Vinner& Hershkowitz(1983) and Hershkowitz et al. (1990), found that for each geometric concept there is at least one prototypical example. For example the square is a prototypical example of the quadrilaterals group. The prototypical examples are acquired first, and are therefore found in the concept image of most learners. Prototypical examples are usually a subset of the concept's examples with the longest “list” of attributes, the critical attributes of the concept and some attributes that are specific to that subset but are non-critical. These non-critical attributes have dominant visual properties, which have an effect on the construction of geometric concepts, and affect the classification and identification abilities, construction, and judgment concerning basic geometrical concepts. This phenomenon is in agreement with construction of concepts in everyday life (Rosch& Mervis, 1975). Hershkowitz et al. (1990) mentioned several ways to judge geometric shapes as examples of a concept: (i) Visual judgment: The student relies on the prototype as a visual frame of reference. For example, the prototype of altitude of a triangle is an altitude inside the triangle. This classification and identification level fits the first van Hiele level (visualization). (ii) Judgment by prototype attributes: Classification into examples and non-examples by checking the existence and non-existence of the special attribute of the prototype. This wrong judgement fits the second van Hiele level (iii) Analytical classification: Student rely on the critical attributes of the concept as they appear in its mathematical definition. This correct judgment fits the second/third van Hiele levels.
The definition implies inclusion relationships between groups of the concepts’ examples on the one hand, and groups of attributes of the same concepts on the other. These inclusion relations have opposite directions (Hershkowitz et al., 1990). For example, the squares included in the parallelograms, but the group of critical attributes of the squares contains the group of critical attributes of the parallelograms.