DUALITY, AMBIGUITY, AND FLEXIBILITY - A PROCEPTUAL VIEW OF SIMPLE ARITHMETIC

In this paper we consider the duality between process and concept in m athematics, 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 proc ess to carry out a mathematical task and the concept to be mentally ma nipulated as part of a wider mental schema. Symbolism that inherently represents the amalgam of process/concept ambiguity we call a ''procep t.'' We hypothesize that the successful mathematical thinker uses a me ntal structure that is manifest in the ability to think proceptually. We give empirical evidence from simple arithmetic to support the hypot hesis that there is a qualitatively different kind of mathematical tho ught displayed by the more able thinker compared to that of the less a ble one. The less able are doing a more difficult form of mathematics, which eventually causes a divergence in performance between them and their more successful peers.