Editor’S Section Commentary

In the preceding four chapters, the authors present us with interesting case studies of people dealing with transformations of various kinds. Edwards describes for us what happens when teenaged students engage with a computer program that allows them to enact geometrical transformations on a two-dimensional plane. The transformations include the commands “slide,” “rotate,” and “reflect” that the students use as part of tasks or “games.” The author notes that unlike mathematicians, who see geometric transformations as mappings of the plane onto itself, students understand what they do as moving objects about the surface. From a purely experiential perspective, given the resources that the students have (the commands slide, rotate, and reflect), we might have expected students—if they consciously thought about their actions at all—to experience what they are doing as moving the given objects using the given commands. Perceptually, the do not map the plane onto itself, for experientially, the plane remains constant; perceptually, they move the objects. If the idea is to allow students to map a plane onto itself and allow students to draw on their embodied experience of the world, the mapping process itself may have to become more salient in the task. For example, a teacher or researcher may provide both source and target domains in different layers, then ask students to transform one layer so that it is identical to the other. In this case, not only the object but also the domain as a whole undergoes the mapping (onto itself). In this case, the embodied experience in and with the world mediates access to the mathematical understanding of geometrical transformation.