Uma visita ao universo matemático de Lewis Carrol e o (re)encontro com sua lógica do nonsense

Notable mathematics teacher, Lewis Carroll, pseudonym of Charles Lutwidge Dodgson (1832-1898), made the mixture of mathematics with literature a ludic environment for learning that discipline. Author of Alice s Adventures In Wonderland and its sequel Alice Through The Looking Glass, he eventually created a real and complex universe which uses what we call the logic of the nonsense as an element to motivate the development of mathematical thinking of the reader, taking it as well, learn by establishing a link between the concrete (mathematics) and the imaginary (their universe). In order to investigate and discuss the educational potential of their works and state some elements that can contribute to a decentralized math education from the traditional method of following the models and decorate formulas, we visited his works based on the studies of archeology of knowledge (FOUCAULT, 2007), the rational thought and symbolic thinking (VERGANI, 2003) and about the importance of stories and narratives to the development of human cognition (FARIAS, 2006). Through a descriptive, analytical study, we used the literary construction and presented part of our study in form of a mathematical novel, to give the mathematical school a particular charm, without depriving it of its basics properties as discipline and content. Our study showed how the works of Carroll have a strong didactic element that can deploy in various activities of study and teaching for mathematics classes

Um estudo sobre as origens da Lógica Matemáitca

The present study has as objective to explaining about the origins of the mathematical logic. This has its beginning attributed to the autodidactic English mathematician George Boole (1815-1864), especially because his books The Mathematical Analysis of Logic (1847) and An Investigation of the Laws of Thought (1854) are recognized as the inaugural works of the referred branch. However, surprisingly, in the same time another mathematician called Augutus of Morgan (1806-1871) it also published a book, entitled Formal Logic (1847), in defense of the mathematic logic. Even so, times later on this same century, another work named Elements of Logic (1875) it appeared evidencing the Aristotelian logic with Richard Whately (1787-1863), considered the better Aristotelian logical of that time. This way, our research, permeated by the history of the mathematics, it intends to study the logic produced by these submerged personages in the golden age of the mathematics (19th century) to we compare the valid systems in referred period and we clarify the origins of the mathematical logic. For that we looked for to delineate the panorama historical wrapper of this study. We described, shortly, biographical considerations about these three representatives of the logic of the 19th century formed an alliance with the exhibition of their point of view as for the logic to the light of the works mentioned above. In this sense, we aspirated to present considerations about what effective Aristotelian´s logic existed in the period of Boole and De Morgan comparing it with the new emerging logic (the mathematical logic). Besides of this, before the textual analysis of the works mentioned above, we still looked for to confront the systems of Boole and De Morgan for we arrive to the reason because the Boole´s system was considered better and more efficient. Separate of this preponderance we longed to study the flaws verified in the logical system of Boole front to their contemporaries' production, verifying, for example, if they repeated or not. We concluded that the origins of the mathematical logic is in the works of logic of George Boole, because, in them, has the presentation of a new logic, matematizada for the laws of the thought similar to the one of the arithmetic, while De Morgan, in your work, expand the Aristotelian logic, but it was still arrested to her

Uma análise histórico-epistemológica do conceito de grupo

This work aims to analyze the historical and epistemological development of the Group concept related to the theory on advanced mathematical thinking proposed by Dreyfus (1991). Thus it presents pedagogical resources that enable learning and teaching of algebraic structures as well as propose greater meaning of this concept in mathematical graduation programs. This study also proposes an answer to the following question: in what way a teaching approach that is centered in the Theory of Numbers and Theory of Equations is a model for the teaching of the concept of Group? To answer this question a historical reconstruction of the development of this concept is done on relating Lagrange to Cayley. This is done considering Foucault s (2007) knowledge archeology proposal theoretically reinforced by Dreyfus (1991). An exploratory research was performed in Mathematic graduation courses in Universidade Federal do Pará (UFPA) and Universidade Federal do Rio Grande do Norte (UFRN). The research aimed to evaluate the formation of concept images of the students in two algebra courses based on a traditional teaching model. Another experience was realized in algebra at UFPA and it involved historical components (MENDES, 2001a; 2001b; 2006b), the development of multiple representations (DREYFUS, 1991) as well as the formation of concept images (VINNER, 1991). The efficiency of this approach related to the extent of learning was evaluated, aiming to acknowledge the conceptual image established in student s minds. At the end, a classification based on Dreyfus (1991) was done relating the historical periods of the historical and epistemological development of group concepts in the process of representation, generalization, synthesis, and abstraction, proposed here for the teaching of algebra in Mathematics graduation course

Reflexões sobre a construção e evolução de conceitos geométricos nas séries intermediárias do ensino fundamental

This thesis represents a didactic research linked to the Post-graduation Programme in Education of the Universidade Federal do Rio Grande do Norte which aimed to approach the construction of the geometrical concepts of Volume of the Rectangular Parallelepiped, Area and Perimeter of the Rectangle adding a study of the Area of the Circle. The research was developed along with students from the 6th level of the Elementary School, in a public school in Natal/RN. The pedagogical intervention was made up of three moments: application of a diagnostic evaluation, instrument that enabled the creation of the teaching module by showing the level of the geometry knowledge of the students; introduction of a Teaching Module by Activities aiming to propose a reflexive didactic routing directed to the conceptual construction because we believed that such an approach would favor the consolidation of the learning process by becoming significant to the apprentice, and the accomplishment of a Final Evaluation through which we established a comparison of the results obtained before and after the teaching intervention. The data gathered were analyzed qualitatively by means of a study of understanding categories of mathematical concepts, in addition to using descriptive statistics under the quantitative aspect. Based on the theory of Richard Skemp, about categorization of mathematical knowledge, in the levels of Relational and Instrumental Understanding were achieved in contextual situations and varied proportions, thus enabling a contribution in the learning of the geometrical concepts studied along with the students who took part in the research. We believe that this work may contribute with reflections about the learning processes, a concern which remained during all the stages of the research, and also that the technical competence along with the knowledge about the constructivist theory will condition the implementation of a new dynamics to the teaching and learning processes. We hope that the present research work may add some contribution to the teaching practice in the context of the teaching of Mathematics for the intermediate levels of the Elementary School

Como vão se formando os professores em física e química: embates entre o ser, o ter e o fazer na formação de licenciandos da UFRN sob a perspectiva da teoria das representações sociais

The purpose of this work is to approach and understand the Social Representations (SR) (MOSCOVICI, 2003) about Physics and Chemistry from people who are major in these courses, as well as their Social Representations about teaching . We took as principle that approaching these representations it would be possible to relate their symbolic contents, in order to show how people who are following the first segments of bachelor degree courses in Physics and Chemistry become teachers, taking into account a psychosocial view. Two source of data was used during this research: Free-association Technique FA (ABRIC, 1994); and Multiple Classification Procedure (MCP) (ROAZZI, 1995). The analytical treatment of the collected data from FA was done according to the proposition of Grize, Vergés and Silem (1987 apud ABRIC, 1994, p. 66). MCP data were analyzed through MSA (Multidimensional Scalogram Analysis) and SSA (Singular Spectrum Analysis) methods associated with the Facet Theory (BILSKY, 2003). The discourses of MCP discussing groups at the moment of explanations were studied by Content Analysis as it was proposed by Bardin (1977) and Franco (2005). Indicative of an approach to the relations with knowledge (CHARLOT, 2000), the connections which aroused from the analyses showed that the group of future majors in Physics thought that this scientific field was based on a rationalist conception, influencing the idealization sense of the phenomena to be explained by Physics. Thus, Physics as school content started to require the student of the fundamental and high school to think abstractly as a cognitive skill of learning. The identifying elements observed in the relations between SR about Physics and Teaching aroused from the antagonism between future majors and their teacher, as well as from the speculation between their fundamental and high school students and themselves, mainly when they had to face the act of teaching due to the obstacles imposed by the own educational system, and by the weakness of the initial preparation. The group of future majors in Chemistry, through its discourses, showed these relations when they conceived empiricist Chemistry and said that teaching was the way of transmission of this knowledge, and didactics of Chemistry teaching was the direction to learning through pedagogic methods in order to lead the students to discoveries. The psychosocial contents which were built and showed from the symbolic relations in the studied SR achieved the relation of identity. This relation revealed identifying elements for these people, resulting from the traffic between their condition as students of Chemistry, and as teachers regarding their work, what placed the current relational contents in the teaching space, named as Knowledge changing and Adaptability . In order to study emerging questions in the discussing environment about formation and teaching professionalization, we focused the psychosocial view on this traffic and managed to observe epistemological practical and pedagogic obstacles that limited a configuration of the teaching work as a professional activity, especially from the particular conditions which led the relations of senses to Physics , Chemistry and Teaching ; and Chemistry and Physics as it was seen in this research. Generally speaking, we noted that these obstacles can denounce such obstacles concerning to the pedagogic doings which mainly impair the learning process of fundamental and high school students

Educação inclusiva e formação continuada: contribuições dos casos de ensino para os processos de aprendizagem e desenvolvimento profissional de professores

This study focuses on processes of learning and professional development experienced by elementary school teachers who have students with special educational needs in their classrooms. Cases and case methods can be used as methodological resource to articulate the continued training of teachers in an inclusive perspective. This research-intervention adopted a constructive-collaborative model for continued teachers formation. The main objective was to investigate the possible contributions of teaching cases, while investigative and formative strategies, for the processes of learning and professional development of teachers who work in the regular school. The data were collected by means of analytical activities, drafting collective discussion and teaching cases, having eight teachers as participants in a regular public school, located in the municipality of Natal/RN, Brasil. The theoretical reference covers the inclusive education, teaching learning, teachers professional development, the knowledge base for teaching and teaching cases as a resource for continued teachers formation in an inclusive perspective. The results indicated that teaching cases allowed description and analysis of educational practices developed by regular education teachers and adoption of reflective processes about situations reported and on their own pedagogical actuation, achieving indications of changes. It also indicates the contribution of cases for the clarification, systematization and extension of professional knowledge about inclusive education process as well as for involvement by the teachers of the study in a pedagogical thinking process. The lessons learned are related mainly to own role as teachers of regular education, to the role of professional support and specialized institutions faced to school inclusion. The analyses showed the methodological option was suitable to develop a schoolcentric training process, allowing teachers to seek in their actual environment alternatives for construction of a new logic of teaching that encloses diversity. The conclusion is that the cases, while bringing educational situations closer to the reality experienced by teachers in their daily professional role, causes relevant improvement on teachers education, because it offers training in conjunction with the experience and knowledge that teachers already have

Educação inclusiva e formação continuada: contribuições dos casos de ensino para os processos de aprendizagem e desenvolvimento profissional de professores

This study focuses on processes of learning and professional development experienced by elementary school teachers who have students with special educational needs in their classrooms. Cases and case methods can be used as methodological resource to articulate the continued training of teachers in an inclusive perspective. This research-intervention adopted a constructive-collaborative model for continued teachers formation. The main objective was to investigate the possible contributions of teaching cases, while investigative and formative strategies, for the processes of learning and professional development of teachers who work in the regular school. The data were collected by means of analytical activities, drafting collective discussion and teaching cases, having eight teachers as participants in a regular public school, located in the municipality of Natal/RN, Brasil. The theoretical reference covers the inclusive education, teaching learning, teachers professional development, the knowledge base for teaching and teaching cases as a resource for continued teachers formation in an inclusive perspective. The results indicated that teaching cases allowed description and analysis of educational practices developed by regular education teachers and adoption of reflective processes about situations reported and on their own pedagogical actuation, achieving indications of changes. It also indicates the contribution of cases for the clarification, systematization and extension of professional knowledge about inclusive education process as well as for involvement by the teachers of the study in a pedagogical thinking process. The lessons learned are related mainly to own role as teachers of regular education, to the role of professional support and specialized institutions faced to school inclusion. The analyses showed the methodological option was suitable to develop a schoolcentric training process, allowing teachers to seek in their actual environment alternatives for construction of a new logic of teaching that encloses diversity. The conclusion is that the cases, while bringing educational situations closer to the reality experienced by teachers in their daily professional role, causes relevant improvement on teachers education, because it offers training in conjunction with the experience and knowledge that teachers already have

Formação e vivências: a representação social do ser professor em comunidades ribeirinhas do Vale do Juruá Acre

Being a teacher is a challenging task, driving action strategies that mobilize technical dimensions of being / doing teaching, but also affection. This research was conducted with teachers / students of the Special Program of formation for Teachers of Basic Education agricultural zone (PROFIR) developed by the Federal University of Acre in partnership with the State Department of Education (SEE) and City Halls of the region; locus typically Amazon and, in this research, comprises five municipalities in the Valley of Juruá - Acre. Northern Brazil. It Aims to capture the social representation of being a teacher in riverside communities. We performed this study from the Theory of Social Representations (MOSCOVICI, 1978, 1990, 2005a, 2005b, 2009, 2011) and (JODELET, 2001), from discussions on teacher formation (LDB No. 9394 of 1996). Work, among others, (TARDIF, 2004), (TARDIF; LESSARD, 2007), (NÓVOA, 1992, 1999), (ARROYO, 2007). We used as a methodological strategy, the multiple classification procedure (PCM), which is organized in two steps: Free Classification and Directed Classification, as well as semi-structured interviews, involving, thus, 137 teachers surveyed. For the analysis of the data, we appealed to software SPSS, version 13.0 for Windows, mapping the elements of the representational content through Multidimensional Analysis (MSA and SSA). Through the different strategies of analysis, we identified from the Free Classification, the regionalization: Beyond forest, teaching Attributes and Mobilized Attributes. With data of the Directed Classification, the analysis was based on the Theory of Facets (BILSKY, 2003) and (BUSCHINI, 2005) and we found: Reference of teacher; Negative Dimension and affective and technicians attributes. Finally, with the Analysis of Content (BARDIN, 2004) e (FRANK, 2005) and of the interviews the categories we found the categories: Urban relation versus agricultural, Knowledge in construction and possible bets. The report of the experiences along the edge of the rivers, in these different methodological strategies and of analysis, demonstrates the existence of symbolic and representational elements that influence and guide the educational conduct and actions of these teachers and that they are deeply anchored and objectified in technical and affective elements of the making teacher. They daily live deeply the concreteness of the Amazonian realism redesigning the Real and the symbolic one as a way to understand themselves as a teacher in adversity of this context. We also observed the historical, social and cultural local marks, framing, thus, the collective personality and generating its guides of action

A implementação do programa Gestão nota 10 no sistema municipal de educação em Mossoró/RN (2005 a 2009)

This research work is focused to show the changes in educational administration from the agreements between the Mossoró / RN and the Ayrton Senna Institute IAS, for education provision. Nowadays, the partnership policy is a constitutive element of the reform of the Brazilian State, which dropped its action on social policies and to strengthen its regulatory role, encouraging private participation in planning, preparation and implementation of public policies, new printing setting the political-social. In this context, the 10 Note Management Programme, developed by the IAS, is part of the neoliberal logic of modernization of public school systems, focusing on results and developing strategies for control and regulation of schools work and its efficiency, effectiveness and greater productivity. The 10 Note focuses on two dimensions: the management of learning and teaching in networking, in a managerial perspective to overcome the culture of failure (expressed as age-grade, dropout and repetition rates in) and implantation of culture of success (as measured in the improvement of the indices). To understanding the process, we have delimited as the object of study, the process of implementing them mentioned program in the city, which its objective is to analyze implications for the school community from the perspective of democratic management, adopting the dimensions of autonomy and participation in institutional processes as a criterion of analysis. From a methodological point of view, the survey was conducted from a literature review and documentary about educational policy developed in the country since the 1990´s, seeking to understand, in a dialectical perspective, the political dimensions of teaching, training and performance of the subjects involved in the school work. Besides the empirical observation, it was also used semi-structured interviews with a methodological tool for gathering information and opinions about the partnership and the implementation of the 10 Note Management Program in the county. The interviewee participants were ex-former education managers, coordinators, school managers, secretaries and teachers. Regarding the dimensions inside the analysis (autonomy and participation), the research led to the conclusion: that GEED, under the guidance of IAS promoted regulation of school autonomy, set up the selection process for exercising the office of school administration and system awards to schools, pupils and teachers, subject to results, there is mismatch between the managerial logic and the democratic management principles, that the ideological discourse of modernization of municipal management coexists with traditional practices, centralizing patronage, which ignores the democratic participation in the school decisions processes, the goals of the partnership were partially achieved, since that the city has improved over the approval and dropouts, although the approval of the Education Municipal Plan of the rules institutional (administrative, financial and educational) and the creation of the Councils observed that the school community participation is still limited, not being characterized as a coordinated intervention, capable of promoting the transformation and improvement its quality in the county. In the same way, the orientation of networking is a limit to the autonomy of schools, given the external definition of goals and strategies to be adopted, along with pressure exerted through the accountability of each school community for their achievements

Nos tempos do blackout: cena musical, práticas urbanas e a ressignificação da Rua Chile, Natal-RN

This study aims to analyze the process of resignification of Chile Street, in Natal, from the development of a music scene in the late 1990s. Chile Street, as part of the Historic Centre of Natal, had its images constructed from the discursive practices and everyday life of its regular visitors, leading to a series of symbolic and imaging transformations throughout the twentieth century. Initially transformed into glamorous space as a result of urban actions of the new republic of Albuquerque Maranhão, in the early twentieth century, Ribeira and Chile Street, specifically, came to be seen as bohemian area, during the war time"; followed by a marginal phase, it was eventually transformed into the main rocker area of Natal, with the development of a musical scene in the second half of the 1990s. This music scene, its practices, economic interests, cultural events and identity ties created among their practitioners made Chile Street, "in the time of Blackout night club", an "alternative" space. As the historic centers, inserted in the logic of postmodern city marketing, both spaces are dynamic in their practices, as in their images, Chile Street also suffered changes in its meanings and symbols, around the year 2000 when the alternative-underground space became a pop space, where people from various parts of the city began attending its events and places, transforming it into a point of very heterogeneous sociability