Dragging as a conceptual tool in dynamic geometry environments

Some ‘drag-to-fit’ solutions given by student teachers to three geometric construction problems in a dynamic geometry environment (DGE) are analysed. The responses of a group of experienced mathematics teachers to the question whether or not such solutions can be considered ‘legitimate’ are then discussed. This raises fundamental questions concerning the concept of legitimacy, the relationship between DGEs and Formal Axiomatic Euclidean Geometry, the nature of ‘conceptual tools’ in different geometric environments, and the functions of dragging in DGEs. The authors argue that, if dragging is viewed as a conceptual tool, then certain drag-to-fit solutions, although soft constructions, may still be considered as conceptually legitimate and therefore valid. Finally, some important questions are raised concerning the impact that teachers’ different attitudes towards legitimacy might have on students’ learning through DGEs.