Helena Rocha

Neste texto apresentamos os jogos no ensino da matemática como uma forma de aprendizagem de conteúdos e não apenas como um recurso que cada professor pode usar nas suas aulas para tornar a aula diferente. Analisamos dois jogos desenvolvidos por nós e que utilizámos com alunos dos 7.º e 10.º anos de escolaridade, procurando não só apresentar os jogos, mas também aspetos da sua implementação em sala de aula, ponderando o contributo que trouxeram à aprendizagem dos alunos.
Aprender matemática depende de um grande número de variáveis, o que torna o ensino um processo complexo, pois é necessário que se desenvolva o raciocínio lógico, além de estimular o desenvolvimento das mais variadas capacidades transversais, tais como o pensamento autónomo, a criatividade, o sentido de estratégia e a capacidade de resolver problemas.
Duas das dificuldades frequentemente encontradas pelos professores passam pela falta de motivação para a aprendizagem e pelo desinteresse pela Matemática. A solução para estes problemas pode passar pela utilização de jogos para complementar o estudo, mas também para a aquisição de novos conteúdos. No entanto, apenas a implementação dos jogos não basta. O papel do professor é de extrema importância e a planificação e orientação da aula são fundamentais para que se alcancem os objetivos pretendidos.

This study aims to understand the functional thinking of 10th-grade students while studying functions. Specifically, we intend to answer the following research questions: what are the functional thinking processes used by 10th-grade students when studying functions? What difficulties do students present while learning functions? In view of the nature of this research objective, we adopted a qualitative and interpretative approach. In order to answer these questions, data were collected from the written records produced by the students while solving the proposed tasks, from records of the oral interactions during discussions and from a questionnaire. The results show that functional thinking processes were implicit in the resolution of the tasks proposed to the students. The students expressed an understanding of how the variables were related, presenting evidence of their functional thinking while working on the new concepts represented by the functions addressed in the proposed tasks. Some students expressed difficulties in interpreting the different types of representations associated with the functions, in retaining the necessary information from a graphical representation that would help them to draw conclusions and establish correspondences, in explaining functional relationships, and in interpreting the information provided by algebraic expressions. These difficulties can reduce the recognition of the relationships between variables and their behavior in the different representations, becoming an obstacle to learning for some students.

Apesar da escrita ter, habitualmente, uma maior expressão no ensino da Matemática que a própria oralidade, os alunos não estão habituados a explicitar raciocínios e a utilizar linguagem matemática apropriada. A comunicação matemática escrita tem algumas particularidades que podem ser diretamente trabalhadas com os alunos. Por exemplo, a escrita ajuda os alunos a dar sentido à Matemática e a melhorar o próprio discurso. As produções dos alunos transportam informações para o professor contribuindo para a planificação e concretização da sua prática profissional. Assim, e apesar de frequentemente ser descurada, a escrita matemática pode ser trabalhada na sala de aula, em particular, com futuros professores. Este artigo reporta parte de uma experiência realizada com uma turma da Licenciatura em Educação Básica, tendo por base a resolução em grupo de um problema de Geometria e o registo escrito do processo de resolução elaborado pelos alunos. Pretendeu-se desta forma caraterizar a comunicação escrita dos alunos e identificar contributos desta para a compreensão por parte do professor dos conhecimentos dos alunos. A análise da escrita matemática dos alunos, tendo por base um conjunto de critérios previamente definidos, permitiu identificar a preferência destes pelo recurso à representação verbal, dificuldades em fundamentar adequadamente as respostas apresentadas e uma forte tendência para desvalorizar as abordagens prévias que não conduziram à resposta ao problema. Permitiu ainda identificar uma tendência para não explicitar o entendimento das questões que lhes eram colocadas. A forma como os conceitos matemáticos surgem nas repostas escritas permite identificar aspetos relevantes do conhecimento dos alunos.

Games are commonly appointed as a methodological tool capable of promoting students’ effective learning. In this context, this study intends to analyze the impact of mathematical discussions developed while
playing a polynomial game. Namely it intends to analyze the impact on the consolidation of mathematical concepts previously worked in the classroom and on the communications skills. Two case studies where developed involving 10th grade students. Data gathering was based on direct observation and an inquiry. The main conclusions suggest that the game encouraged the discussion about the mathematical contents and therefore promoted the development of the mathematical discourse. Besides that, it allowed a deeper apprehension of mathematical concepts, and the overcome of some difficulties.

This chapter aims to make more explicit the grounded or ‘pragmatic theories’ that inform the design of mathematics teachers’ professional development (PD) to exploit technological affordances. It uses aspects of some representative projects that took place in four countries (Colombia, Italy, Mexico, and Portugal) to illustrate lessons learned (e.g., similarities and differences, barriers and opportunities) and provide important insights to inform future PD implementations. To do this, we have identified a set of aspects (and sub-aspects) that emerged in relation to five major themes and reveal our ‘pragmatic theories’ alongside a consideration of the interconnections between these aspects. Our contribution offers a methodological frame to support future PD designs for teachers of mathematics concerning digital technology uses.

Deductive reasoning, being central in mathematics, is also usually a source of difficulties for students, more used to the empirical approaches. In this study we focus on mathematical proof and we try to give attention to how this kind of reasoning is envisaged by the students, to the options they assume when asked to develop a deductive reasoning and to the factors affecting the implementation of this kind of reasoning. The study follows a qualitative and interpretative methodological approach, including the completion of two case studies of students of the 10th grade. Data were collected in work sessions and through interviews. The main findings point to a devaluation of mathematical proof and a strong preference for empirical approaches. Yet students show ability to develop different approaches. The preference for the mathematical subject and the attention given in class to the deduction work, appears to be relevant factors when considering the students' ability to develop a deductive reasoning when involved on a mathematical proof.

For most teachers, the use of games is not a common practice. This limited professional experience forces the teacher to reflect on the use of games, on how to integrate them into class work, and on how to enhance the development of learning based on them. Our objective is to analyse a set of game scenarios and study aspects that characterise each of them, relating these characteristics to aspects of the professional knowledge to teach with technology. The study adopts an interpretative and qualitative methodology that seeks to understand the relationship between teachers' choices and their professional knowledge. The main findings suggest that the different characteristics of the game scenarios imply the creation of different working environments in the classroom; enhance the emergence of different student difficulties, requiring different ways to support their work; allowing/promoting different conceptual approaches; and the work with different representations. These differences between the game scenarios point to different emphases on the teaching and learning knowledge and the chemistry knowledge, suggesting different mobilizations of the teacher's professional knowledge.

This article focuses the choice and use of examples by two students of the 11th grade to prove or refute a set of statements. The use of representations of sequences and functions is also considered. The study adopts a qualitative approach and data were collected by interviews and documental gathering. The conclusions suggest most of the examples used were well-known sequences or functions. However, the students sought different purposes for the use of examples, such as understanding the conjecture, demonstrate the falsity or truthfulness of the statement and conveying a general argument. The students made a satisfactory articulation between the various types of representations but relied mostly in the cartesian graph.

Through writing, students express many of their processes and ways of thinking. Since at high school level some of the activities are carried out with the graphing calculator, we intend to investigate the contribution of this resource to promote the mathematical writing in the learning of continuous nonlinear models at 11th grade. Adopting a qualitative methodology, we collected and analyzed the students’ writing productions. What they write when using the calculator gives evidence about the information valued (when they sketch graphics without any justification); about the strategies used (when they define the viewing window and relate different menus on the graphing calculator); and about the reasoning developed (when they justify the information given by the calculator and the formulation of generalizations and conjectures validation).