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O desenvolvimento do conceito de número racional em alunos do 4º ano de escolaridade
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Orientador(es)
Resumo(s)
Quando os alunos iniciam as suas aprendizagens sobre números fraccionários
enfrentam uma série de dificuldades que muitas vezes se prendem com as situações
didácticas que são implementadas.
Estes números estão associados à multiplicação e divisão, exprimem relações e,
por isso, mobilizam mais o raciocínio multiplicativo do que o aditivo. Assim como o
desenvolvimento do cálculo mental implica um trabalho intenso, intencional e
progressivo, o desenvolvimento do raciocínio multiplicativo também exige a
implementação de estratégias próprias que privilegiem a resolução de problemas e
focalizem a atenção na procura de relações.
Esta dissertação de mestrado desenvolveu-se no contexto do 4° ano de
escolaridade e centrou-se na construção e implementação de um conjunto de situações
didácticas, transversais às áreas temáticas dos números, operações, grandezas e medida,
com o propósito de ajudar os alunos a desenvolver compreensivamente o conceito de
número racional.
Na consecução deste estudo levantaram-se as seguintes questões:
Questão 1 - Quais as tarefas e estratégias a privilegiar em cada uma das
dimensões de número racional: na dimensão parte - todo, na de quociente, na de medida
e na dimensão de razão?
Questão 2 - Como é que a conjugação de vários modos de representação de
números e ideias matemáticas (materiais, símbolos falados, modelos figurativos,
símbolos escritos e linguagem simbólica) associados a problemas da vida real podem
aumentar os níveis de compreensão e a construção de ideias sobre números
fraccionários?
O trabalho de campo foi desenvolvido no ano lectivo de 2001/2002 numa turma
do 4º ano de escolaridade, numa escola da periferia urbana de Lisboa. A professora
participante teve um papel preponderante na realização deste estudo, uma vez que a
planificação e reflexão sobre as tarefas implementadas foram desenvolvidas em
conjunto.
Das tarefas exploradas na sala de aula foram escolhidas as mais significativas
relacionadas com cada uma das dimensões do número racional. Os protocolos dos
alunos, algumas entrevistas realizadas nos momentos de trabalho de grupo e as notas de campo recolhidas durante as apresentações e discussões em grande grupo constituíram a
principal fonte para análise de dados.
Da análise de dados toma-se evidente que as primeiras abordagens ao
desenvolvimento de ideias sobre fracções deram enfoque à linguagem oral e à
manipulação de modelos físicos. A simbologia desenvolveu-se de uma forma natural e
sempre associada às acções realizadas com os vários modelos.
Nos problemas de partilha equitativa e nos problemas de razão, os alunos
mobilizaram estratégias próprias para resolver as suas partições. Alguns usaram o
raciocínio multiplicativo com flexibilidade e demonstraram uma boa capacidade para
descobrir relações e trabalhar com unidades compostas.
O contexto do dinheiro e das medidas foi também mobilizador no
desenvolvimento de ideias sobre fracções e decimais porque permitiu uma diversidade
de transferências que ampliaram o sentido de número e de operação. As estimativas
desempenharam aqui um papel crucial.
Outra dimensão que emergiu dos dados foi a relevância dos vários modos de
representação de números e de situações na resolução dos problemas. Assim, valorizou-se
a interacção entre materiais manipuláveis, desenhos feitos pelos alunos, a linguagem
oral, a construção de tabelas, diagramas e esquemas e a escrita matemática por palavras
e símbolos. É também descrito o papel dos contra-exemplos nas abordagens que
privilegiam a descoberta de relações.
Assim, partindo com a convicção que no 1º ciclo se têm de criar estruturas
numéricas fortes, este estudo aponta algumas estratégias que ajudem os alunos do 1° ciclo a compreender os números fraccionários e a desenvolver o raciocínio
multiplicativo.
When students start learning fractional numbers they will face several difficulties, as a direct consequence of didactical situations that have not been implemented. Fractional numbers are related with multiplication and division, they express relations and therefore imply more the multiplicative reasoning than the additive one. As well as developing mental reasoning implies a strong payload of work, intentionally focused and progressive, the building of a multiplicative reasoning also involves implementing specific strategies and solving problems focused in relationship searching. This study was carried out in the 4th grade and was aimed to build and implement a certain number of didactical situations covering different areas of numbers, operations, weights and measures, aiming to help students to build the genesis of rational numbers understanding its concept. During the development of this study the following questions were raised: 1. Which tasks and strategies should be implemented for each rational numbers dimension: as the part vs. the whole, as a quotient, as a measure, as a ratio? 2. How the combination of several ways of presenting numbers (didactical materials, real life problems, spoken symbols, figurative models and symbolic language) may raise understanding levels and building the idea of fractional numbers. The field work was developed during the academic year 2001/2002 in a 4th grade class at a school around Lisbon. The teacher leading the field work had an important role in this study once planning and analysis of the tasks implemented were developed jointly. From all the tasks done m the classes, the most significant ones, for each of the rational number dimensions, were chosen. Student's reports, some interviews made during group work, notes collected during presentations and discussions in fiill group sessions were the main source for data analysis. From data analysis is quite relevant that, since the very first approaches, the development of knowledge on fractions was preferred with focus on oral language and real models work. Students worked out their own strategies to solve equitative sharing and ratio problems, and some have used a flexible multiplicative reasoning showing good capacity to find relationships and to work with compounded units. Money and measuring environment was also very motivating to develop ideas about fractional and decimal numbers because it allowed a certain number of links that boasted the meaning of number and operation. Guesses/conjectures played a crucial role at this respect. Another dimension revealed by data was the importance of the various ways of representing situations in problem solving. Therefore, the interaction between manipulative materials, student made drawings, oral language, table building, diagrams and schemes, spoken and symbolic mathematical language was valued. The role of counterexamples was also referred in those approaches that privileged finding relationships. So, starting with the conviction that in 1st grade classes is necessary to build strong numerical structures, this study points out some strategies to help grade students to understand fractional numbers and develop multiplicative reasoning.
When students start learning fractional numbers they will face several difficulties, as a direct consequence of didactical situations that have not been implemented. Fractional numbers are related with multiplication and division, they express relations and therefore imply more the multiplicative reasoning than the additive one. As well as developing mental reasoning implies a strong payload of work, intentionally focused and progressive, the building of a multiplicative reasoning also involves implementing specific strategies and solving problems focused in relationship searching. This study was carried out in the 4th grade and was aimed to build and implement a certain number of didactical situations covering different areas of numbers, operations, weights and measures, aiming to help students to build the genesis of rational numbers understanding its concept. During the development of this study the following questions were raised: 1. Which tasks and strategies should be implemented for each rational numbers dimension: as the part vs. the whole, as a quotient, as a measure, as a ratio? 2. How the combination of several ways of presenting numbers (didactical materials, real life problems, spoken symbols, figurative models and symbolic language) may raise understanding levels and building the idea of fractional numbers. The field work was developed during the academic year 2001/2002 in a 4th grade class at a school around Lisbon. The teacher leading the field work had an important role in this study once planning and analysis of the tasks implemented were developed jointly. From all the tasks done m the classes, the most significant ones, for each of the rational number dimensions, were chosen. Student's reports, some interviews made during group work, notes collected during presentations and discussions in fiill group sessions were the main source for data analysis. From data analysis is quite relevant that, since the very first approaches, the development of knowledge on fractions was preferred with focus on oral language and real models work. Students worked out their own strategies to solve equitative sharing and ratio problems, and some have used a flexible multiplicative reasoning showing good capacity to find relationships and to work with compounded units. Money and measuring environment was also very motivating to develop ideas about fractional and decimal numbers because it allowed a certain number of links that boasted the meaning of number and operation. Guesses/conjectures played a crucial role at this respect. Another dimension revealed by data was the importance of the various ways of representing situations in problem solving. Therefore, the interaction between manipulative materials, student made drawings, oral language, table building, diagrams and schemes, spoken and symbolic mathematical language was valued. The role of counterexamples was also referred in those approaches that privileged finding relationships. So, starting with the conviction that in 1st grade classes is necessary to build strong numerical structures, this study points out some strategies to help grade students to understand fractional numbers and develop multiplicative reasoning.
Descrição
Tese de mestrado em Educação (Didáctica da Matemática), apresentada à Universidade de Lisboa através da Faculdade de Ciências, 2005
Palavras-chave
Números racionais Raciocínio Fracções Resolução de problemas Interacção Teses de mestrado - 2005