Atividades de investigação em Geometria: uma experiência no 2º ano de escolaridade

Dissertação apresentada à Escola Superior de Educação de Lisboa para obtenção de grau de mestre em Ciências da Educação, especialidade Educação Matemática na Educação Pré-Escolar e nos 1.º e 2.º Ciclos do Ensino Básico

Os padrões e a resolução de problemas no 1º ciclo do ensino básico: potencialidades para o ensino-aprendizagem da matemática

Mestrado (PES II), Educação Pré-Escolar e Ensino do 1º Ciclo do Ensino Básico, 27 de Junho de 2014, Universidade dos Açores (Relatório de Estágio).

Spontaneous Focusing on Quantitative Relations and the Development of Rational Number Conceptual Knowledge

The aim of the present set of studies was to explore primary school children’s Spontaneous Focusing On quantitative Relations (SFOR) and its role in the development of rational number conceptual knowledge. The specific goals were to determine if it was possible to identify a spontaneous quantitative focusing tendency that indexes children’s tendency to recognize and utilize quantitative relations in non-explicitly mathematical situations and to determine if this tendency has an impact on the development of rational number conceptual knowledge in late primary school. To this end, we report on six original empirical studies that measure SFOR in children ages five to thirteen years and the development of rational number conceptual knowledge in ten- to thirteen-year-olds. SFOR measures were developed to determine if there are substantial differences in SFOR that are not explained by the ability to use quantitative relations. A measure of children’s conceptual knowledge of the magnitude representations of rational numbers and the density of rational numbers is utilized to capture the process of conceptual change with rational numbers in late primary school students. Finally, SFOR tendency was examined in relation to the development of rational number conceptual knowledge in these students. Study I concerned the first attempts to measure individual differences in children’s spontaneous recognition and use of quantitative relations in 86 Finnish children from the ages of five to seven years. Results revealed that there were substantial inter-individual differences in the spontaneous recognition and use of quantitative relations in these tasks. This was particularly true for the oldest group of participants, who were in grade one (roughly seven years old). However, the study did not control for ability to solve the tasks using quantitative relations, so it was not clear if these differences were due to ability or SFOR. Study II more deeply investigated the nature of the two tasks reported in Study I, through the use of a stimulated-recall procedure examining children’s verbalizations of how they interpreted the tasks. Results reveal that participants were able to verbalize reasoning about their quantitative relational responses, but not their responses based on exact number. Furthermore, participants’ non-mathematical responses revealed a variety of other aspects, beyond quantitative relations and exact number, which participants focused on in completing the tasks. These results suggest that exact number may be more easily perceived than quantitative relations. As well, these tasks were revealed to contain both mathematical and non-mathematical aspects which were interpreted by the participants as relevant. Study III investigated individual differences in SFOR 84 children, ages five to nine, from the US and is the first to report on the connection between SFOR and other mathematical abilities. The cross-sectional data revealed that there were individual differences in SFOR. Importantly, these differences were not entirely explained by the ability to solve the tasks using quantitative relations, suggesting that SFOR is partially independent from the ability to use quantitative relations. In other words, the lack of use of quantitative relations on the SFOR tasks was not solely due to participants being unable to solve the tasks using quantitative relations, but due to a lack of the spontaneous attention to the quantitative relations in the tasks. Furthermore, SFOR tendency was found to be related to arithmetic fluency among these participants. This is the first evidence to suggest that SFOR may be a partially distinct aspect of children’s existing mathematical competences. Study IV presented a follow-up study of the first graders who participated in Studies I and II, examining SFOR tendency as a predictor of their conceptual knowledge of fraction magnitudes in fourth grade. Results revealed that first graders’ SFOR tendency was a unique predictor of fraction conceptual knowledge in fourth grade, even after controlling for general mathematical skills. These results are the first to suggest that SFOR tendency may play a role in the development of rational number conceptual knowledge. Study V presents a longitudinal study of the development of 263 Finnish students’ rational number conceptual knowledge over a one year period. During this time participants completed a measure of conceptual knowledge of the magnitude representations and the density of rational numbers at three time points. First, a Latent Profile Analysis indicated that a four-class model, differentiating between those participants with high magnitude comparison and density knowledge, was the most appropriate. A Latent Transition Analysis reveal that few students display sustained conceptual change with density concepts, though conceptual change with magnitude representations is present in this group. Overall, this study indicated that there were severe deficiencies in conceptual knowledge of rational numbers, especially concepts of density. The longitudinal Study VI presented a synthesis of the previous studies in order to specifically detail the role of SFOR tendency in the development of rational number conceptual knowledge. Thus, the same participants from Study V completed a measure of SFOR, along with the rational number test, including a fourth time point. Results reveal that SFOR tendency was a predictor of rational number conceptual knowledge after two school years, even after taking into consideration prior rational number knowledge (through the use of residualized SFOR scores), arithmetic fluency, and non-verbal intelligence. Furthermore, those participants with higher-than-expected SFOR scores improved significantly more on magnitude representation and density concepts over the four time points. These results indicate that SFOR tendency is a strong predictor of rational number conceptual development in late primary school children. The results of the six studies reveal that within children’s existing mathematical competences there can be identified a spontaneous quantitative focusing tendency named spontaneous focusing on quantitative relations. Furthermore, this tendency is found to play a role in the development of rational number conceptual knowledge in primary school children. Results suggest that conceptual change with the magnitude representations and density of rational numbers is rare among this group of students. However, those children who are more likely to notice and use quantitative relations in situations that are not explicitly mathematical seem to have an advantage in the development of rational number conceptual knowledge. It may be that these students gain quantitative more and qualitatively better self-initiated deliberate practice with quantitative relations in everyday situations due to an increased SFOR tendency. This suggests that it may be important to promote this type of mathematical activity in teaching rational numbers. Furthermore, these results suggest that there may be a series of spontaneous quantitative focusing tendencies that have an impact on mathematical development throughout the learning trajectory.

Some aspects of history of Indian mathematics in 18th and early 19th century

This thesis is an attempt to throw light on the works of some Indian Mathematicians who wrote in Arabic or persian In the Introductory Chapter on outline of general history of Mathematics during the eighteenth Bnd nineteenth century has been sketched. During that period there were two streams of Mathematical activity. On one side many eminent scholers, who wrote in Sanskrit, .he l d the field as before without being much influenced by other sources. On the other side there were scholars whose writings were based on Arabic and Persian text but who occasionally drew upon other sources also.

Objetos intencionais e existência objetiva

Neste artigo quero apontar para a possibilidade de uma ontologia da matemática que, mesmo mantendo alguns pontos em comum com o platonismo e com o construtivismo, desliga-se destes em outros pontos essenciais. Por objeto matemático entendo o foco referencial do discurso matemático, ou seja, aquilo sobre o qual a matemática fala. Entendo que a existência destes objetos é meramente intencional, presuntiva, mas, simultaneamente, objetiva, no sentido de ser uma existência comunalizada, compartilhada por todos aqueles engajados no fazer matemático. A existência objetiva das entidades matemáticas não está, entretanto, garantida de uma vez por todas, mas apenas enquanto o discurso matemático for consistente. Este é o espírito do critério de existência objetiva enunciado que, acredito, deve sustentar uma ontologia matemática sem o pressuposto da existência independente de um domínio de objetos matemáticos, sem o empobrecimento que lhe impõem as diferentes versões construtivistas e sem a aniquilação que lhe infringe o formalismo sem objetos.

Características dos Problemas que os Alunos Constroem a partir do Enunciado de uma Questão Aberta de Matemática

The present paper shows characteristics of problems that students in the 4(th), 8(th), and 12(th) grades built in response to the enunciation of an open-ended question, taken from their written work in the common question of AVA-2002 (Parana State Large-Scale Assessment-2002). These problems are characterized, in part, by a linear resolution structure constituted based on students' interpretation of individual bits of information contained in the enunciation, one by one, without establishing relations between them; and partly by a non-linear resolution structure constituted based on relations established among the bits of information in the enunciation. Analysis of students' written work is one alternative for teachers to know students' mathematical activity, as well as their particular ways of interpretating enunciations.

A produção matemática em um ambiente virtual de aprendizagem: o caso da geometria euclidiana espacial

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Análise de um processo de estudo de semelhança

Trata de uma investigação de um processo de estudo de semelhança de figuras realizado por uma comunidade de estudo, em uma turma do ensino médio de uma escola da rede pública estadual da periferia de Belém, buscando responder se as atividades desenvolvidas pelos alunos em sala de aula caracterizam uma atividade matemática a luz da teoria da transposição didática de Yves Chevallard. Isso é realizado por meio de atividades colaborativas, em que busca identificar os movimentos dos saberes matemáticos evocados pelos alunos na construção do conceito de semelhança. A pesquisa é de natureza qualitativa, numa abordagem etnográfica adaptada à educação, segundo Lüdke e André. As análises mostram que as atividades realizadas promovem um fazer matemático e, portanto, uma atividade matemática, por meio dos saberes evocados e as articulações estabelecidas na construção de modelos para a compreensão pelos alunos do conceito de semelhança. São destacadas as dificuldades como elementos importantes na identificação de saberes e articulações destes, bem como a comunidade de estudo colaborativo como facilitador do processo de estudo.

Da prescrição curricular ao currículo como ação compartilhada: dilemas das tentativas de renovação dos programas de ensino de Matemática no estado de São Paulo pós-64

The present study is the result of joint actions between education, research and outreach that is focused on the training of educators and its implications for the renovation of teaching Mathematics. Its objective is to analyze the attempts to reorganize in mathematics curriculum implemented in the state of São Paulo in the post-64. It makes use of documentary analysis and interviews of teachers for reasons of discussion and conclusions. Situated in the theoretical and methodological context of collaborative research and historical-cultural theory. The research results which shows that mathematical activity is the centrality of the discussion about learning mathematics, which has consequences for the organization, of teaching programs. This is a genesis school thinking that motivates the students to the reconstruction of ideas and thinking a production process in the classroom that considers the conditions of the school, other than the conditions that governs the production of knowledge of mathematical science, which requires thinking about the formation of a teacher epistemologically curious.

Perspectivas de estudiantes para profesores sobre el papel de la tecnología para apoyar el aprendizaje matemático de los estudiantes

El objetivo de esta investigación es caracterizar perspectivas de estudiantes para profesores de educación secundaria (EPS) sobre el papel que puede desempeñar la tecnología para apoyar el aprendizaje matemático de los estudiantes. Los datos proceden de la planificación de una lección basada en la resolución de problemas mediante el uso de tecnología. Las perspectivas de los estudiantes para profesor se situaron a lo largo de un continuo considerando la relación entre: (i) lo que se pretendía con el uso de recursos tecnológicos y (ii) la naturaleza de la actividad matemática generada. La relación entre ambos aspectos ayuda a reconocer el papel que pueden desempeñar las perspectivas de los estudiantes para profesor cuando están aprendiendo a integrar la tecnología en la enseñanza de la resolución de problemas.

GeoGebraTUTOR : développement d’un système tutoriel autonome pour l’accompagnement d’élèves en situation de résolution de problèmes de démonstration en géométrie plane et genèse d’un espace de travail géométrique idoine

Travaux d'études doctorales réalisées conjointement avec les travaux de recherches doctorales de Nicolas Leduc, étudiant au doctorat en génie informatique à l'École Polytechnique de Montréal.

GeoGebraTUTOR : développement d’un système tutoriel autonome pour l’accompagnement d’élèves en situation de résolution de problèmes de démonstration en géométrie plane et genèse d’un espace de travail géométrique idoine

Travaux d'études doctorales réalisées conjointement avec les travaux de recherches doctorales de Nicolas Leduc, étudiant au doctorat en génie informatique à l'École Polytechnique de Montréal.

Trabalhar com a matemática em educação de infância

Este estudo tem como principal objetivo compreender e analisar o modo como crianças de creche e jardim-de-infância resolvem problemas matemáticos e o que pode constranger a resolução. Em particular, procurei analisar a atividade matemática que as crianças desenvolvem quando se confrontam com problemas matemáticos e os desafios com que se deparam. Do ponto de vista metodológico, o estudo enquadra-se numa abordagem qualitativa de investigação e num paradigma interpretativo. Além disso, trata-se de uma investigação-ação orientada pela questão “como otimizar a atividade de resolver problemas matemáticos em contextos de educação de infância?”. Neste âmbito, propus a quatro crianças de creche e a 21 de jardim-de-infância um conjunto de tarefas selecionadas para, potencialmente, terem, para si, algum grau de desafio. Os principais métodos de recolha de dados foram a observação participante, a análise documental e um inquérito por questionário realizado às educadoras cooperantes. O estudo ilustra que é possível envolver crianças de creche e de jardim-de-infância numa atividade de resolução de problemas matemáticos e que esta atividade é favorecida se o contexto dos problemas estiver próximo do que fazem no dia-a-dia da sala. Durante o processo de resolução das tarefas propostas, foram mobilizadas e trabalhadas diversas noções matemáticas. Na creche, todas as crianças evidenciaram possuir conhecimentos acerca da noção topológica “dentro de” e “fora de” e algumas foram bem-sucedidas no uso do processo de classificação, tendo em conta um critério. Neste âmbito, recorreram a representações ativas. No jardim-de-infância, todas as crianças conseguiram fazer a contagem sincronizada das letras do seu nome, de indicar a quantidade de letras, o que indicia o conhecimento da noção de cardinal, e de representar esta quantidade recorrendo tanto a numerais como a representações icónicas. Além disso, foram capazes de interpretar uma tabela de modo a construir um gráfico com barras e de elaborar um pictograma, o que revela possuírem conhecimentos ao nível da literacia estatística. Por último, algumas crianças foram bem-sucedidas na descoberta de estratégias de resolução de problemas que lhes permitiram inventariar exaustivamente todas as possibilidades de resolução e contar, organizadamente, estas possibilidades. No decurso desta atividade surgiram tentativas de generalização, embora nem sempre corretas, sobressaindo o recurso a representações ativas nomeadamente à dramatização de situações. Quanto aos desafios com que se depararam destacam-se, no caso da creche, o uso correto do processo de classificação. No caso do jardim-de-infância, as crianças demonstraram dificuldades em distinguir a legenda do pictograma dos dados, em resolver um problema em que estava em jogo o sentido combinatório da multiplicação e em encontrar estratégias de generalização. O estudo indicia, ainda, que é essencial que o educador proponha tarefas diversificadas e desafiantes que, partindo sempre da curiosidade e interesse das crianças, lhes permitam trabalhar com ideias matemáticas importantes e representar adequadamente o conhecimento com que lidam.

Diseño y estudio de situaciones didácticas que favorecen el trabajo con registros semióticos

What’s behind the mistakes and difficulties that appear on the students to understand and study mathematics?are only related to the cognitive complexity of the content or such difficulties are also related to the possible ways to access the different mathematical objects? The mathematical activity generated in many students learning difficulties that are not manifested in cognitive processes related to other areas of knowledge. If something characterizes the processes of teaching and learning of mathematics is that, unlike what happens with the objects of study in the experimental sciences, the only way to access to them is through its different semiotic representations. The coordination among the different systems of representation that refer to the same mathematical concept, needs to move from one register to another (D’Amore, 1998, 2001, 2003, 2004, 2006; Duval, 1993, 1994, 1995, 1996, 2000, 2003, 2004, 2005, 2007, 2008, 2011, 2012; Godino, 2002, 2003, 2012, 2014; Kaput, 1989a, 1989b,1992, 1998; Radford, 1998, 2004a, 2004b, 2004c, 2006a, 2008,2009, 2011, 2013, 2014a). Therefore, the treatments that can be realized within a given register and the conversion of one register into another, play an essential role in the grasp of the object and mathematical concepts. Through this work with representations, students give meanings to the objects of study and are able to understand the underlying mathematical structures, which is the main educational interest of this issue...