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Developing Geometric Thinking: Van Hiele Levels

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Developing Geometric Thinking: Van Hiele Levels

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

Mara Alagic

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

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

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- 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, ...

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

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- 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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- 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

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

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

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- 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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- 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

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- 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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- 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.

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