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Learning to Think Spatially: GIS as a Support System in the K-12 Curriculum http://www.nap.edu/catalog/11019.html SPATIAL THINKING IN EVERYDAY LIFE, AT WORK, AND IN SCIENCE 69 • visualizing a three-dimensional object or structure or process by examining observations collected in one or two dimensions; • describing the position and orientation of objects you encounter in the real world relative to a conceptual coordinate system anchored to Earth; • remembering the location and appearance of previously seen items; • envisioning the motion of objects or materials through space in three dimensions; • envisioning the processes by which objects change shape; • using spatial thinking to think about time; and • considering two-, three-, and four-dimensional systems where the axes are not distance. Describing the Shape of an Object, Rigorously and Unambiguously Untold person-lifetimes have been invested in the task of describing natural objects in a way that is rigorous and unambiguous, and attends to all of the important observable parameters. Faced with the huge range of objects found in nature, mineralogists, petrologists, geomorphologists, structural geologists, sedimentologists, zoologists, and botantists had to begin by agreeing upon words, measurements, and concepts with which to describe these natural objects. Given a collection of objects that intuitively seem related in some way, what should you observe, and what should you measure, in order to capture the shape of each object? After much spatial thinking, crystallographers decided that you should observe how many planes of symmetry the crystal has and whether or not the angles between those planes of symmetry are right angles. Size and color of the crystal are not so important (Figure 3.13). After much spatial thinking, structural geologists decided that you should describe a fold in a sedimentary layer by imagining a plane cutting through the axis of symmetry of the fold, and then measuring the dip and strike of this plane, and the plunge of the intersection between this axial plane and the fold itself (Figure 3.14). Although this descriptive style of geoscience and biology tends to be disparaged nowadays, the worldwide effort, in the eighteenth through twentieth centuries, to develop methodologies describing the objects of nature was crucial to the success of the discipline. The processes on the part of expert observers as they develop a new description methodology include (1) careful observation of the shape of a large number of objects; (2) integrating these observations into a mental model of what constitutes the common characteristics among this group of objects; (3) identifying ways in which individual objects can differ while still remaining within the group; and (4) developing a methodical, reproducible set of observation parameters that can describe the range of natural variability within the group, thus defining a class. Step (4) may include developing a lexicon or taxonomy of terms, developing measuring instruments, developing units of FIGURE 3.13 Describing the shapes of natural objects, rigorously and unambiguously: stereograms of crystals. SOURCE: Hurlbut, 1971. Copyright © 1971 by C. S. Hurlbut. Reproduced by permission of John Wiley & Sons, Inc. Copyright © National Academy of Sciences. All rights reserved.
Learning to Think Spatially: GIS as a Support System in the K-12 Curriculum http://www.nap.edu/catalog/11019.html 70 LEARNING TO THINK SPATIALLY The fold at left is classified based on the orientation of the hinge line and axial surface. Diagrammatic projections below the block diagram show orientation data. FIGURE 3.14 Describing the shapes of natural objects, rigorously and unambiguously: classification of folds. SOURCE: Hobbs et al., 1976. measurement, and/or developing two-dimensional graphical representations of some aspect of the three-dimensional objects. For the novice, developing this descriptive skill involves becoming facile with the terms and techniques used by specialists who have previously studied this class of objects. In some cases, these descriptive techniques call upon projective and Euclidean spatial skills that many learners find extremely difficult. For example, when structural geologists collect data on which to apply their fold classification system, they record the vector that is perpendicular to the surface of the fold at many points on the fold, mentally extend those vectors until they intersect an imaginary hemisphere beneath the fold, and convey this information as points on a lower-hemisphere equal-area projection diagram. Command of projective spatial skills is required either to record or to interpret such data (Figure 3.15). Similarly, many students have difficulties in spotting the subtle planes of symmetry that are required to make use of the crystallographers’ descriptive methodology. Then, when it comes time to describe the angular relationships between crystal faces (using a system called Miller indices in which the three faces of a cube, for example, are labeled 001, 010, and 100) most students find that their Euclidean spatial ability initially fails them (Figure 3.13). Identifying or Classifying an Object by Its Shape Paleontologists or micropaleontologists identify fossils or microfossils according to shape or morphology. A variant of this task is identifying an object according to its shape, texture, and color. Mineralogists or petrologists identify minerals in a hand sample or photomicrograph by their shape, their color (including how their color changes under different lighting conditions), and their texture Copyright © National Academy of Sciences. All rights reserved.
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