Although our world is three-dimensional, most of our media, as it happens, are two-dimensional: blackboards, books, movies, television, and computer screens. We all invest a great deal of effort learning how to interpret such planar visual information, often in order to help us deal with situations in three dimensions. To live in a three-dimensional world, we do have to know how two-dimensional shapes interact: their behavior provides a necessary prelude to understand fully our own dimension.
As it happens, we gain a good deal of insight by investigating the geometry of an even lower dimension — the line — where number and geometry intermix in the most intimate and powerful way. The geometry of the number line translates beautifully into plane geometry, both in its classical form and in the analytic geometry of number pairs. The momentum that we gain in moving from the first to the second dimension can carry us into our home dimension with renewed insight. The dimensional analogy is a very powerful tool.
Here is an exciting theme that is worth recognizing and passing on to our students: the momentum that brings us from one to two and up into three dimensions does not stop there! The invitation is clear: there are other dimensions waiting to be explored. Mathematics is the key to the elevator that makes them accessible.
The fourth dimension, in particular, is one of our nearest neighbors. Just as we learn a good deal about our own language and culture by studying the language and culture of other countries, so we can begin to appreciate new things about our own "real" world by seeing structures that carry forward to the fourth dimension. Although we cannot explore higher dimensions physically, they are accessible to our minds and, thanks to modern technology, more and more to our vision as well.
Research into language acquisition indicates that, although any infant is capable of learning any language, a child will rather quickly settle into the sound patterns of its own particular language, effectively blocking development of other possibilities. If a child is not introduced early to other languages, he or she will experience much more difficulty in learning a second tongue. Might the same be true with respect to mathematical perceptions? If we wait until students have developed a great deal of arithmetic sophistication (and a great many misconceptions) before we encourage them to think about solid objects and the interaction between different dimensions, we may be depriving them of the chance to appreciate the full power and scope of geometry. [an error occurred while processing this directive]