There are two main “classic” aspects of teaching and learning geometry: viewing geometry as the science of space and viewing it as a logical structure, where geometry is the environment in which the learner can get a feeling for mathematical structure (Freudenthal, 1973). At a more advanced stage, this geometry environment acquires a broader sense, without the necessity of a real environment as a basis.

There is a consensus that these two aspects are linked because some levels of geometry as the science of space are needed for learning geometry as a logical structure. This point of view–one that sees the different phases of learning geometry as a developmental process–is intrinsic to most of the theoretical work, research, and instruction that is done in geometry and is the thread that connects the different sections in this chapter.

The various phases of geometry learning raise different kinds of psychological questions. If our concern is geometry as the science of space in general, the initial questions are broad, such as:

• How do children perceive their surroundings?

• What kinds of codes are used in processing visual information?

The questions become narrower if we confine ourselves to visualization; for example:

• What kinds of visual abilities are needed for geometry learning? In particular, how do children create documentation of their surroundings and how do they interpret this documentation; that is, how do children describe (verbally or visually) the three-dimensional world, and how do they interpret such a description?