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As tarefas de ensino e a aprendizagem dos números decimais
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A aprendizagem dos números racionais constitui uma ampliação significativa no
conhecimento do sistema numérico. Tendo em conta as diferentes formas de
representação desses números e as diferenças relativamente ao conjunto dos números
inteiros, a aprendizagem dos números racionais é complexa e, muitas vezes, fonte de
dificuldade para os alunos.
A particularidade do currículo de Matemática em Portugal onde a aprendizagem
dos números racionais se inicia, no ciclo, através da abordagem aos números decimais
e as evidências do insucesso dos alunos nessa aprendizagem justificam a pertinência deste
estudo centrado num contexto de sala de aula onde se introduziram os números decimais.
Trata-se de um estudo de natureza qualitativa, do tipo de estudo de caso
instrumental, realizado no contexto de uma turma de 3° ano de escolaridade de uma
escola urbana do distrito de Setúbal. Foi orientado de acordo com as seguintes questões:
1) Como se caracterizam as tarefas utilizadas para introduzir os números
decimais?
2) Como se caracterizam as aprendizagens realizadas pelos alunos relativamente
aos números decimais?
3) Como se caracterizam as dificuldades dos alunos relativamente aos números
decimais?
4) Que relações se podem estabelecer entre as tarefas utilizadas e as
aprendizagens e dificuldades dos alunos?
Os dados foram recolhidos através da observação naturalista de aulas e de
entrevistas à professora titular de turma e a sete alunos. Da observação das aulas foram
seleccionadas sete tarefas e analisadas de acordo com a utilização pedagógico-didáctica
feita pela professora e do trabalho proporcionado e realizado pelos alunos. Nas
entrevistas aos sete alunos foram também aplicadas tarefas e analisado o desempenho de
cada aluno nas mesmas. Os dados recolhidos foram sujeitos a uma análise de conteúdo e
posterior triangulação com o quadro conceptual e teórico de suporte deste trabalho.
As tarefas de ensino observadas nas aulas caracterizaram-se fundamentalmente
como situações problemáticas centradas na abordagem do sistema monetário e na relação
ente o euro e os cêntimos. Embora esta abordagem pudesse ser utilizada como referência
para o ensino dos números decimais tendo em conta o seu uso quotidiano, a utilização
pedagógico-didáctica das tarefas privilegiou o desenvolvimento de técnicas e
procedimentos e parece não ter permitido o desenvolvimento da compreensão dos
conceitos de base como a relação parte todo. Por outro lado, certas particularidades do
sistema monetário como tratar-se de um sistema finito com um número limitado de
moedas, a representação do valor das moedas ser feita a partir de números inteiros e a
divisão equitativa não ter uma visibilidade física revelaram-se como entraves para a
aprendizagem dos números decimais. As dificuldades dos alunos foram claramente
evidenciadas durante as entrevistas aos sete alunos, especialmente nas tarefas mais
direccionadas para a compreensão dos números decimais. Quatro dos sete alunos não
conseguiram mobilizar as noções de décima e centésima, nem representá-las
numericamente e, embora cumprissem a regra "vírgula debaixo de vírgula", nem sempre
reconheceram a razoabilidade das suas respostas.
Os dados recolhidos por este trabalho permitiram concluir que a introdução dos
números decimais centrada no desenvolvimento de técnicas e procedimentos parece dificultar a compreensão dos conceitos. Sugere-se como recursos pedagógico-didácticos a
utilizar inicialmente para construção dos conceitos de décima e centésima o uso de
materiais manipuláveis como o material Dienes, o incentivo ao cálculo mental para o
estabelecimento de relações e a utilização da recta numérica para representação da
densidade dos números racionais. O recurso aos algoritmos e às regras e procedimentos
rotineiros deverá ser utilizado numa fase posterior, depois de estarem consolidados o
significado dos números e o sentido das operações, para que os mesmos sejam
contextualizados e definidos no seu objectivo principal de tomarem o cálculo mais
rápido.
As far as the numerical system is concerned the process of learning rational numbers constitutes a meaningful amplification of the pupils' mathematical knowledge. Regarding the different ways of representation of these numbers and the differences relatively to the group of whole numbers, the learning of rational numbers is complex and often a source of difficulties to pupils. Regarding the specificity of the Mathematics' curriculum in Portugal, the process of learning the rational numbers begins in the primary school, through the learning of decimal numbers. The low success of pupils in this area justifies this study namely concerning the classroom context. This is a qualitative and instrumental study case, analysed in a third grade class of an urban school in the Setúbal area. It was guided through the following questions: 1) What are the main characteristics of the tasks used at the begin of the learning process of decimal numbers? 2) How is it possible to characterize the level of knowledge obtained regarding the decimal numbers? 3) How are the main difficulties of pupils detected regarding the decimal numbers? 4) What kind of relation can be established between the tasks and the difficulties of the pupils? Tlie data were obtained through the regular observation of the classes and through interviews to the class teacher and seven pupils. Seven tasks were chosen from the regular observations and analysed accordingly to the pedagogical principles applied by the teacher and the school work done by the pupils In the interviews with the pupils there were also placed several tasks and the performance of the pupils was also analyzed. The data gathered was subjected to an analyses framed in the theory and conceptual support of this thesis. The teaching tasks observed in the classroom were mainly characterized with problem-situations on the monetary system and the relationship between the euro and the cent. This approach could be used as a reference to the teaching of decimal numbers regarding its daily use. The pedagogical principles of the tasks favoured the development of techniques and procedures but apparently it didn't allow the development of the comprehension of the base concepts regarding the whole concept. On the other side, certain specificities of the monetary system such as being a closed system with a limited number of coins, also the fact that the representation of the value of the coins is only done by whole numbers and that the equitable division doesn't have a physical visibility revealed themselves as barriers to the learning of decimal numbers. The difficulties were widely noticed during the interviews with the seven pupils, especially in the tasks oriented to the full understanding of the decimal numbers. Four of the seven pupils were not able to mobilise the concepts of decimal and centesimal neither were they able to represent tem as a number and although they obliged to the rule "coma under coma", they didn't even fully recognized the reasonableness of their answers. The data gathered from this thesis allows the conclusion that the begining of the learning process of the decimal numbers centred in the development of techniques and procedures seems to difficult the comprehension of the concepts. It would be a good option to use as pedagogical resources the construction of the concepts of decimal and centesimal through manipulative materials such as the Dienes, also the incentive of the mental calculation to the establishment of relations and the use of the number line for the representation of the density of the rational numbers. The use of algorithms, the rules and daily procedures should be use in a later phase, after the meaning of the numbers and the sense of the operations is fully apprehended, so that they can be defined and put in a context of their main purpose which is a faster calculation.
As far as the numerical system is concerned the process of learning rational numbers constitutes a meaningful amplification of the pupils' mathematical knowledge. Regarding the different ways of representation of these numbers and the differences relatively to the group of whole numbers, the learning of rational numbers is complex and often a source of difficulties to pupils. Regarding the specificity of the Mathematics' curriculum in Portugal, the process of learning the rational numbers begins in the primary school, through the learning of decimal numbers. The low success of pupils in this area justifies this study namely concerning the classroom context. This is a qualitative and instrumental study case, analysed in a third grade class of an urban school in the Setúbal area. It was guided through the following questions: 1) What are the main characteristics of the tasks used at the begin of the learning process of decimal numbers? 2) How is it possible to characterize the level of knowledge obtained regarding the decimal numbers? 3) How are the main difficulties of pupils detected regarding the decimal numbers? 4) What kind of relation can be established between the tasks and the difficulties of the pupils? Tlie data were obtained through the regular observation of the classes and through interviews to the class teacher and seven pupils. Seven tasks were chosen from the regular observations and analysed accordingly to the pedagogical principles applied by the teacher and the school work done by the pupils In the interviews with the pupils there were also placed several tasks and the performance of the pupils was also analyzed. The data gathered was subjected to an analyses framed in the theory and conceptual support of this thesis. The teaching tasks observed in the classroom were mainly characterized with problem-situations on the monetary system and the relationship between the euro and the cent. This approach could be used as a reference to the teaching of decimal numbers regarding its daily use. The pedagogical principles of the tasks favoured the development of techniques and procedures but apparently it didn't allow the development of the comprehension of the base concepts regarding the whole concept. On the other side, certain specificities of the monetary system such as being a closed system with a limited number of coins, also the fact that the representation of the value of the coins is only done by whole numbers and that the equitable division doesn't have a physical visibility revealed themselves as barriers to the learning of decimal numbers. The difficulties were widely noticed during the interviews with the seven pupils, especially in the tasks oriented to the full understanding of the decimal numbers. Four of the seven pupils were not able to mobilise the concepts of decimal and centesimal neither were they able to represent tem as a number and although they obliged to the rule "coma under coma", they didn't even fully recognized the reasonableness of their answers. The data gathered from this thesis allows the conclusion that the begining of the learning process of the decimal numbers centred in the development of techniques and procedures seems to difficult the comprehension of the concepts. It would be a good option to use as pedagogical resources the construction of the concepts of decimal and centesimal through manipulative materials such as the Dienes, also the incentive of the mental calculation to the establishment of relations and the use of the number line for the representation of the density of the rational numbers. The use of algorithms, the rules and daily procedures should be use in a later phase, after the meaning of the numbers and the sense of the operations is fully apprehended, so that they can be defined and put in a context of their main purpose which is a faster calculation.
Descrição
Tese de mestrado em Ciências da Educação (Área de especialização em Teoria e Desenvolvimento Curricular), apresentada à Universidade de Lisboa, através da Faculdade de Psicologia e de Ciências da Educação, 2007
Palavras-chave
Teses de mestrado - 2007 Teoria e desenvolvimento curricular Tarefas escolares Aprendizagem da matemática