1. Developing Geometric Thinking: The Van Hiele Levels
CI 319
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.
Mara Alagic

2. 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
2008
Mara Alagic

3. 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
2008
Mara Alagic

4. 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
2008
Mara Alagic

5. 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
2008
Mara Alagic

6. 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
2008
Mara Alagic

7. 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
2008
Mara Alagic

8. 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
2008
Mara Alagic

9. 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
2008
Mara Alagic

10. 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
2008
Mara Alagic

11. 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
2008
Mara Alagic

12. 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
2008
Mara Alagic

13. 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
2008
Mara Alagic

14. 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
2008
Mara Alagic

15. 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
2008
Mara Alagic

16. The levels: 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
2008
Mara Alagic

17. 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
2008
Mara Alagic

18. 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.
2008
Mara Alagic