Annotationen:Subitizing: The Role of Figural Patterns in the Development of Numerical Concepts
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)
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.[8]
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.
There are dot patterns and other spatial configurations that are obviously not distinguishable on the basis of qualitative motor characteristics of scan-paths. If two dots are recognized as a characteristic configuration by the sole fact that they are connected by a straight scan-path, then three, four, or more dots in linear arrangement would be considered the same. Yet this is not the case. Starkey and Cooper (1980) have elegantly shown that even infants of less than six months discriminate quite reliably three dots in a line from two, whereas they do not seem to discriminate between four and six. We may safely assume that infants of that age cannot count. On the other hand, the experiment made clear that the discrimination does not depend on the distance between the dots or on the over-all extension of the display. Hence there must be some inherent difference in the process of perceiving two as opposed to three dots.[11] That difference, I suggest, arises from the capability of the nervous system to differentiate between the temporal patterns constituted by a succession of two perceptual acts as opposed to one or three. “Perceptual act” in this context means no less and no more than the isolation of any item whatever from the rest of the experiential field which, in the case at hand, can be reduced to the visual field. In other words, I am positing a neural mechanism that distinguishes a dual incidence of a given event from a single or a treble incidence (von Glasersfeld, 1981b). What will be considered a dual event, rather than two single ones in succession, depends of course on temporal parameters which we must assume to be inherent in the system – an assumption that can not be invalidated by the observation that these temporal parameters can, under certain circumstances, be modified by experience, practice, or training.
The best-known instances of non-numerical discrimination and recognition of single, dual, treble, etc., events are probably those that occur when we listen to music, where they form the experiential material that goes by the name of “rhythm”. Recent work with children leaves no doubt that the ability to remember, recognize, and replicate simple rhythms (by clapping or tapping) is manifested prior to any abstract or notational representations of such rhythms (Bamberger, 1975, 1980, 1982). That is to say, a sequence of beats which an adult might capture in a conventional notation, is recognized by the child as a figural pattern in a way analogous to that in which visual shapes are first recognized, but with the difference that the rhythm is a configuration in time, whereas the shapes appear as configurations in space (von Glasersfeld, 1979b). From my point of view, it is particularly interesting that the children in that study had little or no difficulty in “drawing” the rhythms by inventing their own visual representations. This readiness to translate a temporal pattern into a linear spatial one strongly supports my assumption that the visual perception of a linear arrangement of items can, indeed, be characterized (and distinguished from other visual experiences of the kind) by its rhythmic properties. In both vision and audition the temporal pattern is constituted by a sequence of perceptual acts which are given their structure by a succession of attentional pulses. This is rather obvious in auditory experience, which is necessarily perceived as an ordered sequence of events in time. In visual experience the sequentially is obscured because, under most circumstances, the individual perceptual acts that constitute a spatial configuration are not subject to an obligatory order.[12] That is why linear arrangements are something of a special case: There, a definite order is at least strongly suggested by the ease and economy of the linear scan-path. A striking example of the ability to transpose spatial into temporal patterns is the telegraphist’s translation of written Morse code into a sequence of motor acts that make and break an electric contact. Here the conventional order of the left to-right line is turned into a temporal order. The groups of dots and dashes on the source tape are neither counted nor perceived as numerical structures – they are taken as patterns of perceptual acts that can be immediately realized in terms of motor acts because the rhythmic configuration remains experientially the same in both modes. In short, then, I am suggesting that the infants’ ability to distinguish between linear displays of one, two, and three dots (Starkey & Cooper, 1980), young children ‘s facility for depicting auditory patterns graphically (Bamberger, 1975), the telegraphist’s skill of translating visual into motor sequences, and, finally, the general ability to subitize linear arrangements of perceptual items, can all be reduced to an underlying capacity to distinguish, recognize, and represent simple temporal configurations that have a characteristic iterative structure. Such iterative structures have rhythmic patterns that can be empirically abstracted as figural patterns from sensory events, not only in the auditory mode, but also in the kinesthetic, tactual, and visual modes, because, in all these modes of experience, structure is constituted by a patterned succession of attentional moments that was experientially determined by the sequence of sensory signals. Once abstracted, these rhythmic patterns may be semantically associated with number-words. The link to a number word, however, does not turn them into numerical concepts. Conceptually they are still figural patterns. Only “reflective” abstraction – the focusing of attention on their iterative structure rather than on the actual or representational sensory material with which that structure happens to have been implemented – can raise them to the level of “pure” abstraction that is characteristic of the conception of numbers (von Glasersfeld, 1981a).
On the other hand, already as figural patterns consisting of sequences of perceptual items, representations of such rhythmic structures make possible the recognition of specific sequential configurations of dots, checkers, fingers, etc., as instances of patterns that are associated with one of the number-words between one and five. Thus, analogous to the way in which a characteristic scan-path serves as the criterial recognition feature for specific classes of spatial patterns (e.g. triangles), a characteristic rhythmic structure serves as the criterial recognition feature for large classes of sensory events which, experientially, can be reduced to temporal patterns. Since these classes comprise not only auditory and motor events (whose nature is sequential in any case), but also visual and tactual events whose sequentiality is created by a relatively fixed linear order of the perceptual process, the recognition of rhythmic structures plays an important role in the phenomenon of subitizing and as a necessary precondition to the reflective abstraction of numerical concepts from linear configurations of items such as dots and tiles, and from Finger patterns.Number and numerosity, thus, do have a root in the processing of percepts and hence there arises the question of how that involvement comes about. I am proposing the thesis that there are several experientially independent paths that contribute to the generation of the complex conceptual structure that we, as adults, ordinarily call number. All three paths contribute to the adult concept, but none of them entails the pursuit of the others.
Before attempting to delineate these paths, let me justify the emphasis on the complexity of the number concept. In an earlier paper (1981a) I presented an attentional model for the structure of number concepts and concentrated on the material out of which such concepts may be formed, namely attentional pulses. I explicitly stated that although the products of the attentional construction have numerosity, their structure must not be confused with the concept of numerosity. That distinction is particularly important because, ordinarily, when we use the word “number” we do not specify whether we are speaking of a numerical structure or of its numerosity. The difference may become more palpable if we compare it to a logically analogous difference in another area. The colors we perceive can be correlated to the frequency of waves which we postulate as the structure of light. If we then say yellow lies between such and such frequencies, we are talking about the structure of yellow, not about the characteristic quality of yellowness that we distinguish from orange and green. Although we can correlate wave lengths and frequencies with our experience of color, they tell us nothing about that experience, nor why it should change as we move along the frequency scale, why we see blue as we move up from green, and red as we move down from orange. Similarly, the attentional model provides a way of thinking about the structure of number concepts but, by itself, it tells us nothing about what we call “quantity” or “numerosity”.
Unlike the color analogy, where there is, at present, no way at all of relating the qualitative experience to any structural model, I shall suggest in what follows how the conception of numerosity can be related to the attentional model of number. I have stressed the complexity of number concepts because, in the view presented here, they comprise, on the one hand, an attentional structure that characterizes all of them – analogous to the wave notion that characterizes the structure of all colors – and, on the other hand, a conception of numerosity that is itself a composite of the products of several relatively independent conceptual developments.Experientially – i.e. reconstructing the child’s experience not from an observer’s perspective but such as it might be from the child’s own point of view – there are obvious differences between a situation with one item and situations with two or three items. There is no problem about picking up one block; either hand can do it. When there are two, one hand may pick them up one after the other, but both hands are needed if the blocks are to be picked up simultaneously. If there are three, there is no way to pick them up simultaneously, but they can, of course, be picked up in succession. Although such considerations may seem trivial to an adult manipulator, they provide, in the case of the child, some important insights into what I call “protonumerical” processes and notions.
1. The simultaneous use of two hands in the displacement of two objects confers a very palpable difference to the visual and tactual experience of spatial configurations of one and two items rather than three or more. 2. The very fact that the same result of displacing two objects can be achieved by using one hand twice in temporal succession or by using both hands simultaneously, confers a very special status to the visual and tactual experience of two items. Moving both hands at the same time can be seen (whenever it is reflected on) as both a “single” and a “double” act, in that the motor component accentuates the singleness whereas its result accentuates the duality.
Though this experience may and does take place long before the infant has acquired any number words, let alone notions of numerosity, it is an experience that continually recurs during human life, simply because we have two hands and we sometimes use them separately and sometimes together. Hence there is in every (normal) child’s experience a host of situations from which to abstract the notion that configurations called “two” can be constructed or decomposed by taking “one” and then another “one”. As an empirical abstraction from sensorimotor experience, that notion constitutes solid protonumerical knowledge.As the work of Steffe has shown, children learn the sequence of number-names and come to use it appropriately in counting sensory items quite some time before they acquire any awareness of the abstract units that constitute the components of numbers on the conceptual level (Steffe, Richards, & von Glasersfeld, 1979; Steffe & Thompson, 19 79). Thus kindergarten children, and as Gelman and Gallistel (1978) report, even three-year-olds, are able to coordinate the vocal production of the conventionally ordered string of number-words with the sequential tagging of perceptual items. Since this activity is generally fostered and rewarded by adults, who tend to take it as evidence of a far more advanced operating with numbers, children quickly become very good at this “apparent number skill” (Hatano, 1979). Among other things, they learn that this “counting” is considered the proper response to any question that contains the expression “How many?”, and that, in this context, success is greatly enhanced if they particularly stress or repeat the last number word of the string they are coordinating with the perceptual items. Although this has, as yet, nothing to do with a conception of number or numerosity,[13] it enhances the appearance of number skill which, in turn, assures reinforcement and, consequently, the proliferation of the “response”. Once this verbal counting becomes something like a routine, it will interact, as Klahr and Wallace (1976) have pointed out, with the other activity that involves number-words: subitizing. This is quite inevitable in the case of those figural patterns that I have called iconic. These patterns – fingers, dominoes, playing cards – are configurations of countable perceptual items and the child therefore can, and at times will, count them. Whenever that happens the count will end with the very same number-word that has already been semantically associated as name with the figural pattern as a whole. That is to say, a dot pattern that has been called a “five” will yield “five” whenever the component dots are coordinated to the conventional string of number words. Such coincidences are unlikely to remain unnoticed. It will not take the child very long to discover that, while the connection between the word “five” and the numeral “5” cannot be confirmed by counting, the connection between the other (iconic) figural patterns and the names they have been given can be so confirmed. The discovery constitutes the first experiential root of the concept of numerosity or, indeed, cardinality (cf. Klahr & Wallace, 1976). The reason for this is that the coincidence of the number-word as terminal point of an iteractive procedure (i.e. the recitation and coordination of a fixed sequence of words) with the same number-word as result of a figural pattern perceived as a whole, provides an experiential foundation for the conception of number as a unit composed of units. The coordination of number-words to a succession of perceptual items requires that each item be conceived as a unit. No matter what sensory properties of the individual items are focused on, each item, in order to become the occasion for a vocal act in the recitation of number words, must be isolated and set apart from the rest of the experiential field, i.e., the item must be framed by the attentional pulses that constitute the unit structure. Similarly, however, the configuration of items, in order to be recognized as a whole, must also be framed by the attentional pulses that turn it, too, into a conceptual unit and, thus, it cannot but contain the others.
The second path towards the full conception of number is equally dependent on the phenomenon of subitizing but leads to a different yet no less “abstract” aspect. What I want to propose could be explicated with the help of any kind of perceptual items that happen to occur as the recursive element in configurations that are associated with the first few number words and are therefore likely candidates for subitizing. Because of their ubiquitous availability, I shall take the fingers of the hand; but what I say would apply equally to dot patterns or other configurations whose elements are always arranged and expanded according to simple composition rules (e.g. Brownell, 1928). When a child has to associate a certain finger pattern with the word “two” and another pattern with the word “four”, he or she can discover on the sensorimotor level (i.e. without any reference to numbers) that a “two” can be turned into a “four” by producing another “two”-configuration. Similar and even simpler will be the discovery that a “two” can be produced by joining a “one” to a “one” and so can a “three” if one starts with a “two”. For an adult, who cannot retrieve the procedures of his or her own infancy – simply because decades of experience have buried or obliterated them – it is practically impossible to see these combinations in a protonumerical, purely figural fashion; but there is every reason to believe that that is the only way a two or three-year-old sees them. In principle, the completion of a “three” configuration by joining a “one” to a “two” is no different from completing a face by drawing a mouth in a circle that already has a pair of eyes. In other words, some relations analogous to those that characterize the conceptual system of whole numbers can be experientially acquired, without the concept of number, through the manipulation of perceptual patterns, their partition and composition. We have thus another instance of simple experiential situations that provide a basis for the future reflective abstraction of the second salient aspect of the constructs we call numbers: their interrelatedness in a homogeneous conceptual system.
There is one further point I want to stress. No matter how figural the deliberately encouraged activities may be, children are likely to persevere in the development of counting because, even if school makes no effort to foster it, everyday life and the social milieu will.[15] Hence children will count; that is to say, they will apply and coordinate the string of number words in an iterative fashion to the configurations they produce in perception or representation. In doing so, they create another occasion for the transition from the figural to the numerical, because through that activity the traditional order of the number-names can be seen figurally and numerically replicated in the ordered sequence of configurations: they get larger and more numerous as one goes on.[16] Thus also another abstract concept, namely ordinality, arises seemingly without break from a sensory-motor activity. What comes after “five” in the verbal sequence can be visually perceived as larger than “five” in the figural progression, and when the child then begins to count abstract units, the independently developed concepts of “after” and “larger” quite smoothly merge to form the abstract concept of the numerical “more”.
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.[2]
“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).
“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).
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.[3] 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.).[4]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.
The same process has been described and analyzed by Husserl in 1890[7] 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.
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).
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).
“permanent object” (Piaget, 1937)
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).
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).
Piaget and Szeminska characterized it well when they said that, for instance, the corner points of a triangle correspond to the corner points of other triangles, irrespective of the knowledge that their number is the same – just as the parts of one human face correspond to those another (1967, p. 93).
Since such characteristic paths can be abstracted as figural representations of “twos”, triangular “threes”, quadrilateral “fours”, and certain “fives”, they reduce the number of individual figural representations that have to be remembered in the case of spatial patterns consisting of up to five perceptual units. And this economy is achieved without any reference to number, nor is the concept of numerosity involved in the procedure (cf. Mandler & Shebo, 1980).
The work of Brownell, who undertook a monumental study of the “apprehension of visual concrete number” (1928), contains many observations that indicate children’s preferential association of specific configurations with certain numbers and that the fact that figural regularities facilitate the “apprehension” of the correct numerosity. Unfortunately Brownell started from the realist assumption that numbers are “concrete” and can therefore be perceptually “apprehended”. Hence he assumed that, no matter how, say, four elements were arranged in space, the perception of their number would always be the same task, and he took great care to average the measurements obtained with different “sensory material”. Indeed, he criticized the earlier study by Howell (1914) because it used only one type of “number pictures”, namely the quadratic type, which represents numbers by placing dots in the corners of imaginary squares.
Starkey and Cooper (1980) have elegantly shown that even infants of less than six months discriminate quite reliably three dots in a line from two, whereas they do not seem to discriminate between four and six.
Recent work with children leaves no doubt that the ability to remember, recognize, and replicate simple rhythms (by clapping or tapping) is manifested prior to any abstract or notational representations of such rhythms (Bamberger, 1975, 1980, 1982).
In short, then, I am suggesting that the infants’ ability to distinguish between linear displays of one, two, and three dots (Starkey & Cooper, 1980), young children ‘s facility for depicting auditory patterns graphically (Bamberger, 1975), the telegraphist’s skill of translating visual into motor sequences, and, finally, the general ability to subitize linear arrangements of perceptual items, can all be reduced to an underlying capacity to distinguish, recognize, and represent simple temporal configurations that have a characteristic iterative structure.
Only “reflective” abstraction – the focusing of attention on their iterative structure rather than on the actual or representational sensory material with which that structure happens to have been implemented – can raise them to the level of “pure” abstraction that is characteristic of the conception of numbers (von Glasersfeld, 1981a).
Infants and very young children manipulate objects – pebbles, buttons, building blocks, or whatever they are given or find to play with. Frequently such play takes the form of assembling a collection in one place and then transferring it one by one to another place. These manipulations have been extensively observed and analyzed by Forman (1973; Forman, Kuschner & Dempsey, 1975).
As the work of Steffe has shown, children learn the sequence of number-names and come to use it appropriately in counting sensory items quite some time before they acquire any awareness of the abstract units that constitute the components of numbers on the conceptual level (Steffe, Richards, & von Glasersfeld, 1979; Steffe & Thompson, 19 79).
Thus kindergarten children, and as Gelman and Gallistel (1978) report, even three-year-olds, are able to coordinate the vocal production of the conventionally ordered string of number-words with the sequential tagging of perceptual items. Since this activity is generally fostered and rewarded by adults, who tend to take it as evidence of a far more advanced operating with numbers, children quickly become very good at this “apparent number skill” (Hatano, 1979).
Once this verbal counting becomes something like a routine, it will interact, as Klahr and Wallace (1976) have pointed out, with the other activity that involves number-words: subitizing. This is quite inevitable in the case of those figural patterns that I have called iconic.
The discovery constitutes the first experiential root of the concept of numerosity or, indeed, cardinality (cf. Klahr & Wallace, 1976).
It is this non-numerical, figural manipulation of perceptual items that Hatano has exploited in his system of early arithmetic instruction. He uses tiles and combines them to form linear configurations up to five. In his report (Hatano, 1979) he does not use the term “subitizing”, but it is clear that there is no counting; the configurations of two, three, four, and five are subitized by the children, and the transformations from one configuration to another are acquired perceptually and not numerically. Five was chosen as the initially most important upper limit, not because of the popularity of the abacus, but because it was found necessary.
The use of five as an intermediate unit, adopting unsegmented tiles of five instead of continuous, connected or isolated tiles, makes it possible to represent these numbers as 5 + 1, 5 + 2, 5 + 3, (Fig. 4 b) respectively, and thus to grasp them almost intuitively, as different from others” (Hatano, 1979,p. 10; the reference to the Figure was changed to fit the numeration in this paper.)
(b) Easily recognizable configurations, whose figural representations can be associated with number words between “five” and “ten” (from Hatano, 1979).
In my view, the success of Hatano’s method is due above all to the fact that it supplies material that encourages the creation of stable, recursive, figural representations and sensorimotor activities such that they facilitate manipulation and composition and, beyond that, once the child comes to reflect upon them, they supply an optimal ground for the abstraction of numerical concepts and operations.
The process that Hatano engenders by the deliberate insertion of the intermediary unit “five” is analogous to the process Steffe has hypothesized on the basis of his observation of children who could reliably solve counting-on problems when the number to be counted-on was within the subitizing range (i.e. no larger than 5) but were invariably unable to solve those problems when the number was larger. Steffe explained this observation by the assumption that, as long as the child can produce for himself a figural, subitizable representation of the counting-on number, he can proceed to tick off the perceptual items contained in that representation as he continues the number word sequence from the given starting-point. In this way the child comes to an obligatory stop when he reaches the last item in the representation. When, on the other hand, the counting-on number is larger than 5, there is no ready figural representation for it and hence no way of ticking off and nothing to provide a stopping-point (Steffe, personal communication, 1978). Hatano’s method and Steffe’s observation of these counting-on episodes are examples that provide a perfect fit for the theoretical assumptions 1 have made.
