two, and three dimensions is described in the following section). Tables 6-1 and 6-2 summarize development of spatial and geometric thinking, as well as measurement, in two and three dimensions. Ages are grouped in the same was as in the previous chapter in order to illustrate how children’s engagement with mathematics should build and develop over the prekindergarten years.

In the tables, children’s competence within each band is described on the basis of the level of sophistication in their thinking. These levels are called thinking visually/holistically, thinking about parts, and relating parts and wholes.

Step 1 (Ages 2 and 3)
2-D and 3-D Objects

Very young children match shapes implicitly in their play. Working at the visual/holistic level (see Table 6-1), they can describe pictures of objects of all sorts, using the shape implicitly in their recognition. By age 2 to 3, they also learn to name shapes, with 2-D shapes being more familiar in most cultures, beginning with the familiar and symmetric circle and square and extending to at least prototypical triangles. Although they may name 3-D shapes by the name of one of its faces (calling a cube a square), their ability to match 2-D to corresponding 2-D (and similar for 3-D) indicates their intuitive differentiation of 2-D and 3-D shapes.

Children also learn to recognize and name additional shapes, such as triangles and rectangles—at least in their prototypical forms—and can begin to describe them in their own words. With appropriate knowledge of number, they can begin to describe these shapes by the number of sides they have, just starting to learn the concepts and terminology of the thinking about parts level of geometric thinking.

Spatial Relations

From the first year of life, children develop an implicit ability to move objects. They also learn relationship language, such as “up” and “down” and similar vocabulary. They learn to apply that vocabulary in both 3-D contexts and in 2-D situations, such as the “bottom” of a picture that they are drawing on a horizontal surface.

Compositions and Decompositions

At the visual/holistic level, children can solve simple puzzles involving things in the world (e.g., wooden puzzles with insets for each separate ob-

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