This book is an attempt to break that cycle, being designed for students who might later teach children. It has two aspects, one mathematical, and one educational. Mathematically, it expounds the topology of compact surfaces as far as the Classification Theorem, and treats also the Morse Theory of such surfaces. In fact the whole treatment is a 'handle-body' approach. However, a conventional mathematical treatment would begin 4logically' with topological spaces, continuity and homeomorphisms, before settling down to combinatorial ideas. Any student who could not jump the initial conceptual thresholds would then be cut off from the rich intuitive material that underlies the later work. And it is the rich intuitive material that a future teacher will need, for introducing children to 3-dimensional thinking, so that he can stress the meaning rather than the syntax.