Geometry

1. Geometry The Van Hiele Levels of Geometric Thought

2. Level 0 - Visualization • Recognize and name figures • Make measurements and talk about properties of shapes • Not abstracted from shape • Appearance defines shapes • May see square and diamond as different shapes • Begin classification of shapes – similarities and differences

3. Level 1 - Analysis • Consider all shapes within a class (i.e. all rectangles) • Features of all shapes within a class (i.e. what makes a rectangle?) • Observe irrelevant features (size, orientation) • Generalize properties of a class • May be able to talk about squares, rectangles, and parallelograms but not connect them to each other

4. Level 2 – Informal Deduction • Begin to recognize relationships among properties • If-then reasoning • Minimum conditions classification • Logical arguments about properties • Formal deductive arguments about shapes and their properties • “Proof” informal and intuitive

5. Level 3 - Deduction • Conjectures concerning relationships among properties • Analysis of informal arguments move to structured system: axioms, definitions, theorems, corollaries, postulates • Necessary to establish truth • Minimum set of assumptions – derive truth (logical) • Arguments based on more than intuition – logic • Understands that what appears to be true by intuition needs to be proven with logic • Product is deductive axiomatic systems

6. Level 4 - Rigor • Deductive axiomatic systems are objects of study • Appreciation of the distinctions and relationships between different axiomatic systems • Compare and contrast different axiomatic systems