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

O processo ensino-aprendizagem de história no ensino fundamental: seus limites, suas possibilidades

Study about the teaching-learning process in History, which intents to point limits and possibilities to this process, starting from its characterization and analysis and understanding of the concepts of history, time, society and culture, used by teachers and students. The field research was performed in the Municipal School of Basic Education Zumbi dos Palmares, located in the city of João Pessoa, Paraíba State, in the period between 1996 and 2006. To achieve the objective of the study, a number of research instruments were used, amongst which, interviews, questionnaires and exercises emphasizing conceptual learning. The theoretical-methodological premises of the qualitative research justify the use of these various instruments, and serve as base for the interpretation and analysis of the data. This study demonstrated that some limits and possibilities that are found in the teaching-learning process in History are originated in the school context of the 1st phase of basic education and remain in the 2nd phase of this education level, partly, because of the understanding that teachers and students have regarding the concepts of history, time, society and culture

Há possibilidade do ensino da história desenvolver o pensamento teórico?

The expectation that there is a relationship between the teaching of history and the development of theoretical thinking , this study aimed to examine this relationship with the collaboration of two teachers who teach in the fourth year of elementary school in a public school in the city of Mossley - RN . Sues the theoretical and methodological contributions are identified with the history Dialectical Materialism . The methodological approach was collaborative research . Was based on studies of Vygotsky ( 1998, 2001 ) , Rubinstein (1965 , 1973) , Ibiapina (2007 , 2008) , Afanassiev (1985 ) , Goes (2003 ) , Ferreira and Ibiapina (2005 , 2006, 2007 ) , Davydov (1981 ) , Glenisson (1991 ) among others . The methodological procedures consisted of : meetings , questionnaire , interview , cycles of reflective studies , reports of experiences and reflective session. For data analysis , we used the methodology of conceptual elaboration , proposed by Ferreira (2009 ) and the shares of reflection proposed by Magalhães (2000 ) . Data analysis indicates the predominance of empirical thinking of collaborating both with regard to the process of design and ( re ) development of the concepts of history and theoretical thinking on the relationship between history teaching and the development of theoretical thought . However , we see the beginning of the start of the change process in thinking the teaching of history

Mecanismos de desgaste, história térmica e ciclo de vida de componentes de um motor de combustão interna de aeromodelo

We developed an assay methodology that considered the temperature variation and the scanning electron microscopy as a method to quantify and characterize respectively the consumption evolution in three 46 LA machines, with internal combustion and two-stroke engines, 7.64 cm3 cylinder capacity, 23.0 millimeters diameter and 18.4 millimeters course, RPM service from 2.000 to 16.000 rpm, 1.2 HP power, and 272 grams weight. The investigated engines components were: (1) head of the engine (Al-Si alloy), (2) piston (Al-Si alloy) and (3) piston pin (AISI 52100 steel). The assays were carried out on a desktop; engines 1 and 2 were assayed with no load, whereas in two assays of engine 3 we added a fan with wind speed that varied from 8.10 m/s to 11.92 m/s, in order to identify and compare the engine dynamic behavior as related to the engines assayed with no load. The temperatures of the engine s surface and surroundings were measured by two type K thermopairs connected to the assay device and registered in a microcomputer with data recording and parameters control and monitoring software, throughout the assays. The consumed surface of the components was analyzed by scanning electron microscopy (SEM) and microanalysis-EDS. The study was complemented with shape deformation and mass measurement assays. The temperature variation was associated with the oxides morphology and the consumption mechanisms were discussed based on the relation between the thermal mechanical effects and the responses of the materials characterization

Um estudo de práticas matemáticas históricas e sugestões de uso na matemática escolar

This dissertation aims to contribute on teaching of mathematics for enabling learning connected to the relationship among science, society, culture and cognition. To this end, we propose the involvement of our students with social practices found in history, since. Our intention is to create opportunities for school practices that these mathematical arising from professional practice historical, provide strategies for mathematical thinking and reasoning in the search for solutions to problematizations found today. We believe that the propose of producing Basic Problematization Units, or simply UBPs, in math teacher formation, points to an alternative that allows better utilization of the teaching and learning process of mathematics. The proposal has the aim of primary education to be, really forming the citizen, making it critical and society transformative agent. In this sense, we present some recommendations for exploration and use of these units for teachers to use the material investigated by us, in order to complement their teaching work in mathematics lessons. Our teaching recommendations materialized as a product of exploration on the book, Instrumentos nuevos de geometria muy necessários para medir distancia y alturas sem que interuengan numeros como se demuestra em la practica , written by Andrés de Cespedes, published in Madrid, Spain, in 1606. From these problematizations and the mathematics involved in their solutions, some guidelines for didactic use of the book are presented, so that the teacher can rework such problematizations supported on current issues, and thus use them in the classroom

História da ciência como estratégia didática no ensino médio: um breve olhar de conteúdos da óptica

From a critique of aspects of the current teaching of physics, we propose in this paper an investigation into the conceptions of Nature of Science (NOS) submitted by students from the high school level, as well as the inclusion of discussions about some elements of NOS, through the History and Philosophy of Science, understanding them as facilitating strategy for more effective learning of Physics and, more specifically, the contents of optics. Based on the historical period corresponding to Greek antiquity to the mid-nineteenth century, built and applied a teaching unit to a high school class at the Federal Institute of Education, Science and Technology of Rio Grande do Norte (IFRN) Ipanguaçu campus. Our teaching unit involved the reading and interpretation of texts with historical content, observation and assembling five experimental activities, among other activities. Data analysis occurred through questionnaires investigative applied before and after the completion of the teaching unit, as well as questionnaires for content present in the texts. Although not significant, the results indicated that the application of the teaching unit allowed the (re) construction of some conceptions about NOS targets students present in our work, and contribute to a better learning content and greater optical science approach and its nature

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

Dificuldades de aprendizagem sobre conceitos de genética no ensino fundamental

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

Fontes primárias da história do vácuo e da pressão atmosférica na sala de aula: cartas e jornais históricos em articulação com o livro-texto do ensino médio

The use of the History and the Philosophy of Science (HPS) for the teaching of science and scientific subjects has been advocated in recent decades. It has been pointed out that the History of Science could make for a deeper learning of scientific concepts, since it would promote a contact with the problems which that knowledge has set out to solve. Furthermore, historical episodes would serve to put the discussions about the nature of scientific knowledge into context. With a view to explore those potentialities, the literature in the field has sought to identify the challenges and obstacles for the didactic transposition of subjects from the History of Science. Amongst many aspects, the deficiencies in the training of teachers, so that they can work with the insertion of HPS in the classroom have been highlighted. Another aspect that has been mentioned to be a challenge has to do with the didactic transposition of the Primary Sources, that is, of the original texts on the History of Science. The Primary Sources have significant potentialities: making a connection possible between scientists and concepts, showing the difficulties faced during scientific endeavors, perceiving the role of mistakes as obstacles to be surpassed, not as defeat, etc. On the other hand, there has been little exploration of these concepts in an educational context, due to their own peculiarities. The original texts are often hard to understand and their interpretation demands knowledge of the historical and scientific context in which they were written, as well as skills pertaining to the conduction of research in the field of the History of Science. With this scenario in mind, the research towards this Professional MSc degree starts from the challenge of elaborating and discussing proposals which could enable the didactical transposition of the Primary Sources. We have worked specifically with Primary Sources on the History of the Vacuum and of the Atmospheric Pressure, because of the insertion of these subjects in the Brazilian High School curriculum, in connection with the didactical textbooks. "Historic Journals" were made up from clippings of the original historical texts, as was a Didactical Unit, which takes the usual textbooks as a basis and contemplates using the Journals and the entire Primary Sources in High School. At last, we have elaborated and implemented a course designed for the preparation of teachers and for being an opportunity for the discussion of the feasibility of putting these kinds of proposal into practice

As chaves para o mundo do quinto império: as imagens de mundo na história do futuro do padre Antonio Vieira

Iconic historical figure, Antônio Vieira (1608-1697) is regarded as an essential character to the Luso-Brazilian history. Between 1646 and 1667, the priest began writing the History of the Future, the first volume of the celebrated "Clavis Prophetarum , political and theological unity that would leverage the process of spread of Christianity across the globe, recognizing Portugal as forefront and center of all millenarian movement. Them, Jesuit represented the "World" in two metaphors: "theater" and "body" responsible for viewing spaces of abstract Christian truth. Starting from the hermeneutical assumptions of analysis, we investigated the historical relations present in the construction of such representations, by which establish the dialogue between politics, theology and seventeenth-century rhetoric. Therefore, the following study supports the hypothesis that beyond a mere stylistic expressions, spatial metaphors of "World" were formulated by Antonio Vieira as a resource that could sharpen the minds of his readers, engaging words into action, become alive and effective use of rhetoric

A história do intemperismo na Província Borborema Oriental, Nordeste do Brasil: implicações paleoclimáticas e tectônicas

Until some years ago, weathering geochronology was primarily based on the K-Ar and 40Ar/39Ar dating of supergene minerals. Recent advances in the analysis of supergene goethite by the (U-Th)/He method expanded the number of suitable minerals for such purpose, as well as the time of application for weathering geochronology. This study represents the first systematic approach in Brazil, combining both the 40Ar/39Ar e (U-Th)/He methodologies to improve the knowledge on the weathering and the age of nonfossiliferous sediments. Supported by geologic and geomorphologic correlations, we identified different types of weathering profiles occurring in the interior and coastal areas of northeastern Brazil. These profiles were correlated to main regional geomorphological domains: the Borborema Plateau , the Sertaneja Depression , and the Coastal Cuestas and Plains, and respective planation surfaces, which study is fundamental to understand the landscape evolution of the northern portion of the eastern Borborema Province. The depth and stratigraphic organization of the weathering profiles in each of the geomorphological domains permitted to establish that: (i) the profiles on the highlands that cap the Borborema Surface are deeper (up to 100 m) and can be considered as typical lateritic profiles; (ii) on the lowlands that form the Sertaneja Surface , the weathering profiles are shallow and poorly developed (2-5 m deep); (iii) the profiles along the coastal area are moderately developed (up to 25 m deep), and are characterized by thick saprolites and mottle zones. Aiming to establish the timing of the evolution of northeastern Brazil, we studied 29 weathering profiles representing distinct topographic levels of the Borborema Province, from the highlands to the coast, through the analysis of 248 grains of supergene manganese oxides using laser step-heating 40Ar/39Ar geochronology. Additionally, we applied the (U-Th)/He method in 20 weathering profiles, by dating 171 grains of supergene iron oxides and hydroxides. Geochronological results for 248 grains of manganese oxides analyzed by the 40Ar/39Ar method indicate that the weathering profiles in the study area record the history of weathering from the Oligocene to the Pleistocene, with ages in the order of 31.4 ± 1.0 Ma to 0.8 ± 0.4 Ma. Dating of 171 grains of goethite by the (U-Th)/He method yielded ages ranging from 43.2 ± 4.3 Ma to 0.8 ± 0.1 Ma, suggesting the weathering processes last from the Eocene to the Pleistocene. The precipitation of supergene goethite in this interval confirms the age of the weathering processes identified from the manganese oxides record. 105 goethite grains from 8 different occurrences of the Barreiras Formation were dated by the (U-Th)/He method. Five grains collected from the cement in the Barreiras Formation sandstones, in the Lagoa Salgada and Rio do Fogo coastal cuestas, yielded ages of 17.6 ± 1.8 Ma, 17.3 ± 1.7 Ma, 16.3 ± 1.6 Ma, 16.2 ± 1.6 Ma and 13.6 ± 1.4 Ma. Results of 69 goethite grains from authigenic pisoliths collected in 7 different localities also yielded concordant ages, varying from 17.8 ± 1.8 to 7.5 ± 0.8 Ma. Results obtained from 31 detrital grains are concordant in 3 distinct localities (Lagoa Salgada, Praia da Garças e Ponta Grossa); they vary in the range of 43.2 ± 4.3 to 21.6 ± 2.2 Ma, and indicate that the maximum age for the Barreiras Formation deposition is around 22 Ma. 40Ar/39Ar results for 15 manganese oxides grains associated with the Barreiras Formation weathering profiles, in 3 different localities, vary from 13.1 ± 0.9 to 7.7 ± 0.4 Ma, in the same range of ages obtained by the (U-Th)/He method. The systematic application of the 40Ar/39Ar and (U-Th)/He methods, respectively for manganese oxides and goethites, show that the Barreiras Formation sediments were already deposited since ca. 17 Ma, and that the weathering processes were active until ca. 7 Ma ago. The ages obtained from manganese oxides collected in the Cenozoic basalts (Macau Formation) also reveal a weathering history between 19 and 7 Ma, pointing to hot and humid conditions during most of the Miocene. 40Ar/39Ar ages yielded by manganese oxides associated with the Serra do Martins Formation vary from 14.1 ± 0.4 to 10.5 ± 0.3 Ma. On the other hand, (U-Th)/He ages from iron oxides/hydroxides collected in the Serra do Martins Formation mesas vary from 20.0 ± 2.0 to 5.5 ± 0.6 Ma, indicating that those sediments are older than 20 Ma. 40Ar/39Ar and (U-Th)/He results produced in this study are in agreement with paleoclimatic interpretations based on stable isotopes and clay index values measured in the Atlantic Ocean sediments, validating the use of weathering geochronology to investigate paleoclimatic variations. The direct dating of the Barreiras Formation permitted, for the first time, confident inferences on the age of the brittle deformation recorded by this sedimentary unit in the Rio Grande do Norte and Ceará states. The first event, syn-deposition, occurred during the early Miocene; an younger event, related to the post-depositional deformation of the Barreiras Formation, is associated with tectonic activity from the very early Miocene to the Holocene. In agreement with data from other areas, results obtained in this study reveal that the depth and complexity of the weathering profiles reflect the time of exposition of such areas to the weathering agents close to the surface. However, there is no clear relationship between ages vs. altitude. The depth and the stratigraphic organization of weathering profiles in northeastern Brazil, contrary to the southeastern Brazil pattern, do not vary toward the coast. In our study area, field observations reveal the presence of ancient, thick and complex lateritic profiles preserved in the sedimentary mesas on the Borborema Plateau, as younger, narrow and incipient ones occur in the dissected areas. Geochronological results obtained for these profiles yielded older ages on the high altitudes, and younger ages in the lowlands, suggesting the scarp retreatment is the most reliable model to explain the regional landscape evolution. However, in the coastal lowlands, the relatively older ages obtained indicate that more complexes processes were involved in the modeling of the local relief

Usando a história da resolução de alguns problemas para introduzir conceitos: sistemas lineares, determinantes e matrizes.

In this work we studied the method to solving linear equations system, presented in the book titled "The nine chapters on the mathematical art", which was written in the first century of this era. This work has the intent of showing how the mathematics history can be used to motivate the introduction of some topics in high school. Through observations of patterns which repeats itself in the presented method, we were able to introduce, in a very natural way, the concept of linear equations, linear equations system, solution of linear equations, determinants and matrices, besides the Laplacian development for determinants calculations of square matrices of order bigger than 3, then considering some of their general applications