Van Hiele Levels

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Van Hiele Levels

• A Model for Geometric Understanding

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What are the levels?
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There are five levels.

• Visualization
• Analysis
• Informal Deduction
• Deduction
• Rigor

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The Educational Researchers

• Pierre van Hiele
• Dina van Hiele, his wife
• Dutch educators
• Developed theory in 1950s

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Characteristics of Levels

• Levels are sequential move through prior levels
  to get to a level
• Levels not age dependent
• Need appropriate experiences to advance
• Inappropriate experiences inhibit learning

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Visualization

• Students recognize and name shapes by appearance
• Do not recognize properties or if they do, do not
  use them for sorting or recognition
• May not recognize shape in different orientation
  (e.g., shape at right not recognized as square)

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Analysis

• Students can identify some properties of shapes
• Use appropriate vocabulary
• Cannot explain relationship between shape and
  properties (e.g., why is second shape not a
  rectangle?)

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Analysis (continued)

• Understand that size and orientation do not
  determine shape
• Do not make connections between different shapes
  and their properties (e.g., what do 2 shapes have
  in common?)

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Informal Deduction

• Students can see relationships of properties
  within shapes
• Recognize interrelationships among shapes or
  classes of shapes (e.g., where does a rhombus fit
  among all quadrilaterals?)

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Informal Deduction (contd)

• Can follow informal proofs (e.g., every square is
  a rhombus because all sides are congruent)
• Cannot see which steps of proof can be
  interchanged
• Cannot construct a proof

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Deduction

• Usually not reached before high school maybe not
  until college
• Can construct proofs in an axiomatic system
  (e.g., can prove that if two sides and the
  included angle of one triangle are congruent with
  the corresponding sides and angle of another
  triangle, the 2 triangles are congruent)

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Deduction (continued)

• Understand the importance of deduction in
  creating a coherent geometry
• Understand how postulates, axioms, and
  definitions are used in proofs (e.g., how
  definition of angle used in SAS proof)

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Rigor

• Some students attain this level in college
• Can compare different axiom systems (e.g.,
  Euclidean versus spherical geometry)

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Implications for Instruction All Levels

• Use the levels to diagnose where your students
  are
• It is important that students have lots of
  experiences at the appropriate level
• Levels are not age dependent, so you can move
  students along the continuum at any age

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Implications for InstructionVisualization

• Make sure students see shapes in different
  orientations
• Make sure students see different sizes of each
  shape
• Instruction should be informal

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Implications for InstructionVisualization

• Provide activities that have students sort
  shapes, identify and describe shapes (e.g., Venn
  diagrams)
• Have students use manipulates
• Build and draw shapes
• Put together and take apart shapes

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Implications for InstructionAnalysis

• Activities emphasize classes of shapes and their
  properties (e.g., all squares have congruent
  sides, all 4 interior angles are 90 degrees,
  diagonals are perpendicular bisectors, 4 lines of
  symmetry, 90 degree rotational symmetry)

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Implications for InstructionAnalysis

• Work with concrete or virtual manipulatives
• Define properties, make measurements and look for
  patterns
• Explore what happens if a measurement or property
  is changed
• Discuss what is sufficient to define a shape
  (e.g., rectangle)

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Implications for InstructionAnalysis

• Use technology (e.g., Geometers Sketchpad) to
  explore properties
• Classify shapes based on lists of properties
• Solve problems involving properties of shapes

• I have, Who has Game
• Create a rectangle in Geometers Sketchpad
  measure lengths of two diagonals measure
  distances from vertices to point of intersection
  of diagonals

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Implications for InstructionInformal Deduction

• Activities involving if then thinking (e.g.,
  if its a square, then )
• Creating diagrams showing relationships between
  different shapes (see right)

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Implications for InstructionInformal Deduction

• Activities that ask what properties are necessary
  and/or sufficient to be a certain shape
• Use informal deductive language (all, some,
  none, if then)

• If all squares are rectangles, does that mean all
  rectangles are squares?
• If the two diagonals of a quadrilateral bisect
  each other, does that guarantee the shape is a
  rectangle?

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Implications for InstructionInformal Deduction

• Use examples and counterexamples to develop a
  definition (e.g., convex polygon)
• Make and test conjectures about shapes and their
  properties

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Next Steps

• How can you use what you have learned about van
  Hiele levels to improve the teaching and learning
  of geometry in your class?