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• #### Index 371-386

of folding one sheet into two halves, origami practice might facilitate the development of an area concept, which is related to the spontaneous use of the procedure.

To measure, a unit must be established. Teachers often assume that the product of two lengths structures a region into an area of 2-D units for students. However, the construction of a 2-D array from linear units is nontrivial. Young children often cannot partition and conserve area and instead use counting as a basis for comparing. For example, when it was determined that one share of pieces of paper cookie was too little, preschoolers cut one of that share’s pieces into two and handed them both back, apparently believing that the share was now “more” (Miller, 1984).

As with length measurement, children often cover space, but they do not initially do so without gaps or overlapping (i.e., they do not tile the region with units). They also initially do not extend units over the boundaries when a subdivision of that unit is needed to fill the surface (Stephan et al., 2003). Even more limiting, children often choose units that physically resemble the region they are covering; for example, choosing bricks to cover a rectangular region and beans to cover an outline of their hands (Lehrer, 2003; Lehrer, Jenkins, and Osana, 1998; Nunes et al., 1993). They also mix different shapes (and areas), such as rectangular and triangular, to cover the same region and accept a measure of “7” even if the seven covering shapes are of different sizes (84 percent of primary grade children; Lehrer, Jenkins, and Osana, 1998). These concepts have to be developed before children can use iteration of equal units to measure area with understanding. Once these problems have been solved, children need to structure 2-D space into an organized array of units to achieve multiplicative thinking in determining volume, a concept to which we now turn.

##### Volume Measurement

Volume introduces even more complexity, not only in adding a third dimension and thus presenting a significant challenge to students’ spatial structuring, but also in the very nature of the materials that are measured using volume. This leads to two ways to measure volume, illustrated by “packing” a space, such as a 3-D array with cubic units, and “filling” with iterations of a fluid unit that takes the shape of the container. For the latter, the unit structure may be psychologically 1-D for some children (i.e., simple iterative counting that is not processed as geometric 3-D), especially, for example, in filling a cylindrical jar in which the (linear) height corresponds to the volume (Curry and Outhred, 2005). Given the possible complexities, is either of these more or less appropriate for young children, beyond, say, informal experiences?

For children in Grades 1-4, competence in filling volume (e.g., estimating and measuring the number of cups of rice that filled a container) was

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