This paper addresses the developmen

This paper addresses the development of mathematical thinking from elementary beginnings in young children to university undergraduate mathematics and on to mathematical research. It hypothesises that mathematical growth starts from perceptions of, and actions on, objects in the environment. Successful “perceptions of” objects lead through a Van
Hiele development in visuo-spatial representations with increasing verbal support to visually inspired verbal proof in geometry. Successful “actions on” objects use symbolic representations flexibly as “procepts” — processes to do and concepts to think about — in arithmetic and algebra. The resulting cognitive structure in elementary mathematical thinking becomes advanced mathematical thinking when the concept images in the cognitive structure are reformulated as concept definitions and used to construct formal concepts that are part of a systematic body of shared mathematical knowledge. The analysis will be used to highlight the changing status of mathematical concepts and mathematical proof, the difficulties occurring in the transition to advanced mathematical thinking and the difference between teaching and learning the full process of advanced mathematical thinking as opposed to the systematic product of mathematical thought.