see a simple arrangement of pattern blocks, then try to reproduce it. The configuration is shown again for a couple of seconds as many times as necessary. Older children can be shown a line drawing and try to draw it themselves (Yackel and Wheatley, 1990). This often creates interesting discussions revolving around “what I saw.”

Spatial visualization and imagery have been positively affected by interventions that emphasize building and composing with 3-D shapes (Casey et al., in press). Another series of activities described above that develops imagery is the sequence of tactile-kinesthetic exploration of shapes.

Achievable and Foundational Geometry and Spatial Thinking

Although longitudinal research is needed, extant research provides guidance about which geometric and spatial experiences are appropriate for and achievable by young children and will contribute to their mathematical development. First, of the mathematics children engage in spontaneously in child-centered school activities, the most frequent deals with shape and pattern. Second, each of the recently developed, research-based preschool mathematics curricula includes geometric and spatial activities (Casey, Paugh, and Ballard, 2002; Clements and Sarama, 2004; Ginsburg, Greenes, and Balfanz, 2003; Klein, Starkey, and Ramirez, 2002), with some of these featuring such a focus in 40 percent or more of the activities. Third, pilot-testing has shown that these activities were achievable and motivating to young children (Casey, Kersh, and Young, 2004; Clements and Sarama, 2004; Greenes, Ginsburg, and Balfanz, 2004; Starkey, Klein, and Wakeley, 2004), and formal evaluations have revealed that they contributed to children’s development of both numerical and spatial/geometric concepts (Casey and Erkut, 2005, in press; Casey et al., in press; Clements and Sarama, 2007c, in press; Starkey et al., 2004, 2006).

Fourth, previous work has shown that well-designed activities can effectively build geometric and spatial skills and general reasoning abilities (e.g., Kamii, Miyakawa, and Kato, 2004). Fifth, results with curricula in Israel that involved only spatial and geometric activities (Eylon and Rosenfeld, 1990) are remarkably positive. Children gained in geometric and spatial skills and showed pronounced benefits in the areas of arithmetic and writing readiness (Razel and Eylon, 1990). Similar results have been found in the United States (Swaminathan, Clements, and Schrier, 1995). Children are better prepared for all school tasks when they gain the thinking tools and representational competence of geometric and spatial sense.

In this section, we describe teaching-learning paths for spatial and geometric thinking in 2-D and 3-D contexts. For each area outlined below, children should be engaged in activities that cover a range of difficulty, including perceive, say, describe/discuss, and construct (measurement in one,

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