Developing Geometric Thinking: Van Hiele Levels

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Developing Geometric Thinking: Van Hiele Levels. Mara Alagic. Van Hiele: Levels of Geometric Thinking. Precognition Level 0: Visualization/Recognition Level 1: Analysis/Descriptive Level 2: Informal Deduction Level 3:Deduction Level 4: Rigor. Van Hiele: Levels of Geometric Thinking.

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### Developing Geometric Thinking: Van Hiele Levels

Mara Alagic

Van Hiele: Levels of Geometric Thinking
• Precognition
• Level 0: Visualization/Recognition
• Level 1: Analysis/Descriptive
• Level 2: Informal Deduction
• Level 3:Deduction
• Level 4: Rigor

Mara Alagic

Van Hiele: Levels of Geometric Thinking
• Precognition
• Level 0: Visualization/Recognition
• Level 1: Analysis/Descriptive
• Level 2: Informal Deduction
• Level 3:Deduction
• Level 4: Rigor

Mara Alagic

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

Mara Alagic

Van Hiele: Levels of Geometric Thinking
• Precognition
• Level 0: Visualization/Recognition
• Level 1: Analysis/Descriptive
• Level 2: Informal Deduction
• Level 3:Deduction
• Level 4: Rigor

Mara Alagic

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

Mara Alagic

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.

Mara Alagic

Van Hiele: Levels of Geometric Thinking
• Precognition
• Level 0: Visualization/Recognition
• Level 1: Analysis/Descriptive
• Level 2: Informal Deduction
• Level 3:Deduction
• Level 4: Rigor

Mara Alagic

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

Mara Alagic

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

Mara Alagic

Van Hiele: Levels of Geometric Thinking
• Precognition
• Level 0: Visualization/Recognition
• Level 1: Analysis/Descriptive
• Level 2: Informal Deduction
• Level 3:Deduction
• Level 4: Rigor

Mara Alagic

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

Mara Alagic

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

Mara Alagic

Van Hiele: Levels of Geometric Thinking
• Precognition
• Level 0: Visualization/Recognition
• Level 1: Analysis/Descriptive
• Level 2: Informal Deduction
• Level 3:Deduction
• Level 4: Rigor

Mara Alagic

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

Mara Alagic

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

Mara Alagic

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

Mara Alagic

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.

Mara Alagic