Procept Theory

The notion of procept was introduced by Gray & Tall (1991, 1994) to denote the use of symbol both as a process (such as addition) and as a concept (the sum). Suddenly the whole of arithmetic, algebra, and symbolic calculus was seen to be populated by procepts. The processes often begin as step-by-step procedures that are slowly routinised into processes that can be thought of as a whole without needing to carry them out. Symbols are used to allow the mind to pivot between the procedure or process on the one hand and the mental concept on the other. This spectrum of symbol usage can be described more precisely as follows: 

• A procedure consists of a finite succession of actions and decisions built into a coherent sequence. It is seen essentially as a step-by-step activity, with each step triggering the next.

• The term process is used when the procedure is conceived as a whole and the focus is on input and output rather than the particular procedure used to carry out the process. A process may be achieved by n procedures (n ≥ 0) and affords the possibility of selecting the most efficient solution in a given context. (We shall consider the enigmatic case n = 0 later.)

• A procept requires symbols to be conceived flexibly both as processes to do and concepts to think about. This flexibility allows more powerful mental manipulation and reflection to build new theories. Development consists of increasingly sophisticated usage of symbols with differing qualities of flexibility and ability to think mathematically .

Procedural thinking certainly has its value—indeed, much of the power of mathematics lies in its algorithmic procedures. However, a focus on procedures alone leads to increasing cognitive stress as the individual attempts to cope with more and more specific rules for specific contexts. Gray, Pitta, Pinto, and Tall (1999) give evidence to show that there is a continuing bifurcation in mathematical behaviour, with the less successful focusing more on procedures and the more successful students developing a hierarchy of sophistication through encapsulation of process into procepts.

Human thinking capacity is enhanced by a range of activities unique to the human brain. One is the use of language, which allows the mind to conceive of hierarchies of concepts and focus at an appropriate level of generality to solve a given problem. Language also allows communication that shares the corporate knowledge of the species. Another major facility is the use of symbols in mathematics. The invention of the modern decimal number system and the  development of algebra and calculus all build on a use of symbolism that specially fits the configuration of the human brain. The brain is a multi-processing system, with an enormous quantity of activity going on at the same time. The only way that the human brain makes sense of such pandemonium is to focus attention on things that matter and suppress other detail from conscious thought (Crick, 1994, pp. 63- 64). To reduce the strain of cognitive processing, some activities become routinised so that they can be performed largely without conscious attention, although they may be brought to the fore when necessary. To link the comparatively small, but dynamic, focus of attention to other brain activity requires two essential conditions. One is that there is a part of the activity sufficiently compressed to be held, perhaps as a token, in the focus of attention; the other is that this part has rich and immediate links with important related ideas. The notion of procept has precisely these properties. The symbol, as a token for the procept, can be used to focus either on the compressed concepts and their relationships, or attention can easily switch to the process aspects of the activity—preferably using routinised procedures operating in efficient ways that place minimal cognitive strain on overall brain function. 

The phenomenal explosion of science and technology in recent centuries has owed much of its power to the construction of mathematical models followed by the use of mathematical techniques to compute and predict what is going to happen. I believe that the formulation of the notion of procept allows us to describe how this happens and that this notion is as fundamental to the cognitive psychology of symbol manipulation as the notions of set or function are foundational to mathematics itself. Each of these constructs—set, function, procept—has certain properties in common. Firstly, they are all three essentially simple concepts, underlying the theory in such a profound way that, paradoxically, although they are eventually seen as foundational to the theory, they have not been formulated explicitly until long into the practical development of the theory. For instance, although the concept of set may now be seen as being more fundamental than that of a function, it was not formally defined until long after. 

It is the classic tool-object dialectic (Douady, 1986): a concept is used as a tool for a considerable time until it takes on sufficient importance to be studied as an object in its own right. When Eddie Gray and I first formulated the notion of procept, it occurred in a context where we were talking together about children doing arithmetic. We knew the theory of process-object encapsulation (or reification) that had arisen in the theories of Piaget , Dienes, Davis, Dubinsky, Sfard, and others (Tall, D. O., Thomas,

M. O. J., Davis, G., Gray, E. M., & Simpson, A. (2000).). But we found that an addition such as 3 + 2 could equally well be conceived by children as the sum 3 + 2 and we had no word to describe the duality of symbol use. The word “procept” was suggested by one of us and the other quickly saw its power in terms of duality (as process and object), ambiguity (as process or object) and flexibility (to move easily from one to the other).