Duality, Ambiguity, and Flexibility: A "Proceptual" View of Simple Arithmetic

In this paper we consider the duality between process and concept in mathematics, in particular using the same symbolism to represent both a process (such as the addition of two numbers 3+2) and the product of that process (the sum 3+2).The ambiguity of notation allows the successful thinker the flexibility in thought to move between the process to carry out a mathematical task and the concept to be mentally manipulated as part of a wider mental schema.We hypothesise that the successful mathematical thinker uses a mental structure which is an amalgam of process and concept which we call a procept.We give empirical evidence from simple arithmetic to show that this leads to a qualitatively different kind of mathematical thought between the more able and the less able, in which the less able are actually doing a more difficult form of mathematics, causing a divergence in performance between success and failure.