doing, building and iterating units, as well as units of units, helps them develop strong concepts and skills. Teachers should help children closely connect the use of manipulative units and rulers. When conducted in this way, measurement tools and procedures become tools for mathematics and tools for thinking about mathematics (Clements, 1999c; Miller, 1984, 1989). Well before first grade, children have begun the journey toward that end.

Area

Children need to structure an array to understand area as truly 2-D (see Appendix B). Play with structured materials, such as unit blocks, pattern blocks, and tiles, can lay the groundwork for children’s spatial structuring, although achieving the conceptual benchmark will not be achieved until after the primary grades for most children, even with high-quality instruction. In brief, the too-frequent practice of simple counting of units to find area (achievable by preschoolers) leading directly to teaching formulas may not build the requisite foundational concepts (Lehrer, 2003). Instead, educators should build on young children’s initial spatial intuitions and appreciate their need to construct the idea of measurement units—including development of a measurement sense for standard units, for example, finding common objects in the environment that have a unit measure. Children need to have many experiences covering quantities with appropriate measurement units, counting those units, and spatially structuring the object they are to measure, in order to build a firm foundation for eventual use for formulas. For example, children might build rectangular arrays with square tiles and learn to count the number of manipulatives used in each. Eventually, they need to link counting by groups to reflect the structure of rectangular arrays, for example, counting the squares in an array by skip-counting the number in each row.

This long developmental process usually only begins in the years before first grade. However, we should also appreciate the importance of these early conceptualizations. For example, 3- and 4-year-olds’ use of a linear rating scale to judge area, even if using an additive rule, indicates an impressive level of quantitative ability and, according to some, nascent mental structures for algebra at an early age (Cuneo, 1980).

Competencies in the major realms of geometry/spatial thinking and number are connected throughout development. The earliest competencies may share common perceptual and representational origins (Mix, Huttenlocher, and Levine, 2002). Infants are sensitive to both the amount of liquid in a container (Gao, Levine, and Huttenlocher, 2000) and the distance away a toy is hidden in a long sandbox (Newcombe, Huttenlocher, and Learmonth, 1999). Visual-spatial deficits in early childhood are detrimental to children’s development of numerical competencies (Semrud-Clikeman



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