et al., 1999; Hannibal and Clements, 2008). It is time to change the presentation of squares as an isolated set. Instead, recent approaches present many examples of squares and rectangles, varying orientation, size, and so forth, including squares as examples of rectangles. If children say “that’s a square,” teachers respond that it is a square that is a special type of rectangle, using double-naming (“it’s a square-rectangle”). This approach has been shown to be successful with preschoolers and kindergartners (Clements and Sarama, 2007c, in press; Clements, Sarama, and Wilson, 2001; Sarama and Clements, 2002). Kindergarten and first graders can discuss general categories, such as quadrilaterals and triangles, counting the sides of various figures to choose their category. They can then build hierarchical relationships of subsets of these general categories (Kay, 1987).

Children should also learn about composing and decomposing shapes from other shapes. This competence is significant in that the concepts and actions of creating and then iterating units and higher order units in the context of constructing patterns, measuring, and computing are established bases for mathematical understanding and analysis (Clements et al., 1997b; Reynolds and Wheatley, 1996; Steffe and Cobb, 1988). In addition, there is empirical support that this type of composition corresponds with, and supports, children’s ability to compose and decompose numbers (Clements et al., 1996).

The sequence in Table 6-1 is based on a series of developmental studies describing children’s capabilities (Clements, Sarama, and Wilson, 2001; Mansfield and Scott, 1990; Sales, 1994; Sarama, Clements, and Vukelic, 1996). These studies were synthesized into an empirically verified developmental progression that identified skills that are achievable for children at different ages, especially if provided opportunities to learn (Clements, Wilson, and Sarama, 2004). Starting with a lack of competence in composing geometric shapes, they gain abilities to combine shapes into pictures, and finally synthesize combinations of shapes into new shapes (composite shapes). As further evidence, interventions at the preschool level have shown notable gains in this ability for 2-D shapes (Casey and Erkut, in press). Intentional interventions with 3-D shape construction (i.e., building with unit blocks) have also resulted in statistically significant gains (Casey et al., in press).

Many activities develop these abilities. With a variety of groups of shapes, such as pattern blocks, tangrams, or groups with a greater variety of shapes, children can be encouraged to combine shapes creatively to create pictures and designs. Noting children’s developmental level, teachers can make suggestions and pose challenges that will facilitate their learning of more sophisticated thinking.

Outline puzzles that can be filled with those same groups of shapes are also motivating and particularly useful because they can be designed to promote a particular level of thinking. Teachers can then view children’s active



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