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# Why Do People Have Difficulty with Geometry? (page 2)

By Pearson Allyn Bacon Prentice Hall
Updated on Jul 20, 2010

#### The van Hiele Levels of Geometric Thought (as Articulated by Clements and Battista, 1992)

 Level 0: Pre-Recognition Children perceive geometric shapes, but are unable to identify many of them. They can distinguish between broad categories, such as curvilinear and rectilinear shapes. but they cannot recognize different types within these broad categories. They do not construct mental representations, or visual images, of shapes. Level 1: Visual Children recognize basic shapes as wholes. They have mental representations of types of shapes. These mental representations are broadly conceived visual prototypes. For example, any triangular shape would fit the prototype of a triangle, even if the sides were curved.A child at the visual level would call a curved shape a triangle if it had a generally triangular shape. Level 2: Descriptive/Analytic Children use specific properties of shapes, rather than visual wholes, to distinguish between them. Reasoning is in terms of combinations of properties. Level 3: Abstract/Relational Children can begin to follow informal logical reasoning about properties of shapes. Concepts such as class inclusion (squares as special cases of rectangles) are understood. Definitions become logical organizers rather than lists of properties. Level 4: Formal Deduction Students become capable of constructing original meaningful proofs. They can produce a logical argument on the basis of "givens." Level 5: Rigor/Metamathematical Students extend their reasoning power to the elaboration and comparison of alternate axiomatic systems of geometry. They become capable of reasoning in the absence of reference models.