The first step is extracting spatial structures. This process of pattern description involves identifying relations between the components of a spatial representation and understanding them in terms of the parts and wholes that give rise to patterns and coherent wholes. The second step is performing spatial transformations. Translations in space or scale transformations (changes in viewing distance) are easier than rotations or changes of perspective (changes in viewing angle or azimuth). Imagining the motions of different parts in relation to each other—running the object—can be very difficult. The third step, drawing functional inferences, is central to the process of scientific thinking. It requires establishing temporal sequences and cause-and-effect relations.

The difficulty of each of these steps increases with increasing dimensionality: spatial structures in two-space are easier to understand than those in three-space. In scientific applications, difficulty also increases as a function of data quality and quantity. Missing data require extrapolation and interpolation. Data error leads to uncertainty and increasing difficulty. Partial and incomplete data require an even more skilled use of extrapolation and interpolation, as well as more complex inference processes.

People use representations, whether in the mind or external, to comprehend and remember a set of concepts as well as to make inferences and discoveries about those concepts. Understanding the spatial relations and structure of a diagrammed system is relatively straightforward for most learners, because a diagram shows the parts in their spatial relations, using diagrammatic space to map real space. Most people can grasp the essential parts and their spatial relations from a diagram, such as a bicycle pump or a heart. What is much harder to understand is the meaning, interpretation, function, and causal chain that the diagram is meant to convey. While a novice can understand the spatial structure of a bicycle pump or heart from a diagram, only those with some expertise can grasp the functional and causal relations among the parts—that is, understand how the pump or the heart works (Heiser and Tversky, 2002).

For most scientific and engineering contexts, diagrams are meant to convey not just the structure of a system but also its behavior or the causal chain of its parts or the function of its operations. Yet, these are exactly the aspects of diagrams that students of all ages find difficult. Diagrams show structure, but they do not “show” function or behavior or causal relations. Language can compensate by stating this information directly. However, diagrams can also be enriched with extrapictorial devices, notably lines, arrows, boxes, and brackets, to convey abstract information. For example, when asked to describe a diagram of a bicycle pump, students describe the structural relations among the parts. When arrows are added to the diagram that denote the sequence of actions of the pump, students describe the causal, functional actions of the pump (Heiser and Tversky, 2002). Still, even the addition of arrows may not be sufficient to convey the functional information. For understanding bicycle pumps and car brakes, diagrams were sufficient for undergraduates with high mechanical ability but not for those of low ability; for those of low mechanical ability, language compensated (Heiser and Tversky, 2002).

In many educational settings, diagrams are taken for granted. These studies suggest that teaching how to reason from diagrams could reap significant benefits. Such teaching would be needed in a number of domains: geography, arithmetic and mathematics, biology, geology, chemistry, physics, engineering, and so forth. Diagrams are common in history and in the humanities as well. Across the curriculum, students need exercises in interpreting the spatial entities and spatial relations of diagrams, making inferences as well as making discoveries. Constructing diagrams is an integral part of this instruction, especially in groups. Junior high school dyads working together produced diagrams of, for example, plant ecology, that were more abstract and contained less irrelevant pictorial information than those produced by individuals.

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