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A compreensão dos conceitos de limite e continuidade de uma função : um estudo com alunos do 12º ano
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Resumo(s)
Esta investigação tem como objetivo analisar que compreensão evidenciam os alunos do 12.º ano sobre os conceitos de limite e continuidade de uma função. O quadro teórico evidencia a complexidade destes conceitos do Cálculo Infinitesimal, constatando-se que os alunos têm muita dificuldade na compreensão destes entes matemáticos. Para alcançar o objetivo, assume-se que a compreensão dos conceitos de limite e continuidade envolve: (i) os significados que os alunos atribuem ao conceito, considerando as suas conceções intuitivas; (ii) as representações e a visualização do conceito, nomeadamente o uso e a articulação que os alunos lhes dão; e (iii) as dificuldades de aprendizagem e os erros que manifestam os alunos durante a resolução de tarefas. Este estudo envolve alunos de 12.º ano de uma escola secundária, porque constitui a etapa inicial de transição entre o pensamento matemático elementar e o pensamento matemático avançado, na qual os alunos enfrentam procedimentos que já não se levam a cabo através da aritmética ou álgebra simples. A metodologia da investigação segue, em termos gerais, o paradigma interpretativo com uma abordagem qualitativa. O principal procedimento de recolha de dados foi a recolha documental das produções escritas dos alunos nas resoluções de um conjunto de tarefas que foram propostas em sala de aula, além da realização de observações de aula e de uma entrevista a alguns alunos.
Os resultados revelam que os participantes neste estudo evidenciam uma compreensão instrumental dos conceitos de limite e continuidade de uma função, embora se possa identificar duas formas diferentes de desenvolver esta compreensão. Primeiramente, desenvolvem uma compreensão instrumental de caráter mais intuitiva, resolvendo tarefas a partir de experiências particulares mais próximas das suas próprias conceções do que da formalidade matemática, sem terem domínio do conceito em si mesmo. No entanto, depois do estudo das caraterísticas, propriedades, processos e definições dos conceitos, os alunos evidenciam uma compreensão instrumental de caráter mais formal, caraterizada pela aplicação de regras e estratégias pré-estabelecidas pelas convenções matemáticas de sala de aula, bem como o uso, visualização e articulação dos conceitos através de diferentes sistemas de representação, e a formulação de significados mais próximos da conceção dinâmica dos conceitos.
This research aims to analyze the understanding that students of the 12th year show on the concepts of limit and continuity of a function. The theoretical framework highlights the complexity of these concepts of infinitesimal calculus, noting that students have great difficulty in understanding these mathematical entities. To achieve the goal, it is assumed that understanding the concepts of limit and continuity involves: (i) the meaning that students attribute to the concept, considering its intuitive conceptions; (ii) the representations and concept visualization, including the use and articulation that students give them; and (iii) learning difficulties and mistakes that students show while solving tasks. This study involves students from 12th grade of a secondary school because it is the initial stage of transition between elementary mathematical thinking and advanced mathematical thinking, in which students face procedures that are no longer carried out through arithmetic or algebra simple. The research methodology follows, in general terms, the interpretative paradigm with a qualitative approach. The main data collection procedure was the documentary collection of written productions of students in the resolution of a set of tasks that have been proposed in the classroom, in addition to conducting classroom observations and an interview done to a few students. The results showed that participants in this study reveal an instrumental understanding of the concepts of limit and continuity of a function, although one can identify two different ways to develop this understanding. First, they develop an instrumental understanding with a more intuitive character, solving tasks from particular experiences closer to their own conceptions than of the mathematical formality, without having concept of the domain itself. However, after the study of the characteristics, properties, processes and definitions of concepts, students showed an instrumental understanding with a more formal character. This understanding is characterized by the application of rules and pre-established strategies by mathematical conventions of classroom, well as the use, visualization and articulation of concepts across different systems of representation, and the development of closer meanings of the dynamic conception of the concepts.
This research aims to analyze the understanding that students of the 12th year show on the concepts of limit and continuity of a function. The theoretical framework highlights the complexity of these concepts of infinitesimal calculus, noting that students have great difficulty in understanding these mathematical entities. To achieve the goal, it is assumed that understanding the concepts of limit and continuity involves: (i) the meaning that students attribute to the concept, considering its intuitive conceptions; (ii) the representations and concept visualization, including the use and articulation that students give them; and (iii) learning difficulties and mistakes that students show while solving tasks. This study involves students from 12th grade of a secondary school because it is the initial stage of transition between elementary mathematical thinking and advanced mathematical thinking, in which students face procedures that are no longer carried out through arithmetic or algebra simple. The research methodology follows, in general terms, the interpretative paradigm with a qualitative approach. The main data collection procedure was the documentary collection of written productions of students in the resolution of a set of tasks that have been proposed in the classroom, in addition to conducting classroom observations and an interview done to a few students. The results showed that participants in this study reveal an instrumental understanding of the concepts of limit and continuity of a function, although one can identify two different ways to develop this understanding. First, they develop an instrumental understanding with a more intuitive character, solving tasks from particular experiences closer to their own conceptions than of the mathematical formality, without having concept of the domain itself. However, after the study of the characteristics, properties, processes and definitions of concepts, students showed an instrumental understanding with a more formal character. This understanding is characterized by the application of rules and pre-established strategies by mathematical conventions of classroom, well as the use, visualization and articulation of concepts across different systems of representation, and the development of closer meanings of the dynamic conception of the concepts.
Descrição
Tese de mestrado, Educação (Área de especialidade em Didática da Matemática), Universidade de Lisboa, Instituto de Educação, 2016
Palavras-chave
Função Ensino secundário Matemática (Aplicações) Teses de mestrado - 2016