained results with empirical correlations allowed identifying the dominant retention mechanisms. In addition, it was shown that the larger the ratio of particle to pore sizes, the more intensive the straining mechanism and the larger the discrepancies between experimental data and classical model predictions. The classical model and proposed model were compared via statistical analysis. The obtained p values allow concluding that the proposed model should be preferred especially when straining plays an important role. In addition, deep bed filtration with finite retention capacity was studied. This work also involves the study of filtration of particles through porous media with a finite capacity of filtration. It was observed, in this case, that is necessary to consider changes in the boundary conditions through time evolution. It was obtained a solution for such a model using different functions of filtration coefficients. Besides that, it was shown how to build a solution for any filtration coefficient. It was seen that, even considering the same filtration coefficient, the classic model and the one here propposed, show different predictions for the concentration of particles retained in the porous media and for the suspended particles at the exit of the media

Olhares sobre a formação do professor de matemática. Imagem da profissão e escrita de si

The main focus of this thesis is the formation of a mathematical teacher at a college institution. The general aim is to describe and to analyze the formation process of a mathematical teacher which is an undergraduate student in Mathematics at the Instituto de Educação Superior Presidente Kennedy IFESP, in Natal-RN. It is based on a qualitative ethnographic approach, and has its theoretical anchorage in the (auto)biographical narratives, the social representative theories, and the mathematical education. The number of participants in this investigation was 12 undergraduate students, which corresponds to 25% of the total number of students. The corpus utilized in our analysis included 48 (auto)biographical essays, 12 (auto)biographies (formation's memories), and 12 contextualization files, besides the research's diary. The sources were obtained from the whole program of studies, i.e. from November 2003 to December 2006. The analysis revealed that the reminiscences of the 12 students' academic trajectory influenced their professional formation, since their images of a mathematical teacher were intrinsically related to the one they had before. These representations were being either demolished or constructed in a network along the assertive image of their profession, changing afterwards the mathematical representation and the teaching way of this discipline. Our study also shows that the beginning of their teacher career was marked by mechanical practices influenced by their old teachers. The (trans)formation of themselves and their teaching practices happened in a smooth way as soon as they increased their knowledgements in Mathematics, and it reflected upon the way they learned mathematics. The writing of their (auto)biographies helped the set up of new knowledgements, leaving to a self-consciousness as well as a self-formation, and contributed for the construction of a new way to see and to live the profession. Therefore, a mathematical teacher, for the undergraduate students of the IFESP involved in this work, is made at the interface of the familiar, academic, and professional context, besides the reflexive writings about the formation path, the way of life and the relationships among them

Professores em contexto formativo: um estudo do processo de mudanças de concepções sobre o ensino da matemática

El objetivo en esta tesis consistió en estudiar el proceso de los cambios de los conceptos de profesores de la educación infantil y de los años iníciales de la educación básica referente a la enseñanza de la matemática. La investigación se desenvolvió en la escuela Presidente Kennedy, en la ciudad de Natal, en Rio Grande do Norte, teniendo como participante 05 (cinco) profesores del curso normal superior a través de la educación superior del instituto relacionado. El trabajo asocia el programa a él Programa de Pós-Graduação em Educação da Universidade Federal do Rio Grande do Norte, en la base de pesquisa Formação e Profissionalização Docente coordinada de los doctores Betânia Leite Ramalho e Isauro Beltran Núñez. El referencial teórico-metodológico en quien si apoya el trabajo se inserta en la señal conceptual usada por Giordan y de Vecchi (1996), de Carrillo y Contreras (1994), Ramalho; Núñez y Gauthier (2003), Ponte (1998), Guimarães (1988), Ernest (1989). En esta investigación, los conceptos de los profesores habían sido estudiados en el contexto educativo de la formación del nivel superior, usándola reflexiva crítico práctico como estrategia formativa. Estos conceptos se entienden como estructuras subyacentes al pensamiento del profesor. Dado la naturaleza del objeto del estudio, la información, para las intenciones de esta investigación, habían sido cosechados a través de los instrumentos siguientes: cuestionario, plan de la lección, entrevista diaria y del campo. El cuestionario fue constituido de preguntas abiertas y de las entrevistas de la mitad-structuralized. La organización de los datos permitió a La inferencia de los conceptos, usando la técnica de la triangulación de datos. La investigación divulgó que los conceptos de los profesores, a través del proceso formativo, se habían desarrollado de una plataforma para otra, yéndose puesto que los modelos didácticos tradicionales para otros modelos dirigidos a una tendencia didáctica de espontaneísta/investigativa. La reflexión crítica era considerada como elemento catalítico de los cambios de los conceptos de los profesores en la educación de las matemáticas, sin embargo déjenos verifican que estos cambios son difíciles de ocurrir para la naturaleza compleja de estos conceptos. Como facilitadores de los factores de estos cambios, encontramos y el investigativo el trabajo, la dinámica y la naturaleza de las actividades se convirtió en el colaborativo de proceso formativo, entre otros. Como obstáculos a los cambios, identificamos el contexto del trabajo de los profesores, de la cultura de los individualistas prácticos de sus profesores de los colegas, del concepto linear, estático y de los mecánicos de los procesos para enseñar, el conocimiento profesional construído durante la formación inicial, alineación con los modelos didácticos de sus viejos profesores, entre otros

O ensino de matemática no Rio Grande do Norte: trajetória de uma Modernização (1950-1980)

This work aims to describe and analyze the process of the mathematics teacher modernizing in Rio Grande do Norte, in the period from 1950 to 1980. For that, we use as theoretical foundation assumptions of Cultural History and memories of the researchers Maurice Halbwach, Ecléa Bosi and Paul Thompson. As methodological tools, we used bibliographical resources and semi-structured interviews, in order to do a historical reconstruct of the mathematics educational scene of institutions and people who taught mathematics in Rio Grande do Norte, or those who participated in the modernization of the teaching of this subject, recovering their training and its practices in teaching. For the analysis of the bibliographical resources, initially we organized in a systematic way the transcripts of the interviews and documents, which were accumulated during the research, so long our thoughts, returning to the theoretical basis of this research, through questioning of knowledge acquired and that guided the problem of our study. The analysis showed that, important moments to modernize the teaching of mathematics in Rio Grande do Norte happened such: (1) Training Course of Lay Teachers in Rio Grande do Norte, in 1965, (2) Course for Teachers in Normal Schools, in 1971 (3) Satelite Project on Interdisciplinary Advanced Communications (SPIAC) in 1973; (4) Lectures of the teacher Malba Tahan, at Natal, from the end of the 50 s, that could be analyzed through the lessons notes of the teacher Maria Nalva Xavier de Albuquerque and the narrative of teacher Evaldo Rodrigues de Carvalho and (5) Courses of the Campaign for Improvement of Secondary Education and Broadcasting (CISEB). Thereby, the modernization of the school s mathematics teaching in Rio Grande do Norte, in the period from 1950 to 1980, was given mainly by disclosure of the Discovery Method and by the Set Theory contents in Teacher Training Courses

Pedagogia etnomatemática : ações e reflexões em matemática do Ensino Fundamental com um grupo sócio cultural específico

mongst the trends in Mathematics Education, which have as their object a more significant and criticallearning, is the Ethnomathematics. This field of knowledge, still very recent amongst us, besides analyzing an externalist history of the sciences in a search for a relationship between the development of the scientific disciplines and the socio-cultural context, goes beyond this externalism, for it also approaches the intimate relationships betwe_n cognition and culture. In fact, the Ethnomathematics proposes an alternative epistemological approach associated with a wider historiography. It struggles to understand the reality and come to the pedagogical action by means of a cognitive approach with strong cultural basis. But the difficulty of inserting the Ethnomathematics into the educational context is met by resistance from some mathematics educators who seem indifferent to the influence of the culture on the understanding of the mathematics ideas. It was with such concerns in mind that I started this paper that had as object to develop a curricular reorientation pedagogical proposal in mathematics education, at the levei of the 5th grade of the Ensino Fundamental (Elementary School), built from the mathematical knowledge of a vegetable farmers community, 30 km away from the center of Natal/RN, but in accordance with the teaching dimensions of mathematics of the 1 st and 2nd cycles proposed by the Parâmetros Curriculares Nacionais - PCN: Numbers and Operations, Space and Form, Units and Measures, and Information Treatment. To achieve that, I developed pedagogical activities from the mathematical concepts of the vegetable farmers of that community, explained in my dissertation research in the period 2000 through 2002. The pedagogical process was developed from August through Oecember 2007 with 24 students of the 5th Grade of the Ensino Fundamental (Elementary School) of the school of that community. The qualitative analysis of the data was conducted taking into account three categories of students: one made up of students that helped their parents in the work with vegetables. Another one by students whose parents and relatives worked with vegetables, though they did not participate directly of this working process and one third category of students that never worked with vegetables, not to mention their parents, but lived adjacent to that community. From the analyses and results of the data gathered by these three distinct categories of students, I concluded that those students that assisted their parents with the daily work with vegetables solved the problem-situations with understanding, and, sometimes, with enriching contributions to the proposed problems. The other categories of students, in spite of the various field researches to the gardens of that community, before and during the pedagogical activities, did not show the same results as those students/vegetable farmers, but showed interest and motivation in ali activities of the pedagogical process in that period

Um estudo sobre a relação entre matemática e religião na obra de Blaise Pascal

The aim of the present study is to investigate the way through which the relations between Mathematics and Religion emerge in the work of Blaise Pascal. This research is justified by the need to deepen these relations, so far little explored if compared to intersection points between Mathematics and other fields of knowledge. The choice for Pascal is given by the fact that he was one of the mathematicians who elaborated best one reflection in the religious field thus provoking contradictory reactions. As a methodology, it is a bibliographical and documental research with analytical-comparative reading of referential texts, among them the Oeuvres complètes de Pascal (1954), Le fonds pascalien à Clermont-Ferrand (2001), Mathematics in a postmodern age: a cristian perspective by Howell & Bradley (2001), Mathematics and the divine: a historical study by Koetsier & Bergmans (2005), the Anais dos Seminários Nacionais de História da Matemática and the Revista Brasileira de História da Matemática. The research involving Pascal's life as a mathematician and his religious experience was made. A wider background in which the subject matter emerges was also researched. Seven categories connected to the relation between mathematics and religion were identified from the reading of texts written by mathematicians and historians of mathematics. As a conclusion, the presence of four of these seven categories was verified in Pascal's work

Uma reavaliação do pensamento lógico de George Boole à luz da história da matemática

The aim of the present study is to reevaluate the logical thought of the English mathematician George Boole (1815 - 1864). Thus, our research centers on the mathematical analysis of logic in the context of the history of mathematics. In order to do so, we present various biographical considerations about Boole in the light of events that happened in the 19th century and their consequences for mathematical production. We briefly describe Boole's innovations in the areas of differential equations and invariant theory and undertake an analysis of Boole's logic, especially as formulated in the book The Mathematical Analysis of Logic, comparing it not only with the traditional Aristotelian logic, but also with modern symbolic logic. We conclude that Boole, as he intended, expanded logic both in terms of its content and also in terms of its methods and formal elaboration. We further conclude that his purpose was the mathematical modeling of deductive reasoning, which led him to present an innovative formalism for logic and, because the different ways it can be interpreted, a new conception of mathematics

Ensino de matemática, história da matemática e artefatos: Possibilidade de interligar saberes em cursos de formação de professores da Educação Infantil e anos iniciais do Ensino Fundamental.

This work is located at the shield of research that defends the use of Mathematics History, based on the utilization of historical artifacts at teaching activities, at Mathematics classrooms, and at graduation courses for teachers of Elementary School and of the first grades of High School. The general objective is to examine the possibility of the use of historical artifacts, at teaching activities, at graduation courses for teachers of Elementary School and of the first grades of High School. Artifact, at this work, is comprehended as objects, documents, monuments, images and other kinds of materials that make sense to the Human actions at the past and that represent what have been said and done at the Human history. At the construction of the theoretical-methodological way of the research we have based ourselves upon the ideas of the authors that are engaged at the teachers formation; at researchers adherents to the use of Mathematics History (MH) as a methodological resource, and at studies accomplished that elucidate the role of the artifacts at the history and as a mediatory element of learning. We defend the thesis that the utilization of historical artifacts at teaching activities enables the increasing of the knowledge, the development of competencies and essential abilities to the teacher acting, as well as interact at different areas of the knowledge, that provides a conception of formation where the teacher improves his learning, learning-doing and learning-being. We have adopted a qualitative research approach with a theoretical and pratic study disposition about the elements that contribute to the teachers works at the classroom, emphasizing the role of the Mathematics history at the teacher s formation and as a pedagogical resource at the mathematics classroom; the knowledge, the competencies and abilities of the historical artifacts as an integrative link between the different areas of the knowledge. As result, we emphasize that the proposition of using the MH, through learning activities, at the course of teacher graduation is relevant, because it allows the investigation of ideas that originate the knowledge generated at every social context, considering the contribution of the social and cultural, political and economical aspects at this construction, making easy the dialog among the areas and inside of each one The historical artifact represents a research source that can be deciphered, comprehended, questioned, extracting from it information about knowledge of the past, trace and vestiges of the culture when it was created, consisting of a testimony of a period. These aspects grant to it consideration to be explored as a mediatory element of the learning. The artifacts incorporated at teaching activities of the graduation courses for teachers promote changes on the view about the Mathematics teaching, in view of to privilege the active participation of the student at the construction of his knowledge, at the reflection about the action that has been accomplished, promoting stimulus so the teachers can create their own artifacts, and offer, either, traces linking the Mathematics with others knowledge areas.

Música, filosofia, formação: por uma escuta sensível do mundo

La musique, plus qu`um mot et le texte écrit, a le pouvoir de produire le déplacement du sujet en relation aux contingences du temps e de l´espace. Entendre de la musique est plus que de l´amusement. Comme un artifice cognitif qui ultrapasse l´expérience analytique e métonymique, les formes tonales doublent la réalité, expandent les sens, actionent l´imagination, retotalisent les sentiments, amplient les expériences, permettent la rencontre du sujet avec les éssences des phénomènes qui ne sont pas traductibles par les mots. Nous écoutons de la musique pour restituer notre dignité e conférer à la vie plus de vérité, plus de concrétion (Santiago Kovadloff). La mélodie permet l´expérimentation des états de cohésion, de conéction pleine entre sens et intention, entre commencement et fin, de la finitude de la vie (Schopenhauer). Le phénomène musical provoque l´expréssion de la douleur, de la souffrance et, au même temps, du jubile et de la joie, réliant nature et homme (Nietzsche). Tenant comme base cette compréhension, la dissertation tient dans la musique une importante métaphore pour comprendre la complexité humaine, une fois qu´elle propicie une écoute sensible du monde et mobilise dans le sujet l´expérimentation de divers états de l´être. Nous pouvons dire que la musique est un opérateur de la conaissence parce qu´elle fait affleurer l´écoute intérieure, la rencontre du sujet avec lui même. Cet opérateur cognitif actione les pôles de l´ésprit qui font dialoguer sensibilité, éthique et esthétique, ordre et chaos, silence et bruit, mouvement et pause, repétition et inovation. Par l´expérience musicale, nous habitons des formes hibrides de sensibilité et raison. L´inachèvement, notre principal caractéristique comme humains, a, dans la musique, une image impair, parce que la musique est l´expression du dévir. Pour tisser ces arguments, la dissertation part des études de Schopenhauer et Nietzsche sur la musique, esthétique et métaphysique; Expose des fragments de la biographie de trois grands penseurs contemporains (Werner Heinsenberg, Ilya Prigogine et Edgar Morin) accentuant la présence de la musique en ses vies; et, par fin, présente ce que nous appelons biographies sonores de quatre artistes-musiciens brésiliens (Benedito Juarez et Gil Jardim, de São Paulo, et Ronaldo Ferreira de Lima et, Cleudo Freire, du Rio Grande do Norte)

Espaços transversais em educação matemática :uma contribuição para a formação de professores na perspectiva etnomatemática

This PH.D. thesis is an attempt to show the beginning, evolution and unfolding of the making of a pedagogical work proposal based on culturally-built knowings in the heart of a traditional community, having as one of its starting points the knowings and doings experienced by dish-making women from Maruanum living in the city of Macapá, State of Amapá, Brazil. This proposal is strongly associated with the need we have to think about the nature of (ethnological)-mathematical knowledge generated by particular communities and about the way such knowledge can be discussed, worked out, and validated in learning environments, regardless of the level of instruction and the constraints imposed by government programs and educational institutions. Among its theoretical foundations are studies on instrumental activities that are typical of the Maruanum ceramics and investigative studies from the point of view of ethnomathematics. Methodological development took place with the application of activities, where traditional and instrumental knowledge observed in the production of ceramics had been adapted for and brought into the school environment , participative observation, as well as data collecting and organization techniques, such as interviews, statements, and audio an visual recordings. Analysis of the data collected focused on the relationship between the data-generating potential and the purpose of this study. Our aim is to make and estimate of the potential contributions from local situations and/or problems it would possibly bring to the formative learning of people involved in the educational processes of these communities, with a view to a spatial and temporal transformation of reality

A cultura da produção de farinha: um estudo da matemática nos saberes dessa tradição

This research argues about the mathematical knowledge built in the tradition of the cassava flour production, seeking to analyse these mathematical knowledge in the perspective of the categories of time and measure, built and practiced in the flour production, located in Serra do Navio and Calçoene, in Amapá - Brazil. The following work discuss the identification and the description of the mathematics during the production activities of the flour, where is presented elements related to generation and transmission of the traditional knowledge, which is the basis for maintenance of the tradition of the flour, characterizing the research as an Ethnomathematic study. The methodological procedures highlight ethnographical techniques and elements that characterize the participating observation. The results obtained showed us that the flour workers articulate some length, area and volume measure due to own and traditionally acquired systems, which is apprehended and countersigned by other kind of culturally established system; thus they relativism the measures systems and the official calendars. And it lifts as one of the main proposal that the academic mathematics and the tradition establish knowledge make conjunction of the both knowledge, that is important for a possible reflection and application in the construction of a pedagogical practice in mathematical education, trying to establish points of socio-economic and cultural mark

Globos e mapas ao alcance das mãos : ensino de matemática numa perspectiva de alfabetização funcional na EJA

The following dissertation has as its main advantage the privilege of visualizing the literacy processes through the angle of the functional perspective, which does not see the literary process as a practice solely based on the decoding of alphabetical codes, and then allows for the opening of ample spaces for the allocation of mathematical skills in the realms of the functional literacy. The main object of this study was to investigate which are the contributions that a sequence of activities and of methodologies developed for the teaching of Geometry could provide for a part of the functional literacy process in mathematics of youngsters and adults of EJA, corresponding to the acquisition or to the improvement of skills related to the orientation capacity. The focus of the analyses consisted in the practice of these activities with the young and adult students of an EJA class belonging to a municipal public school of Natal/RN. The legacies of Paulo Freire about the redimensioning of the role of the teacher, of the students, of the knowledge and of their connections within the teaching-learning process, prevailed in the actions of the methodology implemented in the classroom and, especially, in the establishing of dialogic connections with the students, which directed all the observations and analyses regarding the collected information. The results indicated that the composition of articulations between the teaching of mathematics and the exploration of maps and the earth globe enabled the creation of multidisciplinary learning environments and situations, where we could observe, gradually, the development of procedures and attitudes indicating the evolution of space-visual type skills

Possibilidades de exploração da história da ciência na formação do professor de matemática: mobilizando saberes a partir da obra de Nicolau Copérnico de revolutionibus orbium coelestium

This study aims to analyze the implications that the knowledge of an important work for the History of Science, De revolutionibus orbium coelestium , by Nicholas Copernicus, can bring for the formation of Mathematics professors. The study focuses on Book I of Copernicus s work, where, in the final part, is found the Table of the Subtense Straight Lines in a Circle, a true sine table constructed by the author. The study considers two theoretical references, the History of Science and of Mathematics, in the professor s formation searched amongst others in Miguel and Miorm, Brito, Neves and Martins, and Radford, and the necessary teaching knowledge professors mst have, on the basis of Gauthier, Schulman and Imbernón amongst others, through which it is established a net of knowledge grouped in dimensions such as mathematical, psycho pedagogical, cultural and practical diversity, that guide the study analysis. In the search for more necessary elements to enrich the analysis, beyond the theoretical research in Book I, it is carried through, with under graduation pupils, future Math professors, the construction of a sine table following the project used in De revolutionibus . The study still makes a description of the life and work of Nicholas Copernicus, detaching the historical context where the author lived and the conceptions about the Universe existing at that time. The research reveals that the studied work is an important source of culture, able to provide to the Mathematics professor in formation, beyond the conceptual and procedural mathematical knowledge, a cultural knowledge that allows him to be opened to the knowledge of other areas that not his specific area, and so to acquire knowledge about the world history, the development of sciences and of the society