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

Abstract

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).

Objects, Actions, and Images: A Perspective on Early Number Development

“…In doing so the paper summarises empirical evidence obtained over a period of ten years. This evidence first signalled qualitative differences in numerical processing (Gray, 1991), and was seminal in the development of the notion of procept (Gray & Tall, 1994 …”

The transition to formal thinking in mathematics

“…Piaget hypothesised that all individuals pass through the same sequence of stages at different rates, but Gray and Tall (1994) observed the proceptual divide in which children develop in different ways, some clinging to the security of known step-by-step procedures, while others compress their knowledge into the flexible use of symbols as process and concept (procepts). Procedures occur in time and work in limited cases but may not be sufficiently compressed into thinkable concepts to be used flexibly for more sophisticated thinking.…”

“…This can lead to a spectrum of outcomes within a single group of learners between those who perform procedurally and those who develop greater flexibility. In arithmetic, Gray & Tall (1994) called this the proceptual divide.…”

“…However, there is a more sophisticated level that involves not only the ability to perform the procedure, but also to think about it as sophisticated entities in their own right, where symbols operate dually as process and concept (procept) to allow us to think flexibly (Gray & Tall, 1994).…”

“…(Figure 1. ) 'Proceptual symbolism' refers to the use of symbols that arise from performing an action schema, such as counting, that become thinkable concepts, such as number (Gray & Tall, 1994). A symbol such as 3+2 or b 2 − 4ac represents both a process to be carried out or the thinkable concept produced by that process.…”