# Geometry levels of development by Pierre and Dina van Hiele

Pierre and Dina van Hiele created a developmental model to help explain how people come to understanding geometry. The first three levels are relevant for elementary students with the fourth level relevant for middle school students and the fifth level for high school geometry students.

## The Five Stages of Geometric Thinking:

### Level One: Visualization

Children at this level can:

• Recognize figures by their physical appearance: Teacher holds up a square pattern block, students go through a pile of pattern blocks and pick out those that have that shape.
• Identify both squares and rectangles
• Think squares are not rectangles
• Are not aware of the properties of these quadrilaterals or other shapes.

### Level Two: Analysis

Children at this level can:

• Classify to some extent: Compares geometric figures to see how they are alike and how they are different.
• Can identify some characteristics of a figure: A square is flat, has four sides, is closed, and its sides have the same length. Both circles and squares are flat, but a square has line segments for sides and a circle does not.
• Canâ€™t see interrelationships between figures

### Level Three: Informal Deduction

Children at this level can:

• Establish interrelationships between figures: A picture of a cube is a cube even though a cube is not flat and the picture is.
• Can identify, describe, and justify relationships among figures: Identifying the results when geometric figures under go change. If a square piece of paper is cut into two parts and the pieces are refitted to make a triangle, then the shape of the paper is changed but its area is not.
• Simple proofs can be followed but not understood completely: If the length of each side of a square is doubled, then the area of the resulting square is four times as great.

### Level Four:  Deduction

Children at this level can:

• Understand the significance of deduction.
• Understand the role of postulates, theorems, and proofs.
• Can write proofs with understanding.

### Level Five:  Rigor

Children at this level can:

• Understand how to work in an axiomatic system.
• Able to make abstract deductions.
• Non-Euclidean geometry can be understood at this highest level.