per day, students noticed immediately that the rates of growth were not the same. However, one student pointed out that the curves for both the roots and the shoots showed the same S-shape. This S-shape appeared again on graphs describing the growth of tobacco hornworms and populations of bacteria on a plate. Students came to recognize this shape as a standard graph pattern that indicated growth. This similarity in patterns would not have been noticeable without the mathematical representation afforded by the graph.

Given the importance of mathematics in understanding science, elementary school mathematics needs to go beyond arithmetic to include ideas regarding space and geometry, measurement, and data and uncertainty. Measurement, for example, 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 and scales—resolve many of the conceptual challenges of measurement for children. As a result, students may fail to understand that measurement entails the use of repeated constant units and that these units can be partitioned. Even upper elementary students who seem proficient at measuring lengths with rulers may believe that measuring merely entails counting the 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 tiles are lined up flush with the end of the object measured, leaving spaces among the units in between. These spaces do not trouble a student who holds this “boundary-filling” conception of measurement.

Data

Data modeling is central to a variety of scientific enterprises, including engineering, medicine, and natural science. Scientists build models with an acute awareness of the data that are required, and data are structured and recorded as a way of making progress in articulating a scientific model or deciding among rival models.

Students are better able to understand data if as much attention is devoted to how they are generated as to their analysis. First and foremost, students need to understand that data are constructed to answer questions, not provided in a



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