Van Hiele model

Theory of how students learn geometry
In mathematics education, the Van Hiele model is a theory that describes how students learn geometry. The theory originated in 1957 in the doctoral dissertations of Dina van Hiele Geldof and Pierre van Hiele (wife and husband) at Utrecht University, in the Netherlands. The Soviets did research on the theory in the 1960s and integrated their findings into their curricula. American researchers did several large studies on the van Hiele theory in the late 1970s and early 1980s, concluding that students' low van Hiele levels made it difficult to succeed in proof oriented geometry courses and advising better preparation at earlier grade levels. Pierre van Hiele published Structure and Insight in 1986, further describing his theory. The model has greatly influenced geometry curricula throughout the world through emphasis on analyzing properties and classification of shapes at early grade levels. In the United States, the theory has influenced the geometry strand of the Standards published by the National Council of Teachers of Mathematics and the Common Core Standards.

The student learns by rote to operate with [mathematical] relations that he does not understand, and of which he has not seen the origin Therefore the system of relations is an independent construction having no rapport with other experiences of the child. This means that the student knows only what has been taught to him and what has been deduced from it. He has not learned to establish connections between the system and the sensory world. He will not know how to apply what he has learned in a new situation. Pierre van Hiele, 1959

The best known part of the van Hiele model are the five levels which the van Hieles postulated to describe how children learn to reason in geometry. Students cannot be expected to prove geometric theorems until they have built up an extensive understanding of the systems of relationships between geometric ideas. These systems cannot be learned by rote, but must be developed through familiarity by experiencing numerous examples and counterexamples, the various properties of geometric figures, the relationships between the properties, and how these properties are ordered. The five levels postulated by the van Hieles describe how students advance through this understanding. The five van Hiele levels are sometimes misunderstood to be descriptions of how students understand shape classification, but the levels actually describe the way that students reason about shapes and other geometric ideas. Pierre van Hiele noticed that his students tended to "plateau" at certain points in their understanding of geometry and he identified these plateau points as levels. In general, these levels are a product of experience and instruction rather than age. This is in contrast to Piaget's theory of cognitive development, which is age dependent. A child must have enough experiences (classroom or otherwise) with these geometric ideas to move to a higher level of sophistication. Through rich experiences, children can reach Level 2 in elementary school. Without such experiences, many adults (including teachers) remain in Level 1 all their lives, even if they take a formal geometry course in secondary school. The van Hieles claim that much of the difficulty experienced by geometry students is due to being taught at the Deduction level when they have not yet achieved the Abstraction level. 2. Adjacency: properties which are intrinsic at one level become extrinsic at the next. (The properties are there at the Visualization level, but the student is not yet consciously aware of them until the Analysis level. Properties are in fact related at the Analysis level, but students are not yet explicitly aware of the relationships.) 3. Distinction: each level has its own linguistic symbols and network of relationships. The meaning of a linguistic symbol is more than its explicit definition; it includes the experiences the speaker associates with the given symbol. What may be "correct" at one level is not necessarily correct at another level. At Level 0 a square is something that looks like a box. At Level 2 a square is a special type of rectangle. Neither of these is a correct description of the meaning of "square" for someone reasoning at Level 1. If the student is simply handed the definition and its associated properties, without being allowed to develop meaningful experiences with the concept, the student will not be able to apply this knowledge beyond the situations used in the lesson. 4. Separation: a teacher who is reasoning at one level speaks a different "language" from a student at a lower level, preventing understanding. When a teacher speaks of a "square" she or he means a special type of rectangle. A student at Level 0 or 1 will not have the same understanding of this term. The student does not understand the teacher, and the teacher does not understand how the student is reasoning, frequently concluding that the student's answers are simply "wrong". The van Hieles believed this property was one of the main reasons for failure in geometry. Teachers believe they are expressing themselves clearly and logically, but their Level 3 or 4 reasoning is not understandable to students at lower levels, nor do the teachers understand their students’ thought processes. Ideally, the teacher and students need shared experiences behind their language. 5. Attainment: The van Hieles recommended five phases for guiding students from one level to another on a given topic:

Information or inquiry: students get acquainted with the material and begin to discover its structure. Teachers present a new idea and allow the students to work with the new concept. By having students experience the structure of the new concept in a similar way, they can have meaningful conversations about it. (A teacher might say, "This is a rhombus. Construct some more rhombi on your paper.") Guided or directed orientation: students do tasks that enable them to explore implicit relationships. Teachers propose activities of a fairly guided nature that allow students to become familiar with the properties of the new concept which the teacher desires them to learn. (A teacher might ask, "What happens when you cut out and fold the rhombus along a diagonal? the other diagonal?" and so on, followed by discussion.) Explicitation: students express what they have discovered and vocabulary is introduced. The students’ experiences are linked to shared linguistic symbols. The van Hieles believe it is more profitable to learn vocabulary after students have had an opportunity to become familiar with the concept. The discoveries are made as explicit as possible. (A teacher might say, "Here are the properties we have noticed and some associated vocabulary for the things you discovered. Let's discuss what these mean.") Free orientation: students do more complex tasks enabling them to master the network of relationships in the material. They know the properties being studied, but need to develop fluency in navigating the network of relationships in various situations. This type of activity is much more open ended than the guided orientation. These tasks will not have set procedures for solving them. Problems may be more complex and require more free exploration to find solutions. (A teacher might say, "How could you construct a rhombus given only two of its sides?" and other problems for which students have not learned a fixed procedure.) Integration: students summarize what they have learned and commit it to memory. The teacher may give the students an overview of everything they have learned. It is important that the teacher not present any new material during this phase, but only a summary of what has already been learned. The teacher might also give an assignment to remember the principles and vocabulary learned for future work, possibly through further exercises. (A teacher might say, "Here is a summary of what we have learned. Write this in your notebook and do these exercises for homework.") Supporters of the van Hiele model point out that traditional instruction often involves only this last phase, which explains why students do not master the material. For Dina van Hiele Geldof's doctoral dissertation, she conducted a teaching experiment with 12 year olds in a Montessori secondary school in the Netherlands. She reported that by using this method she was able to raise students' levels from Level 0 to 1 in 20 lessons and from Level 1 to 2 in 50 lessons.