model pemikiran geometri van hiele
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The van Hiele Model
of GeometricThought
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Define it
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When is it appropriate
to ask for a definition?A definition of a concept is onlypossible if one knows, to some
extent, the thing that is to bedefined.
Pierre van Hiele
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Definition?How can you define a thing beforeyou know what you have to define?
Most definitions are not preconceivedbut the finished touch of theorganizing activity.
The child should not be deprived ofthis privilegeHans Freudenthal
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Levels of Thinking in
GeometryVisual Level
Descriptive Level
Relational Level
Deductive Level
Rigor
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Levels of Thinking in
Geometry Each level has its own network of
relations.
Each level has its own language. The levels are sequential and
hierarchical. The progress from one
level to the next is more dependentupon instruction than on age ormaturity.
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Visual Level
Characteristics
The student
identifies, compares and sorts shapes on thebasis of their appearance as a whole.
solves problems using general properties andtechniques (e.g., overlaying, measuring).
uses informal language. does NOT analyze in terms of components.
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Visual Level Example
It turns!
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Where and how is the
Visual Level representedin the translation andreflection activities?
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Where and how is the Visual
Level represented in thistranslation activity?
It slides!
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Where and how is the VisualLevel represented in thisreflection activity?
It is a flip!
It is a mirror image!
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Descriptive LevelCharacteristics
The student
recognizes and describes a shape (e.g.,
parallelogram) in terms of its properties. discovers properties experimentally by
observing, measuring, drawing and modeling.
uses formal language and symbols.
does NOT use sufficient definitions. Lists manyproperties.
does NOT see a need for proof of generalizationsdiscovered empirically (inductively).
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Descriptive Level
Example
It is a rotation!
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Where and how is theDescriptive Levelrepresented in thetranslation and reflection
activities?
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Where and how is theDescriptive Levelrepresented in thistranslation activity?
It is a translation!B'
A'
F'
E'
D'
C'
C
D
E
F
G
A
B
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Where and how is theDescriptive Levelrepresented in thisreflection activity?
It is a reflection!
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Relational LevelCharacteristics
The student can define a figure using minimum
(sufficient) sets of properties. gives informal arguments, and discovers
new properties by deduction.
follows and can supply parts of a
deductive argument. does NOT grasp the meaning of an
axiomatic system, or see theinterrelationships between networks of
theorems.
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Relational Level
ExampleIf I know how to findthe area of the
rectangle, I can findthe area of thetriangle!
Area of triangle =
h
b
1
2h
1
2bh
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Deductive Level
My experience as a teacher of geometryconvinces me that all too often, students
have not yet achieved this level ofinformal deduction. Consequently, theyare not successful in their study of the
kind of geometry that Euclid created,which involves formal deduction.
Pierre van Hiele
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Deductive Level
CharacteristicsThe student
recognizes and flexibly uses the
components of an axiomatic system(undefined terms, definitions, postulates,theorems).
creates, compares, contrasts differentproofs.
does NOT compare axiomatic systems.
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Deductive Level Example
In ABC, is amedian.
I can prove that
Area of ABM= Areaof MBC.
M
C
B
A
BM
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Rigor
The student
compares axiomatic systems (e.g.,
Euclidean and non-Euclidean geometries). rigorously establishes theorems in
different axiomatic systems in theabsence of reference models.
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Phases of the
Instructional Cycle Information
Guided orientation
Explicitation
Free orientation
Integration
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Information Phase
The teacher holds a conversation with thepupils, in well-known language symbols,
in which the context he wants to usebecomes clear.
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Information Phase
It is called a rhombus.
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Guided Orientation Phase
The activities guide the student toward
the relationships of the next level.
The relations belonging to the context are
discovered and discussed.
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Guided Orientation Phase
Fold the rhombus on its axes of symmetry.What do you notice?
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Explicitation Phase
Under the guidance of the teacher,
students share their opinions about therelationships and concepts they havediscovered in the activity.
The teacher takes care that the correcttechnical language is developed andused.
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Explicitation Phase
Discuss your ideas with your group, andthen with the whole class.
The diagonals lie on the lines of symmetry. There are two lines of symmetry.
The opposite angles are congruent.
The diagonals bisect the vertex angles.
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Free Orientation Phase
The relevant relationships are known.
The moment has come for the studentsto work independently with the new
concepts using a variety of applications.
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Free Orientation Phase
The following rhombi are incomplete.
Construct the complete figures.
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Integration Phase
The symbols have lost their visual content
and are now recognized by their properties.
Pierre van Hiele
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What we do and what wedo not do
It is customary to illustrate newly introduced
technical language with a few examples.
If the examples are deficient, the technicallanguage will be deficient.
We often neglect the importance of thethird stage, explicitation. Discussion helps
clear out misconceptions and cementsunderstanding.
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What we do and what wedo not do
Sometimes we attempt to inform by
explanation, but this is useless. Students
should learn by doing, not be informed byexplanation.
The teacher must give guidance to theprocess of learning, allowing students to
discuss their orientations and by havingthem find their way in the field of thinking.
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Instructional
Considerations Visual to Descriptive Level
Language is introduced to describe figures thatare observed.
Gradually the language develops to form thebackground to the new structure.
Language is standardized to facilitatecommunication about observed properties.
It is possible to see congruent figures, but it isuseless to ask why they are congruent.
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Instructional
Considerations Descriptive to Relational Level
Causal, logical or other relations become
part of the language. Explanation rather than description is
possible.
Able to construct a figure from its knownproperties but not able to give a proof.
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Instructional
Considerations Relational to Deductive Level
Reasons about logical relations betweentheorems in geometry.
To describe the reasoning to someone who doesnot speak this language is futile.
At the Deductive Level it is possible to arrangearguments in order so that each statement,except the first one, is the outcome of theprevious statements.
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Instructional
Considerations Rigor
Compares axiomatic systems.
Explores the nature of logical laws.
Logical Mathematical Thinking
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Consequences
Many textbooks are written with only theintegration phase in place.
The integration phase often coincides with
the objective of the learning. Many teachers switch to, or even begin,
their teaching with this phase, a.k.a. directteaching.
Many teachers do not realize that theirinformation cannot be understood by theirpupils.
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Children whose geometric thinking younurture carefully will be better able to
successfully study the kind ofmathematics that Euclid created.
Pierre van Hiele
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