“next to,” and “between,” which they can apply in 3-D and 2-D spaces. Later, they extend this to terms that involve frames of reference, such as “to the side of,” “above,” and “below.”

Later, at the thinking about parts level, preschoolers recognize “matching” shapes at different orientations. They can learn to check if pairs of 2-D shapes are congruent by using geometry motions intuitively, moving from less accurate strategies, such as side-matching, or using lengths, to the use of superimposition (placing one shape on top of the other). They begin to use the geometric motions of slides, flips, and turns explicitly and intentionally, in discussing their solutions to puzzles or in applying such motions in computer environments to manipulate shapes. They learn to predict the effects of geometric motions, thus laying the foundation for thinking at the relating parts and wholes level.

Children also begin to be able to cover a rectangular space with physical tiles and represent their tilings with simple drawings, although they may initially leave gaps in each and may not align all the squares. This is mainly a competence of spatial structuring but it has close connections to the ability to construct compositions in 2-D space.

Preschoolers also learn about the parts of 3-D shapes, using motions to match the faces of 3-D shapes to 2-D shapes and representing 2-D and 3-D relationships with objects. For example, they may make a simple model of the classroom, using a rectangular block for the teacher’s desk, small cubes for chairs, and so forth.

Compositions and Decompositions

At the thinking about parts level, preschoolers can place shapes contiguously to form pictures in which several shapes play a single role (e.g., a leg might be created from three contiguous squares), but they use trial and error and do not anticipate creation of new geometric shapes. When filling in a frame or picture outline, children use gestalt configuration or one component, such as side length (Sarama et al., 1996). For example, if several sides of the existing arrangement form a partial boundary of a shape (instantiating a schema for it), children can find and place that shape. If such cues are not present, they match by a side length. Children may attempt to match corners but do not understand angle as a quantitative entity, so they try to match shapes into corners of existing arrangements in which their angles do not fit. Rotating and flipping are used, usually by trial and error, to try different arrangements (a “picking and discarding” strategy). Thus, they can complete a frame that suggests placement of the individual shapes but in which several shapes together may play a single semantic role in the picture.

Later, preschoolers begin to develop relating parts and wholes thinking.

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