Numerosity discrimination: Infants discriminate small from large numerosities

Numerosity discrimination: Infants discriminate small from large numerosities
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  Numerosity discrimination: Infants discriminate smallfrom large numerosities Wolfgang Mack Johann Wolfgang Goethe-University, Germany Two experiments investigated numerosity discrimination in 7-month-oldinfants, comparing their performance on numbers within the range of subitizing (2 and 3 elements) with numbers marking the limit of this range,4 elements, or lying outside this range (5 and 6 elements). The first experimentidentified 3 as the upper limit of the small range of numbers by contrasting thediscrimination tasks 3 vs. 4 with 4 vs. 5. The second one found, that infants cannot only discriminate 2 from 4, discrepant from the finding of Xu (2003), butalso 3 from 6 elements. It is discussed, that changing the continuous quantityof small numerosities will bias the infants towards element by elementcomparisons instead of comparing numerosities. Nevertheless, the studycorroborates the finding from Xu (2003), that there are two systems of numerosity representation. INTRODUCTION A basic building-block of cognition is discrimination, which is afundamental constituent of every cognitive process. Cognition is based onthe perceptual systems, which allow for a range of sense modality specificdiscriminations, e.g., brightness, loudness or warmness. A major regularityof discrimination that is reliably found in various senses, various ages of onespecies and across species is the Weber–Fechner law. In order to perceive adifference between two stimuli, not their absolute difference but their relativedifference is effective for noticing such a difference. There are qualities thatare amodal, i.e., they can be discriminated in different sense modalities. Sucha so-called primary quality is numerosity, which is a discrete quantity,because one can count, e.g., tones, light flashes, objects in the dark bytouching them one after another. The difference between discrete and Correspondence should be addressed to Wolfgang Mack, Institute of Psychology, JohannWolfgang Goethe University, PO Box 11 19 32, D-60054 Frankfurt am Main, Germany.Email: EUROPEAN JOURNAL OF DEVELOPMENTAL PSYCHOLOGY 2006, 3 (1), 31–47   2006 Psychology Press Ltd DOI: 10.1080/17405620500347695  continuous quantities is their countability, which in language is reflected bycount nouns and mass terms. It is relative easy to count the rain drops onmy window, but it doesn’t make sense to count the water, say in a lake. Buteven discrete quantities must extend into a certain space–time region inorder to be perceived and discriminated one from another, so the differencebetween discrete and continuous quantities in perception is a matter of degree. Nevertheless, counting together with the concept of number is thebase of arithmetic and mathematics that is valued as one of the majorcultural achievements of mankind. Counting has a developmental courseand it is of special interest to study its earliest precursor competence, whichis to be seen in the discrimination of quantities.In developmental psychology research on human preverbal infants hasbeen done to identify precursor competences of the ability to count andother numerical abilities like using certain symbols such as number signs andapplying them according to counting rules (e.g., Gelman & Gallistel, 1978).A certain problem can be seen in the fact that in the human case counting isa kind of discrimination by assigning each entity to be counted a uniquesign, i.e., a number sign. Human preverbal infants do not discriminatediscrete quantities in this symbolic manner, because a string of fixed orderedcounting signs like the decimal numbers is not available to them and theycannot apply the performance rules of counting. But it has been found inseveral studies that they show sensitivity to differences in small numerosities(Antell & Keating, 1983; Starkey & Cooper, 1980; Strauss & Curtis, 1981;Treiber & Wilcox, 1984), even if the correlation of numerosity withcontinuous quantities such as size, area, or contrast is controlled (vanLoosbroek & Smitsman, 1990). The precision in discriminating smallnumerosities contrasts with the imprecision in discriminating largernumerosities (Lipton & Spelke, 2003; Xu, 2003; Xu & Spelke, 2000).These findings correspond with probabilistic discrimination models relatedto the Weber–Fechner Law (van Oeffelen & Vos, 1982), which holds notonly for continuous quantities like weight, length or contrast, becausediscrete quantities also occupy continuous space area and are perceived for acertain duration.Moreover, the conclusion is premature that the greater precision indiscriminating small numerosities up to four is indicative of counting. Itwould be a misapplication of the concept of counting, if infants are ascribedthe ability to count, because counting is based on a generic symbolic numbersystem that allows the generation of an infinite number of distinct signs andtherefore counting is independent of the set size to be enumerated within thelimits of finite human counting practice. Infants’ ability to discriminatesmall numerosities should not be taken for counting, because this can beexplained by a mechanism of percept formation that gives rise to thephenomenon of subitizing (Kaufman, Lord, Reese, & Volkmann, 1949), 32  MACK  found mostly in adult humans utilizing reaction time (RT) as a dependentvariable. As a function of numerosity, RT does not increase for the range of small numerosities up to three or four, in contrast to a significant increaseafter four. This discontinuity of the RT function is seen as evidence for atleast two different processes of enumerating numerosities.There is some disagreement concerning the representational basis of subitizing. Gallistel and Gelman (1992) described subitizing as a very fastkind of non-verbal counting. Trick and Pylyshyn (1994) interpretedsubitizing as a by-product of processes of segmentation and bindingunderlying object formation in early vision. A preattentive capacity-limitedmechanism, which they dubbed finger instantiation (FINST), is supposed toassign reference tokens in a pointer-variable manner in order to individuatefeatures and to limit the computational costs associated with keeping trackof a larger number of items. The optimal number appears to be four. Acomparable model with largely similar implications is the object–file modelof Kahneman and Treisman (1984). Both accounts of object formation outof a small number of object primitives have been generalized to the indexinghypothesis (Leslie, Xu, Tremoulet, & Scholl 1998): object primitives areindividuated using a small set of indices that work like ‘‘mental pointers’’.Because dissociations between subitizing and counting have been found inpatients, suffering from simultanagnosia (Dehaene & Cohen, 1994), small-numerosity discrimination seems to be based on a non-numericalmechanism. A non-numerical mechanism such as ‘‘indexing’’ is a plausiblecandidate for explaining the difference between subitizing and counting,which should already be functional in newborn infants, because infants cansegment visual scenes, and individuate objects and sounds (Simon, 1997).This implies that in infancy the discrimination of small and largenumerosities depends on different mechanisms, an object-tracking system(‘‘indexing’’) and a numerosity approximation system that works accordingto the Weber–Fechner Law (Feigenson, Dehaene, & Spelke, 2004).In a recent study, Xu (2003) found that infants could not discriminatebetween two and four objects. This is surprising, because infants do notgenerally fail to discriminate between small numbers, as is the case with twoand three (Starkey & Cooper, 1980). This is at odds with conclusions thatcan be drawn from an object-tracking mechanism based on assigning a fixednumber of mental pointers. It can be assumed that this divergence fromsome other numerosity discrimination studies carried out with infants wascaused by the variation of the continuous quantity of the elements. In Xu’sExperiment 2, on average, two dots occupied a comparable space to fourdots, but compared to Experiment 1 in the same study, the individualelements were twice as large. As Xu herself states, this may have biasedinfants to compare single objects rather than numerosities. We hypothesizethat infants can discriminate a small numerosity from the within-subitizing NUMEROSITY DISCRIMINATION IN INFANTS  33  range from a numerosity marking the subitizing limit or lying outside thislimit, if elements are made up of a constant continuous quantity and occupya comparable display area. We reason that if the visual angle of a singleelement and of the whole display respectively is to a large extent madesmaller than the visual angle Xu utilized, differences between the continuousquantities of the numerosities are of less importance. The reasoning is thatthe major accomplishment of an index-limited object formation system isthe simultaneous individuation of features grouped in one whole. If thevisual angle is so small that infants can easily perceive the whole group of elements, the unit of comparison will not be a single object, but the discretequantity of the group of elements. Therefore infants should be able todiscriminate two from four, and three from six elements. EXPERIMENT 1 In accordance with Xu’s assumptions, we hypothesized that the Weber– Fechner Law of discrimination would not be valid within the range of smallnumerosities in contrast to numerosities outside this range. This law of discrimination is based on the assumption that the relative difference of twostimuli necessary to notice a change in one stimulus, called just notabledifference (  jnd  ), is constant over the dimension of discriminanda. By addingthis  jnd   to the value  S   of the discriminandum in question and by dividing thesum  S  þ  jnd   by  S  , a constant ratio is obtained, the Weber fraction. Fechner’slaw is an extension of this regularity by an logarithmic function:  S  0 ¼ c log  S  ,where  S  0 is the magnitude of the sensation,  S   is the logarithm of the physicalmagnitude of the stimulus, and k is a constant that takes into account thespecific Weber fraction for a given dimension of discriminanda. Anequivalent formulation is that the threshold of discrimination between twostimuli increases linearly with stimulus magnitude.If there are at least two different mechanisms in perceiving small andlarge numbers, an indexing mechanism for a numerosity range up to fourand a number approximation system for numbers greater than four, then therelative difference between small numbers is not computed for thediscrimination of small numbers, i.e., the Weber fraction is not valid inthe case of small numbers. This would provide extra evidence that small-number discrimination is based on a different mechanism to thediscrimination of large numbers. A small set of objects is individuatedobject by object, because the indexing mechanism assigns each object apointer. A larger set of objects is individuated as a whole by usinginformation like inter-object distance, area occupied by objects andcontrast. Infants should discriminate small numerosities, e.g., 2 vs. 3, butshould not discriminate 4 vs. 6 elements in spite of the same Weber fraction.For the indexing mechanism there are not enough indices to individuate the 34  MACK  numerosity 6, and the Weber fraction of 33% is not enough for the numberapproximation system to arrive at a reliable discrimination (Xu & Spelke,2000). Indeed, Starkey and Cooper (1980) found that infants with anaverage age of 4 months discriminated 2 vs. 3 dots, but not 4 vs. 6 dots, butthe generalizability of this finding is limited, because they only used lineararrays. Therefore, in the following experiment an infant-controlledhabituation technique was utilized, comparable to the procedure used byStarkey and Cooper (1980) with non-linear arrays. Method General remarks concerning participants.  In general, each infant– caregiver dyad took part in only one experiment dealing with numerositydiscrimination and was not followed up thereafter. The parents werecontacted by visiting baby groups or by advertisements in local newspapers.They were informed that the infants would just have to look at differentstimuli, that the infants would always stay with their mothers by sitting ontheir laps and that the whole situation would be arranged in order to allowboth caregiver and infant a comfortable stay. Additionally, they were toldthat no medical examination, no testing of general psychological functionswould be done and that consequently no kind of diagnosis concerning anyability of the infant would be given. All caregivers were told that at any timeduring the lab visit they could feel free to end participation as a result of anysigns of distress on the part of the infant, the caregiver or both. Finally, theywere told that participation would not be paid. At the beginning of theexperiment, all caregivers had to give their written consent to their infantsbeing filmed, state that they had been informed about the study and that theyunderstood that their infants’ data were only to be used for scientificpurposes. After the experiment a Polaroid picture was taken of the infant andthe caregiver, and they were given a little booklet with some informationabout infancy research and addresses that could be consulted concerningquestions of infant health and care.Because there was no hypothesis formulated concerning the effect of sociodemographic status of the caregivers on numerosity discrimination, nosuch information was collected. But most of the addresses indicating theparents living places, the baby groups they joined and the impressions frominteracting with them allow the inference that most of them were typicalmiddle-class members of society.For each of the three experiments a minimum sample size of 32 infants wasapproached, because there were four habituation groups and due to test powerconsiderations it was thought necessary to have at least eight infants in eachgroup to test for order effects of test stimulus presentations. In each experimentinfants were randomly assigned to each habituation and test order group. NUMEROSITY DISCRIMINATION IN INFANTS  35
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