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This book is about the concept of mathematical maturity. The goal of a mathematics education is to transform the student into a mathematically mature individual who treats mathematical ideas analytically and can control and manipulate them effectively. At the heart of the endeavour to solidify this loosely defined but widely acknowledged concept is the privileged position of mathematics as a pure science. The book provides background, data and analysis for recognising and nurturing aspects of mathematical maturity, such as the ability to analyse and communicate abstract ideas, criticise and construct proofs, and move from intuitive to rigorous reasoning. It turns the idea of mathematical maturity from a topic for coffee-room conversation to a topic for analysis and serious consideration. This book will be of interest to anyone, at any level, who wants to be an effective mathematics teacher.