6: Drag, Drag, Drag: The Impact of Dragging on the Formulation of Conjectures within Interactive Geometry Environments¹

Learning to formulate conjectures is a complex process that typically involves stating an initial conjecture and then revising or refining it. As teacher educators, we need to provide experiences in which teachers can learn to formulate conjectures. This is important for two reasons. First, conjecturing is a critical component of doing mathematics at all educational levels, including teacher education (National Council of Teachers of Mathematics [NCTM], 2000; Polya, 1954). Second, teachers need to know how to formulate conjectures in order to design learning environments in which their students formulate conjectures as part of doing mathematics.

Geometry provides opportunities to engage students in formulating conjectures, facilitated by Interactive Geometry Environments (IGEs), such as Cabri Geometry II (Texas Instruments, 1998) and The Geometer’s Sketchpad^® (Jackiw, 2001). IGEs free the user from performing time-consuming activities such as computations, constructions, and repetitive tasks using paper-and-pencil procedures. More strikingly, they enable the user to manipulate or move (i.e., drag) elements of geometric configurations dynamically while preserving invariant mathematical relationships. However, we have noticed that both preservice and inservice mathematics teachers do not always take full advantage of the construction and dragging capabilities of IGEs, resulting in formulation of conjectures that often fail to include needed restrictions, boundaries, or all plausible instances for which they may be true.