• Set theoretic relations between data sets (union, intersection, etc.) leading to aggregation and disaggregation

  • Interpolation and extrapolation

  • Abstraction (e.g., converting numerical point data to a surface with contour lines)

  • Generalization (e.g., classification, smoothing, simplification, exaggeration [as in different scales for horizontal versus vertical distances on a map])


  • Descriptions of the properties of spatial distributions in terms of density, dispersion, centroids, regions, outliers

  • Pattern analysis (e.g., differentiating random patterns from systematic patterns; classifying systematic patterns in terms of regularity [as in grids or other tessellations] or clustering; identifying deviations; identifying relations between patterns [positive and negative correlations])

  • Structural analysis (e.g., for patterns, calculation of nearest neighbors, or spatial autocorrelation; for networks, calculations of centrality, connectivity, and various paths [shortest, the traveling salesman route, etc.]; for hierarchies, calculation of tree structures)

  • Process modeling (e.g., developing explanations of spatial patterns as a function of time and/or distance [Boxes 1.1 and 1.2])

The capacity to manipulate structural relationships is the essence of spatial thinking.


With the increasing availability of powerful IT systems, it is tempting to see the development of support systems for spatial thinking as necessarily being computer based and, therefore, high tech in nature. This is not the case.

Spatial thinking can also be supported with low-tech systems, essentially involving paper-and-pencil systems allied to the use of simple graphic representations (as in the case of Jerome Bruner’s work using simple outline maps, cited in Chapter 1). Such low-tech systems are important for the following reasons:

  • Despite the increasing penetration of IT into American schools, access to hardware and software remains limited. To build supports for spatial thinking only around IT would be a mistake; it would restrict the opportunities for both teachers and students to learn about and use spatial thinking across the curriculum.

  • Understanding many of the fundamental building blocks of spatial thinking can be more readily achieved through simple, low-tech systems (see Box 6.3). These place fewer demands on teachers, students, and schools. They are easily adapted for use across the curriculum. They have immediacy and face validity that permit students to understand the basic components of spatial thinking. Specific skills can be isolated and practiced. For example, students can use simple transparencies to overlay maps and to understand correlations between patterns of data. They can use graph paper to learn about rescaling and transformations of data.

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