Van Hiele Levels One

8/3/2019 Van Hiele Levels One

1/23

Van Hiele LevelsA Model for Geometric

Understanding


2/23

What are the levels?


3/23

There are five levels.

VisualizationAnalysisInformal DeductionDeductionRigor


4/23

The Educational Researchers

Pierre van HieleDina van Hiele, hiswifeDutch educatorsDeveloped theoryin 1950s


5/23

Characteristics of Levels

Levels are sequential;move through priorlevels to get to a levelLevels not agedependentNeed appropriateexperiences toadvanceInappropriateexperiences inhibitlearning


6/23

VisualizationStudents recognizeand name shapes byappearanceDo not recognizeproperties or if theydo, do not use themfor sorting orrecognitionMay not recognizeshape in differentorientation (e.g.,shape at right notrecognized as square)


7/23

Analysis

Students can identifysome properties of shapesUse appropriatevocabularyCannot explainrelationship betweenshape and properties(e.g., why is secondshape not arectangle?)


8/23

Analysis (continued)

Understand thatsize andorientation do notdetermine shapeDo not makeconnectionsbetween differentshapes and theirproperties (e.g.,what do 2 shapeshave in common?)


9/23

Informal Deduction

Students can seerelationships of properties withinshapesRecognizeinterrelationshipsamong shapes orclasses of shapes(e.g., where does arhombus fit amongall quadrilaterals?)


10/23


11/23

DeductionUsually not reached beforehigh school; maybe notuntil collegeCan construct proofs in anaxiomatic system (e.g.,

can prove that if two sidesand the included angle of one triangle are congruentwith the correspondingsides and angle of anothertriangle, the 2 triangles arecongruent)

B

A C

FD

E


12/23


13/23

Rigor

Some studentsattain this level incollege

Can comparedifferent axiomsystems (e.g.,Euclidean versussphericalgeometry)


14/23

Implications for InstructionAll Levels

Use the levels todiagnose where yourstudents areIt is important that

students have lots of experiences at theappropriate levelLevels are not agedependent, so you canmove students alongthe continuum at anyage


15/23

Implications for InstructionVisualization

Make sure studentssee shapes indifferent

orientationsMake sure studentssee different sizesof each shapeInstruction shouldbe informal


16/23

Implications for InstructionVisualization

Provide activities thathave students sortshapes, identify anddescribe shapes (e.g.,Venn diagrams)Have students usemanipulatesBuild and draw shapesPut together and takeapart shapes


17/23

Implications for InstructionAnalysis

Activities emphasizeclasses of shapes andtheir properties (e.g.,all squares havecongruent sides, all 4interior angles are 90degrees, diagonals areperpendicular

bisectors, 4 lines of symmetry, 90 degreerotational symmetry)


18/23


19/23

Implications for InstructionAnalysis

Use technology(e.g., GeometersSketchpad) toexplore propertiesClassify shapesbased on lists of properties

Solve problemsinvolvingproperties of shapes

I have, Who has GameCreate a rectangle in

GeometersSketchpad; measurelengths of twodiagonals; measuredistances fromvertices to point of intersection of diagonals


20/23

Implications for InstructionInformal Deduction

Activities involving if then thinking(e.g., if its a

square, then ) Creating diagramsshowingrelationshipsbetween differentshapes (see right)


21/23


22/23

Implications for InstructionInformal Deduction

Use examples andcounterexamplesto develop a

definition (e.g.,convex polygon)Make and testconjectures aboutshapes and theirproperties

Convex PolygonsNot Convex Polygons


23/23

Next Steps

How can you use what you havelearned about van Hiele levels toimprove the teaching and learning of geometry in your class?

Documents

Van Hiele Levels One