Developing
Geometric Thinking:
Van Hiele Levels
Mara Alagic
Van Hiele: Levels of
Geometric Thinking
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Precognition
Level 0: Visualization/Recognition
Level 1: Analysis/Descriptive
Level 2: Informal Deduction
Level 3:Deduction
Level 4: Rigor
Fall 2002
Mara Alagic
Van Hiele: Levels of
Geometric Thinking
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Precognition
Level 0: Visualization/Recognition
Level 1: Analysis/Descriptive
Level 2: Informal Deduction
Level 3:Deduction
Level 4: Rigor
Fall 2002
Mara Alagic
Visualization or Recognition
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The student identifies, names compares
and operates on geometric figures
according to their appearance
For example, the student recognizes
rectangles by its form but, a rectangle
seems different to her/him then a square.
At this level rhombus is not recognized
as a parallelogram
Fall 2002
Mara Alagic
Van Hiele: Levels of
Geometric Thinking
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Precognition
Level 0: Visualization/Recognition
Level 1: Analysis/Descriptive
Level 2: Informal Deduction
Level 3:Deduction
Level 4: Rigor
Fall 2002
Mara Alagic
Analysis/Descriptive
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The student analyzes figures in terms of their
components and relationships between
components and discovers properties/rules
of a class of shapes empirically by
– folding
– measuring
– using a grid or diagram, ...
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He/she is not yet capable of differentiating
these properties into definitions and
propositions
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Logical relations are not yet fit-study object
Fall 2002
Mara Alagic
Analysis/Descriptive:
An Example
If a student knows that the
– diagonals of a rhomb are perpendicular,
she must be able to conclude that,
– if two equal circles have two points in
common, the segment joining these two
points is perpendicular to the segment
joining centers of the circles.
Fall 2002
Mara Alagic
Van Hiele: Levels of
Geometric Thinking

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Precognition
Level 0: Visualization/Recognition
Level 1: Analysis/Descriptive
Level 2: Informal Deduction
Level 3:Deduction
Level 4: Rigor
Fall 2002
Mara Alagic
Informal Deduction
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The student logically interrelates
previously discovered
properties/rules by giving or
following informal arguments
The intrinsic meaning of deduction
is not understood by the student
The properties are ordered deduced from one another
Fall 2002
Mara Alagic
Informal Deduction: Examples
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A square is a rectangle because it has
all the properties of a rectangle.
The student can conclude the equality
of angles from the parallelism of lines:
In a quadrilateral, opposite sides being
parallel necessitates opposite angles
being equal
Fall 2002
Mara Alagic
Van Hiele: Levels of
Geometric Thinking



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Precognition
Level 0: Visualization/Recognition
Level 1: Analysis/Descriptive
Level 2: Informal Deduction
Level 3:Deduction
Level 4: Rigor
Fall 2002
Mara Alagic
Deduction (1)
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The student proves theorems
deductively and establishes
interrelationships among networks of
theorems in the Euclidean geometry
Thinking is concerned with the
meaning of deduction, with the
converse of a theorem, with axioms,
and with necessary and sufficient
conditions
Fall 2002
Mara Alagic
Deduction (2)
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Student seeks to prove facts
inductively
It would be possible to develop an
axiomatic system of geometry, but
the axiomatics themselves belong to
the next (fourth) level
Fall 2002
Mara Alagic
Van Hiele: Levels of
Geometric Thinking
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

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Precognition
Level 0: Visualization/Recognition
Level 1: Analysis/Descriptive
Level 2: Informal Deduction
Level 3:Deduction
Level 4: Rigor
Fall 2002
Mara Alagic
Rigor
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The student establishes theorems in
different postulational systems and
analyzes/compares these systems
Figures are defined only by symbols
bound by relations
A comparative study of the various
deductive systems can be
accomplished
The student has acquired a scientific
insight into geometry
Mara Alagic
Fall 2002
The levels are “characterized by
differences in objects of thought”:
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geometric figures
classes of figures & properties of these
classes
students act upon properties, yielding
logical orderings of these properties
operating on these ordering relations
foundations (axiomatic) of ordering
relations
Fall 2002
Mara Alagic
Major Characteristics of the Levels
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the levels are sequential
each level has its own language, set of symbols, and
network of relations
what is implicit at one level becomes explicit at the next
level
material taught to students above their level is subject to
reduction of level
progress from one level to the next is more dependant
on instructional experience than on age or maturation
one goes through various “phases” in proceeding from
one level to the next
Fall 2002
Mara Alagic
References
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Van Hiele, P. M. (1959). Development and learning
process. Acta Paedogogica Ultrajectina (pp. 1-31).
Groningen: J. B. Wolters.
Van Hiele, P. M. & Van Hiele-Geldof, D. (1958).
A method of initiation into geometry at secondary
schools. In H. Freudenthal (Ed.). Report on methods
of initiation into geometry (pp.67-80). Groningen: J.
B. Wolters.
Fuys, D., Geddes, D., & Tischler, R. (1988). The van
Hiele model of Thinking in Geometry Among
Adolescents. JRME Monograph Number 3.
Fall 2002
Mara Alagic