would see from a hypothetical point in space) but also visualizing how these perspectives would change over days and months.

Students are often expected to read or produce diagrams and integrate information from the diagram with accompanying text. Understanding diagrams seems to depend less on a student’s problem-solving abilities than on the specific design and content of the diagram. Diagrams can be difficult to understand for many reasons. Sometimes the desired information is missing. Sometimes a diagram does not appear in a familiar or recognizable context. And sometimes features of a diagram can create confusion. For example, the common misconception that the earth is closer to the sun in the summer than in the winter may be due, in part, to the fact that two-dimensional representations of the three-dimensional orbit make it appear as if the earth is indeed closer to the sun at some points than at others.

Students’ understanding of maps can be particularly challenging, because maps preserve some characteristics of the place being represented—for instance, relative position and distance—but may omit or alter features of the actual landscape. Recall the mapping done by Mr. Walker’s class in the case study on biodiversity in Chapter 2, in which the students learned to develop a more systematic plan for mapping the distribution and density of common species. Young children especially have a much easier time representing objects than representing large-scale space. Students may also struggle with orientation, perspective (the traditional bird’s eye view), and mathematical descriptions of space, such as polar coordinate representations.

In a study involving biological growth, Richard Lehrer and Leona Schauble observed characteristic shifts in the understanding of modeling over the span of the elementary school grades.¹ They developed a learning progression that emphasized different and increasingly complex ideas in different grade bands. Each had a different curriculum and tasks:

• Early elementary: Growth of flowering bulbs: A focus on difference

• Middle elementary: Growth of Wisconsin Fast Plants²: A focus on ratio

• Late elementary: Growth of population: A focus on distribution