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Capacidade de generalização em sequências crescentes com estruturas pictóricas em alunos de 4.º ano
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Orientador(es)
Resumo(s)
Este estudo tem como principal objetivo compreender o contributo de uma
proposta pedagógica para o desenvolvimento do pensamento algébrico em alunos de 4.º
ano, que segue uma abordagem de ensino exploratório e envolve a exploração de tarefas
com sequências pictóricas crescentes que diferem nos elementos pictóricos que as
constituem na estrutura matemática subjacente à sequência numérica que se associa à
sequência pictórica. O estudo tem como base uma metodologia qualitativa e de cariz
interpretativo. Nesta experiência, assumo simultaneamente o papel de professor e de
investigador. Para além da observação participante na sala de aula, pelo meio da
transcrição dos registos áudio e vídeo e da construção de diário de bordo, a recolha de
dados é realizada também através da análise documental dos registos produzidos pelos
treze pares de alunos que participaram nesta investigação.
Os resultados do estudo sugerem que estes alunos, ao longo do seu percurso
escolar, não exploraram de uma forma sequenciada, orientada e com objetivos definidos
este tipo de estrutura matemática. Os resultados mostram também que os alunos, numa
fase inicial do estudo, sentem necessidade em recorrer a representações ativas,
passando, progressivamente, a utilizar representações icónicas e, nas últimas tarefas,
mais alunos utilizam representações simbólicas. Os participantes utilizam diversas
estratégias construtivas, nomeadamente, representação e contagem, aditiva e objeto
inteiro. Muitos alunos recorreram insistentemente às tabelas e respetivas estruturas para
sistematizar a informação recolhida e o raciocínio, privilegiando o uso da estratégia
aditiva. Verifica-se também, que ao longo da exploração das diferentes tarefas, os
momentos de discussão coletiva assumiram um papel fundamental no desenvolvimento
do pensamento algébrico, observando-se, gradualmente, uma maior tendência no uso da
estratégia decomposição dos termos. Em relação à capacidade de generalizar, muitos
alunos privilegiam o uso de estratégias de generalização de natureza aritmética,
manifestando dificuldade em usar a linguagem algébrica para representar
generalizações. Apesar das dificuldades apresentadas, os resultados obtidos revelam que
este estudo, contribuiu para o desenvolvimento do pensamento algébrico,
especificamente na capacidade de generalizar usando representações simbólicas.
This project aims at understanding how 4th grade students analyse, represent and what are the generalisation strategies that are used in tasks that entail increasing sequences within a linear mathematical structure, but with different pictorial arrangements and representations, namely the way they determine the next term or the nth term, the order of a term and how they express their generalisations. This study was carried out using a qualitative methodology of interpretative nature. In this experience, I take on both the role of teacher and researcher. Apart from the participant observation in the classroom, by means of audio and video transcripts and through the keeping of a logbook, data collection was also made by analysing the records produced by the thirteen pairs of students that participated in this research. The results of this study suggest that, during previous studies, these students had not handled this type of mathematical structures in a sequenced, guided and goal-oriented way. These results also show that students feel the need to resort to active representations in the initial tasks, eventually progressing to using pictorial representations and, later on, more students are able to handle symbolic representations. The participants make use of many constructive strategies, namely representation and counting, additive, and whole object. Many students resorted insistently to tables and their respective structure so as to systematise the collected data and mathematical reasoning, favouring the use of the additive strategy. We can also report that, throughout the completion of the tasks, the moments of collective discussion took on a fundamental role in the development of the algebraic thinking to the point where we could observe a gradual tendency to make use of the number decomposition strategy. With respect to generalisation skills, many students favour the use of generalisation strategies of arithmetic nature, showing difficulty in using algebraic language to represent them. Despite these difficulties, the results obtained show that this study contributed to the development of students’ algebraic reasoning, especially regarding the skill of generalising symbolically, as a way of expressing generalisations.
This project aims at understanding how 4th grade students analyse, represent and what are the generalisation strategies that are used in tasks that entail increasing sequences within a linear mathematical structure, but with different pictorial arrangements and representations, namely the way they determine the next term or the nth term, the order of a term and how they express their generalisations. This study was carried out using a qualitative methodology of interpretative nature. In this experience, I take on both the role of teacher and researcher. Apart from the participant observation in the classroom, by means of audio and video transcripts and through the keeping of a logbook, data collection was also made by analysing the records produced by the thirteen pairs of students that participated in this research. The results of this study suggest that, during previous studies, these students had not handled this type of mathematical structures in a sequenced, guided and goal-oriented way. These results also show that students feel the need to resort to active representations in the initial tasks, eventually progressing to using pictorial representations and, later on, more students are able to handle symbolic representations. The participants make use of many constructive strategies, namely representation and counting, additive, and whole object. Many students resorted insistently to tables and their respective structure so as to systematise the collected data and mathematical reasoning, favouring the use of the additive strategy. We can also report that, throughout the completion of the tasks, the moments of collective discussion took on a fundamental role in the development of the algebraic thinking to the point where we could observe a gradual tendency to make use of the number decomposition strategy. With respect to generalisation skills, many students favour the use of generalisation strategies of arithmetic nature, showing difficulty in using algebraic language to represent them. Despite these difficulties, the results obtained show that this study contributed to the development of students’ algebraic reasoning, especially regarding the skill of generalising symbolically, as a way of expressing generalisations.
Descrição
Trabalho de Projeto de Mestrado, Educação (Área de Especialidade em Didática da Matemática), Universidade de Lisboa, Instituto de Educação, 2020
Palavras-chave
Pensamento matemático Sequências (Matemática) Generalização Trabalhos de projeto de mestrado - 2020