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Iniciação ao cálculo diferencial : perspetivas e práticas de professores do ensino secundário português
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O presente estudo insere-se na área de investigação sobre o professor de Matemática, especificamente das conceções do professor sobre o ensino do Cálculo Diferencial no ensino secundário. A investigação é realizada tendo por pressuposto que tais conceções desempenham um papel fulcral no modo como o professor interpreta as situações educativas e também na forma como estrutura a atuação na sua prática de ensino. Assim, a presente investigação possui também uma segunda área de interesse, nomeadamente os aspetos centrais que caracterizam a prática do professor no ensino deste tema. Desse modo, o objetivo do estudo é identificar e compreender as conceções dos professores de Matemática sobre o ensino de Cálculo Diferencial no ensino secundário bem como os aspetos centrais da sua prática profissional no seu ensino.
O enquadramento teórico está organizado em três eixos: (i) as conceções do professor de Matemática; (ii) as práticas profissionais do professor de Matemática; e (iii) o ensino do Cálculo Diferencial. O primeiro eixo discute o interesse e as terminologias utilizadas nos estudos no domínio das conceções do professor, bem como a relação entre as conceções e o conhecimento e entre estas e a prática letiva. O segundo eixo aborda o conceito prática do professor e, tendo em atenção a prática letiva, destaca as tarefas e a comunicação em sala de aula como elementos estruturantes dessa prática. O terceiro eixo discute a presença do Cálculo Diferencial no currículo do ensino secundário e os esforços reformistas no ensino deste tema e apresenta, ainda, alguns estudos empíricos sobre o ensino-aprendizagem do Cálculo que possuem uma estreita relação com o objetivo do estudo.
O estudo segue uma abordagem metodológica de natureza qualitativa inscrita em um paradigma interpretativo, com a realização de estudos de caso de três professores do ensino secundário, sendo a recolha de dados realizada através de entrevistas semi-estruturadas, observações de aulas e recolha documental. A análise dos dados teve início ainda durante o processo de recolha de dados, ao que se seguiu um processo analítico que, por seu turno, se desdobrou em duas fases: uma primeira incidindo em cada caso objetivando a sua construção e uma segunda que considerou os três casos tendo em vista a construção das conclusões do estudo.
Quanto as perspetivas dos professores sobre a presença de tópicos de Cálculo Diferencial no ensino secundário, parece existir uma certa “naturalização” sobre tal presença, o que pode ser explicado pelo longo período de tempo que estes tópicos constam nos programas. No tocante às práticas, as conclusões do estudo apontam para um uso quase que exclusivo do exercício dos manuais sem usar outras tarefas. Outro ponto a evidenciar é o trabalho dos tópicos em contextos estritamente matemáticos sem uma interação com outras áreas do conhecimento e sem a aplicação destes conceitos em situações do mundo real (modelação). Contudo, mesmo nestes contextos estritamente matemáticos os professores dão uma ênfase significativa à conexão entre os diferentes conceitos envolvidos, buscando-se relacionar o conceito tratado com outros que já são do conhecimento dos alunos. Quanto à abordagem adotada pelos professores, esta parece estar mais focada em questões conceituais do que em técnicas procedimentais.
This study is situated in the area of research on the mathematics teacher, specifically of the teacher’s conceptions about the teaching of differential calculus in secondary education. The investigation is carried out under the assumption that such conceptions play a pivotal role in the way teachers interpret educational situations and also in the way they structure their actions in their teaching practice. Thus, the present investigation has also a second area of interest, namely the central aspects that characterize the teacher’s practice in teaching this theme. Thus, the aim of the study is to identify and understand the perspectives of mathematics teachers on the teaching of differential calculus in secondary education as well as the central aspects of their professional practice in their teaching. The theoretical framework is organized into three axes: (i) the conceptions of the mathematics teacher; (ii) the professional practices of the mathematics teacher and (iii) the teaching of differential calculus. The first axis discusses the interest and terminologies used in studies in the field of teacher conceptions, as well as the relationship between conceptions and knowledge and between these and teaching practice. The second axis is a discussion on the concept of teacher practice and, taking into account teaching practices, highlights tasks and communication in the classroom as structural elements of this practice. The third axis discusses the presence of differential calculus in the secondary school curriculum and the reformist efforts in teaching this theme and also presents some empirical studies on the teaching-learning of calculus that have a close relationship with the objective of the study. The study follows a qualitative methodological approach inscribed in an interpretive paradigm, with the case studies of three secondary school teachers, with data collection carried out through semi-structured interviews, class observations and document collection. Data analysis begun during the data collection process, followed by an analytical process, which, in turn, was divided into two phases: a first focusing on each case aiming at its construction and a second which considered the three cases with a view to building the study’s conclusions. As for the teachers’ perspectives on the presence of differential calculus topics in secondary education, there seems to be a certain “naturalization” about such presence, which can be explained by the long period of time that these topics are present in the programs. With regard to practices, the study’s conclusions point to an almost exclusive use of the exercises from manuals with no use of other tasks. Another point to highlight is the work of the topics in strictly mathematical contexts without an interaction with other areas of knowledge and without applying these concepts in real-world situations (modeling). However, even in these strictly mathematical contexts, a significant emphasis is placed on the connection between the different concepts involved, seeking to relate the concepts with others that are already known to students. As for the approach adopted by the teachers, it seems to be more focused on conceptual issues rather than on procedural techniques.
This study is situated in the area of research on the mathematics teacher, specifically of the teacher’s conceptions about the teaching of differential calculus in secondary education. The investigation is carried out under the assumption that such conceptions play a pivotal role in the way teachers interpret educational situations and also in the way they structure their actions in their teaching practice. Thus, the present investigation has also a second area of interest, namely the central aspects that characterize the teacher’s practice in teaching this theme. Thus, the aim of the study is to identify and understand the perspectives of mathematics teachers on the teaching of differential calculus in secondary education as well as the central aspects of their professional practice in their teaching. The theoretical framework is organized into three axes: (i) the conceptions of the mathematics teacher; (ii) the professional practices of the mathematics teacher and (iii) the teaching of differential calculus. The first axis discusses the interest and terminologies used in studies in the field of teacher conceptions, as well as the relationship between conceptions and knowledge and between these and teaching practice. The second axis is a discussion on the concept of teacher practice and, taking into account teaching practices, highlights tasks and communication in the classroom as structural elements of this practice. The third axis discusses the presence of differential calculus in the secondary school curriculum and the reformist efforts in teaching this theme and also presents some empirical studies on the teaching-learning of calculus that have a close relationship with the objective of the study. The study follows a qualitative methodological approach inscribed in an interpretive paradigm, with the case studies of three secondary school teachers, with data collection carried out through semi-structured interviews, class observations and document collection. Data analysis begun during the data collection process, followed by an analytical process, which, in turn, was divided into two phases: a first focusing on each case aiming at its construction and a second which considered the three cases with a view to building the study’s conclusions. As for the teachers’ perspectives on the presence of differential calculus topics in secondary education, there seems to be a certain “naturalization” about such presence, which can be explained by the long period of time that these topics are present in the programs. With regard to practices, the study’s conclusions point to an almost exclusive use of the exercises from manuals with no use of other tasks. Another point to highlight is the work of the topics in strictly mathematical contexts without an interaction with other areas of knowledge and without applying these concepts in real-world situations (modeling). However, even in these strictly mathematical contexts, a significant emphasis is placed on the connection between the different concepts involved, seeking to relate the concepts with others that are already known to students. As for the approach adopted by the teachers, it seems to be more focused on conceptual issues rather than on procedural techniques.
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Palavras-chave
Conceções do professor prática do professor ensino do Cálculo Diferencial Teachers’ conceptions teachers’ practice teaching Differential Calculus