Developing Geometric Thinking: Van Hiele Levels

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

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

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

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

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

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

Developing Geometric Thinking: Van Hiele Levels