Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-06-16T14:36:33Z No. of bitstreams: 1 silviadecastrodebarros.pdf: 3863081 bytes, checksum: ab4a7e88d1525ed1bb94676937c74b2c (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-07-13T14:16:49Z (GMT) No. of bitstreams: 1 silviadecastrodebarros.pdf: 3863081 bytes, checksum: ab4a7e88d1525ed1bb94676937c74b2c (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-07-13T14:17:10Z (GMT) No. of bitstreams: 1 silviadecastrodebarros.pdf: 3863081 bytes, checksum: ab4a7e88d1525ed1bb94676937c74b2c (MD5) / Made available in DSpace on 2016-07-13T14:17:10Z (GMT). No. of bitstreams: 1 silviadecastrodebarros.pdf: 3863081 bytes, checksum: ab4a7e88d1525ed1bb94676937c74b2c (MD5) Previous issue date: 2015-11-10 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O presente trabalho investiga historicamente o ensino de Geometria na formação de professores primários em Minas Gerais, entre as décadas de 1890 e 1940. Procurou-se responder às questões: Quais conteúdos de Geometria eram ensinados aos futuros professores nas Escolas Normais? Como esse saber foi tratado na formação dos normalistas? A quais finalidades respondia a Geometria presente na formação? Para tanto, a História Cultural foi utilizada como ferramental teórico-metodológico, juntamente com aportes da História da educação matemática. As fontes analisadas foram: legislações para as Escolas Normais; cadernos de alunas da professora mineira Alda Lodi, uma das fundadoras da Escola de Aperfeiçoamento de Belo Horizonte; dois livros que abordam a Geometria e que constavam da biblioteca da referida professora; e exemplares da Revista de Ensino de Minas Gerais. No período abrangido por essa pesquisa fervilhava o movimento da Escola Nova, que convivia ainda com o Método Intuitivo materializado nas Lições de Coisas. A pesquisa evidenciou a presença reduzida da geometria plana e espacial na formação de normalistas, sobretudo quando comparada à Aritmética ou ao Desenho. Observou-se também o afastamento da Geometria da prática docente; sendo uma disciplina mais próxima da cultura do secundário, que da cultura profissional, servindo como aplicação para a Aritmética. / This paper historically investigates Geometry teaching of elementary school teachers’ training in Minas Gerais, Brazil, between the decades of 1890 from 1940. It tried to answer some questions: What were the geometry contents taught to those teachers in Normal Schools? How did they deal with this knowledge in elementary school teachers’ training? What were the purposes of geometry at that time? Therefore, it was used cultural history as theoretical and methodological tool, together with resources of the history of mathematics education. The analyzed sources were: Normal Schools legislation; female students’ notebooks from teacher Alda Lodi, one of the founders of Belo Horizonte Teaching Improvement School; two books that mentioned Geometry and were in that teacher’s library; and copies of the Minas Gerais Journal of Education. In the period comprised in this research the Free School Movement boomed, and it also coexisted with the Intuitive Method implemented in Primary Object Lessons. The research showed the poor presence of flat and spatial Geometry in elementary school teachers’ training, especially when compared to Arithmetic or Drawing. We also observed a distance between Geometry and teaching practice, which became a subject of high school culture rather than of professional culture, serving as application for the arithmetic.

Submitted by Geandra Rodrigues (geandrar@gmail.com) on 2018-07-05T11:54:32Z No. of bitstreams: 1 mariadasgracasschinnigerassungarcia.pdf: 1134387 bytes, checksum: 64c073ab58c8c7633340840516ced970 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2018-07-06T14:14:47Z (GMT) No. of bitstreams: 1 mariadasgracasschinnigerassungarcia.pdf: 1134387 bytes, checksum: 64c073ab58c8c7633340840516ced970 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2018-07-06T14:15:06Z (GMT) No. of bitstreams: 1 mariadasgracasschinnigerassungarcia.pdf: 1134387 bytes, checksum: 64c073ab58c8c7633340840516ced970 (MD5) / Made available in DSpace on 2018-07-06T14:15:06Z (GMT). No. of bitstreams: 1 mariadasgracasschinnigerassungarcia.pdf: 1134387 bytes, checksum: 64c073ab58c8c7633340840516ced970 (MD5) Previous issue date: 2018-04-02 / Esta dissertação estuda historicamente a presença do Desenho como matéria escolar no curso primário. Tomou-se como questão norteadora do estudo: Quais as finalidades da matéria Desenho lidas nas diretivas oficiais nacionais e mineiras das décadas de 1940-1950? No recorte temporal adotado – as décadas de 1940 e 1950 – o ensino é regido pela Lei Orgânica do Ensino Primário, de 1946. O estudo se situa no campo da História da educação matemática, tomando como referenciais teóricometodológicos aportes advindos da História Cultural, e considerando-se a cultura escolar e as disciplinas ou matérias como objetos históricos. Foram examinados, além da Lei Orgânica para o Ensino Primário, os Programas em Experiência de Minas Gerais, publicados na década de 1940 e republicados até 1961. Esses Programas trouxeram o Desenho integrado às áreas de ensino, apresentado como matéria auxiliar, que aparece como atividade de expressão, observação e intuição. Assume características rudimentares, ou seja, passa a constituir um ensino consubstanciado com o caráter de iniciação aos saberes escolares; sem nada a dever aos saberes de referência. Pautado nas finalidades dadas para o ensino primário, se mostrou alinhado às matérias escolares constitutivas deste ciclo escolar – aritmética, geometria, língua pátria, história e geografia, ciências e higiene. / This dissertation historically studies the presence of Design as primary school subject matter. It was taken as the guiding question of the study: what are the finities of the drawing matter read in the official national and mining directives of the 1940-1950 decades? In the 1940s and 1950s, the teaching is governed by the Organic Law of Primary Education of 1946. The study is situated in the field of History of Mathematics Education, taking as theoretical-methodological references contributions from Cultural History, and considering school culture and disciplines or subjects as historical objects. In addition to the Organic Law for Primary Education, the Programs in Experience of Minas Gerais, published in the 1940s and republished until 1961, were examined. These programs brought the Integrated Design to teaching areas, presented as an auxiliary material, which appears as an activity of expression, observation and intuition. It assumes characteristics of a discipline based on the perspective of rudiments. By these considerations, it begins to constitute a teaching consubstantiated with the character of initiation to the school knowledge; with nothing to do with reference knowledge. Targeted at the primary education objectives, it was aligned with the school subjects constituting this school year - arithmetic, geometry, mother tongue, history and geography, science and hygiene

When the effects of wind and air pressure combine with a high tide to give unusually high water levels this can lead to severe coastal flooding. This happened in England in early 1953 when 307 people died in the East Coast Flood. In Britain today such events, now called storm surges, are forecast daily using computer models from the National Oceanographic Centre in Liverpool, formerly the Liverpool Tidal Institute (TI). In 1919, when TI was established, such events were considered unpredictable. TI's researchers, Joseph Proudman (1888-1975), Arthur Doodson (1890-1968), Robert Henry Corkan (1906-1952) and Jack Rossiter (1919-1972), did much mathematical work to attempt to change this. In 1959 Rossiter published a set of statistical formulae to forecast storm surges on the East Coast and a national warning system was predicting such events using these formulae. At this point TI believed they had made surges at least as predictable as they could with their existing methods. This thesis provides a narrative of how this perceived rise in the predictability of surges happened, analysing how TI worked to achieve it between 1919 and 1959 by following two interwoven, contingent and contested threads: practices of calculation and patronage. A key aspect of this thesis is the attention I pay to material practices of calculation: the methods, technologies and management practices TI's researchers used in their mathematical work on storm surge forecasting. This is the first study by historians of oceanography or meteorology that pays this detailed level of attention to such practices in the construction of forecasting formulae. As well as using published accounts, I analyse statistical research in the making, through notes, calculations, graphs and tables produced by TI's researchers. They used particular practices of calculation to construct storm surges as calculable and predictable scientific objects of a specific kind. First they defined storm surges as the residuals derived from subtracting tidal predictions from observations. They then decided to use multiple regression, correlating their residuals with pressure gradients, to make surges predictable. By considering TI's practices of calculation the thesis adds to the literature on mathematical research as embodied and material, showing how particular practices were used to make a specific phenomenon predictable. I combine this attention to mathematical practice with analysis of why TI's researchers did this work. US historians have emphasised naval patronage of physical oceanography in this period but there is very little secondary literature for the British case. The thesis provides a British case study of patronage of physical oceanography, emphasising the influence on TI's work not only of naval patronage but also of local government, civil state and industrial patronage. Before TI's establishment Proudman argued that it should research storm surges to improve the Laplacian theory of tides. However, when the new Institute received patronage from the local shipping industry this changed and the work on forecasting surges was initially done as part of a project to improve the accuracy of tidal predictions, earning TI further patronage from the local shipping industry. After a flooding event in 1928 the reasons for the work and the patronage again shifted. Between then and 1959 TI did this work on commission from various patrons, including local government, civil state and military actors, which connected their patronage to national debates about state involvement in flood defence. To understand why TI's researchers worked on forecasting surges I analyse this complex mix of patrons and motivations. I argue that such complex patronage patterns could be fruitfully explored by other historians to further existing debates on the patronage of oceanography.

Students today need to be taught not only the real life context of their mathematics lessons but also the historical context of the theory behind their mathematics lessons. Using history to teach mathematics, makes your lessons not only interesting but more meaningful to a large percentage of your students as they are interested in knowing the who, how and why about certain rules, theorems, formulas that they use everyday in class. Students are captivated by learning the history behind mathematicians, rules, etc. and therefore can link the lesson to something in history and a concept. Even learning the mathematics behind historical events motivates and interests them. They cannot get enough!

This diploma thesis deals with the comparison of historical solutions of word problems with student solutions. Its aim was to describe how students solve historical word problems, while looking for analogies between student and historical solutions. This intention led me to a better understanding of student solutions. The theoretical part of the thesis describes important concepts for algebraic word problems, such as a variable, algebraic expression or algebraic word problem. In the historical part I describe chronologically the development of algebra from antiquity through the Middle Ages and the Renaissance to the Baroque. In each period, I mention important mathematicians of the time and present several solved word problems. In most cases, I analyze these solutions from the perspective of today's mathematics. The theoretical part describes the research that took place at the eight-year grammar school. As part of the research, I gave students 6 historical tasks across historical periods and then analyzed the ways in which students solved problems. I found that for most of the tasks, there were solutions similar to the historical solutions among the student's solutions. Some historical methods appeared very often. An example is the use of addition instead of multiplication, or division, as used by the...

[pt] Este trabalho tem como objetivo principal apresentar e demonstrar os teoremas de Miquel que tratam de retas, círculos e suas interseções, assim como a versão de Clifford para os mesmos. Mais especificamente do teorema referente ao pentágono que afirma que dado um pentágono, o prolongamento dos seus lados formam cinco triângulos e os círculos circunscritos a esses triângulos se intersectam dois a dois e os pontos de interseção distintos dos vértices estão sobre uma mesma circunferência. Os teoremas de Miquel são demonstrados de forma original, com exceção do teorema citado, cuja prova é igual àquela do artigo original, a menos de mudanças de notação e maior detalhamento de argumentos. A versão de Clifford para esse teorema é provada apenas com o uso de argumentos de geometria euclidiana, diferente do proposto em seu artigo, que lança mão de ferramentas da geometria projetiva e das curvas algébricas para chegar à sua tese. Também é feita uma demonstração para a generalização do teorema acima ao se tomar n retas. Além disso, este trabalho apresenta uma proposta de atividades pedagógicas com o uso do software de geometria dinâmica GeoGebra, como ferramenta facilitadora à visualização e dedução dos teoremas mais importantes do trabalho. / [en] This work aims to present and demonstrate Miquel s theorems dealing with straigt lines, circles and their intersections, as well as Clifford s version of the same theorems. More specifically regarding the theorem that makes reference to the pentagon, which asserts that given a pentagon, the extension of its sides form five triangles and the circles circumscribed to these triangles intersect two by two, and the intersection points, not considering the vertices, are on the same circumference. Miquel s theorems are presented in an original way, with the exception of the above theorem, which is equal to the original one, apart from little changes of notation and more detailed arguments. Clifford s version of this theorem is presented with the use of Euclidean geometry arguments differing from the one proposed in his article, which makes use of tools of projective geometry and algebraic curves to get to his thesis. There is also a demonstration for the generalization of the above theorem when n straigt lines are taken. In addition, this work proposes a pedagogical activity using the dynamic geometry software GeoGebra, as a facilitating tool for viewing and deduction of the most important theorems presented in this work.

Notre contexte pratique — nous enseignons à des élèves doués de cinquième année suivant le programme international — a grandement influencé la présente recherche. En effet, le Programme primaire international (Organisation du Baccalauréat International, 2007) propose un enseignement par thèmes transdisciplinaires, dont un s’intitulant Où nous nous situons dans l’espace et le temps. Aussi, nos élèves sont tenus de suivre le Programme de formation de l’école québécoise (MÉLS Ministère de l'Éducation du Loisir et du Sport, 2001) avec le développement, notamment, de la compétence Résoudre une situation-problème et l’introduction d’une nouveauté : les repères culturels. Après une revue de la littérature, l’histoire des mathématiques nous semble tout indiquée. Toutefois, il existe peu de ressources pédagogiques pour les enseignants du primaire. Nous proposons donc d’en créer, nous appuyant sur l’approche constructiviste, approche prônée par nos deux programmes d’études (OBI et MÉLS). Nous relevons donc les avantages à intégrer l’histoire des mathématiques pour les élèves (intérêt et motivation accrus, changement dans leur façon de percevoir les mathématiques et amélioration de leurs apprentissages et de leur compréhension des mathématiques). Nous soulignons également les difficultés à introduire une approche historique à l’enseignement des mathématiques et proposons diverses façons de le faire. Puis, les concepts mathématiques à l’étude, à savoir l’arithmétique, et la numération, sont définis et nous voyons leur importance dans le programme de mathématiques du primaire. Nous décrivons ensuite les six systèmes de numération retenus (sumérien, égyptien, babylonien, chinois, romain et maya) ainsi que notre système actuel : le système indo-arabe. Enfin, nous abordons les difficultés que certaines pratiques des enseignants ou des manuels scolaires posent aux élèves en numération. Nous situons ensuite notre étude au sein de la recherche en sciences de l’éducation en nous attardant à la recherche appliquée ou dite pédagogique et plus particulièrement aux apports des recherches menées par des praticiens (un rapprochement entre la recherche et la pratique, une amélioration de l’enseignement et/ou de l’apprentissage, une réflexion de l’intérieur sur la pratique enseignante et une meilleure connaissance du milieu). Aussi, nous exposons les risques de biais qu’il est possible de rencontrer dans une recherche pédagogique, et ce, pour mieux les éviter. Nous enchaînons avec une description de nos outils de collecte de données et rappelons les exigences de la rigueur scientifique. Ce n’est qu’ensuite que nous décrivons notre séquence d’enseignement/apprentissage en détaillant chacune des activités. Ces activités consistent notamment à découvrir comment différents systèmes de numération fonctionnent (à l’aide de feuilles de travail et de notations anciennes), puis comment ces mêmes peuples effectuaient leurs additions et leurs soustractions et finalement, comment ils effectuaient les multiplications et les divisions. Enfin, nous analysons nos données à partir de notre journal de bord quotidien bonifié par les enregistrements vidéo, les affiches des élèves, les réponses aux tests de compréhension et au questionnaire d’appréciation. Notre étude nous amène à conclure à la pertinence de cette séquence pour notre milieu : l’intérêt et la motivation suscités, la perception des mathématiques et les apprentissages réalisés. Nous revenons également sur le constructivisme et une dimension non prévue : le développement de la communication mathématique. / Our practical context -we teach gifted fifth grade students in an International School- has greatly influenced this research. Indeed, the International Primary Years Programme (International Baccalaureate Organization, 2007) fosters transdisciplinary themes, including one intitled Where we are in place and time. Our students are also expected to follow the Quebec education program schools (Ministry of Education, Recreation and Sport, 2001) with the development of competencies such as: To solve situational problem and the introduction of a novelty: the Cultural References. After the literature review, the history of mathematics seems very appropriate. However, there are few educational resources for primary teachers. This is the reason why we propose creating the resources by drawing upon the constructivist approach, an approach recommended by our two curricula (OBI and MELS). We bring to light the advantages of integrating the history of mathematics for students (increased interest and motivation, change in their perception of mathematics and improvement in learning and understanding mathematics). We also highlight the difficulties in introducing a historical approach to teaching mathematics and suggest various ways to explore it. Then we define the mathematical concepts of the study: arithmetic and counting and we remark their importance in the Primary Mathematics Curriculum. We then describe the six selected number systems (Sumerian, Egyptian, Babylonian, Chinese, Roman and Mayan) as well as our current system: the Indo-Arabic system. Finally, we discuss the difficulties students may encounter due to some teaching practices or textbooks on counting. We situate our study in the research of science of education especially on applied research and the contributions of the teacher research reconciliation between research and practice, the improvement of teaching and / or learning and a reflection within the teaching practice). Also, we reveal the possible biases that can be encountered in a pedagogical research and thus, to better avoid them. Finally, we describe the tools used to collect our data and look at the requirements for scientific rigor. Next, we describe our teaching sequence activities in details. These activities include the discovery of how the different number systems work (using worksheets and old notations) and how the people using the same systems do their additions and subtractions and how they do their multiplications and divisions. Finally, we analyze our data from a daily diary supported by video recordings, students’ posters, the comprehension tests and the evaluation questionnaire. Our study leads us to conclude the relevance of this sequence in our context: interest and motivation, perception of mathematics and learning achieved. We also discuss constructivism and a dimension not provided: the development of mathematical communication.

Almost two thousand years ago, Claudius Ptolemy created a guide to drawing maps of the world, identifying the names and coordinates of over 8,000 settlements and geographical features. Using the coordinates of those cities and landmarks which have been identified with modern locations, a series of best-fit transformations has been applied to several of Ptolemy’s regional maps, those of Britain, Spain, and Italy. The transformations relate Ptolemy’s coordinates to their modern equivalents by rotation and skewed scaling. These reflect the types of error that appear in Ptolemy’s data, namely those of distance and orientation. The mathematical techniques involved in this process are all modern. However, these techniques have been altered in order to deal with the historical difficulties of Ptolemy’s maps. To think of Ptolemy’s data as similar to that collected from a modern random sampling of a population and to apply unbiased statistical methods to it would be erroneous. Ptolemy’s data is biased, and the nature of that bias is going to be informed by the history of the data. Using such methods as cluster analysis, Procrustes analysis, and multidimensional scaling, we aimed to assess numerically the accuracy of Ptolemy’s maps. We also investigated the nature of the errors in the data and whether or not these could be linked to historical developments in the areas mapped.

Orientador: Lourdes de la Rosa Onuchic / Banca: Sergio Roberto Nobre / Banca: Norma Suely Gomes Allevato / Resumo: Como professor de uma Faculdade de Engenharia e responsável por disciplinas de Cálculo Diferencial e Integral, pude vivenciar muitas inquietações no processo de ensino e aprendizagem desse ramo da Matemática e constatar dificuldades encontradas nesse processo e, em especial, no ensino e na aprendizagem de Integrais. Nosso Fenômeno de Interesse naturalmente surgiu dessa inquietação. Apoiados na Metodologia de Pesquisa de Romberg desenvolvemos toda nossa Pesquisa seguindo, de perto, um modelo de desenvolvimento criado por nós. Depois de relacionarmos nossas ideias com ideias de outros, foi criada, a Pergunta da Pesquisa que se tornou então, nosso Problema. Trabalhando com a História da Integral como parte da História da Matemática, com Resolução de Problemas e a Metodologia de Ensino-Aprendizagem-Avaliação de Matemática através da Resolução de Problemas, como metodologia de trabalho, analisamos uma sala de aula de um curso de engenharia onde o ensino e a aprendizagem de Cálculo Diferencial e Integral era nosso objetivo. Foi criado um projeto, aplicado em doze encontros de cem minutos cada. Dessa aplicação coletamos evidências que, confrontadas à Pergunta da Pesquisa puderam nos conduzir à resposta da Pergunta feita. Os alunos nesse processo foram participantes e assumidos como co-construtores de seu próprio conhecimento. / Abstract: As a professor of a College of Engineering and responsible for courses in differential and integral calculus, I could experience many concerns in the teaching and learning of this branch of mathematics and find difficulties in that process, in particular in teaching and learning of Integrals. Our Phenomenon of Interest naturally arose that concern. Supported by Romberg Research Methodology, we developed all our research following closely a development model created by us. After we related our ideas with ideas of others, it was created the research question which then became our problem. Working with the History of Integral as part of the History of Mathematics with Problem Solving Methodology and Teaching-Learning Assessment of Mathematics through Problem Solving, as work methodology, we analyzed a classroom of an engineering course where the teaching and learning of differential and integral calculus was our goal. It was created a project implemented in twelve meetings of a hundred minutes each. This application collected evidences that, faced the Question of the Research, lead us to answer the Question asked. The students were participants in that process and assumed to be co-constructors of their own knowledge. / Mestre

Durant l'ère Meiji, afin d'occuper une position forte dans le concert des nations, le gouvernement japonais engage le pays dans un mouvement de modernisation. Dans le cadre de ce mouvement, les mathématiques occidentales, et en particulier la géométrie euclidienne, sont introduites dans l'enseignement. Cette décision est prise alors que, en raison du succès des mathématiques traditionnelles (wasan), aucune traduction sur le sujet n'est disponible. Mes travaux s'intéressent aux premiers manuels de géométrie élémentaire, qui ont été élaborés, diffusés et utilisés dans ce transfert scientifique. Une grille d'analyse centrée sur les questions du langage et des outils logiques est déployée pour mettre en évidence les différentes phases dans l'importation et l'adaptation des connaissances occidentales / During the Meijing era, the political context in East Asia led the Japanese authorities to embark on a nationwide modernization program. This resulted in the introduction of Western mathematics, and especially Euclidean geometry into Japanese education. However, as traditional mathematics (was an) were very successful at that time, there were no Japanese translations of texts dealing with this new geometry available at this time. My work focuses on the first Japanese textbooks that were developed, distributed and used during this period of scientific transfer. My analysis concentrates on language and logical reasoning in order to highlight the various phases in the importation and adaptation of Western knowledge to the Japanese context