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A aprendizagem do conceito de limite no 11º ano de escolaridade
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O presente estudo foi realizado no âmbito da prática de ensino supervisionada, numa turma do 11.º ano de escolaridade da escola Secundária da Ramada, no ano letivo de 2014/15. A unidade curricular “Taxa de Variação e Derivada” do programa de Matemática A (ME, 2001) foi alvo da minha lecionação, num total de cinco aulas de 90 minutos. No entanto, o estudo integrou também aspetos da minha observação com diferentes níveis de participação nos tópicos de funções racionais e sucessões reais.
O objetivo do estudo é compreender como os alunos desta turma construíram e utilizaram o conceito de limite, numa abordagem intuitiva, em contextos diversos (funções racionais, derivada e sucessões) e quais as dificuldades que evidenciaram.
As tarefas propostas aos alunos, no decorrer da lecionação, foram direcionadas para o desenvolvimento de intuições matemáticas, encaminhando os alunos para a generalização do conceito de derivada, bem como, para a noção do conceito de limite, tirando partido do uso de várias representações - gráfica, numéria, algébrica e linguagem natural.
A análise de dados, recolhidos por meio da observação participante (com registo áudio e vídeo das aulas lecionadas) e recolha documental das produções escritas dos alunos, permite concluir que parte dos alunos construiu um “conceito imagem” de limite pouco consistente, porque o seu “conceito imagem evocado” (Tall & Vinner, 1981) integra algumas dificuldades, tais como, a “inatingibilidade” do limite.
O estudo também permite concluir que alguns alunos tendem a utilizar o conceito de limite, na resolução das tarefas que envolvem o mesmo, por meio de estratégias dominantes de procedimentos algébricos, devido à compreensão instrumental (Skemp, 1978) que detêm do mesmo.
Outra conclusão do meu estudo é que parte dos alunos construíram e utilizaram o conceito de limite evidenciando compreensão relacional (Skemp, 1978) entre as várias representações e entre vários contextos, inferindo, deste modo, que o conceito de limite pode ser introduzido de forma intuitiva e formal.
The study presented below has been developed within the practice of supervised teaching, and applied to a 11thgrade class in the Ramada Secondary School, during the school year 2014/15. The curricular unit “Variation rate and Derivative” included in the Mathematics-Grade A program (ME, 2001) was the subject of five of my 90-minute lectures. Nonetheless, this study has also taken in aspects related to my observations concerning different levels of participation on the topics of rational functions and sequences. The main purpose of this study has been to understand how the high school students of the class assemble and use the concept of limit, in an intuitive manner and in different tasks (rational functions, derivatives, sequences), and what difficulties they come upon. The tasks suggested to the high school students, in class, were directed to the development of mathematical intuition, conducting them to the generalization of the concept of derivative, as well as to the notion of the concept of limit, taking advantage of various methods of representation, namely graphic, numerical, algebraic and natural language. The integrated analysis of data obtained on the basis of direct observation (with audio and video records of lectures) and of written papers produced by the high school students, have led us to the conclusion that the concept image of limit assembled by a number of high school students is not very consistent, because their evoked concept image (Tall & Vinner, 1981) comprises some difficulties, such as the unattainability of the limit. The present study shows that some high school students tend to apply the concept of limit to solve tasks involving it making use of dominant strategies in algebraic procedures, because they retain an instrumental understanding (Skemp, 1978) of that concept. Another conclusion of this study is that part of the high school students assemble and use the concept of limit through relational understanding (Skemp, 1978) between several representations and between several contexts, thus making it possible to infer that the concept of limit can be introduced in an intuitive and formal way.
The study presented below has been developed within the practice of supervised teaching, and applied to a 11thgrade class in the Ramada Secondary School, during the school year 2014/15. The curricular unit “Variation rate and Derivative” included in the Mathematics-Grade A program (ME, 2001) was the subject of five of my 90-minute lectures. Nonetheless, this study has also taken in aspects related to my observations concerning different levels of participation on the topics of rational functions and sequences. The main purpose of this study has been to understand how the high school students of the class assemble and use the concept of limit, in an intuitive manner and in different tasks (rational functions, derivatives, sequences), and what difficulties they come upon. The tasks suggested to the high school students, in class, were directed to the development of mathematical intuition, conducting them to the generalization of the concept of derivative, as well as to the notion of the concept of limit, taking advantage of various methods of representation, namely graphic, numerical, algebraic and natural language. The integrated analysis of data obtained on the basis of direct observation (with audio and video records of lectures) and of written papers produced by the high school students, have led us to the conclusion that the concept image of limit assembled by a number of high school students is not very consistent, because their evoked concept image (Tall & Vinner, 1981) comprises some difficulties, such as the unattainability of the limit. The present study shows that some high school students tend to apply the concept of limit to solve tasks involving it making use of dominant strategies in algebraic procedures, because they retain an instrumental understanding (Skemp, 1978) of that concept. Another conclusion of this study is that part of the high school students assemble and use the concept of limit through relational understanding (Skemp, 1978) between several representations and between several contexts, thus making it possible to infer that the concept of limit can be introduced in an intuitive and formal way.
Descrição
Relatório da Prática de Ensino Supervisionada, Mestrado em Ensino da Matemática, Universidade de Lisboa, 2015
Palavras-chave
Linguagem matemática Conceito Derivadas Relatórios da prática de ensino supervisionada - 2015