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This dissertation explores the use of history of mathematics in middle school mathematics. A rationale for the importance of the incorporation of historical dimensions (HD) of mathematics is provided through a review of the literature. The literature covers pedagogical, philosophical, psychological, and social issues and provides arguments for the use of history. The central argument is that history can help reveal significant aspects regarding the origins and evolutions of ideas that provide contexts for understanding the mathematical ideas. History can be used as a means to reflect on significant aspects—errors, contractions, challenges, breakthroughs, and changes—of mathematical developments. Noting recent NCTM (2000) calls for school math to include so-called process standards, I contend that incorporating the history of mathematics can be considered as part of this standard. This study examines how HD is addressed in a contemporary mathematics curriculum. Specifically, the study examines the Connected Mathematics Project (CMP) as a case. This curriculum has some historical references which triggered further exploration on how seriously the historical aspects are incorporated. The analysis and discussion focus on four CMP units and interviews with three curriculum experts, eight teachers, and 11 middle school students. The analysis of textbooks and interviews with the experts explore the nature and purpose of historical references in the curriculum. The interviews with teachers and students focus on their perspectives on the importance of HD in learning mathematics. This study examines specifically historical incorporations of the concepts of fractions, negative numbers, the Pythagorean Theorem, and irrational numbers. The analysis reveals that CMP exhibits some level of historical awareness, but the incorporation of HD was not systematically or seriously considered in the development of the curriculum. The interviews suggest that the teachers did not seriously use the limited historical aspects available in the textbooks. The experts’ and teachers’ interviews suggest skepticism about the relevance of HD for middle school mathematics. The teachers’ accounts indicate that students are most interested in topics that are related to their experience and to future applications. The students’ accounts do not fully support the teachers’ assessment of students’ interest in history. I contend that incorporating HD can complement instruction in ways that relate to students’ experiences and to applications besides adding an inquiry dimension to instruction. / text

Until recently, Taiwan elementary mathematics curriculum has been changing for several times. The aim of this study is to refer to various curriculum reforms, and focus on the way ¡§Number and Calculation Standards¡¨ changed in the history of reforms. The specific objectives of this study: to refer to one curriculum standards and its subsequent standards and do pair wise comparison. To achieve the above objectives, the investigator referred to 7 target versions of mathematics curriculum standards: 41, 51, 57, 64, 82, 89, 92 (R.O.C year). The comparison was done qualitatively, using historical research methodology. The main research findings are the differences in the above 6 pair wise comparisons. 1. The change from Year 41 to Year 51: In the Year 51, the part on Writing numbers in Chinese characters was de-emphasized. Emphasis was on Ordinal numbers, division thinking, mental arithmetic and written algorithm. The size of numbers reduced to 4-digits (due to a change in currency, 4 dollars to 1 New Taiwan dollar). 2. The change from Year 51 to 57: more focus on symbols, did not require the revision on what was learned in previous year. 3. The change from Year 57 to 64: de-emphasized on mental arithmetic and written calculation; emphasized on Inverses, multiplication/division on ¡§0¡¦ and ¡§1¡¨, ratio, approximation, negative numbers and use of electronic calculators. 4. The change from Year 64 to 82: no need to include negative numbers and abacus. Emphasized on two-step problems, number line, and reading multiplication tables. 5. The change from Year 82 to 89: de-emphasis on odd and even numbers; emphasis on realistic contexts, understanding vertical algorithm. 6. The change from Year 89 to 91: no need to use calculators to check working; emphasis on vertical algorithm, whole number calculations, and the connections of multiples/factors, rate/speed, and, fractions/decimals.

Made available in DSpace on 2014-06-11T19:31:42Z (GMT). No. of bitstreams: 0 Previous issue date: 2005-05-04Bitstream added on 2014-06-13T20:02:45Z : No. of bitstreams: 1 martins_jcg_dr_rcla.pdf: 1204232 bytes, checksum: 1076800f1b73a083b5f84979e3080de1 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Tem sido unanimidade entre os filósofos da Matemática a compreensão de que as revoluções científicas, na forma como são apresentadas em A Estrutura das Revoluções Científicas, de Thomas S. Kuhn, não ocorrem na Matemática. Este trabalho pretende o contrário: fundado no Modelo Teórico dos Campos Semânticos e tendo a história da Matemática como cenário mais especificamente, a história da Álgebra esta tese foi elaborada para mostrar que a obra Kitab al mukhtasar fi hisab al-jabr wa l-muqabalah, de al- Khwarizmi, inaugura o primeiro período de pesquisa normal no desenvolvimento da Álgebra na Europa, um período altamente cumulativo e extraordinariamente bem sucedido em seus objetivos paradigmáticos e que se estendeu até as décadas iniciais do século XIX. Mostramos, ainda, que a demonstração do, hoje denominado, Teorema Fundamental da Álgebra, por Gauss, e a publicação do trabalho Sobre a resolução algébrica de equações, de Abel, trouxe à luz, na forma de um fato, uma anomalia irresolúvel do primeiro paradigma da Álgebra no Velho Continente. A partir daí, abriu-se um período de pesquisa extraordinária no âmbito dessa disciplina um período revolucionário de onde viria emergir um novo período de pesquisa normal, um novo paradigma para a Álgebra os sistemas algébricos abstratos fundado nas realizações matemáticas de Galois, Peacock e Hamilton. / Thus far, all the Mathematical Philosophers have unanimously agreed that the scientific revolutions, as it is presented in The Structures of the Scientific Revolutions, by Thomas S. Kuhn, do not take place in Mathematics. This paper intends to prove just the opposite: founded on The Theoretical Models of the Semantic Fields and considering the History of Mathematics as the scenery in question more precisely, the History of Algebra this thesis was prepared to show that the work Kitab al mukhtasar fi hisab al-jabr wa l muqabalah, by al-Khwarizmi, gives birth to the first period of normal research in the European development of Algebra, a highly cumulative and extraordinarily well succeeded period in its paradigmatic objectives, which extended until the first decades of the Nineteenth Century. We further show that the proof of the so called The Fundamental Theorem of Algebra, by Gauss, and the publication of Abel's work on The Algebraic Solutions of Equations, brought to light, as a fact, an unsolvable anomaly of the first paradigm of Algebra in the Old Continent which, from there on, caused the beginning of an extraordinary research period in this particular field in fact, a revolutionary period from which would surface a new time of normal research, a new algebraic paradigm the abstract algebraic systems based on the mathematical achievements of Galois, Peacock and Hamilton.

This dissertation deals with aspects of axiomatization, intuition and visualization in thehistory of mathematics. Particular focus is put on the end of the 19th century, before DavidHilbert's (1862–1943) work on the axiomatization of Euclidean geometry. The thesis consistsof 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. Athorough analysis of his foundational work is made as well as an investigation into his generalview on science and mathematics. Furthermore, his thoughts on geometry and its nature andwhat consequences his view has for how he proceeds in developing the axiomatic system, isstudied. In the second paper different aspects of visualizations in mathematics areinvestigated. In particular, it is argued that the meaning of a visualization is not revealed bythe visualization and that a visualization can be problematic to a person if this person, due to alimited 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 whichfollowed in the wake of Karl Weierstrass' (1815–1897) construction of a nowheredifferentiable function in 1872. In the third paper certain aspects of the thinking of the twoscientists Felix Klein (1849–1925) and Heinrich Hertz (1857–1894) are studied. It isinvestigated how Klein and Hertz related to the idea of naïve images and visual thinkingshortly before the development of modern axiomatics. Klein in several of his writingsemphasized his belief that intuition plays an important part in mathematics. Hertz argued thatwe form images in our mind when we experience the world, but these images may containelements that do not exist in nature.

Title: Life and Work of Josef Úlehla Author: Lukáš Vízek Department: Department of Mathematics Education Supervisor: prof. RNDr. Martina Bečvářová, Ph.D. Abstract: Josef Úlehla (1852-1933) was an important Czech teacher, he taught mathematics and natural sciences at primary and secondary schools in Moravia. He wrote a number of monographs, textbooks, articles and translations of foreign language publications. This thesis describes Úlehla's life, brings the detail analysis and evaluation of his mathematical works and mentions his other publications. The text contains a lot of illustrations and the thesis is supplemented by factual attachments. Keywords: Josef Úlehla, mathematics, education, history

Du premier congrès des scientifiques italiens de 1839 à la veille de la Grande Guerre, de nombreux mathématiciens italiens ont pris part à la vie politique de leur pays. Cette thèse examine les différentes modalités de cet engagement : le mouvement national, qui se décline dans le domaine scientifique par une forme spécifique de patriotisme dans un contexte d’unification de l’Italie, en est un aspect. Mais il s’agit d’analyser plus généralement la façon dont le statut de mathématicien peut être réinvesti dans le champ politique pour fonder un discours de légitimation, une forme d’expertise, revendiquer un regard spécifique sur le politique. Cela suppose de penser la circulation entre champ mathématique et politique avec les outils de l’histoire des intellectuels : comparer les stratégies d’ascension dans ces deux champs, analyser comment les conflits s’y transposent, comment les acteurs répartissent leur temps entre les différentes activités. Il s’agit donc de comprendre comment les transformations de la vie politique italienne autour de l’unification ont permis l’émergence de nouveaux hommes politiques, de mesurer leur réception par le milieu politique mais aussi dans le champ académique, ainsi que la façon dont leur double appartenance a pu affecter leur façon d’être mathématiciens. / From the first congress of Italian scientists in 1839 to the eve of World War I, many Italian mathematicians took part to the political life of their country. This PhD deals with the different modalities of this involvement: Italian national movement, which results in the scientific field in a specific shape of patriotism in a context of Italian unification, is one aspect. But I intend to draw a more general analysis of the way the position of a mathematician can be used in the political field to found a legitimating discourse, some kind of expertise, or to claim a specific way to consider political questions. In order to do so, I will need to consider circulations between mathematical and political fields with tools the history of intellectuals: I will thus compare the strategies of advancement in those two fields, analyze how the conflicts are transposed and how the actors divide their time between their different activities. So I intend to understand how the transformations of the Italian political life around national unification made possible the emergence of new politicians, to assess their reception in political and academic worlds and the way their double belonging influenced their practice as mathematicians.

This dissertation deals with the development of mathematical concepts from a historical and didactical perspective. In particular, the development of concepts in mathematical analysis during the 19th century is considered. The thesis consists of a summary and three papers. In the first paper we investigate the Swedish mathematician E.G. Björling's contribution to uniform convergence in connection with Cauchy's sum theorem from 1821. In connection to Björling's convergence theory we discuss some modern interpretations of Cauchy's expression x=1/n. We also consider Björling's convergence conditions in view of Grattan-Guinness distinction between history and heritage. In the second paper we study visualizations in mathematics from historical and didactical perspectives. We consider some historical debates regarding the role of intuition and visual thinking in mathematics. We also consider the problem of what a visualization in mathematics can achieve in learning situations. In an empirical study we investigate what mathematical conclusions university students made on the basis of a visualization. In the third paper we consider Cauchy's theorem on power series expansions of complex valued functions on the basis of a paper written by E.G. Björling in 1852. We discuss Björling's, Lamarle's and Cauchy's different conditions for expanding a complex valued function in a power seris. In the third paper we also discuss the problem of the ambiguites of fundamental concpets that existed during the mid-19th century. We argue that Cauchy's and Lamarle's proofs of Cauchy's theorem on power series expansions of complex valued functions are correct on the basis of their own definitions of the fundamental concepts involved.

Submitted by CLÁUDIO MÁRCIO RIBEIRO MAGALHÃES null (claudiormagalhaes@uninove.br) on 2016-06-08T19:54:34Z No. of bitstreams: 1 Luis Antonio Verney O Verdadeiro Método de Estudar - Uma Contribuição para o Ensino em Portugal e no Brasil.pdf: 47738896 bytes, checksum: 33705b028f640da23cadbe4d5f53a20c (MD5) / Approved for entry into archive by Ana Paula Grisoto (grisotoana@reitoria.unesp.br) on 2016-06-09T16:57:00Z (GMT) No. of bitstreams: 1 magalhaes_cmr_dr_rcla.pdf: 47738896 bytes, checksum: 33705b028f640da23cadbe4d5f53a20c (MD5) / Made available in DSpace on 2016-06-09T16:57:00Z (GMT). No. of bitstreams: 1 magalhaes_cmr_dr_rcla.pdf: 47738896 bytes, checksum: 33705b028f640da23cadbe4d5f53a20c (MD5) Previous issue date: 2016-04-25 / Este trabalho se constitui da leitura de algumas cartas, que tratam de Filosofia, Lógica e Física, presentes no polêmico livro O Verdadeiro Método de Estudar. O autor, o controverso Luís António Verney, ao publicar essa obra, acreditou na possibilidade de mudança da Educação em Portugal. Apoiado nas ideias do movimento europeu que ficou conhecido como Iluminismo, Verney elaborou um riquíssimo texto no qual faz duras críticas ao ensino ministrado pelos padres da Companhia de Jesus, naquela época em Portugal. Ao sugerir o rompimento com esse sistema de ensino, o autor procura introduzir um ensino pautado no experimentalismo, colocando a Matemática como conhecimento fundamental para o estudo da Física e simultaneamente como ferramenta social necessária para o bem da nação. / This work consist at reading of some letters dealing with Philosophy, Logic and Physics presented at polemic book, The True Method of Study, authored by the controversial Luís António Verney that by publishing his work believed in the possibility of changes in education of Portugal. Supported on the ideas of European movement known as the Enlightenment Verney produced a rich text in which he harshly criticized the teaching realized by priests of Society of Jesus at that moment in Portugal. To suggest a rupture with that education system, the author tried to introduce teaching based on the experimentalism, introducing mathematics as fundamental knowledge for the physics study and simultaneously as a social tool necessary for the good of the nation.

Orientador: Romulo Campos Lins / Banca: Antonio Vicente Marafioti Garnica / Banca: Francisco César Polcino Miles / Banca: Ligia Arantes Sad / Banca: Marcos Vieira Teixeira / Resumo: Tem sido unanimidade entre os filósofos da Matemática a compreensão de que as revoluções científicas, na forma como são apresentadas em A Estrutura das Revoluções Científicas, de Thomas S. Kuhn, não ocorrem na Matemática. Este trabalho pretende o contrário: fundado no Modelo Teórico dos Campos Semânticos e tendo a história da Matemática como cenário mais especificamente, a história da Álgebra esta tese foi elaborada para mostrar que a obra Kitab al mukhtasar fi hisab al-jabr wa’l-muqabalah, de al- Khwarizmi, inaugura o primeiro período de pesquisa normal no desenvolvimento da Álgebra na Europa, um período altamente cumulativo e extraordinariamente bem sucedido em seus objetivos paradigmáticos e que se estendeu até as décadas iniciais do século XIX. Mostramos, ainda, que a demonstração do, hoje denominado, Teorema Fundamental da Álgebra, por Gauss, e a publicação do trabalho Sobre a resolução algébrica de equações, de Abel, trouxe à luz, na forma de um fato, uma anomalia irresolúvel do primeiro paradigma da Álgebra no Velho Continente. A partir daí, abriu-se um período de pesquisa extraordinária no âmbito dessa disciplina um período revolucionário de onde viria emergir um novo período de pesquisa normal, um novo paradigma para a Álgebra os sistemas algébricos abstratos fundado nas realizações matemáticas de Galois, Peacock e Hamilton. / Abstract: Thus far, all the Mathematical Philosophers have unanimously agreed that the scientific revolutions, as it is presented in The Structures of the Scientific Revolutions, by Thomas S. Kuhn, do not take place in Mathematics. This paper intends to prove just the opposite: founded on The Theoretical Models of the Semantic Fields and considering the History of Mathematics as the scenery in question more precisely, the History of Algebra this thesis was prepared to show that the work Kitab al mukhtasar fi hisab al-jabr wa’l muqabalah, by al-Khwarizmi, gives birth to the first period of normal research in the European development of Algebra, a highly cumulative and extraordinarily well succeeded period in its paradigmatic objectives, which extended until the first decades of the Nineteenth Century. We further show that the proof of the so called The Fundamental Theorem of Algebra, by Gauss, and the publication of Abel's work on The Algebraic Solutions of Equations, brought to light, as a fact, an unsolvable anomaly of the first paradigm of Algebra in the Old Continent which, from there on, caused the beginning of an extraordinary research period in this particular field in fact, a revolutionary period from which would surface a new time of normal research, a new algebraic paradigm the abstract algebraic systems based on the mathematical achievements of Galois, Peacock and Hamilton. / Doutor