Level 1: Visualization (Recognition) – Students recognize figures holistically. They may recognize a rectangle but not view a rectangle as containing right angles.
Level 2: Analysis- Students are able to identify relations within a single figure. They may not interrelate figures or properties of figures.
Level 3: Informal deduction (Ordering) - Students can make sense of definitions and are aware of connections among figures. They can logically order the properties of figures by short chains of deductions and understand interrelationships between figures.
Level 4: Formal deduction- Students are able to construct formal proofs.
Level 5: Rigor – Students are able to compare and
contrast different geometries and axiomatic systems.
Descriptions of Levels
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2. Usually refer to visual prototypes of figures and are easily misled by the orientation of figures. 3. Are unable to think of an infinite variation of a particular type of figure. 4. Inconsistently classify figures 5. Inconsistently describe (define) figures by viewing (often visual) conditions as sufficient conditions. |
2. Avoid class inclusions between different classes of figures; for example, squares and rectangles are considered disjoint. 3. Sort figures only in terms of one property. 4. Exhibit an uneconomical use of properties of figures to define them, instead of just using sufficient conditions. 5. Approach the establishment of truth of a statement empirically, use measurement and observation based on a few sketches. |
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2. Classify figure hierarchically 3. Accept different equivalent definitions for the same concept. 4. Implicitly use logical rules of deduction to formulate and handle conjectures. 5. Are uncertain and lack understanding regarding the functions of axioms, definitions, and proof. |
2. Spontaneously make conjectures and self-initiate efforts to deductively verify them. |