Some Comments on Geometry and Models

In March 1990 COMAP sponsored a workshop with educators and researchers in geometry to address a perceived stagnation in the teaching of geometry. One outcome was the publication Geometry's Future (published by COMAP), which includes the following twelve specific recommendations:

1. Geometric objects and concepts should be more studied from an experimental and inductive point of view rather than from an axiomatic point of view. (Results suggested by inductive approaches should be proved.)
2. Combinatorial, topological, analytical, and computational aspects of geometry should be given equal footing with metric ideas.
3. The broad applicability of geometry should be demonstrated: applications to business (linear programming and graph theory), to biology (knots and dynamical systems), to robotics (computational geometry and convexity), etc.
4. A wide variety of computer environments should be explored (Mathematica, LOGO, etc.) both as exploratory tools and for concept development.
5. Recent developments in geometry should be included. (Geometry did not die with either Euclid or Bolyai and Lobachevsky.)
6. The cross-fertilization of geometry with other parts of mathematics should be developed.
7. The rich history of geometry and its practitioners should be shown. (Many of the greatest mathematicians of all time: Archimedes, Newton, Euler, Gauss, Poincaré, Hilbert, Von Neumann, etc., have made significant contributions to geometry.)
8. Both the depth and breadth of geometry should be treated. (Example: Knot theory, a part of geometry rarely discussed in either high school or survey geometry courses, connects with ideas in analysis, topology, algebra, etc., and is finding applications in biology and physics.)
9. More use of diagrams and physical models as aids to conceptual development in geometry should be explored.
10. Group learning methods, writing assignments, and projects should become an integral part of the format in which geometry is taught.
11. More emphasis should be placed on central conceptual aspects of geometry, such as geometric transformations and their effects on point sets, distance concepts, surface concepts, etc.
12. Mathematics departments should encourage prospective teachers to be exposed to both the depth and breadth of geometry.

In emphasizing the important role of visual examples and models, Hilbert wrote in his preface to Geometry and the Imagination:

...[A] presentation of geometry in large brush-strokes, so to speak, and based on the approach through visual intuition, should contribute to a more just appreciation of mathematics by a wider range of people than just the specialists. For it is true, generally spaking, that mathematics is not a popular subject, even though its importance may be generally conceded. The reason for this is to be found in the common superstition that mathematics is but a continuation, a further development, of the fine art of arithmetic, of juggling with numbers. Our book aims to combat that superstition, by offering, instead of formulas, figures that may be looked at and that may easily be supplemented by models which the reader can construct.

This is echoed by Cundy and Rollett in their book Mathematical Models:

The human mind can seldom accept completely abstract ideas; they must be derived from, or illustrated by, concrete examples. Here the reader will find ways of providing for himself tangible objects which will bring that necessary contact with reality into the symbolic world of mathematics.

These issues can be addressed with the use of models to encounter, prove, and illustrate important concepts in geometry. This is certainly not a novel idea, but the regular use of models in middle and high school geometry courses is still infrequent. That this should be the case for a branch of mathematics which is directly motivated by and lends itself so readily to visual illustration and exploration is indeed disturbing. Some contributing factors are the convenience of traditional classroom instruction, the lack of either the time to construct models during class time or the space to store projects in progress between class periods, expense, and lack of experience and familiarity with three-dimensional geometry and visualization.

The concept of ``model'' should be interpreted liberally. A good model may be

• Physical (e.g., a three-dimensional construction or a two-dimensional drawing),
• Analytic (e.g., appropriate equations, coordinates, or data structure),
• Digital or Computational (e.g., created by or drawn using a computer), or
• Verbal.

Different models offer different degrees of exploratory ease and require different levels of mathematical precision and understanding. The same mathematical object can be illustrated by different types of models.

Models can be used to suggest mathematical results that should then be proved (for example, constructing a dodecahedron out of pentagons does not prove its mathematical existence, but can lead to the derivation of the coordinates of its vertices), and mathematical results can conversely be illustrated by models.