One hundred and fifty years ago, Friedrich Froebel, the inventor of the term "kindergarten," devised a set of "gifts" to introduce children to notions of geometry in several different dimensions. His philosophy was clear: if children could be stimulated to observe geometric objects from the earliest stage of their education, these ideas would come back to them again and again during the course of their schooling, deepening with each new level of sophistication. The rudimentary appreciation of shapes and forms at the nursery school level would become more refined as students developed new skills in arithmetic and measurement and later in more formal algebra and geometry.
Figure 1. Friedrich Froebel, inventor of kindergarten, used geometric objects to stimulate children's imaginations.
In order to capture the imaginations of his young students, Froebel presented them with a sequence of wooden objects for their play in the Children's Garden. Only later would the lessons of that directed set of play experiences be turned into concepts and even later formalized into mathematical expressions. The important thing was to introduce students to forms that they could apprehend and to encourage them to observe and recognize those forms in all of their experiences. In this way they could foster the facility of visualization, so important in applying mathematics to both scientific and artistic pursuits.
Froebel began with objects from the most concrete part of mathematics: balls, cubes, and cylinders. He proceeded to a higher level of abstraction by presenting the children with trays covered by patterns of tiles. Then he moved further into abstraction by introducing collections of sticks of varying lengths, to be placed in designs that would ultimately be related to number patterns.
We can recognize some of Froebel's legacy in materials that we find in today's kindergarten classrooms. There we still have blocks for stacking and tiles for creating patterns on tabletops. Too often, however, these "toys" are left behind when children progress into the serious world of elementary school. A great many rods are used for arithmetic exercises, but a student is lucky to see anything two-dimensional between kindergarten and junior high school. At that time there might be a brief mention of area of plane figures, often merely as an illustration of formulas for measurement. Then the student must wait until high school before any further thought is given to the world of plane geometry.
Two generations ago the hardy souls who made it through the year of formal geometry were permitted to re-enter the third dimension in a still more formalized semester of solid geometry. Then curricula changed. Three-dimensional topics (along with all of analytic geometry) were supposed to be incorporated into a single geometry curriculum. All too often the solid geometry components were treated merely as supplementary topics for the interested student who had a bit of leisure time. Needless to say, solid geometry quickly evaporated from the standard course in geometry. In the present-day rush to prepare students for calculus before they go off to college, we are systematically shortchanging them by ignoring the most practical and useful of all geometry — the geometry of our own dimension. We now have a special opportunity to bring the appreciation of different dimensions back into focus. [an error occurred while processing this directive]