Developing Geometric Thinking: The Van Hiele Levels

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Developing Geometric Thinking The Van Hiele
Levels

• Adapted from
• Van Hiele, P. M. (1959). Development and learning
  process. Acta Paedogogica Ultrajectina (pp.
  1-31). Groningen J. B. Wolters.

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Van Hiele Levels of Geometric Thinking

• Precognition
• Level 0 Visualization/Recognition
• Level 1 Analysis/Descriptive
• Level 2 Informal Deduction
• Level 3Deduction
• Level 4 Rigor

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Van Hiele Levels of Geometric Thinking

• Precognition
• Level 0 Visualization/Recognition
• Level 1 Analysis/Descriptive
• Level 2 Informal Deduction
• Level 3Deduction
• Level 4 Rigor

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Visualization or Recognition

• The student identifies, names compares and
  operates on geometric figures according to their
  appearance
• For example, the student recognizes rectangles by
  its form but, a rectangle seems different to
  her/him then a square
• At this level rhombus is not recognized as a
  parallelogram

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Van Hiele Levels of Geometric Thinking

• Precognition
• Level 0 Visualization/Recognition
• Level 1 Analysis/Descriptive
• Level 2 Informal Deduction
• Level 3Deduction
• Level 4 Rigor

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Analysis/Descriptive

• The student analyzes figures in terms of their
  components and relationships between components
  and discovers properties/rules of a class of
  shapes empirically by
• folding /measuring/ using a grid or diagram, ...
• He/she is not yet capable of differentiating
  these properties into definitions and
  propositions
• Logical relations are not yet fit-study object

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Analysis/Descriptive An Example

• If a student knows that the
• diagonals of a rhomb are perpendicular
• she must be able to conclude that,
• if two equal circles have two points in common,
  the segment joining these two points is
  perpendicular to the segment joining centers of
  the circles

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Van Hiele Levels of Geometric Thinking

• Precognition
• Level 0 Visualization/Recognition
• Level 1 Analysis/Descriptive
• Level 2 Informal Deduction
• Level 3Deduction
• Level 4 Rigor

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Informal Deduction

• The student logically interrelates previously
  discovered properties/rules by giving or
  following informal arguments
• The intrinsic meaning of deduction is not
  understood by the student
• The properties are ordered - deduced from one
  another

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Informal Deduction Examples

• A square is a rectangle because it has all the
  properties of a rectangle.
• The student can conclude the equality of angles
  from the parallelism of lines In a
  quadrilateral, opposite sides being parallel
  necessitates opposite angles being equal

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Van Hiele Levels of Geometric Thinking

• Precognition
• Level 0 Visualization/Recognition
• Level 1 Analysis/Descriptive
• Level 2 Informal Deduction
• Level 3Deduction
• Level 4 Rigor

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Deduction (1)

• The student proves theorems deductively and
  establishes interrelationships among networks of
  theorems in the Euclidean geometry
• Thinking is concerned with the meaning of
  deduction, with the converse of a theorem, with
  axioms, and with necessary and sufficient
  conditions

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Deduction (2)

• Student seeks to prove facts inductively
• It would be possible to develop an axiomatic
  system of geometry, but the axiomatics themselves
  belong to the next (fourth) level

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Van Hiele Levels of Geometric Thinking

• Precognition
• Level 0 Visualization/Recognition
• Level 1 Analysis/Descriptive
• Level 2 Informal Deduction
• Level 3Deduction
• Level 4 Rigor

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Rigor

• The student establishes theorems in different
  postulational systems and analyzes/compares these
  systems
• Figures are defined only by symbols bound by
  relations
• A comparative study of the various deductive
  systems can be accomplished
• The student has acquired a scientific insight
  into geometry

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The levels Differences in objects of thought

• geometric figures gt classes of figures
  properties of these classes
• students act upon properties, yielding logical
  orderings of these properties gt operating on
  these ordering relations
• foundations (axiomatic) of ordering relations

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Major Characteristics of the Levels

• the levels are sequential each level has its own
  language, set of symbols, and network of
  relations
• what is implicit at one level becomes explicit
  at the next level material taught to students
  above their level is subject to reduction of
  level
• progress from one level to the next is more
  dependant on instructional experience than on age
  or maturation
• one goes through various phases in proceeding
  from one level to the next

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References

• Van Hiele, P. M. (1959). Development and learning
  process. Acta Paedogogica Ultrajectina (pp.
  1-31). Groningen J. B. Wolters.
• Van Hiele, P. M. Van Hiele-Geldof, D. (1958).
• A method of initiation into geometry at secondary
  schools. In H. Freudenthal (Ed.). Report on
  methods of initiation into geometry (pp.67-80).
  Groningen J. B. Wolters.
• Fuys, D., Geddes, D., Tischler, R. (1988). The
  van Hiele model of Thinking in Geometry Among
  Adolescents. JRME Monograph Number 3.