Shape is a fundamental idea in mathematics and in development. Beyond mathematics, shape is the basic way children learn names of objects, and attending to the objects’ shapes facilitates that learning (Jones and Smith, 2002).

Steps in Thinking About Shape

Children tend to move through different levels in thinking as they learn about geometric shapes (Clements and Battista, 1992; van Hiele, 1986). They have an innate, implicit ability to recognize and match shapes. But at the earliest, prerecognition level, they are not explicitly able to reliably distinguish circles, triangles, and squares from other shapes. Children at this level are just starting to form unconscious visual schemes for the shapes, drawing on some basic competencies. An example is pattern matching through some type of feature analysis (Anderson, 2000; Gibson et al., 1962) that is conducted after the visual image of the shape is analyzed by the visual system (Palmer, 1989).

At the next level, children think visually or holistically about shapes (i.e., syncretic thought, a fusion of differing systems; see Clements, Battista, and Sarama, 2001; Clements and Sarama, 2007b) and have formed schemes, or mental patterns, for shape categories. When first built, such schemes are holistic, unanalyzed, and visual. At this visual/holistic step, children can recognize shapes as wholes but may have difficulty forming separate mental images that are not supported by perceptual input. A given figure is a rectangle, for example, because “it looks like a door.” They do not think about shapes in terms of their attributes, or properties. Children at this level of geometric thinking can construct shapes from parts, but they have difficulty integrating those parts into a coherent whole.

Next, children learn to describe, then analyze, geometric figures. The culmination of learning at this descriptive/analytic level is the ability to recognize and characterize shapes by their properties. Initially, they learn about the parts of shapes—for example, the boundaries of two-dimensional (2-D) and three-dimensional (3-D) shapes—and how to combine them to create geometric shapes (initially imprecisely). For example, they may explicitly understand that a closed shape with three straight sides is a triangle. In the teaching-learning path articulated in Table 6-1, this is called the “thinking about parts” level.

Children then increasingly see relationships between parts of shapes, which are properties of the shapes. For instance, a student might think of a parallelogram as a figure that has two pairs of parallel sides and two pairs of equal angles (angle measure is itself a relation between two sides, and

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