1. Developing Geometric Thinking: Van Hiele’s Levels Mara Alagic

2. Summer 2004 Mara Alagic2 Van Hiele: Levels of Geometric Thinking n Precognition n Level 0: Visualization/Recognition n Level 1: Analysis/Descriptive n Level 2: Informal Deduction n Level 3: Deduction n Level 4: Rigor

3. Summer 2004 Mara Alagic3 Van Hiele: Levels of Geometric Thinking n Precognition n Level 0: Visualization/Recognition n Level 1: Analysis/Descriptive n Level 2: Informal Deduction n Level 3:Deduction n Level 4: Rigor

4. Summer 2004 Mara Alagic4 Visualization/Recognition n The student identifies, names compares and operates on geometric figures according to their appearance n For example, students recognize rectangles by its form but, a rectangle seems different to them then a square n At this level rhombus is not recognized as a parallelogram

5. Summer 2004 Mara Alagic5 Van Hiele: Levels of Geometric Thinking n Precognition n Level 0: Visualization/Recognition n Level 1: Analysis/Descriptive n Level 2: Informal Deduction n Level 3:Deduction n Level 4: Rigor

6. Summer 2004 Mara Alagic6 Analysis/Descriptive n Students analyze figures in terms of their components and relationships between components; discover properties/rules of a class of shapes empirically by –folding –measuring –using a grid or a diagram,... n They are not yet capable of differentiating these properties into definitions & propositions n Logical relations are not yet fit-study object

7. Summer 2004 Mara Alagic7 Analysis/Descriptive: 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

8. Summer 2004 Mara Alagic8 Van Hiele: Levels of Geometric Thinking n Precognition n Level 0: Visualization/Recognition n Level 1: Analysis/Descriptive n Level 2: Informal Deduction n Level 3:Deduction n Level 4: Rigor

9. Summer 2004 Mara Alagic9 Informal Deduction n Students logically interrelate previously discovered properties/rules by giving or following informal arguments n The intrinsic meaning of deduction is not understood by the student n The properties are ordered - deduced from one another

10. Summer 2004 Mara Alagic10 Informal Deduction: Examples A square is a rectangle because it has all the properties of a rectangle n Students can conclude the equality of angles from the parallelism of lines: In a quadrilateral, opposite sides being parallel necessitates opposite angles being equal

11. Summer 2004 Mara Alagic11 Van Hiele: Levels of Geometric Thinking n Precognition n Level 0: Visualization/Recognition n Level 1: Analysis/Descriptive n Level 2: Informal Deduction n Level 3:Deduction n Level 4: Rigor

12. Summer 2004 Mara Alagic12 Deduction (1) n Students prove theorems deductively and establish interrelationships among networks of theorems in the Euclidean geometry n Thinking is concerned with the meaning of deduction, with the converse of a theorem, with axioms, and with necessary and sufficient conditions

13. Summer 2004 Mara Alagic13 Deduction (2) n Students seek to prove facts inductively n It would be possible to develop an axiomatic system of geometry, but the axiomatics themselves belong to the next (fourth) level

14. Summer 2004 Mara Alagic14 Van Hiele: Levels of Geometric Thinking n Precognition n Level 0: Visualization/Recognition n Level 1: Analysis/Descriptive n Level 2: Informal Deduction n Level 3:Deduction n Level 4: Rigor

15. Summer 2004 Mara Alagic15 Rigor n Students establish theorems in different postulational systems and analyze/compare these systems n Figures are defined only by symbols bound by relations n A comparative study of the various deductive systems can be accomplished n Students have acquired a scientific insight into geometry

16. Summer 2004 Mara Alagic16 Levels: Differences in objects of thought n geometric figures n classes of figures & properties of these classes n students act upon properties, yielding logical orderings of these properties n operating on these ordering relations n foundations (axiomatic) of ordering relations

17. Summer 2004 Mara Alagic17 Major Characteristics of the Levels n the levels are sequential n each level has its own language, set of symbols, and network of relations n what is implicit at one level becomes explicit at the next level n material taught to students above their level is subject to reduction of level n progress from one level to the next is more dependant on instructional experience than on age or maturation n one goes through various “phases” in proceeding from one level to the next

18. Summer 2004 Mara Alagic18 References n 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). n 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. n Fuys, D., Geddes, D., & Tischler, R. (1988). The van Hiele model of Thinking in Geometry Among Adolescents. JRME Monograph Number 3.