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Abstract(s)
A Matemática é unanimemente reconhecida como uma das mais importantes ciências de base, atendendo à sua aplicação de forma transversal em quase todas as outras áreas do conhecimento. Por seu turno, a Didática é uma área do conhecimento científico que se desenvolveu na última metade do século XX, com o objetivo de estudar as diferentes etapas do processo de construção de conhecimento. Assim, a investigação em Didática da Matemática emerge como de capital relevo, especialmente a que está direcionada para as diversas etapas da construção de conceitos matemáticos. Conceitos mal compreendidos levam a que, mais tarde, os seus utilizadores revelem dificuldades de adaptação a novos conceitos e realidades, muitas vezes noutras ciências que utilizam a Matemática como pedra basilar.
No caso concreto deste estudo, procurámos compreender como é que os alunos do primeiro ano de uma licenciatura com uma unidade curricular de Métodos Quantitativos no plano de curso constroem os conceitos de continuidade e de limite. Assim, pretendemos responder às seguintes questões de investigação: que ações epistémicas são possíveis identificar no decurso do processo de abstração dos alunos durante a construção dos conceitos de continuidade e de limite; e como se sequenciam e relacionam essas ações epistémicas?
Com esse objetivo, começámos por trabalhar em aula a noção de continuidade e só posteriormente a de limite. Esta é uma sequência de ensino pouco habitual mas justificável, quer historicamente, quer atendendo a que não existe uma estrutura hierárquica entre estes conceitos. Mais ainda, os alunos revelam habitualmente mais dificuldade em compreender a noção de limite, pelo que definir continuidade à custa de limite representa um problema acrescido. Note-se que a investigadora não era a professora da turma, tendo assumido um papel de observadora ativa.
Adotámos a teoria AiC (Hershkowitz, Schwarz e Dreyfus, 2001) e o consequente modelo teórico e metodológico RBC+Co (Recognizing – Ação-R; Building-with – Ação-B; Constructing – Ação-C; Consolidation – Consolidação). As ações epistémicas, ao serem ações do pensamento, tornam- -se visíveis através de ações externas produzidas pelos alunos permitindo, assim, estudar o desenvolvimento do processo de abstração e a construção do novo conhecimento matemático.
Neste estudo considerámos subcategorias das ações epistémicas, de acordo quer com o que se entende ser cada ação epistémica, quer com o contexto, nomeadamente o matemático. Com as subcategorias pretendemos facilitar a identificação dessas ações epistémicas.
Usámos uma metodologia de investigação qualitativa, inserida no paradigma interpretativo. A recolha dos dados foi efetuada no ano letivo de 2014/2015. Ao nível da implementação do estudo, este foi realizado ao longo das aulas. Os alunos procuraram responder a questões de dificuldade crescente sobre estas temáticas, sendo o seu progresso registado, quer através da recolha de produções escritas em papel, quer através do registo áudio e vídeo das aulas. Na análise dos dados recorremos ao software ATLAS.ti.
As conclusões apresentadas indicam que a Ação-R e a Ação-B se encontram interligadas durante o processo de abstração dos alunos, onde a Interpretação do enunciado em conjunto com as Estruturas adquiridas por ela despoletadas fazem parte integrante das Estratégias formuladas, bem como da Aplicação de construções prévias. A combinação de todas estas subcategorias identificadas, quer na Ação-R, quer na Ação-B, é essencial na construção dos conceitos de continuidade e de limite posteriormente alcançados pelos alunos. Concluímos ainda que a Ação- -C se manifestou após o desenvolvimento simultâneo da Ação-R e da Ação-B, estando essa situação relacionada com a Interpretação dos enunciados e com as Estruturas adquiridas anteriormente, as quais permitiram que os alunos fossem formulando Estratégias e Aplicando construções prévias de modo a obterem Soluções intermédias.
A Consolidação só se manifestou em algumas situações, mormente quando os alunos reconheceram similaridades entre a questão que se propunham trabalhar e questões previamente resolvidas, e quando consideraram que construções anteriores lhes poderiam ser úteis para alcançar a nova construção.
A noção de Vizinhança assumiu-se como um bom contexto para a aprendizagem dos conceitos de continuidade e de limite. Foi um poderoso fio condutor da continuidade para os limites, facilitando claramente a construção dos conceitos lecionados. A alteração da sequência de ensino apresentou-se como prometedora.
Finalmente, foram elencadas algumas recomendações, das quais destacamos a necessidade de estudar o papel do professor com o intuito de promover nos alunos a construção de novos conhecimentos matemáticos e o reflexo desta abordagem de construção dos conceitos de continuidade e de limite na construção posterior das noções de derivada e de integral.
Mathematics is unanimously recognized as one of the most important basic sciences, given its transversal application in almost all other areas of knowledge. Didactics, in turn, is an area of scientific knowledge that has developed in the last half of the 20th century, with the objective of studying the different stages of the knowledge construction process. Thus, the investigation in Mathematics Didactics emerges as of capital importance, especially the one that is directed to the several stages of the construction of mathematical concepts. Misunderstood concepts lead to later difficulties for users when adapting to new concepts and realities, often in other sciences that use Mathematics as a cornerstone. In the specific case of this study, we tried to understand how the first-year students of a degree with a curricular unit of Quantitative Methods in the course plan construct the concepts of continuity and limit. Thus, the research sought to answer the following research questions: what epistemic actions are possible to identify in the students' process of abstraction during the construction of concepts of continuity and limit; and how are these epistemic actions sequenced and related? To this end, we began by working on the concept of continuity and only later on the limit. This is an unusual but justifiable teaching sequence, both historically and because there is no hierarchical structure among these concepts. Moreover, students usually find it more difficult to understand the notion of limit, so defining continuity at the expense of limit represents an added problem. The researcher was not the classroom teacher but took an active role as an observer. We adopted the AiC theory (Hershkowitz, Schwarz and Dreyfus, 2001) and the consequent theoretical and methodological model RBC+Co (Recognizing – R-Action; Building-with – B-Action; Constructing – C-Action; Consolidation). The epistemic actions, being actions of the thought, are visible through external actions produced by the students, allowing to study the development of the abstraction process and the construction of new mathematical knowledge. In this study we considered subcategories of the epistemic actions, according to the meaning of each epistemic action, or with the context, namely the mathematician one. With the subcategories it was intended to facilitate the identification of these epistemic actions. We used a qualitative research methodology, inserted in the interpretative paradigm. The data was obtained in the scholar year of 2014/2015. The study was implemented throughout the lessons. The students sought to answer questions of increasing difficulty on these subjects, and their progress was recorded either by collecting written productions on paper or by recording audio and video from lessons. The data analysis was carried out with the software ATLAS.ti. The main conclusions indicate that the R-Action and the B-Action are interconnected during the students' abstraction process. The Interpretation of the statement together with the Acquired structures by it are an important part of the Strategies formulated as well as the Application of previous constructions. The combination of all these subcategories identified in both the R-Action and the B-Action is essential when building the concepts of continuity and limit, subsequently achieved by students. We concluded that the C-Action was manifested after the simultaneous development of R-Action and B-Action, which is related to the Interpretation of statements and to the previously Acquired structures. Previous relations allowed students to formulate Strategies and to Apply previous constructions in order to obtain Intermediate solutions. Consolidation was only manifested in some situations, especially when students recognized similarities between the question that they intended to work on and those previously resolved, and when they considered that previous constructions might be useful to achieve the new construction. The notion of Neighborhood assumed a suitable context for learning the concepts of continuity and limit. It was a powerful conduit from continuity to limit, making it easier to construct these concepts. The teaching sequence inversion was considered as encouraging. Finally, some recommendations were made, highlighting the need to study the teacher´s role in order to promote students’ constructions of new mathematical knowledge. Furthermore, we suggested to study how this approach to the construction of the concepts of continuity and limit can be reflected in the subsequent construction of derivative and integral notions.
Mathematics is unanimously recognized as one of the most important basic sciences, given its transversal application in almost all other areas of knowledge. Didactics, in turn, is an area of scientific knowledge that has developed in the last half of the 20th century, with the objective of studying the different stages of the knowledge construction process. Thus, the investigation in Mathematics Didactics emerges as of capital importance, especially the one that is directed to the several stages of the construction of mathematical concepts. Misunderstood concepts lead to later difficulties for users when adapting to new concepts and realities, often in other sciences that use Mathematics as a cornerstone. In the specific case of this study, we tried to understand how the first-year students of a degree with a curricular unit of Quantitative Methods in the course plan construct the concepts of continuity and limit. Thus, the research sought to answer the following research questions: what epistemic actions are possible to identify in the students' process of abstraction during the construction of concepts of continuity and limit; and how are these epistemic actions sequenced and related? To this end, we began by working on the concept of continuity and only later on the limit. This is an unusual but justifiable teaching sequence, both historically and because there is no hierarchical structure among these concepts. Moreover, students usually find it more difficult to understand the notion of limit, so defining continuity at the expense of limit represents an added problem. The researcher was not the classroom teacher but took an active role as an observer. We adopted the AiC theory (Hershkowitz, Schwarz and Dreyfus, 2001) and the consequent theoretical and methodological model RBC+Co (Recognizing – R-Action; Building-with – B-Action; Constructing – C-Action; Consolidation). The epistemic actions, being actions of the thought, are visible through external actions produced by the students, allowing to study the development of the abstraction process and the construction of new mathematical knowledge. In this study we considered subcategories of the epistemic actions, according to the meaning of each epistemic action, or with the context, namely the mathematician one. With the subcategories it was intended to facilitate the identification of these epistemic actions. We used a qualitative research methodology, inserted in the interpretative paradigm. The data was obtained in the scholar year of 2014/2015. The study was implemented throughout the lessons. The students sought to answer questions of increasing difficulty on these subjects, and their progress was recorded either by collecting written productions on paper or by recording audio and video from lessons. The data analysis was carried out with the software ATLAS.ti. The main conclusions indicate that the R-Action and the B-Action are interconnected during the students' abstraction process. The Interpretation of the statement together with the Acquired structures by it are an important part of the Strategies formulated as well as the Application of previous constructions. The combination of all these subcategories identified in both the R-Action and the B-Action is essential when building the concepts of continuity and limit, subsequently achieved by students. We concluded that the C-Action was manifested after the simultaneous development of R-Action and B-Action, which is related to the Interpretation of statements and to the previously Acquired structures. Previous relations allowed students to formulate Strategies and to Apply previous constructions in order to obtain Intermediate solutions. Consolidation was only manifested in some situations, especially when students recognized similarities between the question that they intended to work on and those previously resolved, and when they considered that previous constructions might be useful to achieve the new construction. The notion of Neighborhood assumed a suitable context for learning the concepts of continuity and limit. It was a powerful conduit from continuity to limit, making it easier to construct these concepts. The teaching sequence inversion was considered as encouraging. Finally, some recommendations were made, highlighting the need to study the teacher´s role in order to promote students’ constructions of new mathematical knowledge. Furthermore, we suggested to study how this approach to the construction of the concepts of continuity and limit can be reflected in the subsequent construction of derivative and integral notions.
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Didáctica da Matemática - Ensino Superior | Portugal