procedures and concepts and see diverse aspects of knowledge as having the same underlying structure” (Baroody, Feil, and Johnson, 2007, p. 26).

Unitizing

Unitizing—finding or creating a mathematical unit—occurs in numerical, geometric, and spatial contexts. When children count, they must create mental units of what they are going to count: single cats, the paws on several cats, or groups of two cats. To measure length, children must select a unit of length measure (for example, they will lay along a length and then count new crayons, feet stepped heel-to-toe along some distance, or inch lengths). To create repeating patterns, children must select and repeat a unit. For example, they might make a bead necklace by repeatedly stringing two cubes then a sphere (their unit). In designing a block building, they might repeatedly place a square, then a triangular block, repeating that unit around the top of their building. When making designs or pictures with pattern blocks, children might join several shapes to make a unit that they repeat throughout the design. To begin to understand the base 10 place-value system, children must be able to view ten ones as forming a single unit of ten. Research suggests a link between being able to view a collection of shapes as a higher order unit and being able to view two-digit numbers as groups of tens and some ones (Clements et al., 1997; Reynolds and Wheatley, 1996). Because the concept of unit underlies core ideas in number and in geometry and measurement, it has been recommended as a central focus for early childhood mathematics education (Sophian, 2007).

Decomposing and Composing

Decomposing and composing are used throughout mathematics at every level and in all topics. In the realm of numbers and operations (addition, subtraction, multiplication, and division), composing and decomposing are used in recognizing the number of objects in a collection, in the meaning of the operations themselves, and in the place-value system. Children can sometimes quickly determine the number of objects in a small collection by viewing the collection as composed of two immediately recognizable collections, such as seeing four counters as composed of a set of three counters and another counter. Composing and decomposing are the basis for the operations of addition and subtraction and later for the operations of multiplication and division. Some key steps toward developing proficiency with arithmetic involve decomposing and composing. Children must be able to decompose numbers from 1 to 10 into all possible pairs and to recognize numbers from 11 to 19 as composed of a ten and some ones. The base 10 place-value system relies on repeated bundling in groups of ten. Proficiency



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