Spatial Representation of Elementary Algebra

Our understanding of a concept is tightly connected to the way we represent that concept. Traditionally, mathematics is presented textually. As a consequence novice errors, in elementary algebra for example, are due as much to misunderstandings of the nature of tokens as they are to miscomprehensions of the mathematical ideas represented by the tokens. This paper outlines a spatial algebra by mapping the structure of commutative groups onto the structure of space. We interact with spatial representations through natural behavior in an inclusive environment. When the environment enforces the transformational invariants of algebra, the spatial representation affords experiential learning. Experiential algebra permits algebraic proof through direct manipulation and can be readily implemented in virtual reality. The techniques used to create spatial algebra lay a foundation for the exploration of experiential learning of mathematics in virtual environments.