Transcript Developing Geometric Thinking: The Van Hiele Levels

Developing
Geometric Thinking:
Van Hiele’s 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
Summer 2004
Mara Alagic
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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
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Mara Alagic
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Visualization/Recognition
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The student identifies, names compares
and operates on geometric figures
according to their appearance
For example, students recognize
rectangles by its form but, a rectangle
seems different to them then a square
At this level rhombus is not recognized
as a parallelogram
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Mara Alagic
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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
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Mara Alagic
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Analysis/Descriptive
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Students analyze figures in terms of their
components and relationships between
components; discover properties/rules of a
class of shapes empirically by
– folding
– measuring
– using a grid or a diagram, ...
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They are not yet capable of differentiating
these properties into definitions &
propositions
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Logical relations are not yet fit-study object
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Analysis/Descriptive:
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
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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
Summer 2004
Mara Alagic
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Informal Deduction
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Students logically interrelate
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
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A square is a rectangle because it has
all the properties of a rectangle
Students 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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Mara Alagic
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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
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Mara Alagic
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Deduction (1)
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Students prove theorems deductively
and establish 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)
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Students seek 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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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
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Mara Alagic
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Rigor
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Students establish theorems in different
postulational systems and
analyze/compare these systems
Figures are defined only by symbols
bound by relations
A comparative study of the various
deductive systems can be accomplished
Students have acquired a scientific
insight into geometry
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Levels: 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
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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
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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.
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