van Hiele Levels of Geometric Thinking

Objects of Thought / Products of Thought
Level 1 (or 0) Visualization / Shapes
Geometric figures are seen as entities without awareness of parts or properties or relationships between parts of the figure / Classes of shapes
This is a rectangle because it looks like a door, or because it looks like one
Level 2 (or1) Analysis / classes of shapes
  • Properties are noticed
  • They are seen as unrelated to one another
/ properties of shapes
This is a rectangle because is has 4 sides, 4 right angles, opposite sides are parallel, it is closed, opposite sides are congruent, diagonals bisect each other, adjacent sides are perpendicular…..
Level 3 (or 2)
Informal Deduction / Properties of shapes
  • definitions have meaning
  • relationships seen between properties and between figures – can deduce properties from other properties
  • Logical implications are understood so Informal arguments be followed
Hierarchical thinking can be used
  • The role of deductive reasoning is not understood
/ Relationships among properties
It is a rectangle because it is a parallelogram with right angles (uses minimal number of properties)
Objects of Thought / Products of Thought
Level 4 (or 3)
Formal
Deduction / Relationships among properties
  • can construct proofs
  • understand role of axioms and definitions
  • meaning of necessary and sufficient conditions
  • can supply reasons for steps in a proof
/ Deductive systems of properties
Given that this is a parallelogram and one angle is a right angle, I can prove that it is a rectangle

Questions and Activities:

It is important that the level of the activities match the level of the student.

Initially, students need to engage in guided, structured activities

Sorting and classifying shapes

How are they alike, how different?

Put together and take apart shapes in 2 and 3 dimensions

Draw shapes

There needs to be explicit discussion of objects being studied and properties. But the language has to match the level.

Focus on properties of shapes rather than just identifying that you have a rectangle etc.

Determine properties that are true for ALL rectangles – look at ALL/Some types of statements

Later, activities should be more open-ended

Encourage students to make and test conjectures – will an observation hold all of the time?

What properties are necessary and sufficient to guarantee that you have a certain polygon – what properties of diagonals guarantee that you have a rhombus?

Finally the teacher helps students develop an overview of the material and try integrate ideas.

The formal deduction is most likely not developed until high school

Try to have students at level 2 when entering high school geometry

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