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.

Words and meanings

Number words, like other words and conventional means of communication, are embedded in the general semiotic system of the human species which we have all learned to use by growing into it rather than by conceptualizing the way in which it works. The apparent effortlessness with which, as a rule, we acquire language and become more or less proficient communicators without ever becoming aware of how language functions – indeed, the very conception of natural language – has helped to maintain some ingenuous “common sense” notions which, for a long time precluded any detailed understanding of the processes involved. For instance, the idea that words could refer to objects in a user-independent “real” world has been as much of a hindrance to the study of language as the purported link between number words and quantities of objects has been for an understanding of numerical operations.

Much time and futile struggle could have been saved if more attention had been paid to what Ferdinand de Saussure taught about language at the beginning of this century.

“It [language] is a system of signs in which the only essential thing is the union of meanings and sound-images, and in which both parts of the sign are psychological” (de Saussure, 1959, p. 15).
 
“The linguistic sign unites, not a thing and a name, but a concept and a soundimage” (ibid., p. 66).
 

What de Saussure said can be complemented by a simple diagram that shows that the “semantic connection” is always on the experiencer’s side of the experiential interface and not in what is often called the “objective environment” (see Fig. 1). De Saussure’s term “sound-image” refers to conceptual representations abstracted from the experience of spoken words and is, thus, analogous to “concept” which refers to conceptual representations of non-verbal experiences. Reality and the items considered part of it, are put between quotation marks because, from the constructivist perspective, they are an observer’s externalized percepts rather than “real” things or events in an observer-independent ontological world (cf. von Glasersfeld, 1974; Richards & von Glasersfeld, 1979).

In their classic work The Meaning of Meaning, Ogden and Richards (1946) drastically simplified that arrangement by compounding (and thus confounding) “concept” and “sound-image “at the apex of a triangle whose lower corners pointed at “referent” (object) and “symbol” (word) respectively. Strongly influenced by the rise of behaviorism they apparently were uneasy about “mental” constructs such as concepts and sound-images but they still felt the need to put the word “thought” at the apex of their triangle and it is certainly to their credit that they emphasized the fact that the direct connection between symbol and objective referent is an imagined or “purported” one. However, their simplification was an unfortunate step in the direction of radical Behaviorism, the school that later flourished and tried to eliminate thought altogether and to substitute a directly connected “stimulus” and “response” for symbols and referents. It has taken a long time to overcome this categorical elimination of mental operations and meaning, both in linguistics and in psychology. But now the general attitude has changed and we may once more adopt the view held at the beginning of the century when not only de Saussure but also Charles Peirce had realized that symbols and their referents could have no connection other than that formed in the minds of symbol users.
Rather than simplify the schematic arrangement of de Saussure’s “psychological” connection between words and things we must amplify it considerably before we can adopt it as plausible model of the human word processor. For the present purpose it will suffice to say that such a model must include the step from percepts to representations and the step from representations to motor programs for the production of utterances (see Fig. 2).

This break-down discriminates between hearing a word and speaking a word, and it shows the dual semantic connections of auditory word-representations with visual representations of perceptual objects on the one hand, and on the other, with representations of written (or printed) words. The diagram is grossly simplified in that it illustrates only the figural, sensorimotor level of the linguistic skills. The operative, or conceptual, levels would have to be added above it, in a third dimension. It does not show general, more or less “ideal” concepts such as “circularity” or “sphericality” – i.e. constructs that play an important role in the recognition (assimilation) of perceptual patterns (e.g. apples) but are themselves not parts of, but results of prior sensory experience. Abstractions, as von Humboldt said with exemplary clarity, arise from a process that has generally been called “reflection”:

“The essence of thinking consists in reflecting, i.e., in distinguishing the thinking from that which is thought about” (von Humboldt, translation by Rotenstreich, 1974, p. 211). 

The same process has been described and analyzed by Husserl in 1890 and Piaget has used the term “reflective abstraction” all along, distinguishing it from “empirical abstraction”. Recently (Piaget, 1975, p. 63) he has come to the conclusion that the two types of abstraction take place in constant interaction. For the purpose of explication, however, it is helpful to present the two levels of abstraction sequentially, even if in the child’s actual development there is, as Piaget now says, frequent reciprocal interaction between them.

In the context of this paper I shall use “empirical abstraction” to characterize the abstracting of figural patterns from sensorimotor |experience, whereas I shall use “reflective abstraction” to indicate a further level of abstraction that uses the results of empirical abstraction and other operations as raw material. Thus I shall maintain that it is empirical abstraction when the experiencing subject attends, not to the specific sensory content of experience, but to the operations that combine perceptual and proprioceptive elements into more or less stable patterns. These patterns are constituted by motion, either physical or attentional, forming “scan-paths” that link particles of sensory experience. To be actualized in perception or representation, the patterns need sensory material of some kind, but it is the motion, not the specific sensory material used, that determines the patterns’ character. Because of this dependence on some (unspecified) sensory material and motion, they are called figural patterns. Reflective abstraction, on the other hand, takes place when the experiencing subject attends only to the mental operations and abstracts them from whatever sensorimotor context that may have given rise to them. Numerical concepts, as Piaget and Szeminska (1967; Piaget, 1970) and many others have pointed out, are stripped of all sensorimotor properties and, therefore, necessarily involve reflective abstraction. A way to obtain wholly abstract numerical concepts that are independent not only of sensorimotor material but also of figural patterns, was proposed in an earlier paper (von Glasersfeld, 1981a). Here I want to turn to the question of how the more primitive, perceptual concepts (that precede the conception of number) fit into the general semiotic system I have sketched out above and, especially, how their links to number words develop.

Number words without numbers

Number words are words and, as happens with other words, children can learn to say them long before they have formed perceptual representations, let alone abstract concepts to associate with them (in Fig. 2, this corresponds to establishing the straight connections B and G prior to the connections D and E). The learning of empty, as yet meaningless words is easier and more likely when the words have a fixed order in which they frequently occur. That is, of course, the case with number words as well as with the rhymes and prayers which children can learn without the least understanding. Piaget remarked long ago that the reciting of the initial string of number words is usually imposed on children at a very early stage (i.e. before they are four years old) but is then “entirely verbal and without operational significance” (Piaget & Szeminska, 1967, p. 48; cf. also Pollio &: Whitacre, 1970 Potter & Levy, 1968; Saxe, 1979). At an even earlier age, however, children may learn a few isolated number words in the same way in which they learn object-words. It usually happens with the first number words of the conventional sequence, at least from “one” through “five”; and since those are the very ones that are then used in subitizing, we have to ask how words of any kind are initially acquired.

A twelve-month-old may come to associate the auditory experience of the word “spoon” (recurrently uttered by mother) with the global sensorimotor experience of being spoon-fed or trying to feed himself. He may then attempt to reproduce the auditory experience through vocal act of his own. By the age of 24 months, at any rate, the child will have segmented and organized his or her sensorimotor experience quite sufficiently to recognize the spoon in the visual field alone (without tactual or motor complements) and from any angle. Indeed, the spoon will have become a “permanent object” (Piaget, 1937) with characteristic visual pattern and other sensorimotor aspects that can be called up as a representation when the object is not in sight, and that representation will have a firm semantic connection with the sounde-image of the word “spoon” (connection D in Fig. 2).

Once semantic connections are beginning to be formed, any recurrent figural pattern can be semantically associated with a number word. That children actually do this, has been observed quite frequently (e.g. Wirtz,1980,p.2).

Figural patterns can be divided into two groups: Those that are constituted as a spatial configuration (to which corresponds a scan-path) and those that are constituted as temporal sequence (to which corresponds a rhythm). In both groups empirical abstraction from the actual sensory material out of which particular configurations or sequences are built up, yields figural patterns that have a certain general applicability and can be semantically associated with specific names. Thus we have, for instance, the notion of “triangularity” that enables us to see spatial configurations as triangles irrespective of their color, size, angles, or other sensory properties; and in the temporal group we have notions such as “waltz” or “iambus” which enable us to recognize these specific rhythms irrespective of the auditory particulars with which they happen to be implemented. Since subitizing has mostly been studied as a visual phenomenon, I shall first deal with the semantic connections formed between number words and spatial configurations and only then with those involving temporal patterns.

If a child is given a set of wooden or plastic numerals to play with and these toys are occasionally pointed out by the parents as a “three”, a “one”, an “eight”, etc., the child will quickly associate the visual patterns with the appropriate name. If he then gets dominoes, he will add dot-configurations as alternative semantic connections to the number words, and he will do the same for the characteristic arrangements of design elements on playing cards as soon as he is introduced to a card game. In fact, the child will continue to add connections for any pattern that experientially co-occurs consistently with one and the same number word. That is in no way different from what a child has to do, and does, to acquire proficiency in the use of ordinary words such as “dog”. If a poodle happens to be part of the household, a representation of the poodle-percept will be the first meaning of the word. As other dogs enter the child’s experience, new perceptual patterns will be associated with the word. Though there have been theories that suggested it (e.g. Katz & Foder, 1963), it is utterly inconceivable that a child actually forms a universal representation of dog percepts when he or she discovers that adults use the word “dog” to refer, not only to his poodle, but also to a Dachshund, a Great Dane, a St. Bernard, and a bulldog. No common figural representation could cover that variety of canines without erroneously including members of other species as well (cf. Barrett, 1978). Hence children may over-extend the use of the word and say “dog” when first they see a lamb or a calf. But children do learn to use the word “dog” appropriately for visually quite different animals that belong to the class of canines only because zoologists have adopted a taxonomic definition that is based on features which are remote from children’s visual or sensorirnotor experience. The acquisition of appropriate use becomes plausible if we think of it as the result of alternative representations linked to one word by separate associations rather than as a variety of experiences connected to one common representation.

The hypothesis of alternative figural representations that are “equivalent” in that they are all semantically connected to one word, has the virtue that it at once fits how children come to recognize a considerable variety of figural patterns as legitimate referents of a number word long before they have any conception of number or numerosity. The sound image of “three” is easily associated with whatever perceptual configurations are explicitly called “three” by the adults in whose company the child grows up. There will be finger patterns, patterns of dots and lines, arrangements of particular design elements (hearts, diamonds, etc.), and of course the variations of the numeral 3.

Some of these figural patterns area “iconic” in that they are composites of perceptual units which, if considered quantitatively (by a counter who already has numerical concepts) represent the numerosity designated by the number word with which they are associated. Others, like the Arabic numerals (and the Roman numerals above III) are not iconic, because they are not constituted by a collection of items that has the indicated numerosity. Thus I am suggesting that, in subitizing, the child associates figural patterns with number words by a semantic connection and not because of the number of perceptual units of which they are composed. In acts of subitizing, the figural patterns that give rise to it are taken as figural wholes and not as composites of units. In fact, they are recognized as a global configuration, not as a collection of countable items. Perhaps the most convincing demonstration of the fact that subitizing does not involve any perception of number or numerosity is the card player’s ability to recognize and name playing cards even when only part of the individual perceptual units that constitute the configuration are visually accessible to him (see Fig 3). The names of the cards are, of course, number words.