# Developing Geometric Thinking: Van Hiele Levels - PowerPoint PPT Presentation

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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

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

### 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

• 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