same-sized units. This idea is not obvious to children. It involves mentally seeing the object as something that can be partitioned (or cut up) before even physically measuring. Asking children what the hash marks on a ruler mean can reveal how they understand partitioning of length (Clements and Barrett, 1996; Lehrer, 2003). Some children, for instance, may understand “five” as a hash mark, not as a space that is cut into five equal-sized units. As children come to understand that units can also be partitioned, they come to grips with the idea that length is continuous (e.g., any unit can itself be further partitioned).

Units and unit iteration. Unit iteration requires the ability to think of the length of a small unit, such as a block as part of the length of the object being measured, and to place the smaller block repeatedly along the length of the larger object (Kamii and Clark, 1997; Steffe, 1991), tiling the length without gaps or overlaps, and counting these iterations. Such tiling, or space filling, is implied by partitioning, but that is not well established for young children, who also must see the need for equal partitioning and thus the use of identical units.

Accumulation of distance and additivity. Accumulation of distance is the understanding that as one iterates a unit along the length of an object and count the iteration, the number words signify the space covered by all units counted up to that point (Petitto, 1990). Piaget, Inhelder, and Szeminska (1960) characterized children’s measuring activity as an accumulation of distance when the result of iterating forms nesting relationships to each other. That is, the space covered by three units is nested in or contained in the space covered by four units. Additivity is the related notion that length can be decomposed and composed, so that the total distance between two points is equivalent to the sum of the distances of any arbitrary set of segments that subdivide the line segment connecting those points. This is, of course, closely related to the same concepts in composition in arithmetic, with the added complexities of the continuous nature of measurement.

Origin is the notion that any point on a ratio scale can be used as the origin. Young children often begin a measurement with “1” instead of zero. Because measures of Euclidean space are invariant under translation (the distance between 45 and 50 is the same as that between 100 and 105), any point can serve as the origin.

Relation between number and measurement. Children must reorganize their understanding of the items they are counting to measure continuous units. They make measurement judgments based on counting ideas, often based on experiences counting discrete objects. For example, Inhelder, Sinclair, and Bovet (1974) showed children two rows of matches, in which the rows were the same length but each row was comprised of a different number of matches as shown in Figure B-1. Although, from the adult perspective, the lengths of the rows are the same, many children argued that

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