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

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

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

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

Semblanzas Ictiológicas: Matías Pandolfi

través de esta serie intentaremos conocer diferentes facetas personales de los integrantes de nuestra “comunidad”. El cuestionario, además de su principal objetivo, con sus respuestas quizás nos ayude a encontrar entre nosotros puntos en común que vayan más allá de nuestros temas de trabajo y sea un aporte a futuros estudios históricos. Esperamos que esta iniciativa pueda ser otro nexo entre los ictiólogos de la región,ya que consideramos que el resultado general trascendería nuestras fronteras