about Einstein, well-known for his prodigious powers of spatial thinking, that in developing the theory of relativity, he imagined himself hurtling in space at the speed of light. Enactions can underlie the ordering of spatial transformations when several transformations are applied in sequence to the same representation. Thus, when people have to apply several transformations to the same figure in order to solve geometric analogies, they first move the figure, then change its orientation, then determine its size, and finally add parts to the figure. This order corresponds exactly to the order in which people draw figures; they first decide where to begin drawing, then what direction to draw in, and then how far to draw (Novick and Tversky, 1987). Thus, the order of performing the mental transformations in this complex reasoning task corresponds to the order of executing the analogous external task, suggesting that the enactment of drawing is internalized and applied to other mental tasks.

The Process of Complex Spatial Reasoning Representations and transformations are the components that enter into complex spatial reasoning. Combining components enables complex spatial reasoning, such as solving geometric analogies or developing relativity theory. In actual practice, spatial reasoning often uses several representations, several comparisons, and multiple transformations. For example, planning a route requires determining a location, then a direction of movement to the next location, then reorientation at each successive location. Deciding which route is the most efficient entails constructing, then comparing several possible routes on several spatial attributes: distance, complexity (number of turns), and type of path, city street or highway. Determining whether two independent variables interact requires making successive magnitude estimations, first on each variable, then between variables. Imagining chemical bondings requires moving elements into an array, then rearranging them as a consequence of the bonding. For some spatial inferences, the spatial-temporal information may be suggestive but not sufficient, for example, for determining force and mass.

Role of Distortions in Spatial Thinking The processes that establish representations and execute transformations are schematic. That is, they delete some information and add or emphasize other information. The schematization is systematic and not random, driven by perceptual organizing principles and leading to predictable distortions in memory and judgment. For example, one way in which perception is organized is with respect to the frame of reference the world provides, one vertical and two horizontal axes. We localize objects in the world with respect to those axes, in an approximate fashion. Organization with respect to a reference frame carries over to the north-south-east-west canonical axes of geographic space and to the axes provided by a diagram in graphic space. One result is that when an entity’s location is coded relative to the surrounding frame of reference, it is remembered as more aligned with the reference frame than it actually is. Thus, people remember “tilted” entities, such as South America, as more “upright,” or aligned north-south, than they actually are (Tversky, 1981). Motion paths are similarly schematized and hence distorted; slightly oblique motion is coded as vertical or horizontal (Pani et al., 1996; Shiffrar and Shepard, 1991).

Role of Abstract Spatial Thinking People use spatial thinking metaphorically in everyday life as well as in science. We say that we feel “close” to one person, that another has “fallen out” of favor, that a third is “on top of the world,” and that yet another has “lost his center” (Lakoff and Johnson, 1980). The periodic table, flow diagrams in heat transfer and computer programs, and the “solar system” model of atoms are but some of the spatial representations used to summarize and organize abstract information. They have multiple effects on ways that scientists think about these concepts and teach them to their students. They enable spatial reasoning to be applied to complex causal phenomena. Because people have had a lifetime of experience in reasoning spatially (indeed, survival depends on it), they come to science with a spatial toolbox that can be applied to abstract concepts. Applying these spatial thinking tools is by no means automatic; in fact, one of the great challenges of education is facilitating transfer of tools acquired in one domain to another (see



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