Giving a New Sense to van Hiele Theory : Reconsideration from perspectives of informal knowledge and zone of proximal development

Abstract

Van Hiele theory, especially its levels of thinking, is widely accepted by mathematics education communities in many countries, and is used as a framework for the research on geometry learning and teaching. The meanings of his theory, however, is not fully clear, and it sometimes causes controversy. The purpose of this paper is making clear the meanings of the van Hiele's levels of thinking and the five stages for facilitating transitions to higher levels, which are the central ideas of his theory. In order to do this, we first consider the relation between van Hiele theory and the theory about informal knowledge. From this consideration, we find the followings; (i) Recognition of figures at the first level can be taken as informal and situated knowledge about those figures; (ii) Transitions from the first level to the third level can be seen as the transition from informal knowledge to formal knowledge. Based on these results, next we analyze the relation between van Hiele theory and Vygotskian theory. Then we find the followings; (i) Recogniton of figure at the second level corresponds to the pseudconcept or potential concept; (ii) Transitions among the levels cerrespond to the development of scientific concepts based on everyday concepts; (iii) The span between the first level and the third can be considered to generate the zone of proximal development concerning the geometrical knowledge. Consequently, we obtain the following characterization of van Hiele theory; This theory deals with teaching geometry using the zone of proximal development so that children have the access to geometical knowledge and can use it with conscious awareness and volition. This result suggests the new research problems relating to van Hiele thery.