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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:

Teacher and three studentsBelow 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

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

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.

  1. 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.
  2. Pierre M. Van Hiele, "Developing Geometric Thinking Through Activities That Begin with Play," Teaching Children Mathematics 5, no. 6 (February 1999): pp. 310-16.
  3. John A. Van de Walle, Elementary and Middle School Mathematics: Teaching Developmentally, 4th ed. (New York: Addison Wesley Longman, 2001), pp. 310-11.
  4. Van de Walle, Elementary and Middle School Mathematics, pp. 311-12.
  5. Van de Walle, Elementary and Middle School Mathematics, pp. 311-49.

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