From figural to theoretical control (and back): a first proposal for framing the interplay between different controls

Grounding on Fischbein’s studies, our work is intended to deeply investigate the systems of control which influence the dialectic between the figural and the conceptual component of figural concepts. We propose a first attempt for characterizing the systems of control by introducing a fine-grained distinction between three dimensions: visual, spatial, and theoretical. This theoretical frame is used for analyzing some excerpts from a case study which involve graduated students in Mathematics, while solving a geometrical task concerning the reciprocal position and the intersection point of cube diagonals. Through the analyses of these excerpts, we show evidence of the three types of control and of their interplay. Finally, theoretical and educational implications of the study are discussed.