: Students need to know the concepts that are the building blocks for spatial thinking. There are general spatial concepts that are found in many disciplines, such as symmetry, isomorphism, reflection, orientation, rotation, and function, and spatial concepts that are tailored to a particular discipline, such as relative versus absolute distance, small versus large scale, and distance decay in geography. Students learn the meanings and uses of concepts relevant to spatial thinking in the context of specific disciplines or school subjects. Thus, in mathematics, students learn about general concepts, such as minima and maxima, and their specific forms, such as hyperbolas and parabolas. In geometry, they learn about conic sections: hyperbola, parabola, ellipse, and circle. They learn to distinguish among a torus, Mobius strip, and Klein bottle. In physics, they learn that the equilibrium position of a fixed chain is a catenary curve (or hyperbolic cosine).


Even this cursory listing of concepts by discipline illustrates two fundamental educational challenges. First, there is a rich, complex, conceptual structure to the description and explanation of space to be learned within each discipline. Second, rather than coming up with an omnibus list of concepts for spatial thinking, students—and especially teachers—should identify concepts relevant to specific disciplines but should also look for common themes. They should reflect on how concepts of one discipline might inform or interfere with learning about concepts in another discipline. For example, in algebra, geometry, and science, the concept of function has different meanings. Similarly, in geometry, a point is a dimensionless location, whereas in geography, a point in space is a specific place with a small but definite area.