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The van Hiele ModelThe van Hiele Model

Matthew C. Robinson, Summer B 2006

22

Define it …Define it …

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Why do some students Why do some students think that:think that:

This is a right angleThis is a right angle but this is a left but this is a left angleangle

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When is it When is it appropriate to ask appropriate to ask for a definition?for a definition?

A definition of a concept is only A definition of a concept is only possible if one knows, to some possible if one knows, to some extent, the thing that is to be extent, the thing that is to be defined. defined.

Pierre van Hiele Pierre van Hiele

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Definition?Definition?

How can you define a thing How can you define a thing before you know what you have before you know what you have to define?to define?Most definitions are not Most definitions are not preconceived but the finished preconceived but the finished touch of the organizing activity.touch of the organizing activity.The child should not be deprived The child should not be deprived of this privilege…of this privilege…

Hans FreudenthalHans Freudenthal

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Developed by Pierre M. van Hiele and his wife Dina van Hiele-Geldof

BASIC IDEA: The learner, aided by appropriate instruction and experience, passes through five levels of thinking—the learner cannot achieve a level without passing through the previous levels.

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Levels of Thinking in Levels of Thinking in GeometryGeometry Visual Level (Holistic)Visual Level (Holistic) Descriptive Level (Analytic)Descriptive Level (Analytic) Relational Level (Abstract, Relational Level (Abstract,

Informal Deduction)Informal Deduction) Deductive LevelDeductive Level RigorRigor

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Levels of Thinking in Levels of Thinking in GeometryGeometry Each level has its own network of Each level has its own network of

relations.relations. Each level has its own language.Each level has its own language. The levels are sequential and The levels are sequential and

hierarchical. The progress from hierarchical. The progress from one level to the next is more one level to the next is more dependent upon instruction than dependent upon instruction than on age or maturity.on age or maturity.

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Visual Level Visual Level CharacteristicsCharacteristics

The student The student – identifies, compares and sorts shapes on identifies, compares and sorts shapes on

the basis of their appearance as a whole.the basis of their appearance as a whole.– solves problems using general properties solves problems using general properties

and techniques (e.g., overlaying, and techniques (e.g., overlaying, measuring).measuring).

– uses informal language.uses informal language.– does NOT analyze in terms of components.does NOT analyze in terms of components.

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Visual Level ExampleVisual Level Example

It turns!It turns!

1111

Where and how is the Where and how is the Visual Level Visual Level represented in the represented in the translation and translation and reflection activities?reflection activities?

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Where and how is the Where and how is the Visual Level represented Visual Level represented in this translation in this translation activity?activity?

It slides!It slides!

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Where and how is the Where and how is the Visual Level represented Visual Level represented in this reflection activity?in this reflection activity?

It is a flip!It is a flip!

It is a mirror image!It is a mirror image!

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Descriptive Level Descriptive Level CharacteristicsCharacteristics

The studentThe student– recognizes and describes a shape (e.g., recognizes and describes a shape (e.g.,

parallelogram) in terms of its properties.parallelogram) in terms of its properties.– discovers properties experimentally by discovers properties experimentally by

observing, measuring, drawing and observing, measuring, drawing and modeling.modeling.

– uses formal language and symbols.uses formal language and symbols.– does NOT use sufficient definitions. Lists does NOT use sufficient definitions. Lists

many properties.many properties.– does NOT see a need for proof of does NOT see a need for proof of

generalizations discovered empirically generalizations discovered empirically (inductively).(inductively).

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Descriptive Level Descriptive Level ExampleExample

It is a rotation!It is a rotation!

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Where and how is the Where and how is the Descriptive Level Descriptive Level represented in the represented in the translation and reflection translation and reflection activities?activities?

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Where and how is the Where and how is the Descriptive Level Descriptive Level represented in this represented in this translation activity?translation activity?

It is a translation!It is a translation!B'

A'

G'F'

E'

D'

C'

C

D

E

F

G

A

B

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Where and how is the Where and how is the Descriptive Level Descriptive Level represented in this represented in this reflection activity?reflection activity?

It is a reflection!It is a reflection!

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Relational Level Relational Level CharacteristicsCharacteristics

The studentThe student– can define a figure using minimum can define a figure using minimum

(sufficient) sets of properties.(sufficient) sets of properties.– gives informal arguments, and gives informal arguments, and

discovers new properties by discovers new properties by deduction.deduction.

– follows and can supply parts of a follows and can supply parts of a deductive argument.deductive argument.

– does NOT grasp the meaning of an does NOT grasp the meaning of an axiomatic system, or see the axiomatic system, or see the interrelationships between networks interrelationships between networks of theorems.of theorems.

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Relational Level Relational Level ExampleExample

If I know how to If I know how to find the area of the find the area of the rectangle, I can find rectangle, I can find the area of the the area of the triangle!triangle!

Area of triangle = Area of triangle =

h

b

1

2h

1

2bh

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Deductive LevelDeductive Level

My experience as a teacher of My experience as a teacher of geometry convinces me that all too geometry convinces me that all too often, students have not yet achieved often, students have not yet achieved this level of informal deduction. this level of informal deduction. Consequently, they are not successful Consequently, they are not successful in their study of the kind of geometry in their study of the kind of geometry that Euclid created, which involves that Euclid created, which involves formal deduction.formal deduction.

Pierre van HielePierre van Hiele

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Deductive Level Deductive Level CharacteristicsCharacteristics

The studentThe student– recognizes and flexibly uses the recognizes and flexibly uses the

components of an axiomatic system components of an axiomatic system (undefined terms, definitions, (undefined terms, definitions, postulates, theorems).postulates, theorems).

– creates, compares, contrasts creates, compares, contrasts different proofs.different proofs.

– does NOT compare axiomatic does NOT compare axiomatic systems.systems.

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Deductive Level Deductive Level ExampleExample

In ∆In ∆ABCABC, is a , is a median.median.

I can prove thatI can prove that

Area of ∆Area of ∆ABMABM = = Area of ∆Area of ∆MBCMBC..

M

CB

A

BM

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RigorRigor

The studentThe student– compares axiomatic systems (e.g., compares axiomatic systems (e.g.,

Euclidean and non-Euclidean Euclidean and non-Euclidean geometries).geometries).

– rigorously establishes theorems in rigorously establishes theorems in different axiomatic systems in the different axiomatic systems in the absence of reference models.absence of reference models.

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What figure is this?What figure is this?How do you know?How do you know?

-It looks like one or it looks like a door-It looks like one or it looks like a door-Four sides, closed, two long sides, two -Four sides, closed, two long sides, two

shorter sides, opposite sides parallel, shorter sides, opposite sides parallel, four right anglesfour right angles

-It is a parallelogram with right angles.-It is a parallelogram with right angles.-This can be proven if I know this figure -This can be proven if I know this figure

is a parallelogram and that one angle is a parallelogram and that one angle is a right angle.is a right angle.

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From Usiskin…From Usiskin…

Which of these are triangles?

U V W X

a. None of these are triangles

b. V only

c. W only

d. W and X only

e. V and W only

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Memorizing the van Memorizing the van Hiele levels is not Hiele levels is not usefuluseful It is useful toIt is useful to

– Know they existKnow they exist– Understand that experiences can Understand that experiences can

move students from one level to move students from one level to anotheranother

– Know that students will not be able Know that students will not be able to do proof—or other logical to do proof—or other logical activities in high school if they are activities in high school if they are not “ready”not “ready”

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Phases of the Phases of the Instructional CycleInstructional Cycle InformationInformation Guided orientationGuided orientation ExplicitationExplicitation Free orientationFree orientation IntegrationIntegration

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Information PhaseInformation Phase

The teacher holds a conversation The teacher holds a conversation with the pupils, in well-known with the pupils, in well-known language symbols, in which the language symbols, in which the context he wants to use becomes context he wants to use becomes clear.clear.

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Information PhaseInformation Phase

It is called a “rhombus.”It is called a “rhombus.”

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Guided Orientation Guided Orientation PhasePhase

– The activities guide the student The activities guide the student toward the relationships of the next toward the relationships of the next level.level.

– The relations belonging to the The relations belonging to the context are discovered and context are discovered and discussed.discussed.

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Guided Orientation Guided Orientation PhasePhase

Fold the rhombus on its axes of Fold the rhombus on its axes of symmetry. What do you notice?symmetry. What do you notice?

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Explicitation PhaseExplicitation Phase

– Under the guidance of the teacher, Under the guidance of the teacher, students share their opinions about students share their opinions about the relationships and concepts they the relationships and concepts they have discovered in the activity.have discovered in the activity.

– The teacher takes care that the The teacher takes care that the

correct technical language is correct technical language is developed and used.developed and used.

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Explicitation PhaseExplicitation Phase

Discuss your ideas with your group, Discuss your ideas with your group, and then with the whole class.and then with the whole class.– The diagonals lie on the lines of The diagonals lie on the lines of

symmetry.symmetry.– There are two lines of symmetry.There are two lines of symmetry.– The opposite angles are congruent.The opposite angles are congruent.– The diagonals bisect the vertex angles.The diagonals bisect the vertex angles.– ……

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Free Orientation PhaseFree Orientation Phase

– The relevant relationships are The relevant relationships are known. known.

– The moment has come for the The moment has come for the students to work independently with students to work independently with the new concepts using a variety of the new concepts using a variety of applications.applications.

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Free Orientation PhaseFree Orientation Phase

The following rhombi are incomplete. The following rhombi are incomplete.

Construct the complete figures.Construct the complete figures.

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Integration PhaseIntegration Phase

The symbols have lost their visual The symbols have lost their visual contentcontent

and are now recognized by their and are now recognized by their properties.properties.

Pierre van HielePierre van Hiele

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Integration PhaseIntegration Phase

Summarize and memorize the Summarize and memorize the properties of a rhombus.properties of a rhombus.

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What we do and what What we do and what we do not do…we do not do… It is customary to illustrate newly It is customary to illustrate newly

introduced technical language with a introduced technical language with a few examples.few examples.

If the examples are deficient, the If the examples are deficient, the technical language will be deficient.technical language will be deficient.

We often neglect the importance of the We often neglect the importance of the third stage, explicitation. Discussion third stage, explicitation. Discussion helps clear out misconceptions and helps clear out misconceptions and cements understanding.cements understanding.

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What we do and what What we do and what we do not do…we do not do… Sometimes we attempt to inform by Sometimes we attempt to inform by

explanation, but this is useless. explanation, but this is useless. Students should learn by doing, not be Students should learn by doing, not be informed by explanation.informed by explanation.

The teacher must give guidance to the The teacher must give guidance to the process of learning, allowing students process of learning, allowing students to discuss their orientations and by to discuss their orientations and by having them find their way in the field having them find their way in the field of thinking.of thinking.

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Instructional Instructional ConsiderationsConsiderations

Visual to Descriptive LevelVisual to Descriptive Level– Language is introduced to describe figures Language is introduced to describe figures

that are observed.that are observed.– Gradually the language develops to form Gradually the language develops to form

the background to the new structure.the background to the new structure.– Language is standardized to facilitate Language is standardized to facilitate

communication about observed properties.communication about observed properties.– It is possible to see congruent figures, but It is possible to see congruent figures, but

it is useless to ask why they are it is useless to ask why they are congruent.congruent.

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Instructional Instructional ConsiderationsConsiderations Descriptive to Relational Level Descriptive to Relational Level

– Causal, logical or other relations Causal, logical or other relations become part of the language.become part of the language.

– Explanation rather than description Explanation rather than description is possible.is possible.

– Able to construct a figure from its Able to construct a figure from its known properties but not able to known properties but not able to give a proof.give a proof.

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Instructional Instructional ConsiderationsConsiderations Relational to Deductive LevelRelational to Deductive Level

– Reasons about logical relations between Reasons about logical relations between theorems in geometry.theorems in geometry.

– To describe the reasoning to someone who To describe the reasoning to someone who does not “speak” this language is futile. does not “speak” this language is futile.

– At the Deductive Level it is possible to At the Deductive Level it is possible to arrange arguments in order so that each arrange arguments in order so that each statement, except the first one, is the statement, except the first one, is the outcome of the previous statements.outcome of the previous statements.

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Instructional Instructional ConsiderationsConsiderations RigorRigor

– Compares axiomatic systems.Compares axiomatic systems.– Explores the nature of logical laws.Explores the nature of logical laws.

““Logical Mathematical Thinking”Logical Mathematical Thinking”

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ConsequencesConsequences

Many textbooks are written with only Many textbooks are written with only the integration phase in place.the integration phase in place.

The integration phase often coincides The integration phase often coincides with the objective of the learning.with the objective of the learning.

Many teachers switch to, or even Many teachers switch to, or even begin, their teaching with this phase, begin, their teaching with this phase, a.k.a. “direct teaching.”a.k.a. “direct teaching.”

Many teachers do not realize that their Many teachers do not realize that their information cannot be understood by information cannot be understood by their pupils.their pupils.

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Children whose geometric thinking Children whose geometric thinking you nurture carefully will be better you nurture carefully will be better able to successfully study the kind able to successfully study the kind of mathematics that Euclid created.of mathematics that Euclid created.

Pierre van HielePierre van Hiele

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NAEP itemNAEP item

The following two items were on The following two items were on the 8th grade 2003 National the 8th grade 2003 National Assessment of Educational Assessment of Educational Progress. The test is given to a Progress. The test is given to a random sample of 8th graders random sample of 8th graders across the United States. Think across the United States. Think about the items and answer the about the items and answer the questions at the bottom of the questions at the bottom of the page.page.

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Alan says that if a figure has four sides, Alan says that if a figure has four sides, it must be a rectangle. Gina does not it must be a rectangle. Gina does not agree. Which of the following figures agree. Which of the following figures shows that Gina is correct?   shows that Gina is correct?   

A) B) A) B) 

C) D) C) D) 

Item #15, NAEP test Item #15, NAEP test 20032003

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How do you think How do you think students did with this students did with this problem?problem? Estimate the percent of 8th grade Estimate the percent of 8th grade

students in the United States who students in the United States who got the correct answer on this got the correct answer on this item. item.

Estimate the percent of 8th grade Estimate the percent of 8th grade students in students in OhioOhio who got the who got the correct answer on this item. correct answer on this item.

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And the results And the results are…….are…….

National National OhioOhio

A)A) 7%7% 3%3%

B) B) 4%4% 3%3%

C)C) 3%3% 2%2%

D)D) 85%85% 90%90%

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Item #10, NAEP test Item #10, NAEP test 20032003

In the figure above, WXYZ is a parallelogram. Which of the following is NOT necessarily true?  

A) Side WX is parallel to side ZY.

B) Side XY is parallel to side WZ.

C) The measures of angles W and Y are equal.

D) The lengths of sides WX and ZY are equal.

E) The lengths of sides WX and XY are equal.

5252

How do you think How do you think students did with this students did with this problem?problem? Estimate the percent of 8th grade Estimate the percent of 8th grade

students in the United States who students in the United States who got the correct answer on this got the correct answer on this item. item.

Estimate the percent of 8th grade Estimate the percent of 8th grade students in students in OhioOhio who got the who got the correct answer on this item. correct answer on this item.

5353

And the results And the results are…….are…….

NationalNational OhioOhio

A)A) 6%6% 4%4%

B)B) 9%9% 9%9%

C)C) 29%29% 24%24%

D)D) 9%9% 11%11%

E)E) 46%46% 50%50%