Pierre van Hiele and Dina van Hiele-Geldof as teachers in Montessori secondary schools noticed the difficulties that their student had in learning geometry. It became apparent to them that the difference in the level of thinking required from primary school to secondary school was very difficult for individuals who had not grasped the basic concepts. These observations led them to develop a model involving levels of thinking in geometry, through which an individual passes when learning, beginning from simply recognising a form to being able to write a formal geometric proof.
The van Hiele model consists of five levels of thinking processes and includes appropriate teaching instructions to assist the learner to move sequentially from the basic level to the highest level. According to van Hiele, “The attainment of the new level cannot be effected by teaching, but still, by a suitable choice of exercises the teacher can create a situation for the pupil favourable to the attainment of the higher level of thinking.” A brief overview of the model is given below.
Level 0: Visualisation
Children can recognise geometric figures by their shape or physical appearance and not by their parts or properties. For example based on drawings in Figure 1 below, a child would be able to recognise that there are squares in Figure 1a and rectangles in 1b. This is because they can identify similar shaped figures but would not however recognise that the figures have right angles or the opposite lines are parallel.
Children at this level can also learn the vocabulary associated with geometry and identify the specified shape. They are able to label certain plane figures such as circles, triangles, squares and rectangles as well as recognise simple solids such as spheres, cubes, pyramids, and cones, name them with those labels or less formal names such as balls and boxes. By the end of this stage children are able to group shapes or figures into classes because they seem to look ‘alike’.
Figure 1 (a and b): Squares and Rectangles
Level 1: Analysis
Children at this level are able to appreciate that the collection of figures goes together because of properties. They also become more proficient in describing the attributes of two and three dimensional shapes. Their language usually consists of a mixture of mathematical terminology and less precise nouns and adjectives. ‘A ball is round all over; a square has ‘straight’ corners’. Children also benefit much from playing with geometric materials as they are able to learn more about 2 and 3 dimensional figures. By the end of this stage children are able to list as many properties of shapes as they know.
Level 2: Informal Deduction
Children at this stage are able to establish relationships among the properties of the figures. ‘If all four angles are right angles the shape must be a rectangle’. They can also formulate relationships among shapes. ‘If it is a square, it must be a rectangle’. They are able to give logical arguments to justify their reasoning.
Level 3: Deduction
Students at this stage are able to go further than giving informal deductive arguments as in the earlier stage. Here, they begin to appreciate the importance and use of axioms, definitions, corollaries and postulates in order to proof a certain geometric truth. This level of thought is typical of secondary school geometry class.
Level 4: Rigor
At this level the object of attention is the differences and relationships between different axiomatic systems that are used to derive mathematical theories. In the previous stage the students are expected to know how to form deductive arguments to proof a theory. In this stage they go further by questioning the initial assumptions or axioms that are used to form the mathematical theories. This is generally the level of university mathematics majoring in geometry.
Apparently the last level of the van Hiele system is the least developed in the original works and received very little attention from researchers. The majority of secondary school geometry courses are taught up to level 3 and as a result most research work has also been concentrated on lower levels.
Figure 2: The van Hiele Model of Geometric Thought
Properties of the Model
In addition to providing the descriptions of thought processes specific to each level of geometry, the van Hiele’s also made statements that help to characterise the van Hiele model. These statements are especially useful for those who intend to apply this model into their teaching methods.
1. The levels are sequential. In order to move above level 0, one must progress through previous levels. This is necessarily so because one needs to acquire the thinking strategies of the previous level in order to function successfully in the next level.
2. Advancement. The progress from one level to the next is more dependent on the content and methods of instruction given rather than on age. However it is acknowledged by researchers that age is certainly related to the amount and types of geometric experiences one can have. Thus it is reasonable to assume that all children at preschool age are at level 0 and also the majority of children who are in year 3 and 4. This is in contrast to Piaget’s model of development, where he placed a bigger importance on maturity or age being a factor for learning.
3. Intrinsic and extrinsic. The inherent objects at one level become the objects of study at the next level. For example, at level 0 the child is only able to perceive the form of the figure. Only at level 1 does the child begin to analyse and discover the properties of the figure. This is said to be possible by providing a good exposure to geometry through exploration, talking about and interacting with objects.
4. Mismatch. The vocabulary used in instruction has to be in par with the student’s level of comprehension. Otherwise the lack of communication would cause the student to struggle with objects of thought that has not been understood at the previous level, and instead may be led to rote learning and achieve temporary success in the present level. For example one may memorize a geometric proof but fail to understand the rationale behind it. Other forms of mismatch include the use of instructional materials and/or content, where if it is at a higher level than the learner, one may not be able to follow the thought process involved.
1. Van de Walle, J.A. (2001). Geometric Thinking and Geometric Concepts. In Elementarty and Middle. School Mathematics: Teaching Developmentally, 4th ed. Boston: Allyn and Bacon.
3. Crowley, M.L. (1987). The van Hiele Model of the Development of Geometric Thought.
4. Fuys, D., Geddes, D., Lovett, C. J. & Tischler, R. (1988) The Van Hiele model of thinking in geometry among adolescents. Journal for Research in Mathematics Education.