Symbols occupy a pivotal position between processes to be carried out and concepts to be thought about. They allow us both to do mathematical problems and to think about mathematical relationships. In this presentation, the discontinuities that occur in the learning path taken by different students, leading to a divergence between conceptual and procedural thinking, are considered. Evidence will be given from several different contexts in the development of symbols through arithmetic, algebra, and calculus, then on to the formalism of axiomatic mathematics. This is taken from a number of research studies recently performed for doctoral dissertations at the University of Warwick by students from the USA, Malaysia, Cyprus, and Brazil, with data collected in the USA, Malaysia, and the United Kingdom. All the studies form part of a broad investigation into why some students succeed yet others fail. (Contains 26 references.) (Author/MM)