Problemas e modelos que contribuíram com o desenvolvimento do cálculo diferencial e integral: dos gregos à Newton

This article refers to a research which tries to historically (re)construct the conceptual development of the Integral and Differential calculus, taking into account its constructing model feature, since the Greeks to Newton. These models were created by the problems that have been proposed by the history and were being modified by the time the new problems were put and the mathematics known advanced. In this perspective, I also show how a number of nature philosophers and mathematicians got involved by this process. Starting with the speculations over scientific and philosophical natures done by the ancient Greeks, it culminates with Newton s work in the 17th century. Moreover, I present and analyze the problems proposed (open questions), models generated (questions answered) as well as the religious, political, economic and social conditions involved. This work is divided into 6 chapters plus the final considerations. Chapter 1 shows how the research came about, given my motivation and experience. I outline the ways I have gone trough to refine the main question and present the subject of and the objectives of the research, ending the chapter showing the theoretical bases by which the research was carried out, naming such bases as Investigation Theoretical Fields (ITF). Chapter 2 presents each one of the theoretical bases, which was introduced in the chapter 1 s end. In this discuss, I try to connect the ITF to the research. The Chapter 3 discusses the methodological choices done considering the theoretical fields considered. So, the Chapters 4, 5 and 6 present the main corpus of the research, i.e., they reconstruct the calculus history under a perspective of model building (questions answered) from the problems given (open questions), analyzing since the ancient Greeks contribution (Chapter 4), pos- Greek, especially, the Romans contribution, Hindus, Arabian, and the contribution on the Medium Age (Chapter 5). I relate the European reborn and the contribution of the philosophers and scientists until culminate with the Newton s work (Chapter 6). In the final considerations, it finally gives an account on my impressions about the development of the research as well as the results reached here. By the end, I plan out a propose of curse of Differential and Integral Calculus, having by basis the last three chapters of the article

Eram realmente pitagórico(a)s os homens e mulheres catalogado(a)s por Jâmblico em sua obra Vida de Pitágoras?

Pythagoras was one of the most important pre-Socratic thinkers, and the movement he founded, Pythagoreanism, influenced a whole thought later in religion and science. Iamblichus, an important Neoplatonic and Neopythagorean philosopher of the third century AD, produced one of the most important biographies of Pythagoras in his work Life of Pythagoras. In it he portrays the life of Pythagoras and provides information on Pythagoreanism, such as the Pythagorean religious community which resembled the cult of mysteries; the Pythagorean involvement in political affairs and in the government in southern Italy, the use of music by the Pythagoreans (means of purification of healing, use of theoretical study), the Pythagorean ethic (Pythagorean friendship and loyalty, temperance, self-control, inner balance); justice; and the attack on the Pythagoreans. Also in this biography, Iamblichus, almost seven hundred years after the termination of the Pythagorean School, established a catalog list with the names of two hundred and eighteen men and sixteen women, supposedly Pythagoreans of different nationalities. Based on this biography, a question was raised: to what extent and in what ways, can the Pythagoreans quoted by Iamblichus really be classified as Pythagoreans? We will take as guiding elements to search for answers to our central problem the following general objectives: to identify, whenever possible, which of the men and women listed in the Iamblichus catalog may be deemed Pythagorean and specific; (a) to describe the mystery religions; (b) to reflect on the similarities between the cult of mysteries and the Pythagorean School; (c) to develop criteria to define what is being a Pythagorean; (d) to define a Pythagorean; (e) to identify, if possible, through names, places of birth, life, thoughts, work, lifestyle, generation, etc.., each of the men and women listed by Iamblichus; (f) to highlight who, in the catalog, could really be considered Pythagorean, or adjusting to one or more criteria established in c, or also to the provisions of item d. To realize these goals, we conducted a literature review based on ancient sources that discuss the Pythagoreanism, especially Iamblichus (1986), Plato (2000), Aristotle (2009), as well as modern scholars of the Pythagorean movement, Cameron (1938), Burnet (1955), Burkert (1972), Barnes (1997), Gorman (n.d.), Guthrie (1988), Khan (1999), Mattéi (2000), Kirk, Raven and Shofield (2005), Fossa and Gorman (n.d.) (2010). The results of our survey show that, despite little or no availability of information on the names of alleged Pythagoreans listed by Iamblichus, if we apply the criteria and the definition set by us of what comes to be a Pythagorean to some names for which we have evidence, it is possible to assume that Iamblichus produced a list which included some Pythagoreans

Um estudo sobre a apreciação do raciocínio matemático na formação inicial de professores

The present work focused on developing teaching activities that would provide to the student in initial teacher training, improving the ability of mathematical reasoning and hence a greater appreciation of the concepts related to the golden section, the irrational numbers, and the incommensurability the demonstration from the reduction to the nonsensical. This survey is classified itself as a field one which data collection were inserted within a quantitative and qualitative approach. Acted in this research, two classes in initial teacher training. These were teachers and employees of public schools and local governments, living in the capital, in Natal Metropolitan Region - and within the country. The empirical part of the research took place in Pedagogy and Mathematics courses, IFESP in Natal - RN. The theoretical and methodological way construction aimed to present a teaching situation, based on history, involving mathematics and architecture, derived from a concrete context - Andrea Palladio s Villa Emo. Focused discussions on current studies of Rachel Fletcher stating that the architect used the golden section in this village construction. As a result, it was observed that the proposal to conduct a study on the mathematical reasoning assessment provided, in teaching and activity sequences, several theoretical and practical reflections. These applications, together with four sessions of study in the classroom, turned on to a mathematical thinking organization capable to develop in academic students, the investigative and logical reasoning and mathematical proof. By bringing ancient Greece and Andrea Palladio s aspects of the mathematics, in teaching activities for teachers and future teachers of basic education, it was promoted on them, an improvement in mathematical reasoning ability. Therefore, this work came from concerns as opportunity to the surveyed students, thinking mathematically. In fact, one of the most famous irrational, the golden section, was defined by a certain geometric construction, which is reflected by the Greek phrase (the name "golden section" becomes quite later) used to describe the same: division of a segment - on average and extreme right. Later, the golden section was once considered a standard of beauty in the arts. This is reflected in how to treat the statement questioning by current Palladio s scholars, regarding the use of the golden section in their architectural designs, in our case, in Villa Emo

Matemática Védica no ensino das quatro operações

This paper describes a study on the possibilities of teaching Vedic Mathematics for teaching the four operations. For this various literature sources were consulted considering three main aspects. The first of a historical-cultural, in order to gather information about the Mathematics originated from Vedic civilization, which highlight (Plofker, 2009), (Joseph, 1996), (Bishop, 1999), (Katz, 1998), (Almeida , 2009). This sought to emphasize relationships of the development of this culture with the math involved in the book Vedic Mathematics written by Tirthaji and published in 1965. In this respect the work brings notes on the history of mathematics on the development of mathematics in ancient India. The second aspect was related to teaching mathematics through research activities in the classroom, in this sense, I sought a bibliography to assist in the construction of a proposed activity to teach the four operations, based on the sutras of Vedic Mathematics, but within an investigative approach, assisting in the development of mental calculation strongly stimulated by the Vedic Mathematics Sutras. The authors were adopted (Mendes, 2006, 2009a, 2009b), Bridge (2003). The third aspect considered to search for books on teaching Vedic Mathematics, written by other authors, based on the book by Tirthaji. This revealed Vedic Mathematics textbooks adopted in schools and free courses in the UK, USA and India, all based on the book Vedic Mathematics of Tirthaji. From the bibliographical studies were prepared didactic guidelines and suggested activities for the teacher, to assist in teaching the four operations. The educational product, consisting of Chapters 4 and 5, is the body of the dissertation and consists of didactic guidelines and suggestions for activities that aim to contribute to the teachers who teach initial years of elementary school