A PROJECT IN EUCLIDEAN GEOMETRY

One of the most effective instructional approaches in teaching Mathematics is project work, which, in an earlier paper, I connected with active learning, see Klaoudatos (1998). And this approach is going to be more interesting for the students if the project has been developed in collaboration with them. At the same time, these kinds of projects include ‘dangers ’ for the teacher because of unexpected demands that might be found within. In this paper, I will describe such a project, which had been created in a problem solving class during the year 2000-2001 first semester. Through successive generalizations of a simple geometric task, the students developed the following problem: ‘In an ABC triangle, D is a point on BC from which we construct segments that form equal angles at the sides AB, AC, at the points I, K respectively. Which is the position of D so that the length of IK will be minimum?’, see figure. The problem is expressed in terms of classical Euclidean geometry so that, at first glance, there is no evidence of the hidden difficulties. The aim of the paper is twofold: first, to attract the attention of mathematics teachers and educators to the problem above, in order to be considered for project work, especially in teacher training courses. Second, to search for conditions under which the project could be effective