Van Hiele Model - Properties of The Levels

Properties of The Levels

The van Hiele levels have five properties:

1. Fixed sequence: the levels are hierarchical. Students cannot "skip" a level. 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.