The van Hiele Theory of learning geometry

The van Hiele theory of learning geometry also drew heavily from the work of Piaget (1972) and Gestalt theory (Wertheimer, 1912). After studying Piaget’s work, Pierre van Hiele and Dina van Hiele-Geldof thought that students’ geometrical competence might well improve by progressing over a period of time through successive stages of thinking. There are five levels in van Hiele theory of learning geometry as follows (Orton, 2004):

Level (1) Visualization:

Students can recognize figures as whole entities (triangles, squares), but do not recognize properties of these shapes (right angles in a square); visual impression and appearance exert a storing influence, thus a square cannot also be a rectangle; drawings of shapes are based on holistic impressions and not on component parts; names may be invented for shapes according to their appearance, for example, ‘Slanty rectangle’ for parallelogram.

Level (2) Analysis:

Students can analyse figures’ components such as sides and angles but cannot relate between figures and properties logically; properties and rules of a class of shapes may be discovered empirically (for example, by folding, measuring, or by using a grid or diagram); a figure can be identified from its properties; generalizations become possible, for example, all squares have four sides, the angles of triangles total 180º.

Level (3) Informal Deduction:

Students can establish interrelationship of properties within shapes and can make simple deductions, though the intrinsic meaning of deduction is not understood; a shape may be used to establish that a square is a rectangle; a statement cannot be separated from its converse.

Level (4) Deduction:

At this level, students can appreciate the need for definitions and assumptions, and can present proofs within a postulational system; The interrelationship and role of undefined terms, axioms, definitions, theorems and formal proof can be understood; proof as the final authority is accepted; inter-relationships among networks of theorems can be established.

Level (5) Rigor:

Students at this level can work abstractly and can compare systems, can examine the consistency and independence of axioms and generalize a principle or theorem to find the broadest context. Geometry is seen in the abstract with a high degree of rigor, even without concrete examples.

All school geometry courses are taught at Level 3 . The van Hieles also recognized some features of their model, including the fact that a person must proceed through the levels in order. The most important features of van Hieles system as summarized by Fuys et.al (1988) are as follows: (a) the levels are consecutive; (b) each level has its own vocabulary, symbols and network of relations; (c) what is implicit at one level becomes explicit at the next level; (d) material taught to students above their level is dependent on reduction of level; (e) progress from one level to the next is more subject to instructional experience than on age or maturation; and (f) one goes through various ‘phase’ in arising from one level to the next.