Concepts and figures in geometrical reasoning

The paper opens with the theoretical construct of figural concepts. It is argued that geometrical figures are characterized by both conceptual and sensorial properties. A geometrical figure is a mental abstract, ideal entity, the meaning of which is governed by a definition. At the same time, it is an image: it possesses extensiveness (spatiality), shape and magnitude. In geometrical reasoning the two categories of properties should merge absolutely, with the sensorial components providing the dynamics of invention and the conceptual component guaranteeing the logical course of the mathematical process.

The paper investigates the effects of interaction between the conceptual and the figural components of a number of geometrical figures under the impact of age and mathematical competence. It has been found that age does not improve the control of the conceptual component on the interpretation of figures (between ages 14 and 17) in contrast to what one could expect. Usually, as an effect of age (grade) the interpretation gets worse. On the other hand, there are striking differences between the three competence levels considered. Strong mathematical students are characterized by their capacity to overcome the apparent contradiction between the figural, direct, impressions and the definitory properties. Their decision is, ultimately, determined according to the formal constraints (axioms and definitions).