Made available in DSpace on 2015-05-07T15:09:24Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 5058012 bytes, checksum: ee33082851932c601b63cce8361c7d85 (MD5) Previous issue date: 2013-03-08 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The Teaching of Mathematics in the early years of schooling is the context in which our research is inserted. The general objective is to analyze the teaching of multiplication aiming to demonstrate the place / role attributed to semiotic representations in the process of teaching and learning, relating them to the concept of pedagogical mediation. In this context, we advocate the thesis that semiotic representations of multiplication are instruments of mediation in the process of teaching and learning that content. To defend it, we discussed the concepts of teaching, learning and pedagogical mediation, supported on cultural-history theory, with contributions by Vygotsky and Leontiev, the concept of semiotic representation registers by Raymond Duval, and we also draw on authors in Mathematics Education who have been researching the teaching of the subject, particularly, multiplication. The methodology for data collection was based on qualitative research, which follows the investigation of complex social phenomena that require further description and discussion. The chosen methodological approach was a discussion group with teachers of early years (Rubio and Varas, 1999), by means of a course we outlined on Teaching Multiplication of Natural Numbers which served at the same time as the locus of field research and a space for academic qualification for the 08 (eight) participating teachers, which also featured as an intervention. During this process the teachers classes were visited and information collected about students' understanding of multiplication. Content analysis was the procedure used for organizing, systematizing and analyzing data, and thematic analysis constituted a means of identifying 'units of meaning'. The data were systematized in two main areas: multiplication, its teaching and students' understanding; semiotic representations used in the teaching of multiplication and produced by the children. The results showed that the teaching of multiplication is still centered on the idea of adding equal portions, with strong appeal to the use of concrete material; mental calculation, through fundamental facts, is a neglected strategy. On students' understanding, we observed difficulties in all explored meanings of the operation, including those most commonly used by teachers; besides, in problems of reverse and combinatorial multiplication this was most evident. Considering all the classes, we demonstrate the variety of semiotic representations of multiplication in teaching, as well as tables and graphs. Regarding students records, we identified the lack of a systematic pedagogical work, involving the clarification, treatment and conversion, since the children s difficulties in these tasks were significant. We conclude that the semiotic representations of multiplication can become pedagogical mediation tools as long as the teachers work focuses on this purpose, so as to encourage discussion and reflection in the classroom on the different semiotic records and strategies for problem solving / O Ensino de Matemática nos anos iniciais da escolarização é o contexto no qual se insere a nossa pesquisa, tendo como objetivo geral analisar o ensino de multiplicação, buscando evidenciar o lugar/papel atribuído às representações semióticas no processo de ensino e aprendizagem, relacionando-as ao conceito de mediação pedagógica. Nesse contexto, defendemos a tese que as representações semióticas de multiplicação constituem instrumentos de mediação pedagógica no processo de ensino e aprendizagem desse conteúdo. Para defendê-la, problematizamos os conceitos de ensino, de aprendizagem e mediação pedagógica, com a fundamentação da teoria Histórico-cultural, com as contribuições de Vigotski e Leontiev; o conceito de registros de representação semiótica, de Raymond Duval, como também nos baseamos em autores da Educação Matemática que vêm pesquisando o ensino da disciplina e o tema da multiplicação. A metodologia para a coleta de dados baseou-se na pesquisa qualitativa, a qual tem como indicação a investigação de fenômenos sociais e complexos que exigem maior descrição e discussão. O procedimento metodológico escolhido foi o grupo de discussão com professores de anos iniciais (Rubio e Varas, 1999) e para efetivá-lo, montamos um curso sobre o Ensino de Multiplicação de Números Naturais que funcionou ao mesmo tempo como lócus da pesquisa de campo e espaço de formação continuada para as 08 (oito) professoras participantes, o que o caracterizou também, como uma intervenção. Durante esse processo foram visitadas as turmas das professoras e coletadas informações sobre a compreensão dos alunos acerca da multiplicação. A análise de conteúdo foi o procedimento utilizado para organização, sistematização e análise de dados, tendo a análise temática como meio de identificação de núcleos de sentido . Os dados foram sistematizados em dois grandes eixos: a multiplicação, seu ensino e compreensão dos alunos; as representações semióticas de multiplicação utilizadas no ensino e produzidas pelas crianças. Os resultados apontaram que o ensino de multiplicação ainda está centrado na ideia de adição de parcelas iguais, com forte apelo ao uso do material concreto, não sendo explorado o cálculo mental, por meio dos fatos fundamentais. Sobre a compreensão dos alunos, observaram-se dificuldades em todos os significados da operação explorados, inclusive naqueles mais usados pelas professoras, sendo que nos problemas de multiplicação inversa e combinatória isto foi mais evidente. Evidenciamos variedade de representações semióticas de multiplicação no ensino, se considerarmos todas as turmas, podendo ser ainda acrescentado o uso de tabelas e gráficos. Em relação aos registros dos alunos, identificamos a ausência de um trabalho pedagógico sistemático, envolvendo a explicitação, o tratamento e a conversão, uma vez que as dificuldades das crianças nessas tarefas foram significativas. Concluímos que as representações semióticas de multiplicação podem constituir-se em instrumentos de mediação pedagógica à medida que o trabalho dos professores se encaminhe para tal finalidade, fomentando na sala de aula discussão e reflexão sobre diferentes registros semióticos e estratégias de solução de problemas

The present study investigated imagery and visualisation characteristics of undergraduate students’ thinking processes in learning selected concepts of mathematical analysis. The aim was to discover the nature of images evoked by these undergraduate students and the role of imagery and visualisations when students were solving some selected problems related to mathematical analysis. The study was guided by the theory of registers of semiotic representations. Psychological notions on imagery were also fused to cater for a cognitive approach to the study. A sample of 50 undergraduate mathematics students participated in the study. The researcher employed both quantitative and qualitative methods. Before the main study, a pilot study was conducted to account for the reliability and validity of the research instruments. The data were collected through use of a cognitive test that was composed of 12 tasks with items selected from mathematical analysis. These tasks were specially designed to capture the variables of imagery and visualisations. A structured interview was also conducted as a follow-up to the results of the cognitive test. The study found that visual images were noticeable in the thinking processes of undergraduate students in solving problems related to mathematical analysis. The nature of the visual images evoked by the students varied from person to person. The nature of these images was also determined by the nature of the task. The most common types of imagery were diagrams, prototypes and symbols. On rare occasions the students also evoked metaphoric images. It was also observed that these images were used for illustrative purposes and to spark the idea for a proof. It was also interesting to note that some images were used to discover the limit of a converging series. The results confirmed the need to use visualisation with caution, especially when treating concepts which involve infinity. The study recommends that instructors of mathematics should encourage visual thinking in the learning and teaching of mathematical analysis. Knowledge of the students’ concept images helped the researcher to understand the nature of the learning difficulties of the students. Further research should focus on the strengths and weaknesses of visual-mediated learning and also on the relationship between creativity and visual thinking. / Mathematics Education / D. Phil. (Mathematics, Science and Technology Education (Mathematics Education))