The Van Hiele Model of Geometric Thought.
Two Dutch educators, Dina and Pierre van Hiele, suggested that
children may learn geometry along the lines of a structure for
reasoning that they developed in the 1950s. educators in the former
Soviet Union learned of the van Hiele research and changed their
geometry curriculum in the 1960s. During the 1980s there was interest
in the United States in the van Hieles' contributions; the {\it
Standards} of the National Council of Teachers of Mathematics (1989)
brought the van Hiele model of learning closer to implementation by
stressing the importance of sequential learning and an activity
approach.
The van Hiele model asserts that the learner moves sequentially
through five levels of understanding. Different numbering systems are
found in the literature but the van Hieles spoke of levels 0 through
4.
Level 0 (Basic Level): Visualization
Students recognize figures as total entities (triangles, squares),
but do not recognize properties of these figures (right angles in a
square).
Level 1: Analysis
Students analyze component parts of the figures (opposite angles of
parallelograms are congruent), but interrelationships between figures
and properties cannot be explained.
Level 2: Informal Deduction
Students can establish interrelationships of properties within
figures (in a quadrilateral, opposite sides being parallel
necessitates opposite angles being congruent) and among figures (a
square is a rectangle because it has all the properties of a
rectangle). Informal proofs can be followed but students do not see
how the logical order could be altered nor do they see how to
construct a proof starting from different or unfamiliar premises.
Level 3: Deduction
At this level the significance of deduction as a way of establishing
geometric theory within an axiom system is understood. The
interrelationship and role of undefined terms, axioms, definitions,
theorems and formal proof is seen. The possibility of developing a
proof in more than one way is seen.
Level 4: Rigor
Students at this level can compare different axiom systems
(non-Euclidean geometry can be studied). Geometry is seen in the
abstract with a high degree of rigor, even without concrete examples.
The majority of high school geometry courses is taught at Level 3.
The van Hieles also identified some characteristics of their model,
including the fact that a person must proceed through the levels in
order, that the advancement from level to level depends more on
content and mode of instruction than on age, and that each level has
its own vocabulary and its own system of relations.The van Hieles
proposed sequential phases of learning to help students move from one
level to another.
Phase 1: Inquiry/Information
At this initial stage the teacher and the students engage in
conversation and activity about the objects of study for this level.
Observations are made, questions are raised, and level-specific
vocabulary is introduced.
Phase 2: Directed Orientation
The students explore the topic through materials that the teacher has
carefully sequenced. These activities should gradually reveal to the
students the structures characteristic at this level.
Phase 3: Explication
Building on their previous experiences students express and exchange
their emerging views about the structures that have been observed.
Other than to assist the students in using accurate and appropriate
vocabulary, the teacher's role is minimal. It is during this phase
that the level's system of relations begins to become apparent.
Phase 4: Free Orientation
Students encounter more complex tasks - tasks with many steps, tasks
that can be completed in more than one way, and open-ended tasks.
They gain experience in resolving problems on their own and make
explicit many relations among the objects of the structures being
studied.
Phase 5: Integration
Students are able to internalize and unify relations into a new body
of thought. The teacher can assist in the synthesis by giving
''global surveys'' of what students already have learned.
Reference: Teppo, Anne , "Van Hiele Levels of Geometric Thought
Revisited." , Mathematics Teacher , March 1991, pg 210-221.
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