Objects should always be nearby. Awareness of space and volume should be a continuing part of mathematical experience in school at all levels. Refinements such as measuring quantities and relating them with formulas will come in good time. But they should come well after the time when a child first becomes aware of different dimensions of measurement. Too often, the first time a student is encouraged to think about what volume means is the same day that he or she is given a formula for the volume of a sphere or a cone. To encourage fluency in the language of geometry, we need a good deal more "pre-geometry" throughout the school experience, and that should include "pre-solid" as well as "pre-plane" geometry.
Froebel and his colleagues created geometrical gifts from materials available to them, primarily wood, paper, and clay. Today we have the means to improve on the gifts in many ways — with plastic and Velcro, with tape and magnets, not to mention with the powerful computer graphics. The educator's term "manipulatives" — classroom materials takes on new meaning when we can put in front of a young student a tool to manipulate not only simple forms but also the very geometry of higher dimensional space. If we care about educating our children toward the perception of space, we should create truly stimulating manipulatives — geometrical gifts for our day. [an error occurred while processing this directive]