Two-Dimensional Math In a Three-Dimensional World
A few years ago, a high-school student attracted national attention by catching an error in the mathematics portion of the Preliminary Scholastic Aptitude Test. The question involved asked students to find the number of exposed faces on a solid, three-dimensional shape. The test-makers' answer suggested that they were unfamiliar with the properties of some fundamental geometric shapes. The student who found the error, we assume, drew on his intuitive sense of geometric shapes rather than what he, as well as the test-makers, had been taught in the classroom.
Most of the public's attention focused on the furor over an error on the PSAT. But the incident highlights a very real problem in the way mathematics is taught: In this three-dimensional world, we teach two-dimensional math. The study of three-dimensional shapes has been put on the back burner.
Both the natural world and the world created by technology abound with geometrical forms, forms that must be both seen and understood by anyone who wants to deal intelligently with these worlds. Yet ignorance of the elementary properties of spatial geometry is profound. We teach our children a great deal about triangles but almost nothing about the tetrahedron, a principal building block in constructions from the atomic structure of metals to the supports of giant bridges.
Ironically, in an era of "back to basics," geometry courses are being compressed and eliminated from school curricula. As a result, teachers and students alike are often unaware of the properties of the space in which they live.
Geometry is basic to almost everything else. The triangles, squares, and circles of plane geometry are the units of construction of the spatial forms that in turn serve as the framework of the material universe. Many of these forms were discovered thousands of years ago by the Greeks; their simplicity and symmetry have inspired artists and scientists through the ages. Today, new structures, with unforeseen properties, are being discovered and studied. For example, the investigation of tensegrity structures (frameworks with flexible joints) is providing insights into fundamental engineering requirements for rigidity, and toroidal (donut-like) shapes are being envisioned for space stations.
The importance of three-dimensional structures and forms increases as our culture becomes more complex. But college students find themselves handicapped in calculus, chemistry, engineering, and art courses because they have had little experience with three-dimensional shapes. The book, blackboard, television screen, video game, and artist's canvas all present two-dimensional abstractions that can, of course, be represented on a drafting board or a computer monitor. But representations, no matter how excellent, have little meaning unless they call forth from the imagination previous experience with objects.
A child who plays with blocks knows that there is no substitute for hands-on experience in developing an intuition about what will stay up and what will fall down. Every hiker knows that reading a contour map of mountain trails is not the same as climbing the mountain. Yet, in our schools (at least after kindergarten), our three-dimensional world is reduced to the plane. We are giving our children a two-dimensional education at a time when science and technology require a mastery of geometrical constructions in three and even four dimensions.
Let's put geometry back in the classroom. And let's teach about the space in which we live. The significance of planar shapes lies not only in the theorems that can be proved from them but also in the things that can be built from them. Rather than focusing on the plane, we should give at least equal emphasis to the third dimension. There is a wealth of material that could easily be incorporated into many courses from elementary school through college.
We envision, for example, a high-school course on the principles of three-dimensional structure. It would begin with a survey of significant forms, both from nature and those of human design: the spherical shapes (shells and convex polyhedra, for example), crystal and molecular structures, and trusses and other bridge constructions. The next step would be to build some of these forms, focusing on basic structural units and how they fit together. This would lead to important questions about the geometric principles on which all forms are based. Students would be led gradually to systematize and formalize their geometric experience into a coherent body of knowledge. This consolidated knowledge would suggest further questions: Which shapes are rigid, which are flexible, and which would collapse? Which shapes fill space? Which shapes can be transformed into other shapes?
But it is not enough to create a new course--no matter how well designed and taught. Geometry should be integrated with the rest of mathematics. Equally important, geometry should also be integrated with instruction in art, biology, chemistry, and physics. Spatial geometry is one of the small group of subjects that indeed make up "the basics." The geometry of fundamental shapes and forms is a practical tool for the machinist--if he or she, in fact, ends up taking the course; it is also an intellectual tool for anyone wanting to understand the arts and the sciences. Why is this basic subject not widely taught in the schools?
Geometry has been taught as part of mathematics, and the fraction of the K-12 curriculum devoted to "mathematics" has remained reasonably constant. The perception that "Johnny can't add" has brought pressure to spend more time on arithmetic in the early grades, and the perception that colleges want statistics, computing, and calculus for admission has led educators to squeeze in more mathematical subjects in the upper grades. As a result, geometry has been squeezed out.
High-school teachers of plane geometry traditionally have had the task of providing rigorous exercises in proving theorems and constructing logical arguments for college-bound students. The few survivors were then subjected to advanced solid geometry--in which the third dimension was finally discussed, but still as the result of theorems and postulates that skirted hands-on experiences. Neither subject appears to have changed since Euclid. Perhaps such courses in ancient and abstract logic should have been squeezed out instead.
We believe that efforts should be made to give a geometric context to elementary-school and junior-high curricula. These students should build three-dimensional objects and investigate their properties. Children should be encouraged to look for basic forms in their surroundings and should be given the vocabulary for talking about the shapes of things. Such aspects of geometry, because they are informal and concrete, often become interesting to students who have shown little aptitude for the math courses taught in the classroom. Care must be given to what geometry is taught, and when and how it is taught; with imagination and proper planning, basic geometry can be introduced as art, science, or construction play. Later, more formal classwork in geometry should build on these early experiences.
Putting geometry back in the classroom and the early introduction of the study of three-dimensional forms will benefit most students and most teachers. Geometric literacy is an important goal that can be achieved, giving new meaning to questions about "the shape of the world" in which we live.
Vol. 04, Issue 20, Page 40