The van Hiele levels MA418 – Spring 2010 McAllister

Who were the van Hiele’s? Dr. Pierre van Hiele and Dr. Dina van Hiele-Geldof were a husband and wife mathematics education research team in the Netherlands They observed children and developed a theory of how individuals progress through 5 levels of geometric thinking

Level 0 - Recognition Students at this level rely only on their perception of the figure. This level is totally concrete. This level is holistic – all of the following might be perceived as triangles:

Level 1 – Analysis of single shapes Rely on perception and is still concrete Know the names and properties of specific geometric figures Do not see relationships between various types of figures: –May not recognize that a square is a special type of rectangle.

Level 2 – Relationships among shapes Beginning to develop abstract understanding of figures Able to create meaningful definitions and form informal arguments and proofs The role and significance of form deduction is not understood

Level 3 – Deductive reasoning Can reason abstractly about figures – no longer depend upon concrete figures to reason and make deductions Can construct a deductive proof using definitions and axioms – such as those found in a high school geometry class

Level 4 - Rigor Have the ability to understand geometry as an axiomatic system Can construct indirect proofs and proofs by contrapositive Can understand non-Euclidean systems Ability to reason is no longer dependent upon any type of diagram or concrete figure.

Effective instruction to achieve advancement Lecture and memorization alone are not effective Students need to be actively engaged with the figures and objects they are studying Touching, drawing, measuring, constructing, folding, discussing, making deductions, etc. is essential