I learned about the Van Hiele model of geometric understanding from the site http://www.homeschoolmath.net/teaching/geometry.php. I liked the version of the model offered by the site because it seemed very practical. The model is best used as a guide, for elementary teachers to foster development of geometric skills or as a diagnostic tool to help high school teachers to understand what is causing a particular student’s difficulties in high school geometry.
The model was developed by two Dutch researchers, Dina van Hiele-Geldof and Pierre van Hiele as a method for analyzing the difficulties their students were having in higher level geometry. They believe that all learners must progress through a sequence of stages of mastery in order to reach a point where they can undertake the reasoning needed for high school geometry. In their model, one must master each stage in turn before proceeding to the next. The first two levels seem at about a primary school level, the third middle school, the fourth is the level used for high school geometry, and the fifth is the level of upper level undergraduate mathematics. In a nutshell, here are the levels:
Level 1 – visual. Geometric figures are recognized based on their appearance
Level 2 – descriptive/analytic. Students can identify properties of figures and recognize them by their properties. They cannot tell a difference between the necessary and sufficient defining properties of a shape, and extra properties of a shape. For example, they would understand that a recangle has four sides and four right angles.
Level 3 – abstract/relational. Students can now understand and form abstract definitions, distinguish between necessary and sufficient conditions for a concept, and understand relationships between different shapes. For example, they would understand that a recangle has four sides and four right angles and also understand that this description is sufficient to describe a rectangle.
Level 4 – formal deduction. Students can reason formally using definitions, axioms, and theorems. They can construct deductive proofs starting from the givens, and producing statements that ultimately justify the statement they are supposed to prove.
Level 5 – rigor/metamathematical. Students can reason formally and compare different axiomatic systems.
The site stresses that
“This theory is not perfect but based on other research, it seems to model the progress of geometrical thinking.”
“The important point is that a lot of the geometry taught before high school does NOT foster students into higher level of geometrical thinking.”
I like the van Hiele model. If for nothing else, it is the first educational model I’ve come across that applies well to mathematical reasoning. It does seem like a reaonable framework to use to judge someone’s reasoning level.
In my opinion, one danger with theories of education is they end up being used as an inflexible set of rules. This site seems to have the right idea about using the van Hiele model: Use it as a tool to construct a curriculum for children that will prepare them for the rigors of axiomatic geometry.