Producing Conjectures Using Maintaining Dragging: Instrumented Abduction

The model of the "maintaining dragging scheme" seems appropriate for describing the processes of conjecture-generation when MD is used, providing evidence to a correlation between the introduction of the dragging schemes, and MD in particular, and a specific new (with respect to those in literature) cognitive process described by the model. We have referred to such process as a form of instrumented abduction, a new notion that we hope can be generalized to other contexts in which abduction is supported by another instrument. We seem to have captured the key idea which may lead to complete appropriation of the MDS,and described how it resides at a meta-level with respect to each specific exploration in whichMD is exploited as an instrument for conjecture-generation. We described how for expert solvers the MDS might be transformed into a way of thinking. In this sense it may lead to the construction of fruitful “mathematical habits of mind” (Cuoco, 2008) that may be in various mathematical explorations leading to the generation of conjectures. When such way of thinking is developed, the abductive reasoning has the advantage of involving geometrical concepts, like in the case of Francesco and Gianni. The geometrical concepts that emerge in this case can become “bridging elements” with respect to the proving phase, since they can be re-elaborated into the deductive steps of a proof. On the other hand, expert use of the MDS supported byMD seems to lead to conjectures in which no geometrical elements arise to “bridge the gap” between the premise and the conclusion. In other words, although expert use of MD seems to offer the possibility of generating “powerful” conjectures that solvers might have trouble reaching without support of the dragging instrument (since the IOD which becomes the premise may be cognitively “quite distant” from the conclusion), generating conjectures “mechanically” through the MDS associated to the dragging instrument, may hinder the proving phase in which these “bridging elements” are essential.