Visualization is an important part of geometrical thinking. It's the skill you use when you pretend to be somewhere else and imagine how that place looks, or when you fancy how a situation would look if things were just a little bit different.

But visualization is especially problematic in three dimensions—perhaps because math curricula do not emphasize three-dimensional geometry. Some people have a hard time, for example, rotating an object in their minds to see how it would look from a different angle. When looking at a map, others find it hard both to imagine where they are on the map and to grasp the relationships of the map objects around them.

Probe your space visualization skills and get practice with the three activities below. Review the background to each activity for additional ideas and connections.

• In I Took a Trip on a Train, you get a map and some snapshots and have to put the snapshots in order.

• Plot Plans and Silhouettes asks you to design a structure that would have the given silhouettes as seen from the front and the side.

• Shadows gives you a three-dimensional figure and a shape of a shadow, and asks whether the figure could cast the shadow given.

If you think that practicing spatial visualization is the same as fantasizing, you are right—in a sense. But it's practical fantasy. Consider these examples:

• You park your car in the unshaded area of a lot on a sunny morning, but you know the shadow of that big tree will move to cover the car in the hot afternoon.

• You are facing south, looking at a map with north at the top. If you want to get to something that is to your left on the map, you will have to turn right in real life.

• You are giving directions on the telephone, telling someone how to make a paper airplane. You are not actually making the airplane but you can imagine what the steps are and how the plane looks when it is only partly done.

• You are packing presents into a carton for mailing. You use your imagination to picture how you would arrange the presents in the carton before you actually pick them up.

So spatial visualization has everyday application. It's not just preparation for the abstract geometry class you may never take. And if you, for any reason, consider yourself (or your students) not very good at this kind of thing, be assured; it can be learned. You do improve with practice.

The new NCTM Standards explicitly include visualization as an item under Chapter 3: Geometry. In the detailed description for grades 3–5, for example, the Council elaborates:

A great deal of the work students do with three-dimensional shapes involves visualization. Through representing three-dimensional shapes in two dimensions and constructing three-dimensional shapes from two-dimensional representations, students learn to describe these shapes with more precision.

In the overview for grades preK–12, it also explains:

Over the grades, students should become comfortable analyzing and drawing perspective views, counting component parts, and describing attributes that cannot be seen but can be inferred. Students need to learn to physically and mentally change the position, orientation, and size of objects in systematic ways as they develop their understandings about congruence, similarity, and transformations.