Model pemikiran Geometri Van Hiele

8/12/2019 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

Model pemikiran Geometri Van Hiele

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