The Role Mathematical Transformations and Practice in Mathematical Development

Abstract

The focus of this chapter is on the role of repetitive experience in the development of mathematical concepts. It will be argued that young children can and frequently do discover mathematical properties and invent mathematical rules through experience and practice, substantially independently from direct instruction. Further, it will be argued that a crucial role of repetitive experience is that it can cause the construction and reorganization of knowledge which then provides new information to the developing cognitive system. There are two sources of information that are important for the developments discussed here that are not discovered in this way: subitizing and conventional counting systems. Subitizing or direct perception of small numerosities appears to be innate and does not provide evidence for what is conventionally called the “number concept,” nor is it irrelevant to its acquisition, as has sometimes been suggested. Counting is socially transmitted, although I would agree that it is repetitive experience with the counting system that makes it meaningful.