Tuesday, July 6, 2010

Why teach Geometry early?

I think Geometry should be introduced at the earliest possible levels. I understand that we teach shapes in kindergarten, but I am talking about more advanced concepts. I believe we need to work on the definition aspect of geometry. Pierre van Hiele and Dina vanHiele-Geldof developed a theory which explains why many students have difficulties with geometry, especially with the application of proofs. They believed that this higher level of thinking was only possible if simpler levels of geometric concepts were mastered. These levels were in an order and involved recognizing a figure, recognizing the properties of a figure, recognizing the relationships between figures and between properties, and finally being able to construct proofs. There are five levels, which are sequential and hierarchical. They are:
Level 1 (Visualization): Students recognize figures by appearance alone,
often by comparing them to a known prototype. The properties of a figure
are not perceived. At this level, students make decisions based on perception,
not reasoning.
Level 2 (Analysis): Students see figures as collections of properties. They
can recognize and name properties of geometric figures, but they do not see
relationships between these properties. When describing an object, a student
operating at this level might list all the properties the student knows, but not
discern which properties are necessary and which are sufficient to describe
the object.
Level 3 (Abstraction): Students perceive relationships between properties
and between figures. At this level, students can create meaningful definitions
and give informal arguments to justify their reasoning. Logical implications
and class inclusions, such as squares being a type of rectangle, are
understood. The role and significance of formal deduction, however, is not
understood.
Level 4 (Deduction): Students can construct proofs, understand the role of
axioms and definitions, and know the meaning of necessary and sufficient
conditions. At this level, students should be able to construct proofs such as
those typically found in a high school geometry class.
Level 5 (Rigor): Students at this level understand the formal aspects of
deduction, such as establishing and comparing mathematical systems.
Students at this level can understand the use of indirect proof and proof by
contrapositive, and can understand non-Euclidean systems.
Mason, M. (1992). The Van hiele levels of geometric understanding. Professional Handbook for Teachers,
Educators have attempted to fulfill these levels of knowledge through traditional means; this has provided us with traditional results. I believe we should focus on level one as early as possible and insure that students are proficient at this level in order to have success with more complex geometry.

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