When people recall their geometry learning experience, many of them recall it not only as an unpleasant experience, but they often also recall difficulties that they experienced in learning geometry. These two facets of learning, unpleasantness and a lack of depth of understanding, go hand in hand. When we find it difficult or impossible to understand an area of study, we typically resort to rote memorization, and this entire experience is distasteful to most people. So why is it that geometry is so often the school mathematics topic that people remember with unpleasantness and a lack of comprehension? The answer hinges on an inadequate school geometry curriculum.

For decades we have had access to the findings of Pierre and Dina van Hiele, Dutch researchers who examined the question of why so many people have difficulty learning geometry. What they found was that people develop their knowledge and understanding of geometric concepts in a predictable sequence of levels of development. At the earliest level in the van Hiele scheme, known as the PreRecognition Stage, children perceive geometric shapes but they are generally unable to identify them. They can recognize broad categories of shapes, such as straight-sided shapes as compared to curved-sided shapes. They do not construct mental images of shapes or hold shapes in their minds for any length of time. At the next higher level in the van Hiele scheme, children do hold mental images of shapes. However, these images are not based on an analysis of the number of sides and/or angles the shape possesses. Rather, the mental images represent a common appearance of a shape. For example, any shape that has a horizontal base and comes to a point would be likely to fit the mental image of a triangle. Without regard to the number of sides, a child at this level would identify such a shape as a triangle. A three-sided shape in a different orientation would have a different name, such as, perhaps, an ice-cream cone. The child at this level evaluates a shape to see if it fits her mental model for shapes of that kind. The van Hieles named this level the Visual level. Beyond the Visual level, children develop the ability to analyze shapes in terms of their properties. They examine the number of sides and/or angles in order to determine what a shape would be called. At this level a child might examine a three-sided shape that is in an unusual orientation and declare that it is a triangle, "because it has three sides." This level, known as the Descriptive/Analytic level, is common among children in elementary school. Children who have progressed beyond the Descriptive/Analytic level are able to reason with definitions of shapes and they are able to make sense of relationships within and among different shapes. Children at this level are said to be at the Abstract/Relational level because the relationships between shapes and between parts of shapes are abstractions. Children at this level understand categories of shapes and how a shape can belong to more than one category at a time. Beyond this, learners develop the ability to use logic and reasoning in regard to geometric relationships. Since high school geometry curricula are built on geometric proofs, this is the level of development that high school students need to be at in order to understand high school geometry. The van Hieles named this level the Formal Deduction level. The van Hieles identified one level that is beyond the Formal Deduction level and one level that preceeds the Visual level, but these two levels are beyond the scope of interest for this particular text. All of the van Hiele levels and their descriptions are shown in the table below.

Unlike the stages of cognitive development uncovered by Piaget, progress through the van Hiele levels of development is dependent on experiences with geometric concepts rather than on maturation. In other words, people do not naturally progress through the van Hiele levels of geometric learning; rather, they progress as a result of direct experiences that they have with geometric concepts. Without appropriate experiences, this progress through the levels does not occur. Unfortunately, school geometry curricula have, until very recently, included very few of the right kinds of experiences. Elementary and middle school geometry curricula have included too many low-level experiences in which learners are simply asked to learn shape names and names of other geometric objects. Then, in high school, learners are expected to learn geometric reasoning as they work with proofs. The typical elementary school curriculum keeps children at a low level of development, and then the high school curriculum unreasonably expects students to jump to a high level of development. For most people this jump is impossible, and their development of geometric thinking is thwarted. Fortunately, since we now understand this process, we have begun to design school experiences that lead students from one van Hiele level to another and, in doing so, prepare them to learn advanced geometry concepts with understanding. However, until very recently, school curricula were not designed with this knowledge in mind. Therefore, most readers of this text probably had an inadequate experience with geometry in their early years. Adult learners can easily overcome this deficit by engaging in appropriate geometry-learning activities. Such activities are a major focus of this chapter. In this sense, the material in this chapter is significantly different from most of the rest of this text. The material here is not primarily meant to inform you about what you need to know. Rather, it is intended to provide you with the actual experiences that will help you to grow and develop as a geometric thinker.

The van Hiele Levels of Geometric Thought (as Articulated by Clements and Battista, 1992)


Level 0: Pre-Recognition Children perceive geometric shapes, but are unable to identify many of them. They can distinguish between broad categories, such as curvilinear and rectilinear shapes. but they cannot recognize different types within these broad categories. They do not construct mental representations, or visual images, of shapes.
Level 1: Visual Children recognize basic shapes as wholes. They have mental representations of types of shapes. These mental representations are broadly conceived visual prototypes. For example, any triangular shape would fit the prototype of a triangle, even if the sides were curved.A child at the visual level would call a curved shape a triangle if it had a generally triangular shape.
Level 2: Descriptive/Analytic Children use specific properties of shapes, rather than visual wholes, to distinguish between them. Reasoning is in terms of combinations of properties.
Level 3: Abstract/Relational Children can begin to follow informal logical reasoning about properties of shapes. Concepts such as class inclusion (squares as special cases of rectangles) are understood. Definitions become logical organizers rather than lists of properties.
Level 4: Formal Deduction Students become capable of constructing original meaningful proofs. They can produce a logical argument on the basis of "givens."
Level 5: Rigor/Metamathematical Students extend their reasoning power to the elaboration and comparison of alternate axiomatic systems of geometry. They become capable of reasoning in the absence of reference models.