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.

Fall 2005Mara 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

Fall 2005Mara 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

Fall 2005Mara 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

Fall 2005Mara 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

Fall 2005Mara 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,... 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

Fall 2005Mara 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

Fall 2005Mara 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

Fall 2005Mara 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

Fall 2005Mara 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

Fall 2005Mara 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

Fall 2005Mara 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

Fall 2005Mara 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

Fall 2005Mara 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

Fall 2005Mara 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

Fall 2005Mara Alagic 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

Fall 2005Mara 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

Fall 2005Mara 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.