plant parts take the same form on the graph? When was the growth of the roots and shoots the fastest, and what was the functional significance of those periods of rapid growth?

Students became competent at using a variety of representational forms as models. For example, students noted that growth over time x,y-coordinate graphs of two different plants looked similar in that they were equally “steep.” Yet the graphs actually represented different rates of growth, because the students who generated the graphs used different scales to represent the height of their plants. The discovery that graphs might look the same and yet represent different rates of growth influenced the students’ interpretations of other graphs in this and other contexts throughout the year.

Fifth-Grade Representations

In the fifth grade, children again investigated growth, this time in tobacco hornworms (Manduca), but their mathematical resources now included ideas about distribution and sample. Students explored relationships between growth factors: for example, different food sources and the relative dispersion of characteristics in the population at different points in the life cycle of the hornworms.

Questions posed by the fifth graders focused on the diversity of characteristics within populations—for example, length, circumference, weight, and days to pupation—rather than simply shifts in central tendencies of attributes (see Figure 6-6 on page 120). As the students’ ability to use different forms of representation grew, so, too, did their consideration of what might be worthy of investigation.

Shifts in Understanding

In sum, over the span of the elementary school grades, these researchers observed characteristic shifts from an early emphasis on models that used literal depiction toward representations that were progressively more symbolic in character. Increased competence in using a wider range of representational types both accompanied and helped promote conceptual change.

As students developed and used new mathematical means for characterizing growth, they understood biological change in increasingly dynamic ways. For example, once students understood the mathematics of changing ratios, they began to conceive of growth not as a simple linear increase but as a patterned rate of change. These shifts in both conceptual understanding and forms of

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