Investigating mathematics and learning to teach mathematics

This paper deals with an idea that plays an increasing role in teaching and in teacher education—investigating as a powerful paradigm of knowledge construction. Investigations may be carried out both in learning mathematics and in learning how to teach mathematics at preservice and inservice levels. I look into investigations in mathematics and in the mathematics curriculum, pointing out some issues that teachers face proposing them in the classroom. Then, I discuss teacher education and professional development, stressing the value of investigations about practice as a means of developing knowledge. I conclude with examples of work done by preservice and inservice teachers and by teams of teachers and researchers focusing on pupils’ investigative work in mathematics classes that illustrate the educational value of this activity and discuss the roles of the teacher.Este artigo baseia-se numa ideia que desempenha um papel crescente no ensino e na formação de professores – investigar constitui um paradigma poderoso de construção do conhecimento. Tanto podem ser realizadas investigações no ensino da Matemática como na formação inicial e contínua do professor de Matemática. Assim, analiso o papel das investigações em Matemática e no currículo de Matemática, apontando algumas questões que os professores enfrentam quando as propõem na sala de aula. De seguida, discuto a formação de professores e o desenvolvimento profissional, dando ênfase ao valor das investigações sobre a prática como meio de desenvolver novo conhecimento. Concluo com exemplos de trabalho realizado por professores em formação inicial e contínua e por equipas de professores e investigadores que se centram no trabalho investigativo dos alunos realizado nas aulas de Matemática, exemplos esses que ilustram o valor educacional desta actividade e permitem discutir os papéis do professor

What is an expert mathematics teacher?

In Portugal, the introduction of new mathematics curriculum for basic education generated new images about the activity of the mathematics classroom and about the role of the teacher, based on the notions of “explorations” and “discussion”. This had strong implications to the perspective on teacher expertise accepted by teachers and also by researchers in this country. This paper provides a brief overview of the national context, in terms of curriculum and teacher education, and describes this perspective of mathematics teachers’ expertise

Investigations and explorations in the mathematics classroom

In Portugal, since the beginning of the 1990s, problem solving became increasingly identified with mathematical explorations and investigations. A number of research studies have been conducted, focusing on students’ learning, teachers’ classroom practices and teacher education. Currently, this line of work involves studies from primary school to university mathematics. This perspective impacted the mathematics curriculum documents that explicitly recommend teachers to propose mathematics investigations in their classrooms. On national meetings, many teachers report experiences involving students’ doing investigations and indicate to use regularly such tasks in their practice. However, this still appears to be a marginal activity in most mathematics classes, especially when there is pressure for preparation for external examinations (at grades 9 and 12). International assessments such as PISA and national assessments (at grades 4 and 6) emphasize tasks with realistic contexts. They reinforce the view that mathematics tasks must be varied beyond simple computational exercises or intricate abstract problems but they do not support the notion of extended explorations. Future developments will show what paths will emerge from these contradictions between promising research and classroom reports, curriculum orientations, professional experience, and assessment frameworks and instruments

External, internal and collaborative theories of mathematics teacher education

Theories about mathematics teacher education depend critically on how one responds to questions such as: What is the nature of teacher knowledge? And how is such knowledge to be appraised? By the external views of the researcher, by the internal views of the teacher group, or by any other way? The external perspective of the researcher is able to mobilize important theoretical and empirical tools, however, this external perspective is often unable to grasp essential elements of the complexity of teacher knowledge, practice, and identity. Just capturing the views of teachers brings with it the concern that these views are often contradictory, assuming different values, orientations, and agendas from one teacher group to another, depending on grade level, school system, country, world region, and so forth. Another approach is to combine the experiences and perspectives of teachers and researchers. This paper explores the question of relevant theories in mathematics teacher education from all three perspectives

The history of the concept of function and some educational implications

Several fields of mathematics deal directly or indirectly with functions: mathematical analysis considers functions of one, two, or n variables, studying their properties as well as those of their derivatives; the theories of differential and integral equations aim at solving equations in which the unknowns are functions; functional analysis works with spaces made up of functions; and numerical analysis studies the processes of controlling the errors in the evaluation of all different kinds of functions. Other fields of mathematics deal with concepts that constitute generalizations or outgrowths of the notion of function; for example, algebra considers operations and relations, and mathematical logic studies recursive functions. It has long been argued that functions should constitute a fundamental concept in secondary school mathematics (Klein, 1908/1945) and the most recent curriculum orientations clearly emphasize the importance of functions (National Council of Teachers of Mathematics, 1989). Depending on the dominant mathematical viewpoint, the notion of function can be regarded in a number of different ways, each with different educational implications. This paper reviews some of the more salient aspects of the history of the concept of function,1 looks at its relationship with other sciences, and discusses its use in the study of real world situations. Finally, the problem of a didactical approach is considered, giving special attention to the nature of the working concept underlying the activities of students and the role of different forms of representation

Mathematics teachers’ professional knowledge

This paper addresses the study of teachers’ knowledge, beliefs, conceptions and practices, presenting some illustrations from the area of problem solving. In mathematics education, the teacher has attracted much less work than the student. This may be due, in part, to the different knowledge base of interest in each case. Regarding students, we are concerned with their learning of mathematics. The nature of mathematical knowledge is itself problematic, yet that does not seem to raise too many difficulties for our work. Regarding teachers, it is much less clear what is the specific knowledge (nec-essary for teaching mathematics) that we should be looking at. Is it knowledge of math-ematics content? Of mathematics pedagogy? Of students’ cognitive processes? Some mixture of several of these? In the first part of the paper I will briefly review work done on teachers’ professional knowledge and related concepts within and outside PME. Then, I will present cases taken from empirical research and discuss a few concepts used in our investigations. And in the final part I will contrast some general frameworks to study mathematics teachers’ professional knowledge and draw some perspectives for future work

Mathematics teachers’ professional development and identity in a distance education setting

This paper discusses the influence of an in-service distance education course in the construction of mathematics teachers’ professional identity, especially regarding their views and practices of reflection and collaboration and their relation with information and communication technology. The course was based in open-learning pedagogy and focused on conducting exploratory and investigative work in the mathematics classroom. Evaluation results show that the perspectives and involvement of the participant teachers depend very much on their previous professional experience and relationship with the Internet. Teachers that use e-mail for collaborative work found this a very stimulating experience whereas those with less professional involvement had some difficulty in assuming the roles and values required for this kind of activity

Research and practice: Bridging the gap or changing the focus?

Bridging the gulf that tends to persist between research in mathematics education and mathematics teaching practice is a timely issue. This comment addresses the impact of research not only on teachers’ practices and the curriculum, but also on students’ practices, teacher education practices, the educational market, and the society at large. It argues that for research to bring about changes in mathematics teaching and learning we need to act at a systemic level. It also argues that if we want to have a real influence on practice, we need to see that as a problem on itself. It concludes indicating that our con-fidence in the power of research to understand phenomena and intervene in practice must be combined with an attitude of social responsiveness, working closely with different social partners and being critical and reflective about what we do.Tapar o fosso que tende a persistir entre a investigação na educação matemática e a prática de ensino é uma questão urgente. Este comentário debruça-se sobre o impacto da investigação não apenas nas práticas de ensino dos professores e no currículo, mas também nas práticas dos alunos, nas práticas de formação de professores, no mercado educacional, e na sociedade em geral. Argumenta que, para que a investigação traga mudanças no ensino e na aprendizagem da Matemática, é necessário agirmos ao nível sistémico. Também argumenta que, se quisermos ter uma influência real na prática, precisamos de ver que isso constitui um problema em si mesmo. O artigo conclui indicando que a nossa confiança no poder da investigação para compreender os fenómenos e intervir na prática deve ser combinado com uma atitude de responsabilidade social, trabalhando estreitamente com diferentes parceiros sociais e sendo críticos e reflexivos em relação ao nosso próprio trabalho

Proportion in school mathematics textbooks: A comparative study

This paper analyses how proportion is introduced and developed in selected mathematics textbooks for middle school students of Portugal, Spain, Brazil, and USA. The analysis focuses on the nature of the approach and on the cognitive demand, structure, and context of the tasks. The results show that the textbooks tend to present tasks at an intermediate level of cognitive demand and with a closed structure. Nonmathematical contexts predominate in three of the four textbooks. However, there are marked differences in the way textbooks approach the conceptual and procedural aspects of proportion. The way the students are addressed also varies, ranging from a questioning/problem solving style, to an explaining/practicing style, each of these styles supporting a rather different kind of activity.Este artigo analisa como proporção é introduzida e desenvolvida nos livros didáticos matemática para alunos do ensino médio (Ensino Fundamental II) em Portugal, Espanha, Brasil e EUA. A análise incide sobre a natureza da abordagem e na demanda cognitiva, estrutura e contexto das tarefas. Os resultados mostram que os livros didáticos tendem a apresentar as tarefas em um nível intermediário de demanda cognitiva e com uma estrutura fechada. Contextos não-matemáticos predominam em três dos quatro livros didáticos analisados. No entanto, há diferenças marcantes no modo como os livros abordam os aspectos conceituais e procedimentais de proporção. A forma didática de tratar o assunto também varia, indo de um estilo de questionamento / resolução de problemas, a um modo explicativo/prático; cada um deles ampara-se em um tipo diferente de atividade

Analysing practice in preservice mathematics teacher education

This paper presents the case of a yearlong course based on fieldwork activities provided to secondary school mathematics preservice teachers, just before their student teaching practicum. The activities are a first experience in investigating professional practice—some concern the school as a whole, while others focus on the mathematics class. Using a qualitative and collaborative methodology, we discuss the implications of this work for preservice teachers’ education. We argue that such fieldwork activities may help prospective teachers in developing a professional discourse and in assuming a professional identity, acquiring new ways of expressing new educational ideas and assuming a new point of view about educational phenomena