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A aprendizagem do conceito de limite de uma sucessão : uma experiência no 11º ano
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Este trabalho de projeto tem como objetivo compreender como alunos do 11.º ano podem mobilizar os seus conhecimentos matemáticos e intuitivos de convergência para construir uma definição formal de limite de uma sucessão. Este foi orientado pelas seguintes questões: Que conhecimentos matemáticos são utilizados pelos alunos no estudo intuitivo do limite de uma sucessão?; De que forma o recurso a representações numéricas e gráficas contribui para a aprendizagem de conceitos ligados ao limite de sucessões?; e Quais as principais dificuldades manifestadas pelos alunos relativamente à noção de limite?.
O trabalho foi desenvolvido com uma turma de 11.º ano, no âmbito do tema Sucessões, lecionado no final do 2.º período e início do 3.º período do ano letivo 2017/2018 numa escola secundária do concelho de Oeiras.
As tarefas propostas aos alunos no decorrer da intervenção foram voltadas para o desenvolvimento do conceito de limite de uma sucessão, tanto do ponto de vista intuitivo, como a definição formal. A construção de conjeturas, ao longo das tarefas, foi auxiliada pela utilização de tecnologias, calculadora e computador, de modo a representar, tanto gráfica como numericamente um grande número de sucessões.
Seguindo uma metodologia de natureza qualitativa assente num paradigma interpretativo, a recolha de dados baseou-se, essencialmente, na observação participante, através de registos áudio e escritos, e na recolha documental das resoluções dos alunos, tendo sido também aplicado um questionário a todos os alunos.
Da implementação da unidade de ensino verificou-se que a maioria dos alunos evidenciou saber aplicar a definição formal de limite de uma sucessão do ponto de vista operacional. Numa fase inicial, estes revelaram conceções erróneas já esperadas na construção da noção de limite, tais como a ideia de que uma sucessão constante não tem limite e de que o limite é um valor que não é alcançado. Com o auxílio das tarefas implementadas, estas conceções erróneas iniciais foram sendo discutidas e esclarecidas. No entanto, o estudo evidencia que os alunos têm alguma dificuldade em conceber sucessões não monótonas, o que, no caso da noção de limite de uma sucessão, pode contribuir para as tais conceções erróneas.
This project work aims to understand how students of the 11th grade can mobilize their mathematical and intuitive convergence knowledge to construct a formal definition of the limit of a sequence. This was oriented by the following questions: What mathematical knowledge are used by students in the intuitive study of the limit of a sequence?; How does the use of numerical and graphical representations contribute to the learning of concepts linked to the limit of sequences?; and What are the main difficulties expressed by the students regarding the notion of limit?. This work was developed with a class of the 11th grade, under the theme Sequences, taught at the end of the second period and beginning of the third period of the 2017/2018 school year in a secondary school in the county of Oeiras. The tasks proposed to the students during the intervention were aimed at developing the concept of limit of a sequence, both from the intuitive point of view and from the formal definition. The construction of conjectures, along the tasks, was aided by the use of technologies, calculator and computer, in order to represent both graphically and numerically a large number of successions. Following a methodology of a qualitative nature based on an interpretative paradigm, the data collection was essentially based on participant observation, through audio and written records, and in the documentary collection of students' resolutions, and a questionnaire was also applied to all the students. From the implementation of the teaching unit it was verified that the majority of the students evidenced how to apply the formal definition of the limit of a sequence from the operational point of view. At an early stage, they revealed misconceptions already expected in the construction of the notion of limit, such as the idea that a constant succession has no limit and that the limit is a value that is not reached. With the aid of the implemented tasks, these initial misconceptions were discussed and clarified. However, the study shows that students have some difficulty in conceiving non-monotonous sequences, which, in the case of the notion of the limit of a sequence, can contribute to such misconceptions.
This project work aims to understand how students of the 11th grade can mobilize their mathematical and intuitive convergence knowledge to construct a formal definition of the limit of a sequence. This was oriented by the following questions: What mathematical knowledge are used by students in the intuitive study of the limit of a sequence?; How does the use of numerical and graphical representations contribute to the learning of concepts linked to the limit of sequences?; and What are the main difficulties expressed by the students regarding the notion of limit?. This work was developed with a class of the 11th grade, under the theme Sequences, taught at the end of the second period and beginning of the third period of the 2017/2018 school year in a secondary school in the county of Oeiras. The tasks proposed to the students during the intervention were aimed at developing the concept of limit of a sequence, both from the intuitive point of view and from the formal definition. The construction of conjectures, along the tasks, was aided by the use of technologies, calculator and computer, in order to represent both graphically and numerically a large number of successions. Following a methodology of a qualitative nature based on an interpretative paradigm, the data collection was essentially based on participant observation, through audio and written records, and in the documentary collection of students' resolutions, and a questionnaire was also applied to all the students. From the implementation of the teaching unit it was verified that the majority of the students evidenced how to apply the formal definition of the limit of a sequence from the operational point of view. At an early stage, they revealed misconceptions already expected in the construction of the notion of limit, such as the idea that a constant succession has no limit and that the limit is a value that is not reached. With the aid of the implemented tasks, these initial misconceptions were discussed and clarified. However, the study shows that students have some difficulty in conceiving non-monotonous sequences, which, in the case of the notion of the limit of a sequence, can contribute to such misconceptions.
Descrição
Trabalho de Projeto de Mestrado, Educação (Área de especialidade de Didática da Matemática), Universidade de Lisboa, Instituto de Educação, 2018
Palavras-chave
Sucessões (Matemática) Limites (Matemática) Ensino secundário (11º ano) Trabalhos de projeto de mestrado - 2018