Giving a New Sense to van Hiele Theory:
Reconsideration from Perspectives of Informal Knowledge and Zone of Proximal Development.


Research Journal of Educational Methods, 19, 37-46.


Summary

Van Hiele thoery, 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, nexe we analyze the relation between van Hiele theory and Vygotskian theory. Then we find the followings; (i) Recognition of figure at the second level corresponds to the pseudconcept or potential concept; (ii) Transitions among the levels correspond 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 geometrical knowledge and can use it with conscious awareness and volition. This result suggests the new research problems relating to van Hiele theory.

A Consideration on van Hiele Levels
from the Perspective of Recognizing Figures


Bulletin of Institute of Education (University of Tsukuba), 16 (2), 139-152.


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Summary

Van Hiele levels of thinking are accepted as a standard framework in many countries, and used in research and curriculum development. His description of the levels is not, however, based on a explicit principle. So interpretation of the levels by other researcheres are slightly different each other, and sometimes include what contradict the original description by van Hiele. The aim of this paper is to present a principle for re-constructing van Hiele levels. This principle can make the relationships among the levels clearer.

We start by analyzing the concept "symbol," which is one of the most important concepts in van Hiele's theory. It is found that according to van Hiele, symbols have three different roles in one's thought and change from one role to the next as the thought develops. Applying this idea to the concept of figure, we can find three different ways in recognizing figures. First, people see a figure totaly as a kind of visual images. Then the figure gets to be recognized as a form having a certain set of characteristics. Finally it is recognized as a junction of the network of the geometry system. Taking the geometry system as a new object of thinking and applying the same argument to it, we can also get three ways of recognizing the geometry system.

We then relate these ways of recognition to other important concepts in van Hiele's theory. We find we can fully re-construct van Hiele levels using the way of recognition above mentioned. This discussion leads us to the following new framework;
levelrecognition of figuresrecognition of geometry system
level 1 recognizing a figure as an indifferential image
level 2 recognizing a figure by a set of characteristics
level 3 recognizing a figure by its definition recognizing a geometry system as an indifferential network
level 4 recongnizing a geometry system by a set of characteristics
level 5 recognizing a geometry system by its definition

As the result ofour discussion, "the change in ways of recognizing the object" is presented as the principle for re-constructing van Hiele levels.

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