will be better able to interpret children’s understanding and ask questions that will lead them to construct these ideas. Both research with children and interviews with teachers support the claims that (a) the principles of measurement are difficult for children, (b) they require more attention in school than is usually given, (c) time needs to first be spent in informal measurement, in which the use of measurement principles is evident, and (d) transition from informal to formal measurement needs much more time and care, with instruction in formal measure always returning to basic principles (see Irwin, Vistro-Yu, and Ell, 2004).

The sequence in Table 6-3 summarizes achievable goals in linear measurement that have been employed in pilot-testing of research-based curricula (Casey et al., 2004; Clements and Sarama, 2004; Greenes et al., 2004; Starkey et al., 2004). Again, evaluations confirm the appropriateness of the sequencing (Clements and Sarama, 2007c, in press; Starkey et al., 2004, 2006).

Area Measurement and Spatial Structuring

Area is an amount of 2-D surface that is contained within a boundary. Area measurement assumes that a suitable 2-D region is chosen as a unit, congruent regions have equal areas, regions do not overlap, and the area of the union of two regions that do not overlap (disjoint union) is the sum of their areas (Reynolds and Wheatley, 1996). Thus, finding the area of a region can be thought of as tiling (or equal partitioning) a region with a 2-D unit of measure. Such understandings are complex, and children develop them over time. These area understandings do not develop well in traditional U.S. instruction (Carpenter et al., 1975), not only for young children, but also for preservice teachers (Enochs and Gabel, 1984). A study of children from Grades 1, 2, and 3 revealed little understanding of area measurement (Lehrer, Jenkins, and Osana, 1998). Asked how much space a square (and a triangle) cover, 41 percent of children used a ruler to measure length. Although area measurement is typically emphasized in the intermediate grades, the literature suggests that some less formal aspects of area measurement can be introduced in earlier years. Concepts that are essential to understanding and learning area measurement are described in Appendix B. One especially important one, spatial structuring, is discussed next.

Nascent awareness of area is often noticed in informal observations, such as when a child asks for pieces of colored paper to cover their table. A way to more formally assess children’s understanding of area is through comparison tasks. Some researchers report that preschoolers use only one dimension or one salient aspect of the stimulus to compare the area of two surfaces (Bausano and Jeffrey, 1975; Maratsos, 1973; Mullet and Paques, 1991; Piaget et al., 1960; Raven and Gelman, 1984; Russell, 1975; Sena



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