1. CI 319 Fall 2007 Mara Alagic 1 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.

2. CI 319 Fall 2006Mara Alagic2 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

3. CI 319 Fall 2006Mara Alagic3 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

4. CI 319 Fall 2006Mara Alagic4 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,... 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

5. CI 319 Fall 2006Mara Alagic5 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

6. CI 319 Fall 2006Mara Alagic6 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

7. CI 319 Fall 2006Mara Alagic7 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

8. CI 319 Fall 2006Mara Alagic8 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

9. CI 319 Fall 2006Mara Alagic9 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

10. CI 319 Fall 2006Mara Alagic10 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

11. CI 319 Fall 2006Mara Alagic11 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

12. CI 319 Fall 2006Mara Alagic12 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

13. CI 319 Fall 2006Mara Alagic13 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.