For example, they might combine pattern block shapes (angles that are multiples of 30°) to make composites that they recognize as new shapes and to fill puzzles with growing intentionality and anticipation (“I know what will fit”). Shapes are chosen using angles as well as side lengths. The equilateral triangle world of pattern blocks provides a microworld, in which matching by sides (all of which are equal in length or double the unit length), fitting angles (multiples of 30°), and composing (two equilateral triangles can “make” the blue rhombus, a rhombus and a triangle make a trapezoid, etc.) are facilitating at this beginning step. Eventually, children consider several alternative shapes with angles equal to the existing arrangement. Rotation and flipping are used intentionally (and mentally, i.e., with anticipation) to select and place shapes (Sarama et al., 1996). Children can fill complex frames (Sales, 1994) or cover regions (Mansfield and Scott, 1990).
Related to their ability to tile the rectangular section of a plane, children can copy designs made from squares (and, for some, also isosceles right triangles) and place these shapes onto squared-grid paper. This square-based microworld is simple and not only facilitates composition, but also develops the foundations of much of mathematics (spatial structuring, multiplication, area, volume, coordinates, etc.).
Using 3-D shapes, preschoolers combine building blocks using multiple spatial relations, extending in multiple directions and with multiple points of contact among components, showing flexibility in integrating parts of the structure. Thus, they can reliably produce arches, enclosures, corners, and crosses, including enclosures that are several blocks in height. Later, they can learn to compose building blocks with anticipation, understanding what 3-D shape will be produced with a composition of 2 or more other (simple, familiar) 3-D shapes.
Kindergartners learn to recognize additional shapes, such as parallelograms, and, more importantly, learn to describe why a certain figure is classified into a given class of shapes (at the relating parts and wholes level). They may therefore discuss that parallelograms have two pairs of sides that are equal in length and two pairs of angles of equal size. This remains just the beginning of this type of thinking, as concepts of parallelism, perpendicularity, and angle measure develop over many years thereafter.
Kindergartners also learn the names of more 3-D shapes, such as spheres, cylinders, prisms, and pyramids. They describe congruent faces of such shapes and begin to understand and discuss such properties as parallel faces in some contexts (e.g., building with blocks).