Vídeo-aula de história da matemática: uma possibilidade didática para o ensino de matemática

In this work we are disagreeing with the possibility of production and of the use of video-class for the disciplines of history of mathematics by the teachers of elementary e middle school as a way to contribute to the development of their classes. Our goal is to provide to the mathematics teachers the option of connecting social, scientific, cognitive, and didactic aspects of topics in math thoughts to their students. That shall be based in the presence of mathematics in the history of humanity. Thus, we consider possible that teachers and their students can link and relate mathematics to other sciences, education culture, and reflect about the many ways of represent them, as well as the patters of organization of nature and of culture. In this way, they shall be able to observe and interpret situations that involve mathematical questions associated to the various means of historic-epistemological studies already done by other researchers and scholars in the field of history of mathematics who works in creating video-classes. In addition to that, we can use all the available media in order to give edifying dynamics to the mathematical formulations established throughout history. In this sense, we are based and focused on the objectives, which are sustained by educational computer technology, techniques for video making, as well as mathematical teaching proposals and the historical inquiring made by Mendes (2001, 2009a, 2009b). The validating experimentation allowed us to conclude that the techniques we used in the production of the history of mathematics video-classes proved they to be valid ones. They are able to be executed with the minimum of technological resources. In addition, they have the same efficacy as far as classroom use

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

Elaboração de uma sequência didática sobre os conceitos geométricos preliminares ao estudo da trigonometria

This study is qualitative, including literature search and preparation of teaching materials. Your goal is to report the study of geometric problems of character presented in the application of trigonometry and work on the preparation of detailed activities that help in overcoming these difficulties. For this, we read some papers on teaching and learning of trigonometry in order to identify the difficulties encountered during their journey. Then separate the geometric difficulties of character and prepare a list of geometric content and procedures associated with them. Thus, we can organize a notebook with activities that would address most of these concepts. Finally we present the specification of activities called Activity on introductory concepts to the study of trigonometry

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

Utilizando as ciências espaciais e a astronáutica na construção de atividades práticas em ensino de física

In this work we consider the use of new technologies fron the Space Science and Astronautical, in the learning process, incorporating technical and conceptions of physics Spatial what can take on the response of some problems what there are a lot insistent in populate the mind of professors and followers regarding agreement of these conceptions into the level Average. Intending to bring another contribution to Physics teaching practice, especially Classical Mechanics, but introducing many introductory concepts from Modern Physics, since this topic is considered of great interest for teachers and students, for it involves high technology. It s advanced manufacturing demand certain processes that make possible the application of those concepts

Uso pedagógico do ábaco romano para ensino do algoritmo de multiplicação

This dissertation describes the construction of a alternative didactic incorporating a historical approach with the use of the Roman abacus for teaching multiplication to students of 2nd year of elementary school, through activities ranging from the representation of numbers to multiplying with the Roman abacus, for learning the multiplication algorithm. Qualitative research was used as a methodological approach since the research object fits the goals of this research mode. Concerning the procedures, the research can be seen as a teaching experiment developed within the school environment. The instruments used for data collection were: observation, logbook, questionnaires, interviews and document analysis. The processing and analysis of data collected through the activities were classified and quantified in tables for easy viewing, interpretation, understanding, analysis of data and then transposed to charts. The analysis confirmed the research objectives and contributed to indicate the pedagogical use of the Roman abacus for teaching multiplication algorithm through several activities. Thus, it can be considered that this educational product will have important contributions for the teaching of this mathematical content, in Basic Education, particularly regarding to the multiplication process

Ocorrência de compostos de interesse emergente no aquífero Dunas-Barreiras e nos esgotos de Natal/RN

The detection of emerging interest microcontaminants in environmental samples of surface water, groundwater, drinking water, wastewater and effluents from water and sewage treatment plants (WTP and STP), in many countries, suggests these pollutants are widespread in the environment, mainly in urban areas. This is a reason for great concern, since many of these compounds are potentially harmful for humans other living beings, and they are not efficiently removed in the majority of WTP and STP, which is exacerbated by precariousness of water supply and sanitation services. In Natal, like other Brazilian cities, the sewage system serves only part of the urban area (about 30%), so that the rest of the wastewater is infiltrated in the sandy soil of the region in cesspool-dry well systems. This has resulted in contamination of groundwater in the area (sand-dune barrier aquifer, which supplies more than 50% of the city population), which has been observed by the increase in nitrate concentration in supply wells. The vulnerability of the sanddune barrier aquifer, combined with reports of the presence of emerging interest microcontaminants in Brazil and worldwide, led to this research, which investigated the occurrence of fifteen microcontaminants in Natal groundwater and sewage. Samples were collected at five wells used for water supply, the raw sewage and the effluents from biological reactors from STP (UASB and activated sludge reactors). Two samples of each sample were taken, with one week apart between the samples. To determine the contaminants, extraction of aquifer water, and raw and treated sewage samples were performed, through the technique of using SPE Strata X cartridge (Phenomenex®) to the aquifer water, and Strata SAX and Strata X (Phenomenex® ) for samples of raw and treated sewage. Subsequently the extracts were analyzed using GC-MS technique. Much of the analyzed microcontaminants were detected in groundwater and sewage. The concentrations in groundwater are generally lower than those found in the sewers. Some of the compounds (estrone, estradiol, bisphenol A, caffeine, diclofenac, naproxen, paracetamol and ibuprofen) are partially removed at STP.