The role of geometrical figures in geometrical thinking: making students blind or explorers?

Presented by Athanasios Gagatsis, University of Cyprus

The way of “looking” at any geometrical figure is a crucial cognitive factor in reasoning and proving in geometry. There are many factors that can inhibit or favor discriminating the proper way of looking at geometrical figures. Our research was organized mainly in reference to the distinction between four kinds of figural apprehension as proposed by Duval (1988) within the register of geometrical visualization: perceptual apprehension, sequential apprehension, operative apprehension and discursive apprehension.

The general aim of this research study was to investigate the structure and the cognitive processes of the geometrical figure apprehension of the lower and the upper secondary school students. In particular, the students’ perceptual, operative, sequential and discursive apprehensions were explored. The data were collected using a test comprising of 16 geometrical tasks, which was administered to 881 students, aged 15 to 17, of lower (Grade 9) and upper (Grade 10, Grade 11) urban and rural secondary schools, in Cyprus.

A structural model was constructed and verified, which determined the importance of the perceptual, the operative, the sequential and the discursive apprehension for the apprehension of a geometrical figure. The structure of the geometrical figure apprehension was found invariant for the students of the two educational levels and of each grade. Interrelations were traced between the different types of apprehension. Strong relations were found between the operative and the discursive apprehension, revealing the importance of visualization in geometrical reasoning. Furthermore, the role of the perceptual apprehension occurred very important for the mobilization of the discursive apprehension and the operative apprehension.