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.