18: Geometry and Spatial Reasoning

School geometry is the study of those spatial objects, relationships, and transformations that have been formalized (or mathe- matized) and the axiomatic mathematical systems that have been constructed to represent them. Spatial reasoning, on the other hand, consists of the set of cognitive processes by which mental representations for spatial objects, relationships, and transformations are constructed and manipulated. Clearly, geometry and spatial reasoning are strongly interrelated, and most mathematics educators seem to include spatial reasoning as part of the geometry curriculum. Usiskin (1987), for instance, has described four dimensions of geometry: (a) visualization, drawing, and construction of figures; (b) study of the spatial aspects of the physical world; (c) use as a vehicle for representing nonvisual mathematical concepts and relationships; and (d) representation as a formal mathematical system. The first three of these dimensions require the use of spatial reasoning.