Obstáculos superados pelos matemáticos no passado e vivenciados pelos alunos na atualidade : a polêmica multiplicação de números inteiros

In Mathematics literature some records highlight the difficulties encountered in the teaching-learning process of integers. In the past, and for a long time, many mathematicians have experienced and overcome such difficulties, which become epistemological obstacles imposed on the students and teachers nowadays. The present work comprises the results of a research conducted in the city of Natal, Brazil, in the first half of 2010, at a state school and at a federal university. It involved a total of 45 students: 20 middle high, 9 high school and 16 university students. The central aim of this study was to identify, on the one hand, which approach used for the justification of the multiplication between integers is better understood by the students and, on the other hand, the elements present in the justifications which contribute to surmount the epistemological obstacles in the processes of teaching and learning of integers. To that end, we tried to detect to which extent the epistemological obstacles faced by the students in the learning of integers get closer to the difficulties experienced by mathematicians throughout human history. Given the nature of our object of study, we have based the theoretical foundation of our research on works related to the daily life of Mathematics teaching, as well as on theorists who analyze the process of knowledge building. We conceived two research tools with the purpose of apprehending the following information about our subjects: school life; the diagnosis on the knowledge of integers and their operations, particularly the multiplication of two negative integers; the understanding of four different justifications, as elaborated by mathematicians, for the rule of signs in multiplication. Regarding the types of approach used to explain the rule of signs arithmetic, geometric, algebraic and axiomatic , we have identified in the fieldwork that, when multiplying two negative numbers, the students could better understand the arithmetic approach. Our findings indicate that the approach of the rule of signs which is considered by the majority of students to be the easiest one can be used to help understand the notion of unification of the number line, an obstacle widely known nowadays in the process of teaching-learning

Uma análise dos questionamentos dos alunos nas aulas de números complexos

This work presents a contribution for the studies reffering to the use of the History of Mathematics focusing on the improvement of the Teaching and Learning Process. It considers that the History of Matematics, as a way of giving meaning to the discipline and improve the quality of the Teaching and Learning Process. This research focuses on the questions of the students, classified in three categories of whys: the chronological, the logical and the pedagogical ones. Therefore, it is investigated the teaching of the Complex Numbers, from the questions of the students of the Centro Federal de Educação Tecnológica do Rio Grande do Norte (Educational Institution of Professional and Technology Education from Rio Grande do Norte). The work has the following goals: To classify and to analyse the questions of the students about the Complex Numbers in the classes of second grade of the High School, and to collate with the pointed categories used by Jones; To disccus what are the possible guidings that teachers of Mathematics can give to these questions; To present the resources needed to give support to the teacher in all things involving the History of Mathematics. Finally, to present a bibliographic research, trying to reveal supporting material to the teacher, with contents that articulate the Teaching of Mathematics with the History of Mathematics. It was found that the questionings of the pupils reffers more to the pedagogical whys, and the didatic books little contemplate other aspects of the history and little say about the sprouting and the evolution of methods of calculations used by us as well

O uso pedagógico de uma seqüência didática para a aquisição de algumas idéias relacionadas ao conceito de números complexos

The aim of the present work is to contribute to the teaching-learning process in Mathematics through an alternative which tries to motivate the student so that he/she will learn the basic concepts of Complex Numbers and realize that they are not pointless. Therefore, this work s general objective is to construct a didactic sequence which contains structured activities that intends to build up, in each student s thought, the concept of Complex Numbers. The didactic sequence is initially based on a review of the main historical aspects which begot the construction of those numbers. Based on these aspects, and the theories of Richard Skemp, was elaborated a sequence of structured activities linked with Maths history, having the solution of quadratic equations as a main starting point. This should make learning more accessible, because this concept permeates the students previous work and, thus, they should be more familiar with it. The methodological intervention began with the application of that sequence of activities with grade students in public schools who did not yet know the concept of Complex Numbers. It was performed in three phases: a draft study, a draft study II and the final study. Each phase was applied in a different institution, where the classes were randomly divided into groups and each group would discuss and write down the concepts they had developed about Complex Numbers. We also use of another instrument of analysis which consisted of a recorded interview of a semi-structured type, trying to find out the ways the students thought in order to construct their own concepts, i.e. the solutions of the previous activity. Their ideas about Complex Numbers were categorized according to their similarities and then analyzed. The results of the analysis show that the concepts constructed by the students were pertinent and that they complemented each other this supports the conclusion that the use of structured activities is an efficient alternative for the teaching of mathematics

Euler e os números pentagonais

The present investigation includes a study of Leonhard Euler and the pentagonal numbers is his article Mirabilibus Proprietatibus Numerorum Pentagonalium - E524. After a brief review of the life and work of Euler, we analyze the mathematical concepts covered in that article as well as its historical context. For this purpose, we explain the concept of figurate numbers, showing its mode of generation, as well as its geometric and algebraic representations. Then, we present a brief history of the search for the Eulerian pentagonal number theorem, based on his correspondence on the subject with Daniel Bernoulli, Nikolaus Bernoulli, Christian Goldbach and Jean Le Rond d'Alembert. At first, Euler states the theorem, but admits that he doesn t know to prove it. Finally, in a letter to Goldbach in 1750, he presents a demonstration, which is published in E541, along with an alternative proof. The expansion of the concept of pentagonal number is then explained and justified by compare the geometric and algebraic representations of the new pentagonal numbers pentagonal numbers with those of traditional pentagonal numbers. Then we explain to the pentagonal number theorem, that is, the fact that the infinite product(1 x)(1 xx)(1 x3)(1 x4)(1 x5)(1 x6)(1 x7)... is equal to the infinite series 1 x1 x2+x5+x7 x12 x15+x22+x26 ..., where the exponents are given by the pentagonal numbers (expanded) and the sign is determined by whether as more or less as the exponent is pentagonal number (traditional or expanded). We also mention that Euler relates the pentagonal number theorem to other parts of mathematics, such as the concept of partitions, generating functions, the theory of infinite products and the sum of divisors. We end with an explanation of Euler s demonstration pentagonal number theorem

Quando as mulheres se tornam mães: a história reprodutiva feminina numa visão evolucionista

In this thesis we tested evolutionary hypotheses, with empirically collected data, in a sample composed of pregnant Brazilian women. We consider that during pregnancy and soon after the baby's birth fundamental reproductive decisions take place, given the complete feminine involvement with the reproduction phenomenon. The results are presented in four empirical articles related to the history of female reproduction. The topics approached were mate selection, the life-history theory, the strategies of parental investment and postpartum depression. Data collection was accomplished through interviews with pregnant women and after the baby s birth, with a sample composed of women from two income classes (low income and middle class), in Natal, Brazil. With respect to mate selection, the results suggest that a real situation of reproductive mate selection shows significant differences when compared to the results obtained in studies involving potential mate selection (Article I). Considering the life-history theory, we have partially confirmed the hypothesis of the father`s absence influencing the development of the young female syndrome (Article II). In regard to parental investment strategies and the decrease of fatherhood uncertainty, we identified a larger attribution of the baby's resemblance after birth with the father, confirming our hypothesis (Article III). The results related to postpartum depression occurrence partially support the hypothesis that it is an evolutionary adaptation (Article IV). This thesis is part of a consolidation movement of Evolutionary Psychology in Brazil and it presents results on female reproductive history hitherto unpublished.

Rosa Luxemburg: marxismo e história

Ao interpretar o marxismo como unidade entre teoria e prática, Rosa Luxemburg lança os fundamentos da sua teoria da ação revolucionária, palavras com que poderíamos sintetizar o seu pensamento político, elaborado numa polêmica ininterrupta com o determinismo economicista da II Internacional.

História da ciência, interdisciplinaridade e ensino de física: o problema do demônio de Maxwell

Propomos que uma evolução de idéias científicas seja usada como instrumento de aprendizagem de conteúdos específicos e, em particular, para ressaltar como os conteúdos se articulam entre as disciplinas. Como exemplo, apresentamos um estudo sobre a proposta do demônio de Maxwell e discussões sobre sua exorcização, isto é, um estudo sobre a compreensão da natureza de um ser inteligente que atua dentro de um sistema físico e de como seria essa atuação. Estão envolvidos nesse problema fenômenos relacionados com várias teorias - Termodinâmica, Física Molecular, Mecânica Estatística, Teoria da Informação - dentro das disciplinas de Física, Química, Biologia, Computação. Entre diversas questões epistemológicas e conceituais aí contidas, será enfatizada a questão do objeto limitado de uma eoria científica, isto é, da limitação de seu significado aos fenômenos por ela compreendidos. A delimitação dos fenômenos estudados e as teorias e técnicas caracterizam a compreensão que vai realizar sua emergência concreta nos laboratórios. Essa compreensão vai dar também a possibilidade de atuação interdisciplinar.

O conceito de racionalidade em Habermas: a 'guinada lingüística' da teoria crítica

Este ensaio originalmente foi uma comunicação apresentada na UNICAMP. Foi escrito a propósito da visita de Habermas ao Brasil, esperada para o segundo semestre do ano passado. Seu objetivo imediato era fornecer algumas informações sobre as reflexões habermasianas mais recentes. Para isto, tentou-se inserir historicamente a proposta habermasiana de uma razão comunicativa no atual contexto de generalizada irrupção de formas discursivas fragmentárias relativistas e irracionalistas (o pós-estruturalismo francês e o pensiero debole italiano são os exemplos estudados). Sem pretender esgotar um tema tão complexo, tentou-se também levantar algumas questões sobre as possibilidades crítico-cognoscitivas da guinada linguística da filosofia de Habermas e de Apel.

História e culturas políticas: as concepções jurídicas evocadas pelos governos militares enquanto instrumento de obtenção de legitimidade

Este artigo se propõe a analisar o período dos governos militares brasileiros (1964-1985) sob a ótica da cultura da legalidade. Pretendemos demonstrar como a tomada do poder político em 1964, longe de se caracterizar apenas pelo emprego da força e do arbítrio, foi pautada por um esforço jurídico, produzido com base numa determinada teoria do direito constitucional, com ênfase no pensamento de Carl Schmitt e Hans Kelsen.

A teoria educacional no Ocidente: entre modernidade e pós-modernidade

O artigo trata das principais forças em filosofia da educação no Ocidente Moderno e Contemporâneo. Particularmente, destaca a posição do Brasil nos últimos anos, dado que localiza na contribuição de Paulo Freire, junto com o alemão Herbart e com o norte-americano Dewey, a formulação das linhas mestras da pedagogia moderna. O texto também faz menção ao trabalho atual do neopragmatismo, sob o qual nasce a filosofia da educação inspirada nos filósofos norte americanos Richard Rorty e Donald Davidson, que revolucionam atualmente não só a filosofia mas seus campos aplicados, como o direito, a religião, a política e, como não poderia deixar de ser, a educação.

Perspectivas para uma teoria geral dos novos direitos: uma leitura crítica sobre a biodiversidade e os conhecimentos tradicionais associados

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Ensino de eletrostática: a história da ciência contribuindo para a aquisição de subsunções

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

História e filosofia da biologia na formação inicial de professores: reflexões sobre o conceito de evolução biológica

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

As origens da teoria dos invariantes na Inglaterra e o Mécanique Analytique de Lagrange (1788)

Pós-graduação em Educação Matemática - IGCE

Informática na EJA: contribuições da teoria histórico-cultural

Pós-graduação em Educação - FFC