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

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Developing geometric thinking van hiele levels

Developing Geometric Thinking: Van Hiele Levels

Mara Alagic


Van hiele levels of geometric thinking

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 thinking1

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

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 thinking2

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

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

    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 thinking3

    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

    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

    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 thinking4

    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

    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

    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 thinking5

    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

    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


    The levels are characterized by differences in objects of thought

    The levels are “characterized by differences in objects of thought”:

    • 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

    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

    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


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