Cognitive & Developmental Issues

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. Through their research they have identified five levels of understanding spatial concepts through which children move sequentially on their way to geometric thinking. There are four characteristics of these levels of thought:

	The levels 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.

Below are listed the five levels with links to more detailed information. Note that only the first three van Hiele Levels and activities are generally appropriate for students in elementary school. 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.

	Visualization
	Analysis
	Informal Deduction
	Deduction
	Rigor

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 explore, to build and take apart, to create and make drawings, and to make observations about shapes in the world around them. 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 his or her students. Diagnostic assessment will help to determine the developmental level in geometry for each of your students.