How To Develope Geometrical Thinking

For children, geometry begins with play. Rich and stimulating instruction in geometry can be provided through playful activities with mosaics, such as pattern blocks or design tiles, with puzzles like tangrams, or with the special seven-piece mosaic shown in figure 1. Teachers might ask, How can children use mosaics, and what geometry do they learn? Before addressing these questions and exploring the potential of the mosaic puzzle for teaching geometry, I note some misconceptions in the teaching of mathematics and present some of my ideas about levels of thinking in geometry.

Misunderstandings in Teaching Mathematics

The teaching of school mathematics - geometry and arithmetic - has been a source of many misunderstandings. Secondary school geometry was for a long time based on the formal axiomatic geometry that Euclid created more than 2000 years ago. His logical construction of geometry with its axioms, definitions, theorems, and proofs was for its time - an admirable scientific achievement. School geometry that is presented in a similar axiomatic fashion assumes that students think on a formal deductive level. However, that is usually not the case, and they lack prerequisite understandings about geometry. This lack creates a gap between their level of thinking and that required for the geometry that they are expected to learn.

A similar misunderstanding is seen in the teaching of arithmetic in elementary school. As had been done by Euclid in geometry, mathematicians developed axiomatic constructions for arithmetic, which subsequently affected the arithmetic taught in schools. In the 1950s, Piaget and I took a stand against this misunderstanding. However, it did not help, for just then, set theory was established as the foundation for number, and school arithmetic based on sets was implemented worldwide in what was commonly called the "new math." For several years, this misconception dominated school mathematics, and the end came only after negative results were reported. Piaget's point of view, which I support affectionately, was that "giving no education is better than giving it at the wrong time." We must provide teaching that is appropriate to the level of children's thinking.

Levels of Geometric Thinking

At what level should teaching begin? The answer, of course, depends on the students' level of thinking. I begin to explain what I mean by levels of thinking by sharing a conversation that two of my daughters, eight and nine years old at the time, had about thinking. Their question was, If you are awake, are you then busy with thinking? "No," one said. "I can walk in the woods and see the trees and all the other beautiful things, but I do not think I see the trees. I see ferns, and I see them without thinking." The other said, "Then you have been thinking, or you knew you were in the woods and that you saw trees, but only you did not use words."

I judged this controversy important and asked the opinion of Hans Freudenthal, a prominent Dutch mathematician and educator. His conclusion was clear: Thinking without words is not thinking. In Structure and Insight (van Hiele 1986), I expressed this point of view, and psychologists in the United States were not happy with it. They were right: If nonverbal thinking does not belong to real thinking, then even if we are awake, we do not think most of the time.