Chapter 4.4). The key to effective transfer of skills, such as spatial ones, from one domain to another is going from concrete content to the abstractions that cut across domains. (Ways to facilitate transfer are discussed in Chapter 4.5.)

Role of External Spatial Representations in Spatial Thinking Note that many of the previous examples (e.g., the periodic table, flow diagrams, molecular models) are external spatial representations. That is, they are visible on paper or some other physical medium, though they do exist in some form in the mind as well. People have been creating external representations since before recorded history, from maps in the sand to bent trees as trail markers to notches on wood to record heads of cattle. The advantages of externalizing spatial representations are multiple (Tversky, 2001). They provide a semipermanent record that can be examined by a community, unlike mental representations that may be forgotten and are accessible only to an individual. For science and engineering, external representations have the advantage of being visible to other members of a community. They can be referred to by gesture as well as language in explaining, inferring, and discovery. They relieve the burden of limited working memory to maintain a representation, freeing it to perform transformations and operations on an external representation. Thus, one important function of external spatial representations is to augment working memory. Another is to focus attention on critical aspects of the conceptions. Like internal spatial representations, external spatial representations schematize; they omit irrelevant information and highlight the relevant. External representations may also be supplemented with diagrammatic devices that focus attention, such as arrows, guidelines, boxes, brackets, and boldface large type. Many external representations are pictorial. All other things equal, pictures are easier to remember than comparable words, so spatial representations also facilitate long-term memory for the concepts they convey.

Effective External Representations We have already mentioned the key to creating effective external representations: they must convey the essential conceptual information and eliminate the irrelevant information that can clutter and distract. This is harder than it sounds. For example, what information should be included in a tourist map? Which facilities of use to tourists—historic buildings, museums, restaurants, hotels, and so on—should be included, without cluttering the map with so much information that none of it is legible? Similar problems arise in scientific visualizations. Another issue in designing effective visualizations is deciding how to depict elements and mapping spatial relations. Ideally elements should be easy to recognize and decipher, associated in some way with what they represent. Spatial relations among elements, especially distance, should reflect conceptual relations in the abstract domain (Tversky, 2001). Animated diagrams have their appeal, yet they turn out to be no more effective than comparable still diagrams across a broad range of content areas in conveying conceptual information. This is partly because animations are typically complex and fleeting, so learners have trouble knowing what to attend to. Perhaps more important is that animations are continuous but people think about continuous processes as discrete steps. A well-designed sequence of still diagrams can convey the information essential to each step (Tversky et al., 2002).

Role of Expertise in Spatial Thinking As with all cognitive competencies, there are significant differences among people as to how, how quickly, and how well they do something. Spatial thinking is no exception. Within domains of knowledge, there are experts and novices. Differences between experts and novices can be accounted for by training and experience. Across domains of knowledge, there are disciplines within which spatial thinking is emphasized and taught and those within which it plays a hidden and relatively minor role. Across groups, there are also significant variations in how people approach spatial thinking. Across age, for example, children and adults do not think spatially in the same way. Those differences can be accounted for by maturation, education, and experience. All of these issues are addressed in Chapter 4 and Appendix C.

If, for the moment, the effects of domain are ignored, the expert-novice distinction can be used to understand some of the major differences in the ways in which people think spatially. A key goal,



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