Developing Geometric Thinking: The Van Hiele Levels
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Transcript Developing Geometric Thinking: The Van Hiele Levels
Developing Geometric
Thinking: The Van Hiele
Levels
Adapted from
Van Hiele, P. M. (1959). Development and learning
process. Acta Paedogogica Ultrajectina (pp. 1-31).
Groningen: J. B. Wolters.
CI 319 Fall 2006
Mara Alagic
1
Van Hiele: Levels of
Geometric Thinking
Precognition
Level
0: Visualization/Recognition
Level 1: Analysis/Descriptive
Level 2: Informal Deduction
Level 3:Deduction
Level 4: Rigor
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Visualization or Recognition
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
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3
Analysis/Descriptive
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, ...
He/she
is not yet capable of
differentiating these properties into
definitions and propositions
Logical relations are not yet fit-study
object
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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
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Mara Alagic
5
Informal Deduction
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
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Informal Deduction: Examples
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
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Deduction (1)
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
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Deduction (2)
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
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Rigor
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
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The levels: Differences in
objects of thought
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
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Major Characteristics of the Levels
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
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References
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.
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