about equivalent to their competence in corresponding length tasks (Curry and Outhred, 2005). The relationship is consistent with the notion that the structure of the task is 1-D, exemplified by some students’ treating the height of the rice in the container as if it were a unit length and iterating it, either mentally or using their fingers, up the side of the container. Some students performed better on length, others on filling volume, giving no evidence of a relationship between the two. The task contained some extra demands, such as creating equal measurements; even many first graders made sure that the cup was not over- or underfilled for each iteration. In another study, 3- and 4-year-olds understood that unit size affects the measurement of the object’s volume (Sophian, 2002). Thus, simple experience with filling volume may be appropriate for young children.

On the other hand, packing volume is more difficult than length and area (Curry and Outhred, 2005). Most children had little idea of how to estimate or measure on packing tasks. There were substantial increases from Grades 2 to 4, but even the older students’ scores were below the corresponding scores for the area task. Furthermore, there was a suggestion that understanding of area is a prerequisite to understanding packing volume. Therefore, children should have many experiences building with blocks and filling boxes with cubes. A developmental progression is provided in Table 6-2. A full conceptual understanding of 3-D space will develop only over several years for most children.

Achievable and Foundational Measurement in One, Two, and Three Dimensions

In this section, we describe children’s development of measurement in one, two, and three dimensions. We do not consider measurement of nongeometric attributes, such as weight/mass, capacity, time, and color, because these are more appropriately considered in science and social studies curricula. Again, for each area outlined below, children should be engaged in activities that cover a range of difficulty, including perceive, say, describe/discuss, and construct. Table 6-3 outlines the path for measurement of length.

Step 1 (Ages 2 and 3)
Objects and Spatial Relations

Young children naturally encounter and discuss quantities in their play (Ginsburg, Inoue, and Seo, 1999). They first learn to use words that represent quantity or magnitude of a certain attribute. Facilitating this language is important not only to develop communication abilities, but for the development of mathematical concepts. Simply using labels such as

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