with multidigit addition and subtraction requires being able to compose ten ones as one ten and to decompose one ten as ten ones.

In geometry, shapes can be viewed as composed of other shapes, such as viewing a trapezoid as made from three triangles, or viewing a house shape as made from a triangle placed above a square. Children can compose rows of squares to make rectangles (see Figure 2-6). Many 3-D shapes seen in everyday life can be viewed as composed of shapes that are found in sets of building blocks (or at least approximately so). A juice box might look like a rectangular prism with a (sideways) triangular prism on top. Children can compose layers of cubes to make larger cubes and rectangular prisms.

In measurement, units are composed to make larger units and decomposed to make smaller units. Measurement itself requires viewing the attribute to be measured as composed of units. In effect, using a unit of measure to partition a continuous quantity, such as a length or area, into discrete and equal size pieces transforms it into a countable quantity.

Relating and Ordering

Relating and ordering allow one to decide which is more and which is less in various domains: number, length, area. Having children see and discuss relating and ordering across domains can deepen mathematical understanding. By broadening the ways in which things can be compared, children are led to the idea of different measurable attributes. For example, two stacks of blocks might be made from the same number of blocks, but one stack might be taller than the other. Relating is a first step toward measurement, because measurement is a quantified form of relating. A measurement specifies how many of one thing (the unit) it takes to make the other thing (the attribute that is measured). When relating and number are joined via measurement, both realms are extended. On one hand, relating becomes more precise when it becomes measurement, and, on the other hand, numbers extend into fractions and decimals in the context of measurement. For example, a bucket of sand might be filled with 2½ smaller pails of sand.

Looking for Patterns and Structures and Organizing Information

Looking for patterns and structures and organizing information (including classifying) are crucial mathematical processes used frequently in mathematical thinking and problem solving. They also have been viewed as distinct content areas in early childhood mathematics learning. Such pattern “content” usually focuses on repeated patterns, such as abab or abcabc, that are done with colors, sounds, body movements, and so forth (such as the bead and block patterning examples discussed in the section on unitizing). Such activities are appropriate in early childhood and can

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