# 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.