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O uso de representações matemáticas pelos alunos do 7º ano no tópico das funções
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O presente trabalho apresenta um estudo efetuado durante a prática letiva supervisionada no tópico de funções, com alunos do 7.º ano, utilizando diferentes tipos de representação matemática. O seu objetivo é compreender as ligações efetuadas pelos alunos entre as diferentes representações e a sua aplicação na realização de tarefas. A prática letiva supervisionada decorreu durante o 2.º semestre do ano letivo de 2021/2022, numa turma de 23 alunos 7.º ano, de uma escola do distrito de Lisboa onde foi abordado o tema Álgebra, com o tópico Funções, e os subtópicos: Representação gráfica; Diferentes formas de representar uma função; Operar com funções; Função linear; e Proporcionalidade direta como função. Anteriormente a professora cooperante tinha lecionado os subtópicos Referencial Cartesiano; Correspondência entre conjuntos; e Relações entre variáveis. No presente estudo participaram todos os alunos da turma, tendo sido selecionados quatro alunos para um conhecimento mais aprofundado. Esses quatro alunos apresentavam níveis de conhecimento distintos, de insuficiente, suficiente, bom, e muito bom. Durante as aulas lecionadas, através do contributo oral e escrito, procurou-se avaliar a evolução dos alunos. Na recolha dos dados foram utilizados os seguintes processos: Observação de aulas; Apontamentos com conversas informais com os alunos; e Recolha de produções escritas dos alunos. Os alunos ao longo das tarefas que lhes foram propostas, realizaram diversas mudanças de representação no tópico sobre funções. Começaram pela representação gráfica e algébrica, sendo que o pedido da expressão algébrica nas tarefas provocou sempre algum desconforto numa parte da turma. Nas diferentes formas de representar uma função, foi apresentado o diagrama sagital e a representação em tabela. Estas duas formas de representação são as que mais facilmente foram aprendidas pelos alunos. O estudo da variação de uma função linear crescente, decrescente, e a representação da função constante em gráfico cartesiano, foi uma aprendizagem conseguida e houve alunos que consideraram a tarefa que envolvia estes conceitos como a mais interessante. A representação da função linear não decorreu como esperado. Eu tinha construído um gráfico cartesiano em larga escala, utilizando cartão e na origem dos eixos coloquei uma cana, que rodava em torno da origem dos eixos. A ideia era, que os alunos rodassem a cana e escolhessem uma função linear, da qual iam representar a sua expressão algébrica. Mas o termo “expressão algébrica”, fez com que uma parte da turma sentisse desconforto com o que lhes era pedido. E o artefacto acabou por não ter a utilidade desejada. Na identificação da função linear como sendo uma função de proporcionalidade direta, a ideia teve uma aceitação mais generalizada, pois é um tema que tem muitas aplicações práticas no dia-a-dia dos alunos. Nas últimas duas aulas continuou-se a apresentação do tópico sobre proporcionalidade direta como função, e foi apresentada uma tarefa que, procurava que os alunos aplicassem novos conhecimentos recentemente adquiridos. Foram trabalhadas as noções de objeto, imagem, expressão algébrica, constante de proporcionalidade, gráfico de uma função, pontos desse gráfico, a representação dos pontos em diagrama sagital e em tabela, e saber justificar porque é que a função () é de proporcionalidade direta. Durante a aula, a questão que mais solicitou a minha intervenção, foi a justificação de que uma função dada, é de proporcionalidade direta. Fazendo uso da tabela anteriormente construída, foi relativamente simples levar toda a turma perceber que o quociente , relativo a um qualquer ponto do gráfico, (com exceção do ponto que representa a origem dos eixos), era a constante que se pretendia encontrar. No seu todo, os alunos mostraram gostar da experiência.
This work presents a study carried out during supervised teaching practice on the topic of functions, to a class of grade 7, using different types of mathematical representation. Its objective is to understand the connections made by students between different mathematical representations, and their application in carrying out tasks. All students in the class participated in the present study, and four students were selected for deeper knowledge. These four students have different levels of knowledge, each with a different level of knowledge, insufficient, sufficient, good, and very good. During the classes taught, through oral and written contributions, we sought to evaluate the students' progress within the scope of this study. The supervised teaching practice took place during the 2nd semester of the 2021/2022 school year, in a class of 23 grade 7 students from a school in the district of Lisbon where the theme Algebra was addressed, with the topic Functions, and the subtopics: Graphic representation; Different ways of representing a function; Operations with functions; Linear function; and Direct proportion as a function. Previously, the cooperating teacher had taught the subtopics Cartesian graphs; Mapping between sets; and Relationships between variables. In the present study, all students in the class participated, and four students were selected for a more in-depth knowledge. These four students had different levels of knowledge, insufficient, sufficient, good, and very good. During the classes taught, through oral and written contributions, I tried to evaluate the evolution of the students. The following processes were used for data collection: Class observation; Notes with informal conversations with students; and Collection of students written productions. Throughout the tasks that were proposed to them, the students made several changes in representation in the topic of functions. They started with graphic and algebraic representations, and the request for algebraic expression in the tasks always caused some discomfort from the class. In the different ways of representing a function, the sagittal diagram and the table representation were presented. These two forms of representation are the ones that were most easily learned by the students. The study of the variation of an increasing and decreasing linear function, and the representation of the constant function in a Cartesian graph, was a learning achievement and there were students who considered the task involving these concepts as the most interesting. The representation of the linear function did not go as expected. I had built a large-scale Cartesian graph using cardboard, and at the origin of the axes I placed a cane, which rotated around the origin of the axes. The idea was that the students would rotate the cane and choose a linear function, from which they would represent its algebraic expression. But the term "algebraic expression" made a part of the class feel uncomfortable with what was asked of them. And the artifact turned out not to achieve the desired use. In the identification of the linear function as being a function of direct proportion, the idea had a more widespread acceptance, as it is a topic that has many practical applications in the daily lives of students. In the last two classes, the presentation of the topic of direct proportion as a function continued, and it was presented a task seeking to have the students apply the new knowledge recently acquired. The notions of object, image, algebraic expression, proportion constant, graph of a function, points of that graph, the representation of points in a sagittal diagram and in a table, and knowing how to justify why the function f(x) is of direct proportion were worked on. During the lessons, the issue that most requested my intervention was the justification that a given function is a direct proportion. Making use of the previously constructed table, it was relatively simple to get the whole class to realize that the quotient , relative to any point on the graph (apart from the point representing the origin of the axes), was the constant that was intended to be found. Overall, the students showed that they enjoyed the experience.
This work presents a study carried out during supervised teaching practice on the topic of functions, to a class of grade 7, using different types of mathematical representation. Its objective is to understand the connections made by students between different mathematical representations, and their application in carrying out tasks. All students in the class participated in the present study, and four students were selected for deeper knowledge. These four students have different levels of knowledge, each with a different level of knowledge, insufficient, sufficient, good, and very good. During the classes taught, through oral and written contributions, we sought to evaluate the students' progress within the scope of this study. The supervised teaching practice took place during the 2nd semester of the 2021/2022 school year, in a class of 23 grade 7 students from a school in the district of Lisbon where the theme Algebra was addressed, with the topic Functions, and the subtopics: Graphic representation; Different ways of representing a function; Operations with functions; Linear function; and Direct proportion as a function. Previously, the cooperating teacher had taught the subtopics Cartesian graphs; Mapping between sets; and Relationships between variables. In the present study, all students in the class participated, and four students were selected for a more in-depth knowledge. These four students had different levels of knowledge, insufficient, sufficient, good, and very good. During the classes taught, through oral and written contributions, I tried to evaluate the evolution of the students. The following processes were used for data collection: Class observation; Notes with informal conversations with students; and Collection of students written productions. Throughout the tasks that were proposed to them, the students made several changes in representation in the topic of functions. They started with graphic and algebraic representations, and the request for algebraic expression in the tasks always caused some discomfort from the class. In the different ways of representing a function, the sagittal diagram and the table representation were presented. These two forms of representation are the ones that were most easily learned by the students. The study of the variation of an increasing and decreasing linear function, and the representation of the constant function in a Cartesian graph, was a learning achievement and there were students who considered the task involving these concepts as the most interesting. The representation of the linear function did not go as expected. I had built a large-scale Cartesian graph using cardboard, and at the origin of the axes I placed a cane, which rotated around the origin of the axes. The idea was that the students would rotate the cane and choose a linear function, from which they would represent its algebraic expression. But the term "algebraic expression" made a part of the class feel uncomfortable with what was asked of them. And the artifact turned out not to achieve the desired use. In the identification of the linear function as being a function of direct proportion, the idea had a more widespread acceptance, as it is a topic that has many practical applications in the daily lives of students. In the last two classes, the presentation of the topic of direct proportion as a function continued, and it was presented a task seeking to have the students apply the new knowledge recently acquired. The notions of object, image, algebraic expression, proportion constant, graph of a function, points of that graph, the representation of points in a sagittal diagram and in a table, and knowing how to justify why the function f(x) is of direct proportion were worked on. During the lessons, the issue that most requested my intervention was the justification that a given function is a direct proportion. Making use of the previously constructed table, it was relatively simple to get the whole class to realize that the quotient , relative to any point on the graph (apart from the point representing the origin of the axes), was the constant that was intended to be found. Overall, the students showed that they enjoyed the experience.
Descrição
Relatório da Prática de Ensino Supervisionada, Mestrado em Ensino de Matemática, 2024, Universidade de Lisboa, Instituto de Educação
Palavras-chave
Matemática - Estudo e ensino Representações matemáticas Funções (Matemática) Tarefas Relatórios da prática de ensino supervisionada - 2024