Many people find mathematics an impenetrable subject. It is a subject where it seems possible to spend many years practising skills and notations without having any substantial understanding of the underlying concepts. This book describes one systematic attempt to intervene and transform this situation. It documents the difficulties experienced by teachers and students as they attempt to adopt new approaches to teaching and learning - approaches based on collaborative discussion and reflection. The book describes an iterative design approach to research and development in which theoretical arguments and reviews of existing research studies are brought together in the design of innovative teaching approaches. These are evaluated in typical classrooms and the outcomes lead to the further refinement of the theories and approaches. Revised approaches are then tested further on a wider scale. The emerging results reveal ways in which teaching methods in mathematics may be designed to become more effective.