for the learning of mathematics

an international journal of mathematics education

David Tall - Vol. 31 Num. 1 (2011)
 Crystalline concepts in long-term mathematical invention and discovery


This paper introduces the notion of 'crystalline concept' as a focal idea in long-term mathematical thinking, bringing together the geometric development of Van Hiele, process-object encapsulation, and formal axiomatic systems. Each of these is a strand in the framework of 'three worlds of mathematics' with its own special characteristics, but all three have an underlying structure that has deductive relationships constrained by the context. In mathematics we may make choices and construct ideas for ourselves, but once a mathematical context is established, it has a crystalline structure of relationships that we discover as a consequence of the structure.