studied produced an emergent S-shaped curve. However, such seeing depended on developing a “disciplined perception” (Stevens and Hall, 1998), a firm grounding in a Cartesian system. Moreover, the shape of the curve was determined in light of variation, accounted for by selecting and connecting midpoints of intervals that defined piece-wise linear segments. This way of representing typical growth was contentious, because some midpoints did not correspond to any particular case value. This debate was therefore a pathway toward the idealization and imagined qualities of the world necessary for adopting a modeling stance. The form of the growth curve was eventually tested in other systems, and its replications inspired new questions. For example, why would bacteria populations and plants be describable by the same growth curve? In this case and in others, explanatory models and data models mutually bootstrapped conceptual development (Lehrer and Schauble, 2002).

It is not feasible in this report to summarize the extensive body of research in mathematics education, but one point is especially critical for science education: the need to expand elementary school mathematics beyond arithmetic to include space and geometry, measurement, and data/ uncertainty. The National Council of Teachers of Mathematics standards (2000) has strongly supported this extension of early mathematics, based on their judgment that arithmetic alone does not constitute a sufficient mathematics education. Moreover, if mathematics is to be used as a resource for science, the resource base widens considerably with a broader mathematical base, affording students a greater repertoire for making sense of the natural world.

For example, consider the role of geometry and visualization in comparing crystalline structures or evaluating the relationship between the body weights and body structures of different animals. Measurement is a ubiquitous part of the scientific enterprise, although its subtleties are almost always overlooked. Students are usually taught procedures for measuring but are rarely taught a theory of measure. Educators often overestimate children’s understanding of measurement because measuring tools—like rulers or scales—resolve many of the conceptual challenges of measurement for children, so that they may fail to grasp the idea that measurement entails the iteration of constant units, and that these units can be partitioned. It is reasonably common, for example, for even upper elementary students who seem proficient at measuring lengths with rulers to tacitly hold the theory that measuring merely entails the counting of units between boundaries. If these students are given unconnected units (say, tiles of a constant length) and asked to demonstrate how to measure a length, some of them almost always place the units against the object being measured in such a way that the first and last tile are lined up flush with the end of the object measured. This arrangement often requires leaving spaces between units. Diagnosti-

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