Developing Geometric Thinking: Van Hiele’s Levels Mara Alagic

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.