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Avaliação reguladora no Ensino Secundário processos usados pelos alunos em investigações matemáticas
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O presente estudo teve por objectivo estudar os processos que os alunos, do ensino secundário, usam como forma de dar resposta às dificuldades, obstáculos e raciocínios erróneos com que se confrontam nas investigações matemáticas. Acredito que este conhecimento é uma forma de contribuir para a concretização de uma avaliação reguladora. Assim, pretendi conhecer os processos de interpretação, os processos de investigação, as dificuldades, os erros e a sua função no desenvolvimento da actividade, as interacções e outros recursos que os alunos do ensino secundário utilizam quando se confrontam com a necessidade de desenvolver uma investigação matemática, num contexto de sala de aula.
Para a concretização deste trabalho foi adoptada a metodologia qualitativa e foram estudados quatro alunos do ensino secundário (estudos de caso). Os alunos foram confrontados com a necessidade de desenvolver quatro investigações matemáticas, duas de geometria e duas de funções. Os dados foram recolhidos através da gravação, da observação, do diário de bordo, da entrevista semi-estruturada, do relatório e da reflexão escrita produzidos pelos alunos sobre o trabalho desenvolvido.
A análise dos dados recaiu sobre a actividade de investigação e avaliação dos alunos, nomeadamente sobre a interpretação da tarefa, o desenvolvimento da tarefa e a auto-avaliação do trabalho e a sua compreensão à luz das perspectivas manifestadas pelo aluno, face ao processo de ensino e aprendizagem.
Relativamente aos resultados, são identificados e caracterizados os processos que os alunos usam no primeiro confronto e no desenvolvimento de uma investigação matemática, os recursos, os erros e a reflexão. Perante a necessidade de definir um percurso de investigação, os alunos usam processos como traduzir, experimentar inicial, interiorizar e ancorar. Durante a investigação os processos identificados são: aprofundar, ziguezaguear, conjecturar, experimentar e solucionar e conjecturar, experimentar e generalizar. Ao nível dos recursos, os alunos usam as interacções com o professor para ultrapassar dificuldades, para reflectir e para validar o seu trabalho, as interacções com os seus pares para ultrapassar dificuldades, para identificar erros e para identificar semelhanças e diferenças no trabalho realizado e outros recursos, como é o caso da interacção crítica consigo próprio e de materiais pedagógicos ou outros. A função que os alunos atribuem aos raciocínios erróneos que cometem situa-se no orientar, ensinar e desviar. Quanto às reflexões, os alunos reflectem sobre as interacções que estabelecem no seio da actividade investigativa, os erros e dificuldades surgidas e a apreciação global do trabalho realizado (auto-avaliação).
Apesar deste estudo depender da escolha dos alunos e das experiências situadas que os mesmos têm da vivência da escola, acredita-se que os resultados encontrados são a realidade de muitos alunos na aprendizagem da matemática. Recomenda-se, por isso, a continuação do estudo da problemática da avaliação, principalmente na vertente da harmonização entre a avaliação e o currículo.
The present study intended to deepen the processes that the students, in the secondary school, use as form of giving answer to the difficulties, obstacles and wrong reasoning with that the students are confronted in the mathematical investigation. I believe that this knowledge is a form of contributing for the materialization of a formative evaluation. Like this, I intended to know the interpretation processes, the investigation processes, the difficulties, the mistakes and the function that have in the development of a work, the relationships and other resources that seek in the seconadry school wthen they are confronted with the need of developing a mathematical investigation, in a classroom context. For the materialization of this work the qualitative methodology was used and four students of the secondary school were studied (case studies). The students were confronted with the need of developing four mathematical investigations, two of geometry and two of functions. The data were collected through the recording, of the observation, of the log book, of the semi-structured interview, of the report ando f the reflection written produced by the students on the developed work. The analysis of the data relapsed about the investigation work and the evaluation of the students’ learning, namely on the interpretation of the task, the development of the task and the solemnity-evaluation of the work and his/her understanding to the light of the perspectives manifested by the student, face to the teaching process and learning. Relatively to the results, they are identified and characterized the processes tha the students use in the first confrontation and in the development of a mathematical investigation, the resources, the mistakes and the reflection. Before the need to define and investigation course, the students use processes how to translate, to try initial, to internalize and to anchor. During the investigation the identified processes are: to deepen to zigzag, to conjecture, to try and to solve and to conjecture, to try and to generalize. At the level of the recources, the students use the help with the teacher to cross difficulties, to rethink and to validate the work, the relationship with their pairs to cross difficulties, to identify mistakes and to identify similarities and diferences in the accomplished work and other resources, as it is the case of the solemnity-critic with itself and of materials pedagogic or other. The function that the students atribute to the wrong reasoning that commit locates in guiding, to teach and to deviate. As for reverse-setting out the accomplished work, the students think on the relationships that establish in the breast of the investigation work, the mistakes and appeared difficulties and the global appreciation of the accomplished work (solemnity-evaluation). In spite of this study to depend on the students’ choice and of the located experiences that the same ones have of the existence of the school, it is believed that the found results are the many students’ reality in the learning of the mathematics. It is recommended, for that, the continuation of the study of the problem of the evaluation, mainly in the slope of the harmonization between the evaluation and the curriculum.
The present study intended to deepen the processes that the students, in the secondary school, use as form of giving answer to the difficulties, obstacles and wrong reasoning with that the students are confronted in the mathematical investigation. I believe that this knowledge is a form of contributing for the materialization of a formative evaluation. Like this, I intended to know the interpretation processes, the investigation processes, the difficulties, the mistakes and the function that have in the development of a work, the relationships and other resources that seek in the seconadry school wthen they are confronted with the need of developing a mathematical investigation, in a classroom context. For the materialization of this work the qualitative methodology was used and four students of the secondary school were studied (case studies). The students were confronted with the need of developing four mathematical investigations, two of geometry and two of functions. The data were collected through the recording, of the observation, of the log book, of the semi-structured interview, of the report ando f the reflection written produced by the students on the developed work. The analysis of the data relapsed about the investigation work and the evaluation of the students’ learning, namely on the interpretation of the task, the development of the task and the solemnity-evaluation of the work and his/her understanding to the light of the perspectives manifested by the student, face to the teaching process and learning. Relatively to the results, they are identified and characterized the processes tha the students use in the first confrontation and in the development of a mathematical investigation, the resources, the mistakes and the reflection. Before the need to define and investigation course, the students use processes how to translate, to try initial, to internalize and to anchor. During the investigation the identified processes are: to deepen to zigzag, to conjecture, to try and to solve and to conjecture, to try and to generalize. At the level of the recources, the students use the help with the teacher to cross difficulties, to rethink and to validate the work, the relationship with their pairs to cross difficulties, to identify mistakes and to identify similarities and diferences in the accomplished work and other resources, as it is the case of the solemnity-critic with itself and of materials pedagogic or other. The function that the students atribute to the wrong reasoning that commit locates in guiding, to teach and to deviate. As for reverse-setting out the accomplished work, the students think on the relationships that establish in the breast of the investigation work, the mistakes and appeared difficulties and the global appreciation of the accomplished work (solemnity-evaluation). In spite of this study to depend on the students’ choice and of the located experiences that the same ones have of the existence of the school, it is believed that the found results are the many students’ reality in the learning of the mathematics. It is recommended, for that, the continuation of the study of the problem of the evaluation, mainly in the slope of the harmonization between the evaluation and the curriculum.
Descrição
Tese de mestrado em Educação (Didáctica da Matemática), apresentada à Universidade de Lisboa através da Faculdade de Ciências, 2005
Palavras-chave
Interacção na aula Investigação matemática Auto-regulação Aprendizagem da matemática Teses de mestrado - 2005