1 Investigating mathematical investigations1

There are good reasons why we may involve the students in doing mathematics investigations. Recent curricula encourage this sort of activity but we notice that its application in the classroom is not a simple matter. We discuss the issues that arise when students are presented with investigative tasks, with special interest in the dynamics of the classroom and in the role of the teacher. Our aim is to derive suggestions for classroom practice as well as for further research and teacher development. Why mathematical investigations in the classroom? New perspectives on the philosophy of mathematics, stressing the consideration of mathematical activity, brought attention to the investigative processes involved in the creation of new mathematical knowledge (Davis and Hersh, 1980; Ernest, 1991; Lakatos, 1976; Tymoczkco, 1986). However, the idea is not new. It may be seen behind P—lya’s proposal to make problem solving an essential element of students’ mathematical experience: “mathematics has two faces; it is the rigorous science of Euclid but it is also something else (...) mathematics in the making appears as a