Maintaining dragging in a dynamic geometry environment: the interplay between a psychological model and task design

In a dynamic geometry environment (DGE) maintaining dragging (MD) consists of identifying a geometrical property that a constructed figure could have and inducing the invariance of such a property through dragging. MD can be used as a powerful tool to generate conjectures in open problems in Euclidean Geometry. A model describing psychological aspects associated to its use has been described in detail as the result of qualitative studies carried out over the last decade. In this contribution, I highlight key features of this model, focusing on how it describes a process of abductive reasoning, and discuss relationships between it and the design of tasks for students learning Euclidean Geometry. Specifically, the model shows how conjecture generation through MD “offloads” theoretical control onto the DGE, leaving a “theoretical gap” for the student between the premise and the conclusion of the conjecture. This hinders the proving process when it comes to proving the conjecture generated through MD. However, we have found that students who know how to use MD to generate conjectures, but who are not able to successfully perform MD on a specific dynamic figure, can use MD as a psychological tool and generate a conjecture as if they had used the tool physically. Interestingly, this process leads to a conjecture in which the “theoretical gap” is now filled, and a proof can be reached quickly by reorganizing the theoretical pieces discovered during the process. I will conclude discussing implications of these qualitative findings for the design of educational tasks, specifically in relation to the questions: Should we teach maintaining dragging and foster students’ use of MD for conjecture generation? If so, with which kinds of tasks? With what caveat?