Multiple Representations Help Teachers and Students Understand a Geometry Problem

This narrative account begins in a high school classroom as we describe how students were mostly unengaged with a "Problem of the Week." As observers in this setting, we sat in the back of the classroom and attempted to solve the problem: Choose any three vertices of a cube at random.What is the probability that any three vertices will form a right triangle? Because of our different answers to the problem and the struggles we experienced as we attempted to visualize a cube with triangles on the faces and in the interior space we later created concrete and virtual manipulatives. Additionally, we posed this problem in a mathematics methods course with preservice high school teachers and then discussed the use of enactive (concrete), iconic (pictorial), and symbolic representations (Bruner, 1966). The significance of using concrete manipulatives for some mathematics problems cannot be overstated