The National Council of Teachers of Mathematics recognizes the importance of geometry and spatial sense in its publication Curriculum and Evaluation Standards for School Mathematics (1989).

Spatial understandings are necessary for interpreting, understanding, and appreciating our inherently geometric world. Insights and intuitions about two- and three-dimensional shapes and their characteristics, the interrelationships of shapes, and the effects of changes to shapes are important aspects of spatial sense. Children who develop a strong sense of spatial relationships and who master the concepts and language of geometry are better prepared to learn number and measurement ideas, as well as other advanced mathematical topics. (p. 48)

And in its Principles and Standards for School Mathematics, NCTM has placed the Standard for Geometry at every grade level from preK to 12.

Arithmetic is an important corner of mathematics, but too often we neglect the rest of the field. Geometry suffers because we have the mistaken impression that it doesn't become real, serious mathematics until it gets abstract and we deal with proof. But geometry is important, even in its less formal form. Here's why.

• First, the world is built of shape and space, and geometry is its mathematics.

• Second, informal geometry is good preparation. Students have trouble with abstraction if they lack sufficient experience with more concrete materials and activities.

• Third, geometry has more applications than just within the field itself. Often students can solve problems from other fields more easily when they represent the problems geometrically.

• And finally—a related point—many people think well visually. Geometry can be a doorway to their success in mathematics.

Consider this: Children who play with Tinkertoy^®, the construction system, develop informal experience and understanding of isosceles right triangles. They know that if the legs are blue, the hypotenuse is red. When they study geometry or learn the Pythagorean theorem, they already have the background textbook writers and teachers may unconsciously take for granted. Children who miss out on playing with triangles—for whatever reason—must get this experience and understanding somewhere else.

So teachers, be watchful. When you see a student who "just doesn't get it," you might ask yourself, is it a lack of talent or a lack of experience? Think about the out-of-school experiences that might have given the student the needed background—and try to provide something that serves the same purpose in the classroom.

The activities in this lab will help you bring this practice to your teaching. Before you try them, read the introduction to each category of activities—shape and space. It outlines the rationale for teaching the topic, briefly describes the activities, explains how the activities relate to different grade levels or to daily life, and connects the topic to national standards. Then follow the links to the activities themselves. There you can access a background page that elaborates on the rationale and the grade-level information. You may also find additional connections to standards for that specific activity as well as related resources for investigating the topic further.

Collectively, the activities explore sophisticated mathematics without using formal geometry. All you have to do is think about shape and space—and maybe do a little calculation.

Are you ready? Then start your exploration with either activities about shape or about space.