Van Hiele Levels of Geometric Reasoning

The work of two Dutch educators, Pierre van Hiele and Dina van Hiele-Geldof, has given us a vision around which to design geometry curriculum.3 Through their research they have identified five levels of understanding spatial concepts through which children move sequentially on their way to geometric thinking.4 There are four characteristics of these levels of thought:

The Van Hiele levels of geometric reasoning are sequential. Students must pass through all prior levels to arrive at any specific level.
These levels are not age-dependent in the way Piaget described development.
Geometric experiences have the greatest influence on advancement through the levels.
Instruction and language at a level higher than the level of the student may inhibit learning.5

Teacher and three students Below are listed the van Hiele levels and activities that would be appropriate for students at each level. Most students in grades K-3 will be at Level 1 (visualization) while students in grades 4-5 may be at Level 2 (analysis) and some possibly at Level 3 (informal deduction). It is important for elementary school teachers to provide their students with experiences that will help them move from Level 1 to Level 3 by the end of the eighth grade.

(Note that the five van Hiele levels are sometimes numbered 0-4, as opposed to 1-5, which is used in this document.)

1. Visualization

Students can name and recognize shapes by their appearance, but cannot specifically identify properties of shapes. Although they may be able to recognize characteristics, they do not use them for recognition and sorting.

Suggestions for instruction using visualization6

sorting, identifying, and describing shapes
manipulating physical models
seeing different sizes and orientations of the same shape as to distinguish characteristics of a shape and the features that are not relevant
building, drawing, making, putting together, and taking apart shapes.

2. Analysis

Students begin to identify properties of shapes and learn to use appropriate vocabulary related to properties, but do not make connections between different shapes and their properties. Irrelevant features, such as size or orientation, become less important, as students are able to focus on all shapes within a class. They are able to think about what properties make a rectangle. Students at this level are able to begin to talk about the relationship between shapes and their properties.

Suggestions for instruction using analysis

shifting from simple identification to properties, by using concrete or virtual models to define, measure, observe, and change properties
using models and/or technology to focus on defining properties, making property lists, and discussing sufficient conditions to define a shape
doing problem solving, including tasks in which properties of shapes are important components
classifying using properties of shapes.

3. Informal Deduction

Students are able to recognize relationships between and among properties of shapes or classes of shapes and are able to follow logical arguments using such properties.

Suggestions for instruction using informal deduction

doing problem solving, including tasks in which properties of shapes are important components
using models and property lists, and discussing which group of properties constitute a necessary and sufficient condition for a specific shape
using informal, deductive language ("all," "some," "none," "if-then," "what if," etc.)
investigating certain relationships among polygons to establish if the converse is also valid (e.g., "If a quadrilateral is a rectangle, it must have four right angles; if a quadrilateral has four right angles, must it also be a rectangle?")
using models and drawings (including dynamic geometry software) as tools to look for generalizations and counter-examples
making and testing hypotheses
using properties to define a shape or determine if a particular shape is included in a given set.

Note: Students usually do not reach Levels 4 and 5 until high school or college, but teachers should be aware of these levels nonetheless.

4. Deduction

Students can go beyond just identifying characteristics of shapes and are able to construct proofs using postulates or axioms and definitions. A typical high school geometry course should be taught at this level.

5. Rigor

This is the highest level of thought in the van Hiele hierarchy. Students at this level can work in different geometric or axiomatic systems and would most likely be enrolled in a college level course in geometry.

Implications of van Hiele for instruction

Geometry taught in the elementary school should be informal. Such informal geometry activities should be exploratory and hands-on, in order to provide children with the opportunity to investigate, to build and take apart, to create and make drawings, and to make observations about shapes in the world around them.7 This provides the basis for more formal activities at higher levels.

Teaching a geometry lesson at one van Hiele level when students are functioning at a lower level may hinder student learning. For example, a teacher asks his or her students to play the "What am I?" game with properties of geometric figures, saying, "I have four sides and all of my interior angles are right angles. What am I?" To answer this question, a student must be functioning at Level 2 (analysis) in van Hiele's model of geometric reasoning.

If the students in this class are functioning at Level 1 (visualization), where they recognize a figure by its appearance, they will not be able to play the game. If students are at different levels in one class, the teacher must use differentiated instruction to meet the needs of all of his or her students.

Diagnostic assessment will help to determine the developmental level in geometry for each student.

Mary L. Crowley, "The Van Hiele Model of the Development of Geometric Thought," in Learning and Teaching Geometry, K-12, ed. Mary M. Lindquist (Reston, Va.: National Council of Teachers of Mathematics, 1987), pp. 1-16.
Pierre M. Van Hiele, "Developing Geometric Thinking Through Activities That Begin with Play," Teaching Children Mathematics 5, no. 6 (February 1999): pp. 310-16.
John A. Van de Walle, Elementary and Middle School Mathematics: Teaching Developmentally, 4th ed. (New York: Addison Wesley Longman, 2001), pp. 310-11.
Van de Walle, Elementary and Middle School Mathematics, pp. 311-12.
Van de Walle, Elementary and Middle School Mathematics, pp. 311-49.