 by Beth Walker Ph.D. Mathematics Education
What do you know about space; not the final frontiers of cosmic explorers but our most immediate environment? We live in three-dimensional space, eat off two-dimensional tabletops, and drive or walk in one-dimension at a time. Like a riddle, we don’t create space, but often try to contain it. We assign space measures, but often don’t understand what those measures mean. In mathematics, the study of one-, two- and three-dimensional space is often referred to as geometry. What are dimensions and are there really more than three?

Mathematical understanding is built from a collection of related facts, which contribute to a conceptual understanding based on observed characteristics. The next step in cognitive development is generalization of conceptual understanding so that knowledge can be applied to new situations. For example, students learn to recognize certain shapes as triangles. Conceptual understanding would be that ALL closed shapes with three sides or three angles are triangles. A generalization is that closed shapes are named by the number of sides (and therefore the number of angles) they contain.

A common vocabulary is helpful in constructing conceptual understanding and generalizations so that ideas can be shared with others. In the above example, names of shapes were crucial in forming a generalization.

So what is a dimension? How is it defined? For the sake of simplicity, discussion will be restricted to one-, two- and three- dimensions, which is enough to describe the world that most of us live in. If many one-dimensional examples are encountered, a conceptual idea can be formed about what makes something one-dimensional. This conceptual understanding may result in a generalization or definition. Ideally, this definition will extend to higher dimensions.

Here are some examples of one-dimensional objects from a theoretical perspective and, ignoring the fact that all physical objects have some thickness however small, some examples from real life:

• a line
• a ray
• a line segment