La thèse met en lumière les conditions d’enseignement et d’apprentissage des mathématiques en langue seconde en étudiant avec précision l’articulation des savoirs mathématiques et des savoirs linguistiques. Elle traite le cas spécifique de l’enseignement des mathématiques en anglais dans un contexte CLIL et les séances expérimentales ont lieu en classes européennes de lycée. Le thème commun à ces séances est celui des preuves visuelles et multimodales. La Théorie des Situations Didactiques (TSD) offre un cadre théorique privilégié – notamment pour la construction des situations expérimentales - cadre qu’il a fallu compléter par des approches théoriques sémiotiques et linguistiques. Ainsi l’approche adoptée s’est révélée en adéquation avec la perspective actionnelle et la phraséodidactique a apporté de nombreux éléments permettant de mettre en relief le rôle de la phraséologie dans un enseignement intégré. Une focalisation particulière a dû être opérée sur les objets mathématiques et les processus d’abstraction mais aussi sur certains faits de langue. Les investigations ont permis d’affiner les descriptions des raisonnements produits tout en conservant une référence aux niveaux de milieux, au sens de la TSD. L’étude a nécessité de développer le concept de représentation et de décliner les représentations produites dans le contexte de la L2. Ce sont ces concepts et celui d’adidacticité, central dans la TSD, qui ont permis d’organiser les séances de manière optimale, en faisant apparaître le rôle essentiel joué par la perception active dans les processus de conceptualisation. / The purpose of this thesis is to investigate learning and teaching conditions of mathematics in English as a second language by closely examining how mathematical and language knowledge can fit together. This study deals with the specific case of CLIL teaching and the related experimental situations are performed in European classes in a French high school. The situations have a common topic, namely that of visual and multimodal proof. The theory of Didactical Situations is the central theoretical framework but our study has proven to be compatible with task-based pedagogy. Besides, phraseodidactics provided a useful and adequate auxiliary framework by shedding some light onto the essential role played by4phraseology. We particularly kept focused on mathematical objects and processes of abstraction but also on some specific language features. The concept of representation is central in our research works and thus had to be precisely defined. The success of our experimental situations owes a lot to the use of adidacticity, a central concept in TSD, and our focusing on the crucial part played by active perception during processes of conceptualisation. The purpose of one of the experimental situations (conducted in a second language) was to ensure that pupils divised, by themselves, a visual proof of an arithmetic property previously conjectured, carried out on the very level of schematisation an explicit generalisation and used real cubes to perform another type of proof, thus making the inductive step of the induction explicit.

Made available in DSpace on 2017-07-21T20:32:04Z (GMT). No. of bitstreams: 1 Priscila Kabbaz Alves da Costa.pdf: 3528333 bytes, checksum: dcfd880a8a16525a38d82354ac6cf9f9 (MD5) Previous issue date: 2013-09-20 / This study presents an analysis of the learning evaluation process developed in the Distance Teaching Mathematics curse, offered by the Ponta Grossa State University (UEPG), in partnership with the Brazil Open University (UAB). The qualitative nature of this multiple case study, aimed to analyze the learning evaluation process developed in subjects taught in the course and was developed from the following problem: how does the learning evaluation developed at the Distance Mathematics teaching Course – EaD can contribute to the initial instruction of teachers? The characterization of the historical background of EaD in the world and in the Brazilian context, the constitution of such model of distance teaching in UEPG and the partnership UEPG/MEC/UAB to offer graduation courses, mainly Mathematics teaching were discussed in the first chapter. The theoretical background of this study regarding educational, learning and distance learning evaluation was built on the contributions of Hadji (2001), Fernandes (2009), Brandalise (2010), Dias Sobrinho (2003), Cappelleti (2010), Kenski (2010), Valadares (2011), Sousa (1991), Silva (2010), Silva (2011), Behar and Notare (2009), Álvarez Méndez (2002) and Luckesi (2011), Okada( 2003,2006) amongst others. The methodological approach was based on Flick (2007), André (2005), Oliveira (2007) and Lüdke (1998). In order to study multiple cases, the procedure chosen was the analysis of documents as follows: The Course Pedagogical project (CPP), printed teaching material, the learning virtual environment (LVE) of the subjects selected: Instrumentation to the teaching of Mathematics II, Geometry II, Fundaments of Mathematics II and Introduction to Integral Differential calculus, together with evaluation activities, forums, tasks, questionnaires, course book and tests. The research results revealed that although the EaD teaching model has particularities in the evaluation process, the epistemological and methodological fundaments of learning evaluation are relevant for both the attendance and the distance teaching. The learning evaluation analyses showed that the implementation of a diagnostic, formative and student centered process evaluation, as proposed by the course CPP is still incipient and a great challenge for all individuals involved in the evaluation process - teachers, tutors and students – once the author/instructor teacher‟s conception of teaching and learning is usually what defines the students‟ instruction trajectory, together with the LVE organization in order to enable the construction of the mathematics knowledge by the student, and the choice of evaluation instruments. As the summative perspective with a rationalist and traditional base of learning evaluation still prevails, towards the results obtained by the students rather than the learning process, the number of drop outs and failure are significant. The weaknesses of learning evaluation processes, however, should be discussed and improved by their practitioners considering that there is a relevant social value in the offer of distance learning courses which represents the possibility of qualifying professionals to work with mathematics teaching in places where there are no higher education institutions. / Este trabalho apresenta uma análise do processo de avaliação da aprendizagem desenvolvido no curso de graduação de Licenciatura em Matemática a distância, ofertado pela Universidade Estadual de Ponta Grossa (UEPG), em parceria com a Universidade Aberta do Brasil (UAB). A pesquisa de natureza qualitativa, do tipo estudo de casos múltiplos, objetivou analisar o processo de avaliação da aprendizagem desenvolvido em disciplinas do curso e foi desenvolvida a partir do seguinte problema: como o processo de avaliação da aprendizagem desenvolvido na Licenciatura em Matemática - EaD pode contribuir na formação inicial dos professores? A caracterização da trajetória histórica da EaD no mundo e no contexto brasileiro, a constituição da modalidade de ensino a distância na UEPG e a parceria UEPG/MEC/UAB para oferta de cursos de graduação, em particular a Licenciatura em Matemática foi contemplada nas discussões do primeiro capítulo. A fundamentação teórica do trabalho em avaliação educacional, da aprendizagem e da aprendizagem na EaD foi construída a partir das contribuições de Hadji (2001), Fernandes (2009), Brandalise (2010), Dias Sobrinho (2003), Cappelleti (2010), Kenski (2010), Valadares (2011), Sousa (1991), Silva (2010), Silva (2011), Behar e Notare (2009), Álvarez Méndez (2002) e Luckesi (2011), Okada (2003, 2006) entre outros. O percurso metodológico foi traçado com fundamento em Flick (2007), André (2005), Oliveira (2007) e Lüdke (1998). Para o estudo dos casos múltiplos o procedimento escolhido foi a análise de documentos, entre eles: o Projeto Pedagógico do Curso (PPC), os materiais didáticos impressos, o Ambiente Virtual de Aprendizagem (AVA) das disciplinas selecionadas: Instrumentação para o Ensino de Matemática II, Geometria II, Fundamentos da Matemática III, Introdução ao Cálculo Diferencial e Integral, com as atividades avaliativas, fóruns, tarefas, questionários, livro texto, provas, nele contidos. Os resultados da pesquisa revelaram que embora a modalidade de ensino EaD tenha suas especificidades nos processos avaliativos os fundamentos epistemológicos e metodológicos da avaliação da aprendizagem são pertinentes tanto para o ensino presencial como para o a distância. As análises da avaliação da aprendizagem evidenciaram que a efetivação de uma avaliação diagnóstica, formativa e processual centrada no aluno, como explicitado no PPC do curso, é ainda insipiente e um grande desafio para todos os sujeitos envolvidos no processo avaliativo - professores, tutores e alunos - porque a concepção de ensino e aprendizagem do professor autor/formador geralmente é que define o percurso formativo dos alunos no curso, a organização do AVA a fim de propiciar construção do conhecimento matemático pelo aluno, e a escolha que faz dos instrumentos avaliativos. Como ainda prevalece a avaliação da aprendizagem numa perspectiva somativa, de base racionalista e tradicional, mais voltada aos resultados obtidos pelos alunos que ao processo de aprendizagem são expressivos os índices de evasão, desistência e reprovação dos alunos. As fragilidades nos processos de avaliação da aprendizagem, no entanto, são passíveis de discussão e aprimoramento pelos seus atores considerando que há um valor social relevante na oferta da Licenciatura a distância que é a possibilidade de formar profissionais qualificados para atuarem com o ensino de Matemática, em locais desprovidos de instituições de ensino superior.

A presente investigação tem como ponto de partida o constante desafio e desejo do professor de matemática em construir estratégias que sejam promotoras de aprendizagens de matemática na escola básica. Com essa preocupação, inicia-se destacando a necessária centralidade da pesquisa no estudante e seus processos construtivos do conhecimento. A partir de discussões iniciais a respeito da aprendizagem na Cultura Digital e de indicações da potencialidade da arquitetura pedagógica de Projetos de Aprendizagem enquanto alternativa aos desafios educacionais próprios do século XXI, propõe-se investigar a intervenção do professor, de modo a levantar categorias que subsidiem a ação pedagógica com estudantes dos anos finais do ensino fundamental para a promoção da aprendizagem de matemática na Cultura Digital a partir de Projetos de Aprendizagem. Como trajetória de investigação, distinguem-se duas perspectivas: a pesquisa teórica e a experimental. Busca-se, então, fundamento na Epistemologia Genética piagetiana, especificamente nos processos de Equilibração Majorante – como essenciais para a compreensão estrutural da construção de conhecimento –, nas relações entre a Tomada de Consciência e o Fazer e Compreender – fundamentais para a análise da ação do estudante – e no Método Clínico – imprescindível para a discussão sobre a intervenção do professor dentro de uma perspectiva construtivista do conhecimento. Além disso, discutem-se as transformações advindas da Cultura Digital, suas implicações na cognição do jovem aprendiz e na construção de uma escola da Cultura Digital. Por fim, ainda na faceta teórica do estudo, apresenta-se e analisa-se a arquitetura pedagógica de Projetos de Aprendizagem à luz da teoria piagetiana. Com base no construto teórico produzido, propõe-se, então, o modelo de intervenção que se pretende experimentar A segunda perspectiva da investigação, é desenvolvida por meio de três experiências, as quais pretendem analisar (i) a viabilidade potencial, (ii) a viabilidade prática, a eficácia, e (iii) a validade da proposta pedagógica de promover aprendizagens de matemática via Projetos de Aprendizagem nos anos finais do Ensino Fundamental e das categorias de intervenção levantadas. Em consonância com os dados produzidos nas três experiências, as análises permitiram, então, a validação da arquitetura pedagógica de Projetos de Aprendizagem como uma intervenção do tipo 1, adequada a construção de uma escola da Cultura Digital, e das três categorias que identificam a ação do professor e definem as intervenções do tipo 2, denominadas: intervenções exploratórias, desequilibradoras e informativas. Considerando, por fim, os resultados e as proposições da presente investigação, aliados à experiência docente no viés da educação matemática e à importância da continuidade da pesquisa do professor como prática constante da sua ação pedagógica, levantam-se algumas possibilidades para a qualificação das investigações sobre a intervenção do professor e suas implicações na aprendizagem de estudantes ao longo da sua vida escolar. / The present research has as starting point the constant challenge and desire of the mathematics teacher to construct strategies that are incentivetor of mathematics learning in the basic school. With this concern, it begins by highlighting the necessary centrality of student research and its constructive processes of knowledge. From initial discussions about learning in the Digital Culture and indications of the potentiality of the pedagogical architecture of Learning Projects as an alternative to the educational challenges of the 21st century, it is proposed to investigate the teacher’s intervention, in order to identify categories that subsidize the pedagogical action with students of the final years of elementary school to promote the learning of mathematics in Digital Culture from Learning Projects. As a research trajectory, we distinguish two perspectives: theoretical and experimental research. We seek, therefore, a foundation in Piagetian Genetic Epistemology, specifically in the processes of Major Balance – as essential for the structural understanding of the knowledge construction – in the relations between the Taking of Consciousness and the Making and Understanding – fundamental for the analysis of the action of the student – and in the Clinical Method – essential for the discussion about the intervention of the teacher within a constructivist perspective of knowledge. In addition, the transformations arising from Digital Culture, its implications on the young learners cognition and the construction of a school of Digital Culture are discussed. Finally, still in the theoretical facet of the study, the pedagogical architecture of Learning Projects is presented and analyzed in light of the Piagetian theory Based on the theoretical construct produced, it is proposed, then, the intervention model to be tried. The second research perspective is developed through three experiments, which aim to analyze (i) the potential feasibility, (ii) practical feasibility, effectiveness, and (iii) the validity of the pedagogical proposal to promote mathematics learning by Learning Projects in the final years of Elementary School and the categories of intervention identified. In agreement with the data produced in the three experiments, the analyzes allowed the validation of the Learning Projects pedagogical architecture as a type 1 intervention, adequate to the construction of a School of Digital Culture, and of the three categories that identify the action of the teacher and define type 2 interventions, namely: exploratory, unbalanced and informative interventions. Considering, finally, the results and propositions of the present investigation, combined with the teaching experience in the bias of mathematics education and the importance of the continuity of teacher research as a constant practice of its pedagogical action, there are some possibilities for the qualification of the investigations On the intervention of the teacher and its implications in the learning of students throughout their school life. / La presente investigación tiene como base un constante desafío y deseo del profesor de matemática en construir estrategias que sean promotoras del aprendizaje de matemática en la escuela. Con esa preocupación, se inicia destacando la necesaria centralidad de la pesquisa en el estudiante y sus procesos constructivos del conocimiento. A partir de discusiones iniciales respecto al aprendizaje en la Cultura Digital y de indicaciones de la potencialidad de la Arquitectura Pedagógica de Proyectos de Aprendizaje en cuanto a alternativas a los desafíos educacionales proprios del siglo XXI, se propone investigar la intervención del profesor, con la finalidad de levantar categorías que subsidien a la acción pedagógica con estudiantes de los años finales de la enseñanza fundamental para la promoción del aprendizaje de matemática en la Cultura Digital a partir de Proyectos de Aprendizaje. Como trayectoria de investigación se distinguen dos perspectivas: la pesquisa teórica y la experimental. Busca-se, entonces, fundamento en Epistemología Genética Piagetana, específicamente en los procesos de Equilibración Mallorante – como esenciales para la comprensión estructural de la construcción de conocimiento –, en las relaciones entre la Tomada de Consciencia y el Hacer y Comprender – fundamentales para el análisis de la acción del estudiante – y en el Método Clínico – imprescindible para la discusión sobre la intervención del profesor dentro de una perspectiva constructivista del conocimiento. Ademas, se discute las transformaciones venidas de la Cultura Digital, sus implicaciones en la cognición del joven aprendiz en la construcción de una escuela de Cultura Digital Aun en la perspectiva teórica del estudio, se presenta y se analiza a la arquitectura pedagógica de Proyectos de Aprendizaje a la luz de la teoría piagetana. Con base en el constructo teórico producido, se propone, entonces, el modelo de intervención que se pretende experimentar. La segunda perspectiva de investigación, es desarrollada por medio de tres experiencias, las cuales pretenden analizar (i) la viabilidad potencial, (ii) la viabilidad práctica, la eficacia, y (iii) la validad de la propuesta pedagógica de promover aprendizajes de matemática vía Proyectos de Aprendizaje en los años finales de la Enseñanza Fundamental y de las categorías de intervención levantadas. En consonancia con los datos producidos en las tres experiencias, las análisis permitieron, entonces, la validación de la arquitectura pedagógica de Proyectos de Aprendizaje como una intervención del tipo 1, adecuada a la construcción de una escuela de Cultura Digital, y de las tres categorías que identifican a la acción del profesor y definen las intervenciones del tipo 2, denominadas: intervenciones exploratorias, desequilibradoras e informativas. Considerando, por fin, los resultados y las proposiciones da la presente investigación, aliados a la experiencia docente en la educación Matemática y la importancia de la continuidad de la pesquisa del profesor como práctica constante da su acción pedagógica, se levantan algunas posibilidades para la cualificación de las investigaciones sobre la intervención del profesor y sus implicaciones en el aprendizaje de estudiantes en su vida escolar.

<p>One of the big problems of the Swedish nine-year compulsory school is the large number of pupils who fail to achieve a satisfactory standard in mathematics. One explanation that has been increasingly considered over the last ten years is that the pupils have dyscalculia. Some research suggests that 6 per cent of compulsory school pupils suffer from this dysfunction, which would in that case make it one of the Swedish school’s greatest teaching problems.</p><p>The purpose of this thesis is to examine this problem area from two aspects. First of all by examining the concept of dyscalculia by means of a review of the literature from 1992 onwards. The second perspective has as its starting point a case study where the purpose was to give a detailed picture of the pupil with mathematics problems. The latter part of the study was carried out over a six-year period when 200 pupils, 13 of them with particular mathematics problems, were studied in detail.</p><p>A point of departure for the study was provided by a large database where as much information as possible was collected about pupils from Year 5 of the nine-year compulsory school to Year 2 of the three-year upper secondary school. The pupils were asked to fill in regular questionnaires and classroom observations were made of roughly 100 mathematics lessons, 40 of which were recorded on video. Finally there were in-depth interviews of the 13 pupils on two occasions, the final one being during Year 2 of the upper secondary school.</p><p>The review of the research showed a series of dubious and indistinct circumstances surrounding the dyscalculia concept, and also ambiguity with regard to the diagnosis of dyscalculia. The conclusion of the review was that the concept of dyscalculia ought at present to be used with great caution, or perhaps not at all. Admittedly the review does not provide grounds for totally dismissing the dyscalculia concept, but as long as it remains impossible to determine the concept unambiguously, and I have not been able to do this in the course of this study, there are no good scientific grounds for using the term dyscalculia in practice.</p><p>The empirical study shows the complexity of the problem area. Both the causes suggested by the pupils as the origin of the problem and the measures that helped them to obtain their mathematics grades form a complex pattern. The low work input of the pupils during mathematics lessons, an unsettled working environment, large classes, problems of stress and anxiety prior to tests, and obstructive gender patterns are among the causes suggested by the pupils as explanations of the occurrence of the mathematics problems. Good teachers, in other words teachers who can explain, set limits and give encouragement, were a significant factor in reversing the downward trend. Positive experiences of school changes, where the pupil felt that he or she could start again from the beginning, were also mentioned as significant by several pupils. Collaboration with fellow-pupils and the fact that the pupils themselves decided to get to grips with the problems were other important reasons for the change. The prospects of students with specific problems in mathematics nevertheless being able to leave compulsory school with satisfactory grades appear, however, from the results of this study, to be bright. All the pupils left the compulsory school with satisfactory mathematics grades and also completed mathematics studies at upper secondary school, despite major problems in the subject at intermediate school (age 10-13) stage.</p><p>The study indicates the need for research closer to the actual practical situation and to the importance of emphasizing good examples in practice. As the students themselves emphasize discrete communication between them as significant in the subject of mathematics, this is also an important area for future research.</p>

I vilken grad ett barn är fonologiskt medvetet är i många fall avgörande för hur barnets läsinlärning kommer att fortlöpa, varför det är av stor vikt att redan tidigt arbeta för att stärka den fonologiska medvetenheten. En tillfredsställande läsförmåga är central för inlärningen även i andra skolämnen än svenska. Forskning har på senare tid visat att det finns ett tydligt samband mellan läs- och skrivsvårigheter och svårigheter i matematik. I denna studie undersöks huruvida det finns någon koppling mellan tidiga fonologiska svårigheter och matematiksvårigheter två år senare hos en grupp elever. Med kvantitativa metoder och genom analyser av bedömningar i olika diagnos- och kartläggningsmaterial har ett resultat kunnat presenteras som visar att majoriteten av de elever som i förskoleklassen uppvisade tecken på fonologiska svårigheter även hade matematiksvårigheter i år 2. Sambandet är dock inte entydigt, då det samtidigt påvisats att svårigheterna i matematik i år 2 var mer utbredda än väntat och långt ifrån alla dessa elever hade fonologiska svårigheter i förskoleklassen. / The degree to which a child is phonologically aware is in many cases crucial to the child's ability to learn how to read. Therefore it is vital, at a young age, to work towards strengthening the phonological awareness among young children. Satisfactory reading skills are central to learning in other school subjects besides reading and writing. Recent research has shown that there is a clear link between reading problems and difficulties in mathematics. This study examines whether there is any connection within a group of students between early phonological difficulties and difficulties in mathematics two years later. With quantitative methods, and through analysis of assessments of different diagnostic and mapping materials, results have been presented showing that the majority of the students in the preschool class that showed signs of phonological difficulties also had difficulties in mathematics in second grade. However, the relationship is not unambiguous, since the results also demonstrated that the difficulties in mathematics in second grade were more widespread than expected, and far from all of these students had phonological difficulties in the preschool class.

One of the big problems of the Swedish nine-year compulsory school is the large number of pupils who fail to achieve a satisfactory standard in mathematics. One explanation that has been increasingly considered over the last ten years is that the pupils have dyscalculia. Some research suggests that 6 per cent of compulsory school pupils suffer from this dysfunction, which would in that case make it one of the Swedish school’s greatest teaching problems. The purpose of this thesis is to examine this problem area from two aspects. First of all by examining the concept of dyscalculia by means of a review of the literature from 1992 onwards. The second perspective has as its starting point a case study where the purpose was to give a detailed picture of the pupil with mathematics problems. The latter part of the study was carried out over a six-year period when 200 pupils, 13 of them with particular mathematics problems, were studied in detail. A point of departure for the study was provided by a large database where as much information as possible was collected about pupils from Year 5 of the nine-year compulsory school to Year 2 of the three-year upper secondary school. The pupils were asked to fill in regular questionnaires and classroom observations were made of roughly 100 mathematics lessons, 40 of which were recorded on video. Finally there were in-depth interviews of the 13 pupils on two occasions, the final one being during Year 2 of the upper secondary school. The review of the research showed a series of dubious and indistinct circumstances surrounding the dyscalculia concept, and also ambiguity with regard to the diagnosis of dyscalculia. The conclusion of the review was that the concept of dyscalculia ought at present to be used with great caution, or perhaps not at all. Admittedly the review does not provide grounds for totally dismissing the dyscalculia concept, but as long as it remains impossible to determine the concept unambiguously, and I have not been able to do this in the course of this study, there are no good scientific grounds for using the term dyscalculia in practice. The empirical study shows the complexity of the problem area. Both the causes suggested by the pupils as the origin of the problem and the measures that helped them to obtain their mathematics grades form a complex pattern. The low work input of the pupils during mathematics lessons, an unsettled working environment, large classes, problems of stress and anxiety prior to tests, and obstructive gender patterns are among the causes suggested by the pupils as explanations of the occurrence of the mathematics problems. Good teachers, in other words teachers who can explain, set limits and give encouragement, were a significant factor in reversing the downward trend. Positive experiences of school changes, where the pupil felt that he or she could start again from the beginning, were also mentioned as significant by several pupils. Collaboration with fellow-pupils and the fact that the pupils themselves decided to get to grips with the problems were other important reasons for the change. The prospects of students with specific problems in mathematics nevertheless being able to leave compulsory school with satisfactory grades appear, however, from the results of this study, to be bright. All the pupils left the compulsory school with satisfactory mathematics grades and also completed mathematics studies at upper secondary school, despite major problems in the subject at intermediate school (age 10-13) stage. The study indicates the need for research closer to the actual practical situation and to the importance of emphasizing good examples in practice. As the students themselves emphasize discrete communication between them as significant in the subject of mathematics, this is also an important area for future research.

A presente dissertação traz uma pesquisa sobre a aprendizagem da operação de divisão, com crianças do 4º ano do Ensino Fundamental. O objetivo da pesquisa era verificar se, através de uma proposta de ensino em que as crianças pudessem vivenciar a operação de divisão em variados contextos e situações, seria possível favorecer a (re)construção de seus esquemas e provocar sua aprendizagem. A metodologia adotada foi o estudo de caso, com a aplicação da sequência didática elaborada em uma turma de uma escola municipal da cidade de Gravataí, Rio Grande do Sul. A elaboração da sequência didática, assim como a análise dos registros orais e escritos de sua implementação, apoiou-se nos estudos realizados sobre a teoria dos campos conceituais, de Gérard Vergnaud, e em trabalhos de outros autores que estudam a construção das estruturas multiplicativas. Foi possível verificar que houve avanços na aprendizagem da operação de divisão por parte das crianças daquela turma. Através dos registros coletados e dos diálogos estabelecidos, foi possível compreender as maneiras como as crianças compreendiam e lidavam com situações de divisão e observar a mobilização e a reformulação de seus esquemas frente às situações vivenciadas em sala de aula. / The present dissertation presents a research on the division operation learning with children of the 4th grade of Elementary School. The purpose of this research to verify if, through a teaching proposal in which children could experience the division operation in various contexts and situations, it would be possible to favour the (re) construction of their schemes and to develop their learning. The methodology adopted was the case study, with the application of the teaching sequence elaborated in a group of a municipal school of Gravataí, Rio Grande do Sul. The elaboration of the teaching sequence, as the analysis of the oral and written records of its implementation, was supported by the studies accomplished with the Theory of Conceptual Fields by Gérard Vergnaud, and by papers from other authors that study the construction of multiplicative structures. It was possible to verify that there were advancements in the learning of the division operation by the children of that group. Throughout the collected records and the dialogs established, it was possible to understand the ways children understood and dealt with the division situations and to observe the mobilization and the reformulation of their schemes faced to the situations experienced in the classroom.

A presente investigação tem como ponto de partida o constante desafio e desejo do professor de matemática em construir estratégias que sejam promotoras de aprendizagens de matemática na escola básica. Com essa preocupação, inicia-se destacando a necessária centralidade da pesquisa no estudante e seus processos construtivos do conhecimento. A partir de discussões iniciais a respeito da aprendizagem na Cultura Digital e de indicações da potencialidade da arquitetura pedagógica de Projetos de Aprendizagem enquanto alternativa aos desafios educacionais próprios do século XXI, propõe-se investigar a intervenção do professor, de modo a levantar categorias que subsidiem a ação pedagógica com estudantes dos anos finais do ensino fundamental para a promoção da aprendizagem de matemática na Cultura Digital a partir de Projetos de Aprendizagem. Como trajetória de investigação, distinguem-se duas perspectivas: a pesquisa teórica e a experimental. Busca-se, então, fundamento na Epistemologia Genética piagetiana, especificamente nos processos de Equilibração Majorante – como essenciais para a compreensão estrutural da construção de conhecimento –, nas relações entre a Tomada de Consciência e o Fazer e Compreender – fundamentais para a análise da ação do estudante – e no Método Clínico – imprescindível para a discussão sobre a intervenção do professor dentro de uma perspectiva construtivista do conhecimento. Além disso, discutem-se as transformações advindas da Cultura Digital, suas implicações na cognição do jovem aprendiz e na construção de uma escola da Cultura Digital. Por fim, ainda na faceta teórica do estudo, apresenta-se e analisa-se a arquitetura pedagógica de Projetos de Aprendizagem à luz da teoria piagetiana. Com base no construto teórico produzido, propõe-se, então, o modelo de intervenção que se pretende experimentar A segunda perspectiva da investigação, é desenvolvida por meio de três experiências, as quais pretendem analisar (i) a viabilidade potencial, (ii) a viabilidade prática, a eficácia, e (iii) a validade da proposta pedagógica de promover aprendizagens de matemática via Projetos de Aprendizagem nos anos finais do Ensino Fundamental e das categorias de intervenção levantadas. Em consonância com os dados produzidos nas três experiências, as análises permitiram, então, a validação da arquitetura pedagógica de Projetos de Aprendizagem como uma intervenção do tipo 1, adequada a construção de uma escola da Cultura Digital, e das três categorias que identificam a ação do professor e definem as intervenções do tipo 2, denominadas: intervenções exploratórias, desequilibradoras e informativas. Considerando, por fim, os resultados e as proposições da presente investigação, aliados à experiência docente no viés da educação matemática e à importância da continuidade da pesquisa do professor como prática constante da sua ação pedagógica, levantam-se algumas possibilidades para a qualificação das investigações sobre a intervenção do professor e suas implicações na aprendizagem de estudantes ao longo da sua vida escolar. / The present research has as starting point the constant challenge and desire of the mathematics teacher to construct strategies that are incentivetor of mathematics learning in the basic school. With this concern, it begins by highlighting the necessary centrality of student research and its constructive processes of knowledge. From initial discussions about learning in the Digital Culture and indications of the potentiality of the pedagogical architecture of Learning Projects as an alternative to the educational challenges of the 21st century, it is proposed to investigate the teacher’s intervention, in order to identify categories that subsidize the pedagogical action with students of the final years of elementary school to promote the learning of mathematics in Digital Culture from Learning Projects. As a research trajectory, we distinguish two perspectives: theoretical and experimental research. We seek, therefore, a foundation in Piagetian Genetic Epistemology, specifically in the processes of Major Balance – as essential for the structural understanding of the knowledge construction – in the relations between the Taking of Consciousness and the Making and Understanding – fundamental for the analysis of the action of the student – and in the Clinical Method – essential for the discussion about the intervention of the teacher within a constructivist perspective of knowledge. In addition, the transformations arising from Digital Culture, its implications on the young learners cognition and the construction of a school of Digital Culture are discussed. Finally, still in the theoretical facet of the study, the pedagogical architecture of Learning Projects is presented and analyzed in light of the Piagetian theory Based on the theoretical construct produced, it is proposed, then, the intervention model to be tried. The second research perspective is developed through three experiments, which aim to analyze (i) the potential feasibility, (ii) practical feasibility, effectiveness, and (iii) the validity of the pedagogical proposal to promote mathematics learning by Learning Projects in the final years of Elementary School and the categories of intervention identified. In agreement with the data produced in the three experiments, the analyzes allowed the validation of the Learning Projects pedagogical architecture as a type 1 intervention, adequate to the construction of a School of Digital Culture, and of the three categories that identify the action of the teacher and define type 2 interventions, namely: exploratory, unbalanced and informative interventions. Considering, finally, the results and propositions of the present investigation, combined with the teaching experience in the bias of mathematics education and the importance of the continuity of teacher research as a constant practice of its pedagogical action, there are some possibilities for the qualification of the investigations On the intervention of the teacher and its implications in the learning of students throughout their school life. / La presente investigación tiene como base un constante desafío y deseo del profesor de matemática en construir estrategias que sean promotoras del aprendizaje de matemática en la escuela. Con esa preocupación, se inicia destacando la necesaria centralidad de la pesquisa en el estudiante y sus procesos constructivos del conocimiento. A partir de discusiones iniciales respecto al aprendizaje en la Cultura Digital y de indicaciones de la potencialidad de la Arquitectura Pedagógica de Proyectos de Aprendizaje en cuanto a alternativas a los desafíos educacionales proprios del siglo XXI, se propone investigar la intervención del profesor, con la finalidad de levantar categorías que subsidien a la acción pedagógica con estudiantes de los años finales de la enseñanza fundamental para la promoción del aprendizaje de matemática en la Cultura Digital a partir de Proyectos de Aprendizaje. Como trayectoria de investigación se distinguen dos perspectivas: la pesquisa teórica y la experimental. Busca-se, entonces, fundamento en Epistemología Genética Piagetana, específicamente en los procesos de Equilibración Mallorante – como esenciales para la comprensión estructural de la construcción de conocimiento –, en las relaciones entre la Tomada de Consciencia y el Hacer y Comprender – fundamentales para el análisis de la acción del estudiante – y en el Método Clínico – imprescindible para la discusión sobre la intervención del profesor dentro de una perspectiva constructivista del conocimiento. Ademas, se discute las transformaciones venidas de la Cultura Digital, sus implicaciones en la cognición del joven aprendiz en la construcción de una escuela de Cultura Digital Aun en la perspectiva teórica del estudio, se presenta y se analiza a la arquitectura pedagógica de Proyectos de Aprendizaje a la luz de la teoría piagetana. Con base en el constructo teórico producido, se propone, entonces, el modelo de intervención que se pretende experimentar. La segunda perspectiva de investigación, es desarrollada por medio de tres experiencias, las cuales pretenden analizar (i) la viabilidad potencial, (ii) la viabilidad práctica, la eficacia, y (iii) la validad de la propuesta pedagógica de promover aprendizajes de matemática vía Proyectos de Aprendizaje en los años finales de la Enseñanza Fundamental y de las categorías de intervención levantadas. En consonancia con los datos producidos en las tres experiencias, las análisis permitieron, entonces, la validación de la arquitectura pedagógica de Proyectos de Aprendizaje como una intervención del tipo 1, adecuada a la construcción de una escuela de Cultura Digital, y de las tres categorías que identifican a la acción del profesor y definen las intervenciones del tipo 2, denominadas: intervenciones exploratorias, desequilibradoras e informativas. Considerando, por fin, los resultados y las proposiciones da la presente investigación, aliados a la experiencia docente en la educación Matemática y la importancia de la continuidad de la pesquisa del profesor como práctica constante da su acción pedagógica, se levantan algunas posibilidades para la cualificación de las investigaciones sobre la intervención del profesor y sus implicaciones en el aprendizaje de estudiantes en su vida escolar.

Made available in DSpace on 2014-06-11T19:31:43Z (GMT). No. of bitstreams: 0 Previous issue date: 2007-03-30Bitstream added on 2014-06-13T18:42:46Z : No. of bitstreams: 1 zulatto_rba_dr_rcla.pdf: 1418316 bytes, checksum: 8cfc1b5fe211e92399513a0cac71bc8b (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / A presente pesquisa analisa a natureza da aprendizagem matemática em um curso online de formação continuada de professores, denominado Geometria com Geometricks. Nele, alunos-professores de uma mesma rede de escolas, situadas em diferentes localidades do país, desenvolveram atividades de Geometria utilizando-se do software Geometricks, e se encontravam para discuti-las. Esses encontros aconteceram a distância, em tempo real, por chat ou videoconferência. Nessa proposta pedagógica, a telepresença condicionou a comunicação e oportunizou o estar-junto-virtual-com-mídias. De modo singular, os recursos da videoconferência permitiram que construções geométricas fossem compartilhadas visualmente e realizadas por todos os envolvidos, fomentando a interação e a participação ativa, constituindo, por meio do diálogo, uma comunidade virtual de aprendizagem. Os resultados levam a inferir que, nesse contexto, a aprendizagem matemática teve natureza colaborativa, na virtualidade das discussões, tecidas a partir das contribuições de todos os participantes; coletiva, na medida em que a produção matemática era condicionada pelo coletivo pensante de seres-humanos-com-mídias; e argumentativa, uma vez que conjecturas e justificativas matemáticas se desenvolveram intensamente do decorrer do processo, contando para isso com as tecnologias presentes na interação ocorrida de forma constante e colaborativa. / This study was conducted to analyze the nature of mathematical learning in an online continuing education course for teachers entitled Geometry with Geometricks. Teachers employed in a nation-wide network of privately-supported schools developed geometry activities using the software Geometricks and discussed them in virtual meetings, in real time, via chat or video-conference. In this pedagogical proposal, tele-presence conditioned the communication and provided the opportunity for virtual-togetherness-with-media. In a unique way, the resources of the videoconference made it possible for everyone to participate in and visually share geometrical constructions, encouraging interaction and active participation and constituting a virtual learning community through dialogue. The results indicate that, in this context, mathematical learning nature was characterized by: collaboration, in the virtual discussions that were woven from the contributions of all the participants; collectivity, to the degree to which mathematical production was conditioned by the humans-with-media thinking collective; and argumentation, as the development of mathematical conjectures and justifications was intense throughout the process, aided by the technologies that were present in the constant, collaborative interaction.

Made available in DSpace on 2016-06-02T20:30:57Z (GMT). No. of bitstreams: 1 5918.pdf: 1237684 bytes, checksum: 8d371b67a7fbbba3e74f76754d8822ef (MD5) Previous issue date: 2014-02-26 / Financiadora de Estudos e Projetos / This master&#8223;s thesis consists of a study, reported a paper format, about crucial variables present in the establishment of relations between elements of first degree functions. This study was conducted from the theoretical framework of Behavior Analysis and fits in a growing body of research on mathematical behavior. The teaching and learning of mathematics have been characterized as a difficult process, with unsuccessful results that generates a number of by-products, such as anxiety and aversion to mathematics, worsening a quite problematic panorama. In this context, this master&#8223;s thesis reports a research which sought to analyze the effectiveness of the application of Behavior Analysis technology, specifically derived from the stimulus equivalence paradigm, for the establishment of relations between elements of first degree functions. The participants were submitted to a Matching to Sample procedure, though which relations AB and BC were trained, where A, B and C represent sets of elements related to first degree functions (graph, table, speech, etc). After training, tests of symmetry, transitivity and equivalence (BA, CB, BC, CA) were conducted in order to assess potential formation of equivalent stimuli classes, as well as generalization tests. The results showed that all participants formed equivalence classes and obtained 100% of correct responses in the generalization tests. This research sustains the possibility of effective application of stimulus equivalence paradigm to the establishment of complex mathematical repertoires. / A presente dissertação compõe-se de um estudo, relatado em forma de artigo, acerca de variáveis cruciais presentes no estabelecimento de relações entre elementos de funções do primeiro grau. Esse estudo foi conduzido a partir do referencial da Análise do Comportamento e enquadra-se em um conjunto crescente de pesquisas sobre comportamento matemático. O ensino e a aprendizagem de matemática têm se caracterizado como um processo difícil, com resultados de insucesso geradores de uma série de subprodutos, como ansiedade e aversão à matemática. Nesse contexto, esta dissertação relata uma investigação que procurou analisar a efetividade da aplicação de tecnologias da Análise do Comportamento, especificamente derivadas do paradigma da equivalência de estímulos, para o estabelecimento de relações entre elementos de funções do primeiro grau. Os participantes foram submetidos ao procedimento de escolha de acordo com o modelo (MTS), por meio do qual foram treinadas, através de discriminações condicionais, as relações AB e BC, onde A, B e C representam conjuntos referentes a elementos de funções do primeiro grau (gráfico, tabela, expressão, etc). Após etapa de treino os participantes foram submetidos a testes de simetria, transitividade e equivalência (relações BA, CB, AC, CA) a fim de verificar potencial formação de classes de estímulos equivalentes e ainda testes de generalização. Os resultados mostraram que todos os participantes formaram classes de equivalência e obtiveram 100% de acertos nos testes de generalização. Esta pesquisa indica a possibilidade efetiva da aplicação do paradigma da equivalência de estímulos para o estabelecimento de repertórios matemáticos complexos.