1. Geometry The Van Hiele Levels of Geometric Thought

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

3. Level 1 - Analysis Consider all shapes within a class (i.e. all rectangles) Consider all shapes within a class (i.e. all rectangles) Features of all shapes within a class (i.e. what makes a rectangle?) Features of all shapes within a class (i.e. what makes a rectangle?) Observe irrelevant features (size, orientation) Observe irrelevant features (size, orientation) Generalize properties of a class Generalize properties of a class May be able to talk about squares, rectangles, and parallelograms but not connect them to each other 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 Begin to recognize relationships among properties If-then reasoning If-then reasoning Minimum conditions classification Minimum conditions classification Logical arguments about properties Logical arguments about properties Formal deductive arguments about shapes and their properties Formal deductive arguments about shapes and their properties “Proof” informal and intuitive “Proof” informal and intuitive

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

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