Text:Subitizing: The Role of Figural Patterns in the Development of Numerical Concepts

Glasersfeld E. von (1982) Subitizing: The Role of Figural Patterns in the Development of Numerical Concepts. Archives de Psychologie 50: 191–218. Available at http://vonglasersfeld.com/cgi-bin/index.cgi?browse=journal

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The Role of Figural Patterns in the Development of Numerical Concepts

No thinking, not even the Purest, can take place  but with the aid of the universal forms of our sensua1ity;  only in them can we comprehend it and, as it were, hold fast.  – Wilhelm von Humboldt, 1796.

Whenever we hear a number word, we know, as Euclid already knew, that it refers to a conceptual unit that is itself composed of units. Under normal, everyday circumstances there is no problem about the component units. We know what they are, because we are told by the speaker, or we can see them in front of us, or we can recall them from some specific experiental situation. That is to say, under ordinary circumstances number words are used with reference to actual or represented (imagined) perceptual items. In mathematics, however, that is not the case, even if, as in this paper, we consider nothing but whole numbers and leave aside all the fancier items which mathematicians have come to call “numbers”. Thus, when children are taught arithmetic they are expected to “abstract” the meaning of number words from the perceptual situations in which we ordinarily use them. They are asked to figure out what, for instance, 7 + 5 is, and – though at first they may he given a crutch in the form of tangible beads, checkers, or cookies – they are supposed to solve that kind of problem eventually without the help of any perceptual or representational material. Even if that expectation is fulfilled by many children, the fact that some fulfill it slowly and others not at all, raises the question of how one could specify more precisely what it is that has to be done.
Two earlier papers from our research group dealt with some aspects of that question. The first presented an analysis of counting types that explicated the development of the ability to count abstract unit items that may have been derived from, but are no longer dependent on, sensorimotor material (Steffe, Richards, & von Glasersfeld, 1979), the second a theoretical model for the abstract conceptual structures called “unit” and “number” (von Glasersfeld, 1981 a). In the pages that follow I shall focus on the phenomenon of “subitizing” which has been known a long time but was usually treated as an oddity that is at best marginal in the acquisition of numerical skills. Steffe’s large-scale investigations of children’s progress towards arithmetic competence, however, strongly suggest that perceptual recognition and subsequent representation of small lots up to four or five elements play an indispensable role in the development of arithmetic operations (Steffe, von Glasersfeld, Richards & Cobb, in prep.). The lines of thought I am here pursuing all spring from that interpretation of children’s behavior in their attempts to solve simple number problems and, in particular, from Steffe’s suggestion that they use “subitizing” in a representational mode when no perceptual items are available. I shall argue, however, that perception of composite figural patterns plays an even more fundamental role as an essential building block in the genesis of the concept of number. In order to substantiate this claim, I shall begin by presenting a model developed earlier in the context of psycholinguistic investigations and intended to explicate the connection between words and their meaning. I shall then apply that model to the special case of number words that become associated with spatial as well as temporal configurations of perceptual items. Finally, I shall try to show how the model, in conjunction with Piaget’s notion of “empirical” (or “generalizing”) and “reflective” abstraction, enables us to integrate the perceptual recognition of small lots (i.e. “subitizing”) into a more general theory of numerical concepts and operations.

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