Level 0 (Recognition)

*primarily visual understanding of shapes
*not actively aware of properties of figures

Level 1 (Analysis)

*reference to visual prototypes to characterize shapes
(doors, balls, building, etc.)
*inclusion of irrelevant attributes when identifying shapes such as orientation, or color, or size
*inability to conceive of an infinite variety of shapes

Level 2 (Ordering)

*explicit reference to component parts when identifying shapes, sides, angles, diagonals, etc.
*single attribute sorting of shapes, neglect of other attributes
*prohibiting class inclusions of shapes
*application of a "litany" of necessary properties when defining a shape, instead of sufficient properties

Level 3 (Deduction)

*form complete definitions, i.e. sufficient conditions, for types of shapes
*ability to use equivalent forms of definitions
*explicit use of if - then statements, ability to form informal deductive arguments
*confusion on the role of theorems and axioms

Level 4 (Rigor)

*understanding of the importance of precision in dealing with foundations and interrelationships between structures

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