A obra de triangulis omnimodis libri quinque de Johann Müller Regiomontanus (1436 1476): uma contribuição para o desenvolvimento da trigonometria

Trigonometry, branch of mathematics related to the study of triangles, developed from practical needs, especially relating to astronomy, Surveying and Navigation. Johann Müller, the Regiomontanus (1436-1476) mathematician and astronomer of the fifteenth century played an important role in the development of this science. His work titled De Triangulis Omnimodis Libri Quinque written around 1464, and published posthumously in 1533, presents the first systematic exposure of European plane and spherical trigonometry, a treatment independent of astronomy. In this study we present a description, translation and analysis of some aspects of this important work in the history of trigonometry. Therefore, the translation was performed using a version of the book Regiomontanus on Triangles of Barnabas Hughes, 1967. In it you will find the original work in Latin and an English translation. For this study, we use for most of our translation in Portuguese, the English version, but some doubt utterance, statement and figures were made by the original Latin. In this work, we can see that trigonometry is considered as a branch of mathematics which is subordinated to geometry, that is, toward the study of triangles. Regiomontanus provides a large number of theorems as the original trigonometric formula for the area of a triangle. Use algebra to solve geometric problems and mainly shows the first practical theorem for the law of cosines in spherical trigonometry. Thus, this study shows some of the development of the trigonometry in the fifteenth century, especially with regard to concepts such as sine and cosine (sine reverse), the work discussed above, is of paramount importance for the research in the history of mathematics more specifically in the area of historical analysis and critique of literary sources or studying the work of a particular mathematician

Cosmoeducação: uma abordagem transdisciplinar no ensino de astronomia

This work proposes a transdisciplinary approach that integrates transpersonal psychology exercises with astronomy teaching, seeking to allow one to reintegrate the sky in his/her daily life, expand his/her environmental awareness and eventually experiment the unity between human and cosmos. This proposal intends to collaborate with the supplying of education, which lacks initiatives of this kind, with the promotion of an integration of the scientific knowledge with the human experience that transcends the materialistic and fragmentary objectives of the current educational system. As a result of that lack, the teachers formation is also poor as for an integral and transdisciplinary approach. Besides, we also approached in this research the necessity to propose alternatives so that the educators may work in a more assertive way with the environmental and anthropological crisis in which we are living. Our working hypothesis is that the contents of astronomy, when they are dealt in a holisticanthropological focus and are related with transpersonal psychology practices, can come to be an efficient cultural-academic vehicle, capable of propitiating an expansion of consciousness and changes in the way one conceives the world. Such changes are necessary so that a more solidary, fair and ecologically balanced life may come to exist and prevail in the planet. Part of the collection of data was done through the ethnographic method, once an anthropological interpretation is inextricably associated with this kind of educational intervention, which will naturally include ethno-visions of the universe as well as specific cultural elements. In the beginning the scope of this research was a group of students attending the Astronomy assignment in an undergraduate Geography course (UFRN), in which we accomplished participant observation, half-open interviews and the first experimental practices mentioned. After the evaluation of the first data collected from that initial group, we elaborated an academic extension course, Laboratory in Cosmoeducation, and we offered it to teachers of the 1st and 2nd cycles of the fundamental level of the Alceu Amoroso Lima State School, located in the North Zone of Natal. We prized self-experimentation in that course, so that the teachers could enrich their repertoire of personal experiences, stimulating meditative reflections and eventual changes in the ways of conceiving the world and in their pedagogical practice. The transdisciplinary attitude permeated all our educational action, because this approach transcends the boundaries of disciplines, seeking essentially the integral development of the human being. The process has made us realize that the practice of looking at the sky , as a way of reintegrating it into daily life, provokes a process of expansion of the consciousness and of reintegration of the self in a wider level of environmental interrelation. According to the results, the occurrence of conceptual and existential changes of the world vision of the participant teachers was evident, reassuring ourselves of the idea that the interface between astronomy teaching and the practices of transpersonal psychology can contribute to the recovery of a holistic relationship between the human being and the cosmos and to inspire the arising of a more wide-ranging ethics, based on universal, impartial and sustainable values

Investigação histórica nas aulas de matemática: avaliação de duas experiências

This study reflects on some procedural aspects about the development of mathematics learning from the experience with investigative activities concerning the resolution of second degree equation, which was tested a proposal for education, supported the use of texts in history of mathematics. The survey was conducted in two stages, taking the first-served basis for the second, which was carried out with a study group remainder of the first experiment. The intention was to investigate how the group participant, known as the study group, involved in the implementation of activities of research in mathematics, supported the use of the history of mathematics. Based on the results achieved during the study, it was possible to understand that the activities of research enable the development of students, range of learning mathematics and the development of skills and expertise for research as a vehicle for construction of their mathematical knowledge. This approach proposed research into the classroom is important, both for prospective teachers of mathematics and for students from elementary school, bringing a new phase for mathematical education that will come to schools

Investigação histórica na formação de professores de matemática: um estudo centrado no conceito de função

In this study we analyzed the development of a teaching experience, involving students with a bachelor s degree in mathematics from UFRN, based on the history of mathematics and mathematical investigations with the aim of contributing to the improvement of the teaching-learning of mathematics. The historical investigation tasks were planned and applied in the classroom, focusing on functional thought. The results obtained during the experience were described and evaluated based on authors who support the assumption of investigation and history as an alternative to the learning of mathematics. We emphasize that the material of analysis consisted of a work diary, audio recordings, questionnaires with testimony of the students involved, and, in addition, the assessment of the teacher of that subject. With regard to the mathematical content, the study was restricted to the concept of function, forms of representation and notation. It was evident that students showed great improvement with regard to the necessary formalization of the mathematical contents which were focused on, and to the active involvement of the students at different stages of the study. We can affirm that the completed study certainly represents significant contributions to an approach in the teaching-learning of functional thought

Uma investigação histórica sobre os logaritmos com sugestões didáticas para a sala de aula

This study was conducted from a preliminary research to identify the conceptual and didactic approach to the logarithms given in the main textbooks adopted by the Mathematics teachers in state schools in the School of Natal, in Rio Grande do Norte. I carried out an historical investigation of the logarithms in order to reorient the math teacher to improve its educational approach this subject in the classroom. Based on the research approach I adopted a model of the log based on three concepts: the arithmetic, the geometric and algebraic-functional. The main objective of this work is to redirect the teacher for a broad and significant understanding of the content in order to overcome their difficulties in the classroom and thus realize an education that can reach the students learning. The investigative study indicated the possibility of addressing the logarithms in the classroom so transversalizante and interdisciplinary. In this regard, I point to some practical applications of this matter are fundamental in the study of natural phenomena as earthquakes, population growth, among others. These practical applications are connected, approximately, Basic Problematization Units (BPUs) to be used in the classroom. In closing, I offer some activities that helped teachers to understand and clarify the meaningful study of this topic in their teaching practice

Ensinando sobre a natureza da ciência: uma abordagem explícita e contextualizada a partir da história do vácuo

In last decades, the importance of including the contents of the Nature of Science (NOS) in Science Education has been emphasized. Several studies have focused on investigating the conceptions of NOS, supported by students and teachers, as well as design, implement and evaluate proposals that aim to provide a reflection on this theme in the educational context. Considering the complexity of such content, studies indicate the need for explicit and contextualized approaches and the History of Science (HS) is one of the possible paths to this inclusion. We started from the premise that, through a historical study, that aims to discuss the meaning and the basis of our "beliefs", we can know the process of building on what we "believe" and better understand its meaning. This thesis is part of this perspective, proposing to explore the History of the Vacuum, a themed high didactic potential still little used, in order to collaborate with the teaching content of NOS. We present actions on different fronts that originated three products in the context of this research. On the first front, we insert the research and production of instructional materials (three historical texts) to subsidize people interested in the implementation of HPS for educational context through this material. The relevance of this front is justified by the existence of gaps with regard to the production of such material for the context of teacher training. However, we consider that the preparation of instructional material of good quality and accessible does not guarantee that these resources will be used, if they will not be accompanied by discussions in teacher training, on how to use them, contexts and obstacles to be faced. The second part presented refers to the organization and implementation of a workshop for undergraduate students in physics and physics teachers, considering the instrumentalization of these individuals to the preparation and use of teaching strategies to approach aspects of NOS through episodes of History of the Vacuum, as well as the preparation of the text orientation for people interested in implementing the instructional material for secondary education. This guidance text contemplates the difficulties anticipated by the literature of the area and the main challenges faced by the participants about the didactic transposition of HPS for the educational context they were noted during the workshop. The relevance of this second front, in particular, is justified by the existence of gaps with regard to the inclusion of the theme of NOS and the HPS teacher training

Visões da cidade : da cidade histórica aos bairros da Cidade Alta e Ribeira em Natal-RN

The present work is an initiative of undertaking a perspective that values the aspects of the city in its diversity. The one that t intends is to discuss the urbanity starting from the aspect of the image, of the history, and about main point, to value the point of the individual's view that lives the urban space in its more several configurations: the house, the street, the neighborhood, and the city in wider scale. The described historical research defines the city as space of the human accomplishments. When analyzing the moments of the most recent history of the urban life the modernity/post-modernity notions they are explored in the sense of illuminating the wealth, continuity and it breaks of patterns of behavior, and on the other hand, to describe a I overfly in the several times of the city. The prominences of aspects of the archaeology and of the descriptive resource they are related, for its time, the most punctual perspective of the practical research, the instance of the neighborhood, specifically High City and Riverside, configuring an immersion in the daily more punctual of the city, valuing forms of being, aiming the perspective of the personal computer as point-of-view. The images of the city assume the paper of representing the documental about the current features of the urbanity. It is without a doubt that at the present time the photographic image already possesses a paper of prominence it is as research instrument, be as main source for a work on files of photos. The photographic images in the present work possess a prominence place, focusing specific places of the city in the time. Finally, the individual's voice valued under the perspective on as the person's knowledge that inhabits and live the urbanity he/she understands the knowledge of the life in the city and, more than in any other place, a middle where the loss of action models is pointed very easily tends in view the dynamic movement of the practices and renewal of new indications; in the same space in that becomes urgent to build marks starting from which the individual can shimmer the future. And in that space the individual's voice is valued as rich source of information on living the scenery of conflicts of the present time

Em busca do tempo partido: a consciência histórica e o MST

The work that follows is dedicated to the study of the historic time experience by the political militancy from our time. The political militant from the left-wing is the one that denies the state of current things, recovering a historic experience located far before its time and projecting a future beyond the incessant reproduction of the present relations. We chosen the Landless Workers Movement, not as study object, but as specific place where this consciousness is made and can be comprehended. The historic consciousness study of the landless militancy is, for us, the best starting point to understand the magnification of the historical time operated on the change of social relations lived inside an organization. The time division between before and after, as well as the history being understood as a progress isn t a natural given data, but a construction that obeys the contradictions of the present. It, therefore, must understand how the present lived by the militancy operates changes on the consciousness of time. From documents, reference books, formation notebooks and several materials produced by MST, we try to understand the way that the movement tells his history and lists with this wider experience of the struggle for the change of the current social order. Similarly, when we hear reports of history of several militants, we try to comprehend how this wider narrative re-orients the sight over history himself, over the experience of contradictions on before and after making the landless militant

A relação histórica entre democracia e liberalismo. Um diálogo crítico com Norberto Bobbio.

This work seeks to examine the historical relationships established between liberalism and democracy, questioning the apparent inseparability between the two ideologies. Methodologically construct a hypothetical dialogue with the Italian thinker Norberto Bobbio, one of the most important systematizers of liberal democracy, defending a theoretical and conceptual complementarity between the two ideologies. Following the Bobbio theoretical propose, it presents the political contributions of classical liberalism that the Italian thinker identify as logical and axiological antecedent of the modern democracy, naturalizing and universalizing the principles and ideals of classical liberalism. Going counter, it problematizes the political contributions of classical liberalism, emphasizing the tension between liberal theory and its practice, between the declared political principles and their translation into concrete historical reality, reserving rights and freedoms to property minority and severe restrictions to the majority. The critical analysis of classical liberalism allows questioning the privilege position that Bobbio reserves to the liberalism in the democracy history, to restore the important contribution of illiberal politics currents in the civil, political and social rights history, advocating the democracy with its social character, inclusive and participatory

Anayde Beiriz e a escrita de si: (Educação, história e relações de gênero)

Le thème de cette dissertation est l Histoire de l éducation de la femme. Une analyse des pratiques culturelles de la professeur Anayde da Costa Beiriz (1905-1930), dans le contexte paraibanais du début du XXe siècle, est proposée. Le référentiel théoricométhodologique de l Histoire Culturelle est utilisé, pour aller contre l histoire passée sous silence dans le canon traditionnel. Basée sur le concept de représentation (Chartier, 1990), rompant avec les anciennes idées de sens intrinsèque, absolu, unique, liée aux pratiques complexes, multiples et différenciées, qui construisent le monde comme représentation. Basée aussi sur le concept de configuration (Elias, 1980), en comprenant que apparemment individu et société sont deux objets qui existent indépendamment, mais en vérité, se rapportent à deux niveaux inséparables du monde social, sont unis par des liens d interdépendance. Un dialogue est recherché entre le passé et le présent et les différentes dimensions: historique, sociale, politique et culturelle vécues par la professeur Anayde Beiriz. Pour obtenir des informations sur ses pratiques culturelles, une recherche des marques de cette histoire dans de multiples sources est effectuée: dans la bibliographie disponible parmi les domaines de la littérature, de l histoire, dês sciences sociales et de l éducation; dans des archives de l Institut Historique et Géographique de la Paraiba et de Natal; dans les Bibliothèques de l Université Fédérale de la Paraiba et du Rio Grande do Norte et chez des bouquinistes. L inventaire comprend: des périodiques, la législation en vigueur, des écrits de contemporains, dês témoignages oraux de parents, des lettres et des photos. On comprend de cette façon, que l idée d éplucher au travers de cette étude, des aspects qui n ont pas encore été vus et rappelés, peut non seulement répondre à nos doutes, mais, peut à partir de la réflexion qui se fait, se traduire par une nouvelle compréhension de cette histoire. En conclusion, des significations multiples sont contenues dans le processus d insertion de la femme dans l éducation à partir des Écoles Normales, qui englobent les relations de genre, l histoire du pouvoir, de la morale, de surpassement, des luttes et insatisfactions. On perçoit l héritage de résistances et de conquêtes. Cela a été sous l inspiration des airs de la modernité avec la Belle Époque, en répercutant dans le monde et aussi dans la Parahyba du Nord, que s est concrétisée la formation sociale de la professeur Anayde Beiriz et des femmes de son temps. L analyse de l écriture de soi confirme le discours normatif du contrôle moral social, le joug et les préjugés soufferts par Anayde Beiriz et l éducation différencié de genre qui ont déterminé ses habitudes et coutumes. C est-àdire, pour la femme, la permanence dans l espace privé, la retenue, l anoblissement des tâches ménagères et de la maternité comme sa plus haute aspiration, le support moral de la famille, la préservation de la tradition et la perpétuation des règles religieuses

Quadrinhos nas aulas de ciências: narrando uma história de formação continuada

Located at the intersection of Science Education, Teacher Training and use of didactic Comics (HQ), the text aims to discuss the problem which thus sets: - What is the contribution of the incorporation of comics in the continuing education of teachers Natural Sciences of the early years of elementary school? - May be the comics for enabling science teaching? - How to contribute to a reflexive training of teachers of the early years? - In order to answer these questions, there was a collaborative action research in a school located in the city of Natal / RN, with three teachers who teach natural science in the early years of elementary school. The study involved the conduct of 13 meetings of in-service, under the focus on reflective practice, with the purpose of discussing topics underlying the teaching of science and the use of comics as a methodological strategy. All meetings were recorded on audio and transcripts. The teachers finally recorded the meetings by writing a portfolio. The teachers made use of 10 sessions with reading comics in science classes, which were video recorded and later watched by the educators in autoscopy sessions followed by individual interviews. From the collected data, several aspects that have emerged can be grouped into three categories: The difficult concept of science, teachers' work and the obstacles and possibilities of comics as a teaching strategy. In this sense, are woven assessments taking as its founding, look for the reflective and dialogic practices exercised in the classroom. In the analysis of the data, we can see the conceptual difficulties, methodological and pistemological questions of teachers for teaching science, but also the limitations of comic books for teaching. Learning more relevant appointed by the teachers is related to the importance of collective work to mitigate the training needs of teaching. Finally, signals that the HQ has a great potential for use in science classes, provided that the teacher conduct a careful planning for this, but that the meetings of continuing education promote moments of reflection on teaching practices that are capable of giving rise to new attitudes before the difficulties they depict

Formação continuada de professores de ciências para o ensino de astronomia nos anos iniciais do ensino fundamental

Situated at the intersection of Continuing Formation, Science Teaching and content of astronomy, the research aims to discuss this problematic : What are the challenges encountered in a continuous formation in service of science teachers of the early years of elementary school with contents of astronomy? Aiming to answer this question, we carried out a collaborative action research in a school of Natal / RN, with 6 teachers who teach or lectured Natural Sciences in the early years of elementary school. The study involved 14 encounters of continuos formation in service, to enable a better understanding criticism and propositive concerning the teaching science, especially some content of astronomy for the initial years, through a shared discussion. To this end, we consider that more profound understanding of astronomical knowledge allows the establishment of relationships and connections between theoretical knowledge and the daily teaching practice. All discussions were recorded on audio and transcripts. This research was realized using questionnaires and diary, however, the audio recording was its principal working instrument. About the data collected, several issues emerged as the little field of conceptual content, several misconceptions among teachers about the content of astronomy, lack of adequate space in the school environment to study and the need for collective reflection on the practice. Finally, is indicated that the contents of astronomy should be used in classrooms of Sciences, provided that teaching plan of action consider the existence of diverse alternative conceptions about the subject, among both teachers and students, and the need of permanente continuing education in the teaching

Atribuição de significado ao conceito de proporcionalidade: contribuições da história da matemática

This study is the result of a work which approaches the Mathematics History how source of the meaning s attribution in the proportionality concept. We adopt the methodology of the source qualitative and we work with a group of teachers from instruction s public system of the fundamental and medium level from Pocinhos City Paraíba. For the data collection, we use the field notes, the questionnaire, a sequence of activities and the interview semistructured like instruments. The study had how objective to know the significates attributeds to proportionality concept through of the activity mediate from Mathematics History, besides to investigate if a approach of the nature enables modification according to this sense. The results obtaineds though the data analysis indicate that the activities bring contributions which refer to achieve objectives. On the other hand they also showed that we have a long trajectory to be trailed in the meaning of to turn the Mathematics History a subsidy effective in the teachers practice, in view of the formation absence in the knowledge area, besides the necessity of the approach adequated of the Mathematics History in the didatics books of Mathematic

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

Abordagem de temas em saúde pública por meio de atividades lúdicas com professoras da educação infantil

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