Geometrias não-eucludianas como anomalias: implicações para o ensino de geometria e medidas

This present research the aim to show to the reader the Geometry non-Euclidean while anomaly indicating the pedagogical implications and then propose a sequence of activities, divided into three blocks which show the relationship of Euclidean geometry with non-Euclidean, taking the Euclidean with respect to analysis of the anomaly in non-Euclidean. PPGECNM is tied to the line of research of History, Philosophy and Sociology of Science in the Teaching of Natural Sciences and Mathematics. Treat so on Euclid of Alexandria, his most famous work The Elements and moreover, emphasize the Fifth Postulate of Euclid, particularly the difficulties (which lasted several centuries) that mathematicians have to understand him. Until the eighteenth century, three mathematicians: Lobachevsky (1793 - 1856), Bolyai (1775 - 1856) and Gauss (1777-1855) was convinced that this axiom was correct and that there was another geometry (anomalous) as consistent as the Euclid, but that did not adapt into their parameters. It is attributed to the emergence of these three non-Euclidean geometry. For the course methodology we started with some bibliographical definitions about anomalies, after we ve featured so that our definition are better understood by the readers and then only deal geometries non-Euclidean (Hyperbolic Geometry, Spherical Geometry and Taxicab Geometry) confronting them with the Euclidean to analyze the anomalies existing in non-Euclidean geometries and observe its importance to the teaching. After this characterization follows the empirical part of the proposal which consisted the application of three blocks of activities in search of pedagogical implications of anomaly. The first on parallel lines, the second on study of triangles and the third on the shortest distance between two points. These blocks offer a work with basic elements of geometry from a historical and investigative study of geometries non-Euclidean while anomaly so the concept is understood along with it s properties without necessarily be linked to the image of the geometric elements and thus expanding or adapting to other references. For example, the block applied on the second day of activities that provides extend the result of the sum of the internal angles of any triangle, to realize that is not always 180° (only when Euclid is a reference that this conclusion can be drawn)

Elaboração de atividades didáticas para o ensino de matemática a partir de livros antigos: o exemplo do leçons élémentaires de Lagrange

De nombreuses études sur l`utilisation pédagogique de l`histoire des mathématiques viennent a identifier les arguments qui sous-tiennent ces actions éducatives comme une façon d`aborder les mathématiques scolaires afin de mener les élèves à un apprentissage réflexif et significatif des mathématiques. Cherchant a vérifier, de manière pratique, comment ces relations entre histoire des mathématiques et l`enseignement des mathématiques peuvent se matérialiser sous la forme d`activités didactiques, nous avons effectué un sondage sur les oeuvres du mathématicien Joseph Louis Lagrange (1736-1813) et identifié le potentiel d`exploration éducatif, de l`oeuvre Leçons élémentaires sur les mathématiques données a l`École Normale en 1795, de cet mathématicien. L`objectif principal de notre étude était de faire des recherches sur le potentiel d`une oeuvre antique dédié à l`enseignement des mathématiques et de la considérer comme support conceptuel et didactique pour la création d`un modèle d`activités didactiques pour l`enseignement des mathématiques, dans la formation des enseignants de mathématiques et aussi en ce qui concerne l`apprentissage des mathématiques des élèves de l`école primaire. Nous avons fait une lecture, la traduction et l`ajout de notes et commentaires sur le travail et une recherche bibliographique sur la relation entre l`histoire des mathématiques et l`enseignement des mathématiques, de façon a comprendre les aspects conceptuels et didactiques pour l`élaboration d`um module activités didactiques pour l`enseignement des mathématiques basée sur certains chapitres du livre de Lagrange. À cette fin, l`oeuvre a été utilisé comme source primaire et a été étudié sous un fondement théorique appuyer sur les travaux des Institut de recherche sur l`enseignement des mathématiques IREM. Dans le module élaboré, les activités apportent les contenus dans une suite integrée à une logique de classe, à partir de la lecture directe des découpages du texte original, disposés entre les questions et les situations-problémes , historiquement mis en contexte avec la période et associés à des contenus spécifiques. Comme il s`agit d`une recherche basée sur l`exploitation de livres anciens, nous croyons que des modules d`activités basées sur des source primaires peuvent être utilisées comme un matériel pédagogique pour la formation des enseignants de mathématiques ainsi que pour les dernières années de l`école élémentaire, reformulées ou accrues d`autres questions telles l`intérêt de chaque enseignant qui l`utilise

Contribuições de uma unidade de ensino potencialmente significativa UEPS para o ensino de ecologia em escola pública da educação básica

The research aims to evaluate the contributions of a teaching unit to enhance the learning contents ecological green areas in elementary school. The work was conducted with elementary students in a public school in Natal-RN. We sought to identify the students´ previous knowledge about the contents of ecology, develop and implement a Potentially Meaningful Teaching Units PMTU; assess learning of ecological concepts in Teaching Unit Potentially Significant; assess the contribution of interpretative trail as strategy teaching to learning content ecology. The survey of students´ previous knowledge through the pre-test was essential to identify the appropriate subsumers and partially correct, since they served as "anchor" for further expansion of scientific concepts contained in this research. The green areas of the school as an educational, contributed the motivational aspect, as students were protagonists throughout the entire process of teaching and learning. The method of stimulated memory was effective to evidence learning ecological concepts in interpretive trails. The use of diverse activities organized on a PMTU promoted intellectual autonomy of the students and facilitated the acquisition of new meanings through progressive differentiation, and integrative reconciliation consolidation of ecological content and concepts related to biotic and abiotic factors on the basis of the central ideas thematic Life and Environment. The development of procedural skills to capture and share meanings, observe, collect and record data, hypotheses, ability to explain, to apply knowledge to solve problem situations, argue, identify, compare, differentiate and relate concepts, negotiate meanings, reflecting Critically, systematize data was evidenced. Collaborative activities promoted the incorporation of attitudinal contents as developing respect for differences, learn to work in teams to plan, develop and implement actions together for citizenship and environmental responsibility. The application of the Potentially Meaningful Teaching Units PMTU showed evidence of Critical meaningful learning ecological concepts covered in elementary school.

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

Saberes de astronomia no 1º e 2º ano do ensino fundamental numa perspectiva de letramento e inclusão

This work suggests a discussion about methodologies and didactic-pedagogical activities for the teaching of Astronomy in first and second cycles of the primary school, in a perspective of literacy and inclusion. The presented proposals have been developed in a group of the Public School Alceu Amoroso Lima, North of Natal city, with children since six years old, amongst which two considered as being with special needs . This research aims to demonstrate that it is possible to develop with those children the contents of Astronomy, while they participate of the process of literacy and inclusion. From this, we are searching a theoretical-practical contribution so that the Parâmetros Curriculares Nacionais (Brazilian Nacional Curricular Guideness) include the referred contents in first cycle of primary school. For the accomplishment of this research, Experiential Astronomy was initially proposed. Later on, many workshops had been carried through (clay, ripping , crepon paper, plasticine, cardboard and gastro-lunar ). All the proposed activities were based on the conjunction of contents, which characterized the interdisciplinarity. Through the approach we adopted and the practices we proposed, we could evidence that not only children considered as being normal , but also that ones who carries special needs could appropriate themselves of our writing code, develop and incorporate a daily relationship close to the stuffs of the sky, learn many information about all of this, besides constructing attitudinal, procedural and conceptual contents

Propostas e discussões para o ensino de astronomia nos 1º e 2º ciclos do nível fundamental e na educação de jovens e adultos

This work aims to propose and to discuss methodologies and practical activities for Astronomy teaching in the 1st and 2nd cycles of the primary education and in the adult education. The proposals presented here were applied to students from the metropolitan region of Natal (RN), including students of the called normal education (formerly magisterial education) and of the undergraduate formation in pedagogy at the Instituto de Formação Superior Presidente Kennedy , and also, in particular, to teachers and students of public municipal school Escola Municipal Djalma Maranhão at the district of Felipe Maranhão II, also analyzing some didactic books used by these institutions. Several elements which we confronted with during this didactic-pedagogical experience were systematized, indicating principles, contents, reflections and procedures related to Astronomy teaching to students of those levels of education. Doing this we aim to make such an experience accessible to those interested in developing a similar approach involving the themes treated here as well as other ones related to Astronomy for those levels of education. The resources and practices implemented here aim to contribute to the effective realization of an interdisciplinary and contextualized education according to orientations proposed by the Parâmetros Curriculares Nacionais (Brazilian National Curricular Guidelines). In order to guarantee a broad accessibility to what we propose in this work, we intend to make available in printed form and also in an Internet page the procedures, instruction materials and texts we developed

Uma proposta metodológica do ensino da química para alunos da Zona Rural :O estudo do leite

La enseñanza desarrolló, así como los métodos y las técnicas de enseñar. Aquí el he/she piensa un método instrucción, en que el estudiante está más de un oyente simple: además la parte íntegra, él se llama para intervenir continuamente, mientras tocando una parte ancho, interactivo e involucrando, del contextualização del tema para ser trabajado -inclusivo, asocie día a día al his/her y contextual al programa de la disciplina de Química, hasta la manera como ese tema debe explorarse. La leche, el tema pone en correlación al origen del grupo de los estudiantes seleccionados, fue explorado, dentro de una visión sistémica, bajo el punto de vista de la composición de las propiedades de la industrialización y de la comercialización. Buscando prever los caminos de la construcción y organización del conocimiento de los estudiantes, se usó el modelo de administración mental de Antoine de La Garranderie y la teoría de aprender de Ausubel. El conocimiento de los estudiantes se supervisó antes, durante y al final de las actividades desarrolladas, práctica que permitió verificar la evolución y el nivel de asimilación conceptual de éstos con respecto al tema en el estudio. El análisis de los resultados obtenidos, después de la aplicación de una encuesta que contempla veinte asuntos, todo pertinentes al tema, demostró un nivel del uso sobre el promedio, lo que merecido la pena el método de enseñanza propuesta

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

O ensino de óptica no nível fundamental: uma proposta de ensino aprendizagem contextualizada para a oitava série

This dissertation presents the investigation and possible interference in the current teaching of physics, specially of optics, in the eighth grade/ ninth year / fourth cycle of junior high school. The context of the current science teaching is also presented. Besides, the following aspects are discussed: the preparation of the eighth grade teachers, their professional needs, the pedagogical practices and the theoreticmethodological difficulties faced by them. Another important issue in Science Teaching, which is also discussed in this dissertation, is the need for an efficient scientific literacy so that the citizen may express value judgment about Science and Technology issues. The data about the context were gathered from questionaires answered by the teachers. In value of this information, lesson plans were elaborated and implemented. The goal was to point out alternatives for the development of a teaching-learning evaluation which would be both pleasant for the teachers in relation to the implemented methodology, end meaningful for the students by adding elements such as: interdisciplinary approach, contextualization and preparation by competences. Furthermore it was carried out a deeper analysis of the Brazilian curriculum references (LDB, DCN, PCN, RCB) and content aspects (concepts, procedures e attitudes) to be executed by that branch of human knowledge, aiming at a really Contextualized Science Teaching, qualified and pleasant for the eighth grade level. It is also presented in this paper the profile or the Professional which teach this level at state public schools in Russas, state of Ceará, Vale do Jaguaribe Region. The efficacy of the lesson plans were also analysed by using evaluation procedures with the eighth grade students of the following schools: CAIC Senador Carlos Jereissati and EEF Manuel Matoso Filho, from which the concepts acquired during the implementation of the classes became evident in the students

Refletindo dificuldades de aprendizagem de alunos do ensino médio no estudo do equilíbrio químico

This research intends to identify the main conceptual mistakes committed by High School students when studying Chemical Equilibrium and also how the didactic books can influence in their formation. For the data - collecting we applied 150 questionnaires with 11 open questions e 4 restricted ones. Students from three different private schools in Natal-RN were used in this survey. The questionary about the book analysis tried to understand how the subject has been boarded, making it easier or not for the knowledge building. The books were chosen by being usually the most used ones at the chemistry classes in the schools in town. On the didactic book analysis we evaluated 12 items that tried to identify the clarity with which the books would bring up the subjects, the contextualization of the exercises, the analogies used, the amount and the quality of exercises, the connections between the subjects, and the retaken of important concepts. The questionary given to the students tried to get their knowledge about the concepts related to Chemical Equilibrium. The main mistakes found were: the definition of the state of equilibrium in a chemical reaction, the constant, the double headed arrow, confusions between mass and concentration, the representation of a microscopic reaction in equilibrium, partial pressure and total pressure, and the function of the catalyser. The results demonstrated a lot of insecurity or lack of knowledge about the subject when it was realized that there was a big number of blank questions and a considerable percentage of questions with conceptual errors. We concluded that those errors could be eased or even avoided if the didactic books and the teachers boarded the contents in a more conceptual form with the application of a largest number of good quality exercices, watching the building of definitions by the students, avoiding this way the formation of wrong concepts

O ensino de matemática no Rio Grande do Norte: trajetória de uma Modernização (1950-1980)

This work aims to describe and analyze the process of the mathematics teacher modernizing in Rio Grande do Norte, in the period from 1950 to 1980. For that, we use as theoretical foundation assumptions of Cultural History and memories of the researchers Maurice Halbwach, Ecléa Bosi and Paul Thompson. As methodological tools, we used bibliographical resources and semi-structured interviews, in order to do a historical reconstruct of the mathematics educational scene of institutions and people who taught mathematics in Rio Grande do Norte, or those who participated in the modernization of the teaching of this subject, recovering their training and its practices in teaching. For the analysis of the bibliographical resources, initially we organized in a systematic way the transcripts of the interviews and documents, which were accumulated during the research, so long our thoughts, returning to the theoretical basis of this research, through questioning of knowledge acquired and that guided the problem of our study. The analysis showed that, important moments to modernize the teaching of mathematics in Rio Grande do Norte happened such: (1) Training Course of Lay Teachers in Rio Grande do Norte, in 1965, (2) Course for Teachers in Normal Schools, in 1971 (3) Satelite Project on Interdisciplinary Advanced Communications (SPIAC) in 1973; (4) Lectures of the teacher Malba Tahan, at Natal, from the end of the 50 s, that could be analyzed through the lessons notes of the teacher Maria Nalva Xavier de Albuquerque and the narrative of teacher Evaldo Rodrigues de Carvalho and (5) Courses of the Campaign for Improvement of Secondary Education and Broadcasting (CISEB). Thereby, the modernization of the school s mathematics teaching in Rio Grande do Norte, in the period from 1950 to 1980, was given mainly by disclosure of the Discovery Method and by the Set Theory contents in Teacher Training Courses

A tabela periódica: um recurso para a inclusão de alunos deficientes visuais nas aulas de química

Nowadays there are many reasons that aim to include people with special necessities, like those with visual deficiency, in the world of work, education, and in the society as a whole. However it is observed that when we talk about schooling inclusion, especially in High School, there is a huge gap between the theory and the practice. The lack of didactic resources, the inadequate installations, unprepared teachers, the families´ lack of information, are some of the factors that hinder the process of inclusion. Furthermore, the educators also have to deal with the roughness of the disciplinary contents and, refering to the study of Chemistry, with the use of signals related to this subject´s language. So, the objective of our research is to reflect about the apprehension of this language by the visually handicapped people, and try to contribute with their process of inclusion in the school life. On this perspective we work with the Periodic Table, which constitutes one of the indispensable tools necessary to the Chemistry learning. In order to acomplish it, the way followed by us happened in three passages. Initially, by means of a semistructured interview, we tried to get acquainted with the blind students opinion, who were participating in the research about the Periodic Table used by them throughout High School, as well as the dificulties felt when using it. After getting the answers, the Table was reelaborated to fill those students´necessities. Here, two new Tables were designed, one in Braille which shape is more compacted, and another made with high printed dots, built with sand and glue. On the third moment, the new designed Tables were tested by the students and, by means of a semi-structured interview, we tried to identify if this new resource would solve the problems concerned to the old Table. The students showed that the compacted Tables would facilitate the touch reading of the chemical elements simbols, making it clear and fast. We hope that, with the elaboration of this learning tools we can contribute with one of the elements to favor the effective participation of blind students in Chemistry classes, when studying the Periodic Table

Aprendendo e ensinando o sistema de numeração decimal: uma contribuição à prática pedagógica do professor

The central question of the present study is to identify the epistemological knowledge that the teachers-trainees possess regarding the characteristics (properties) of the decimal numbering system; its purpose is to offer a contribution to the pedagogic practice of the teachers who work within the Basic Literacy Cycle, in terms of what concerns both the acquisition of contents and the development of the knowledge that helps them in the elaboration of adequate strategies to working with the Decimal Numbering System in the classroom. The study is based on the constructivist sociointeractionist approach to teaching Mathematics and it constitutes, in itself, a methodological intervention with the teachers-trainees engaged in the Professional Qualification Program in Basic Education of the Federal University of Rio Grande do Norte. The foundations of the study were found in investigations of researchers who had carried out studies on the construction of numerical writing, showing, for instance, that the construction process of ideas and procedures involved in groupings and changes to base 10 take a lot longer to be accomplished than one can imagine. A set of activities was then elaborated which could not only contribute to the acquisition of contents but that could also make the teachers-trainees reflect upon their teaching practices in the classroom so that in this way they will be able to elaborate more consistent didactic approaches, taking into consideration the previous knowledge of the students and also some obstacles that often appear along the way. Even when teachers have access to the most appropriate dicactic resources, the lack of knowledge of the content and of the real meaning of that content make the Decimal Numbering System, a subject of fundamental importance, be taught most times in a mechanical way. The analisys of the discussions and behaviours of the teachers-trainees during the activities reavealed that they made them reflect upon their current practices in the classroom and that, as a whole, the aims of each of the activities carried out with the teachers-trainers were reached

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

Reestruturação da disciplina "Física I" do Curso de Tecnólogo em Gerência de Obras de Edificações do CEFET/PB

The aim of the present paper is to reorganize a discipline on technological physics so that the construction site of civil engineering becomes a natural environment of learning, providing the learner with the association between theory and practice as well as allowing the subject to process, in real time, information generated from his cognitive constructions and his contextualizations. Thus, a sequence of actions was taken into account: firstly, the programme was developed in the classroom, sharing with its contextualized information through experiments done under supervision by the learners in laboratories; secondly, the data which associate physics with construction were collected and, to do so, technical visits to construction sites were realized, providing the learner the association between the theory and the practice in a suitable site to the constructivist approach. As a result, the first discipline on physics of the Curso de Tecnólogos em Gerência de Obras de Edificações do CEFET/PB was re-structured in terms of syllabus, methodology, application and evaluation. In fact, this work deals with a dynamic process that gathers and gives emphasis to teaching, learning, technology, information, creativity, competence and abilities in a constructivist learning process and, as a consequence, having allowed institutional engagement