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

A difícil aceitação dos números negativos: um estudo da teoria dos números de Peter Barlow (1776-1862)

The present study seeks to present a historico-epistemological analysis of the development of the mathematical concept of negative number. In order to do so, we analyzed the different forms and conditions of the construction of mathematical knowledge in different mathematical communities and, thus, identified the characteristics in the establishment of this concept. By understanding the historically constructed barriers, especially, the ones having ontologicas significant, that made the concept of negative number incompatible with that of natural number, thereby hindering the development of the concept of negative, we were able to sketch the reasons for the rejection of negative numbers by the English author Peter Barlow (1776 -1862) in his An Elementary Investigation of the Theory of Numbers, published in 1811. We also show the continuity of his difficulties with the treatment of negative numbers in the middle of the nineteenth century

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

Os artigos de Euler sobre os números amigáveis

Among the many methodological resources that the mathematics teacher can use in the classroom, we can cite the History of Mathematics which has contributed to the development of activities that promotes students curiosity about mathematics and its history. In this regard, the present dissertation aims to translate and analyze, mathematically and historically, the three works of Euler about amicable numbers that were writed during the Eighteenth century with the same title: De numeris amicabilibus. These works, despite being written in 1747 when Euler lived in Berlin, were published in different times and places. The first, published in 1747 in Nova Acta Eruditorum and which received the number E100 in the Eneström index, summarizes the historical context of amicable numbers, mentions the formula 2nxy & 2nz used by his precursors and presents a table containing thirty pairs of amicable numbers. The second work, E152, was published in 1750 in Opuscula varii argument. It is the result of a comprehensive review of Euler s research on amicable numbers which resulted in a catalog containing 61 pairs, a quantity which had never been achieved by any mathematician before Euler. Finally, the third work, E798, which was published in 1849 at the Opera postuma, was probably the first among the three works, to be written by Euler

O circuito espacial produtivo e os círculos de cooperação da carcinicultura do Rio Grande do Norte

Suponiendo que la dinámica territorial del período histórico actual se caracteriza por la nueva división territorial del trabajo, por la especialización regional productiva y por la racionalidad que preside el movimiento de las mercancías y de la información, la teoría de los circuitos espaciales de producción se convierte fundamental para la comprensión de la organización, regulación y del uso de territorios por las actividades productivas. Así, teniendo en cuenta este supuesto y aplicando esta teoría para analizar el área de la producción de camarón en el estado de Rio Grande do Norte, se llega a comprender la dinámica del territorio Potiguar. Por lo tanto, en nuestro trabajo pretendemos entender el movimiento, la secuencia de los casos que geográficos que separan la producción, distribución, intercambio y consumo de camarón Potiguar, en este momento, sobre todo a partir de la década de 1990, sucede la expansión del cultivo de camarón en la costa norte Riograndense. La investigación ha demostrado que a pesar de la presencia de un gran número de agentes en la actividad, hay un control y una concentración del circuito en las manos de grandes empresas. Este control se pasa por el reducido número de agentes de venta de larvas, alimentación, maquinaria e insumos. Y con respecto a la concentración, esto ocurre sobre todo en la etapa de producción en las haciendas de engorde. En este caso, podemos decir que hay un predominio de los pequeños productores en números absolutos en el cultivo de camarón potiguar, ya que gran parte de los emprendimientos tienen dimensiones inferiores a 10 ha. Sin embargo, cuando analizamos las empresas de gran escala, éstos, a pesar de representar un porcentaje menor del total, concentran casi la mitad de las áreas cultivadas, además de tener en el control de gran parte del movimiento establecido por el circuito espacial productivo de la producción de camarón. Así, aparecen en el circuito espacial productivo de la producción de camarón de Rio Grande do Norte, las diferencias cualitativas de los usos del territorio entre los agentes que participan de este circuito. Porque un número limitado de agentes tienen las mayores áreas productivas, son dotados de un mayor nivel de tecnificación y tienen, hasta cierto punto, el control del movimiento de su producto, mientras que, eso no se reproducen por otros agentes, especialmente aquellos dedicados a la producción, efectiva, en las haciendas de engorde

Gestión del agua a nivel de cuencas: teoría y práctica

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Curso sobre teoría y programación del desarrollo

Curso de Treinamento em Problemas de Desenvolvimento Econômico: Teoria e programação do desenvolvimento econômico

La reforma a la seguridad social en salud de Colombia y la teoría de la competencia regulada

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Subdesarrollo latinoamericano y la teoría del desarrollo

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Divergencia y colapsos del crecimiento: teoría y evidencia empírica

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Teoría económica y pobreza

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El subdesarrollo latinoamericano y la teoría del desarrollo

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El subdesarrollo latinoamericano y la teoría del desarrollo

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