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Números naturais e subtracção : um estudo no 1º ciclo
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Orientador(es)
Resumo(s)
Este estudo pretende analisar as ideias e procedimentos numéricos que os
alunos, do 1.° ciclo, usam na resolução de problemas associados à operação
subtracção e ao conhecimento dos números e o modo como eles evoluem.
A fundamentação teórica incide em dois tópicos: Números e operações e
Sentido de número. No primeiro, aprofundo alguns aspectos relativos à
investigação sobre as operações com números naturais, concretamente
relacionadas com o ensino/aprendizagem dos números e operações e
estratégias de cálculo associadas às operações adição e subtracção com
números-dígito e números multidígitos. No segundo, discuto alguns aspectos
estruturantes relacionados com o sentido de número, em particular, o seu
significado, o modo como o conceito é abordado no currículo e os aspectos a
ter em conta no seu desenvolvimento, nomeadamente: o cálculo mental, os
algoritmos e o papel do professor.
O estudo segue uma abordagem de investigação qualitativa. A recolha de
dados decorreu entre Dezembro de 2008 e janeiro de 2009, com a
realização de entrevistas clínicas, pela investigadora, a dezoito alunos do 1.°
ciclo (seis do 2° ano, seis do 3.° ano e seis do 4.° ano). Os dados recolhidos
são de natureza descritiva. Incluem produções escritas dos alunos relativas
às tarefas propostas, registos áudio das entrevistas de tipo clínico e notas de
campo resultantes da observação das mesmas.
Os resultados do estudo permitem identificar, na resolução de problemas que
envolvem subtracção, as seguintes estratégias de cálculo usadas pelos
alunos: (i) contar, (ii) saltar, (iii) decompor, (iv) cálculo relacional, (v) usar
factos conhecidos e (vi) algoritmo. Identificam-se, também, formas globais
de relacionar e decompor os números: conhecimento sequencial dos
números, conhecimento sequencial usando saltos de 10, conhecimento de estruturas de decomposição decimal dos números, factos conhecidos e
algoritmo. É feita a análise de como evoluem os alunos a respeito de
conhecimento dos números e os erros que cometem.
Os resultados do estudo permitem concluir que: (1) no uso de estratégias de
subtracção é evidente a existência de dois pólos extremos: por um lado o
uso da estratégia contar, por outro, o uso do algoritmo. As estratégias de
nível intermédio de "sofisticação" como o saltar, decompor e uso de factos
conhecidos são pouco usadas; (2) nas questões associadas ao conhecimento
dos números o uso de estratégias progressivamente mais "sofisticadas"
parece estar relacionado com a progressão dos alunos por ano de
escolaridade.
Finalmente, este estudo reforça a ideia de que a introdução precoce dos
algoritmos inibe a compreensão de um conjunto de relações entre os
números e as operações associada ao desenvolvimento do sentido de
número. Pelo que, se evidencia a importância das propostas de ensino dos
professores contemplarem a prática de estratégias de cálculo mental de nível
intermédio no ensino/aprendizagem da subtracção.
This study aims to examine the ideas and numerical procedures that students of Primary School use in solving problems associated with the subtraction operation and the knowledge of numbers they have and how they evolve. The theoretical framework of the study focuses on two topics: Numbers and operations and Number sense. In the first section I discuss some aspects of research operations with natural numbers, specifically related to the teaching and learning of numbers and operations and calculation strategies associated to numbers-digit and multidigit numbers addition and subtraction. In the second section, I discuss some aspects related to structuring the number sense, in particular, its meaning, how the concept is approached in the curriculum and the aspects taken into account in its development, namely: mental calculation, algorithms and the teacher's role. The study follows a qualitative research approach. Data collection took place between December 2008 and January 2009 with the completion of clinical interviews, by the researcher, to eighteen students from Primary School (six from 2nd grade, six from 3rd grade and six from 4th grade). The data collected are descriptive. They include students' written production on the proposed tasks, audio recordings of interviews of clinical type and field notes resulting from the observation of the same. The study results allow me to identify, in solving problems involving subtraction, the following calculation strategies used by students: (i) counting, (ii) jumping (iii) decomposition, (iv) relational calculus, (v) known facts and (vi) algorithm. It also identifies global ways of relating and decompose numbers: knowledge of sequential numbers, knowledge of sequential jumps of 10, knowledge of structures of decimal decomposition of numbers, known facts and algorithm. An analysis of how students evolve, regarding the knowledge of numbers and mistakes they make, is accomplished. The study results show that: (1) in the use of strategies for subtraction it is clear that there are two extremes, in one hund the use of the counting strategies and in the other hand the use of algorithm. The strategies of intermediate level of "sophistication" as jumping , decomposing and known facts are not much used; (2) in issues related to knowledge of the numbers, the use of strategies progressively more "sophisticated" seems to be in connection with the progress of pupils per school year. Finally, this study strengthens the idea that the early introduction of algorithms inhibits the understanding of a set of relationships between numbers and operations associated with the development of number sense. Therefore, it highlights the importance of teachers' education proposals to contemplate the practice of mental calculation strategies in middle-level teaching and learning of subtraction.
This study aims to examine the ideas and numerical procedures that students of Primary School use in solving problems associated with the subtraction operation and the knowledge of numbers they have and how they evolve. The theoretical framework of the study focuses on two topics: Numbers and operations and Number sense. In the first section I discuss some aspects of research operations with natural numbers, specifically related to the teaching and learning of numbers and operations and calculation strategies associated to numbers-digit and multidigit numbers addition and subtraction. In the second section, I discuss some aspects related to structuring the number sense, in particular, its meaning, how the concept is approached in the curriculum and the aspects taken into account in its development, namely: mental calculation, algorithms and the teacher's role. The study follows a qualitative research approach. Data collection took place between December 2008 and January 2009 with the completion of clinical interviews, by the researcher, to eighteen students from Primary School (six from 2nd grade, six from 3rd grade and six from 4th grade). The data collected are descriptive. They include students' written production on the proposed tasks, audio recordings of interviews of clinical type and field notes resulting from the observation of the same. The study results allow me to identify, in solving problems involving subtraction, the following calculation strategies used by students: (i) counting, (ii) jumping (iii) decomposition, (iv) relational calculus, (v) known facts and (vi) algorithm. It also identifies global ways of relating and decompose numbers: knowledge of sequential numbers, knowledge of sequential jumps of 10, knowledge of structures of decimal decomposition of numbers, known facts and algorithm. An analysis of how students evolve, regarding the knowledge of numbers and mistakes they make, is accomplished. The study results show that: (1) in the use of strategies for subtraction it is clear that there are two extremes, in one hund the use of the counting strategies and in the other hand the use of algorithm. The strategies of intermediate level of "sophistication" as jumping , decomposing and known facts are not much used; (2) in issues related to knowledge of the numbers, the use of strategies progressively more "sophisticated" seems to be in connection with the progress of pupils per school year. Finally, this study strengthens the idea that the early introduction of algorithms inhibits the understanding of a set of relationships between numbers and operations associated with the development of number sense. Therefore, it highlights the importance of teachers' education proposals to contemplate the practice of mental calculation strategies in middle-level teaching and learning of subtraction.
Descrição
Tese de mestrado, Educação (Didáctica da Matemática), Universidade de Lisboa, Instituto de Educação, 2011
Palavras-chave
Matemática - Estudo e ensino Cálculo matemático Números naturais Ensino básico (1º ciclo) Teses de mestrado - 2011