Symposium
Chair: Susan Carey
Discussants: Elizabeth Spelke and Susan Carey
In spite of almost twenty years of research on infant sensitivity tonumber, several crucial questions remain unanswered. Most fundamentally,recent results from two laboratories, represented here by Mix andFeigenson, question whether experiments that putatively reveal sensitivityto number adequately controlled for responses based on overall spatialextent (e.g., total surface area, total contour). And those studies whichadequately control for this variable, represented here by Xu and Wynn,raise an equally fundamental question: what is the format of infantrepresentation of number, and what computations are computed over thoserepresentations? Two classes of models for infant number representation have been proposedin the literature: 1) object-file representations and 2) analog magnituderepresentations. According to the former class of models, infantsensitivity to number is subserved by the attentional mechanisms thatestablish representations of individuated objects and track them throughtime. In such models, number is represented only implicitly; what isexplicitly represented is each object in an event. According to thelatter class of models, number is explicitly represented by a magnitude,akin to a number line representation. Each of the symposium papers willaddress how the data presented constrain our theories of the format ofinfant representation of the stimuli. Finally, all four papers, plus the two discussants, will consider thefollowing questions: what are the relations between object filerepresentations and analog magnitude representations of number? What arethe relations between analog magnitude representations of number and analogmagnitude representations of continuous quantities? When arerepresentations of each type deployed by the infant?
Details of individual items:
paper
It is well established that infants and young children are sensitive toquantity before they learn to count. What remains unclear is how it ispossible to represent quantity without benefit of conventional counting.In other words, what form does this preverbal representation take?Several models have been proposed, including pattern recognition (Mandler &Shebo, 1982; VonGlasersfeld, 1981), preverbal counting (Gallistel & Gelman,1992; Wynn, 1995), and object tokens (Huttenlocher, Levine, & Jordan, 1994;Simon, 1997; Uller et al., in press). All three of these models applyequally well to simultaneously presented sets. Because the sets in moststudies have been presented all at once, there has been no basis fordeciding which model is correct. However, the models make distinctpredictions regarding sequential sets. In my talk, I will first presentevidence from a sequential matching study with preschool children thatlends support to the object tokens view (Mix, in press). The key findingis that non-counting children can match sequentially presented sets ofobjects but not sequential sets of events.Although there is reason to believe that preschool children use objecttokens to represent number, this may not be the case with infants.Huttenlocher, Jordan and Levine (1994) found evidence of a shift fromapproximate to exact representations of quantity between 2 and 3 years ofage. Children were given nonverbal calculation problems in which theyproduced a set of objects that was equivalent to a hidden set following anaddition or subtraction. Children did not begin to produce the exactsolutions until nearly 3 years of age. The responses of younger childrenclustered around the correct solutions but were not reliably accurate.Huttenlocher et al. attributed this shift to the emergence of a mentalmodel, or object tokens representation. They argued that the appearance ofthis representation is related to the development of other kinds ofsymbolic thought.This conclusion is bolstered by recent studies with infants. A bedrockfinding in the quantitative development literature is that infantdiscriminate between small sets. Specifically, infants gradually loseinterest when sets of the same number are shown but regain interest when anovel set size is shown. It has been widely assumed that thesediscriminations are based on perception of discrete number. However,because these studies used items that were roughly the same size, it ispossible that infants discriminated between different amounts of stuffinstead. To test whether this was so, Clearfield and Mix (1999) habituatedinfants to displays of items that were the same size and number. At test,they presented alternating displays that were either the same overallamount or the same number as the habituation displays. Infants respondedto the change in amount but not the change in number. Similar results havebeen reported when object size was varied in Wynn's (1992) calculationexperiment (Feigenson & Spelke, 1998).In summary, these findings indicate that although an object tokensrepresentation may be used eventually, it may not be available to infants.Instead, infants appear to start out with an undifferentiated sense ofquantity based on overall amount.
paper
A variety of research has served as the empirical basis for claims thatinfants represent number. For instance, infants dishabituate to changes inthe number of elements comprising an array, and anticipate the results ofnumerical transformations. However, an important confound preventsisolating number as the single property driving infants responding. Invirtually all of these studies, number is confounded with the spatialextent of the array. Infants observing a numerical change from 1 to 2objects also witness a doubling of the total object volume present.Because various dimensions of spatial extent (e.g.: total surface area orvolume) covary with number, it is unclear whether infants have beenresponding to changes in number or to changes in spatial extent. In a series of 7 experiments, we explored the basis for claims thatinfants represent number, while systematically manipulating spatial extent. The first 5 experiments tested 7-month old infants in ahabituation/dishabituation paradigm. Experiment 1 extended the standarddishabituation finding to a 1 vs. 2 comparison with 3-D objects. Here,when number and spatial extent were confounded, infants dishabituated tothe change from 1 to 2 and 2 to 1. In Experiment 2, number and spatialextent were pitted against each other. Infants dishabituated to outcomesrepresenting a change in spatial extent, but not those representing achange in number. In Experiments 3-5, changes in spatial extent were heldconstant to test for any presence of numerical response. In Experiment 3,infants saw a test outcome that was novel in number and one that wasfamiliar in number, with both outcomes equally novel in spatial extent.Experiment 4 used the same procedure, but also varied spatial extent overhabituation. Experiment 5 extended this to a 2 vs. 3 discrimination. Innone of these experiments did infants dishabituate to the numerical change. In Experiments 6 and 7, we asked whether infants would demonstrate anumerical response in Wynns (1992) transformation paradigm. In Experiment6, 7-month old infants showed success in a 1+1 2 or 1 and 2-1 1 or 2task. When number and spatial extent were confounded, infants lookedlonger at the unexpected transformation outcomes. Experiment 7 askedwhether number or spatial extent was driving this pattern. Infants sawoutcomes that were expected in number but unexpected in spatial extent, vs.outcomes expected in spatial extent but unexpected in number. Infants againlooked longer at the incorrect spatial extent than at the incorrect number. Infants responded to spatial extent, but not to number, throughout thisseries of experiments. This finding supports the hypothesis that infantsmay rely on mechanisms of object-based attention in the tasks exploredhere. However, unlike the object-file models proposed by Simon (1997) andUller et. al. (in press), we suggest that the basis of comparison is notalways one-to-one correspondence between object-files. Rather, infants cancompare the property information bound to open object-files, includingtheir spatial extent. Therefore, infants may rely on multiple mechanisms,some non-numerical, in tasks that have been interpreted as addressing number.
paper
Many studies have documented that infants as young as 5 months are able todiscriminate small numbers and to perform rudimentary arithmetic operations(Starkey & Cooper, 1980; Wynn, 1992, among others). Wynn (1995) argued thatthe accumulator model accounts for these data and the infant's competenceis comparable to that of other animals, e.g., rats. However, some studies provide evidence that infants fail on similar taskswith set size larger than 3 or 4, supporting the alternative hypothesisthat tracking individual objects, but not number, underlies the successes.Furthermore, recent studies have shown that when continuous variables(e.g., spatial extent, contour length) have been controlled for, infants nolonger showed any response to small numerosities (Mix & Clearfield, 1999;Feigenson et al., 1998). These limitations are particularly puzzling sinceother animals do not show a limit of 3 or 4 and they still exhibitnumerical knowledge when continuous variables are controlled for. We report four experiments with 6-month-old infants investigating theirdiscrimination of large sets (8 vs. 16 and 8 vs. 12) and small sets (2 vs.4 and 2 vs. 3). Experiment 1 tests for discrimination of 8 vs. 16. Infants were habituatedto displays with 8 (or 16) dots, then tested on new displays with 8 dots or16 dots. The experiment was designed to assure that successfuldiscrimination would depend on number rather than other features of thestimuli that correlate with number (e.g., brightness, density of elements,and spatial frequency). First, in each habituation trial, dots werepresented in a different random pattern and size of the dots varied.Second, the two test displays were equated for brightness, element size,and element density by using a 16-dot display that was twice the overallarea as the 8-dot display. The element size and display were selected soas to present values equi-distant from the average value for the 8- and16-dot habituation displays. We found that infants were able to discriminate 16 vs. 8 (a 2:1 ratio). Ontest trials, the infants looked longer at displays with the novel number ofdots, F (1, 15) 5.002, p < .05. However, in Experiment 2, when a smallerratio difference was presented to the infants, i.e., 8 vs. 12, the infantsfailed, F(1,15) .940, p .348 (Xu & Spelke, in press). In Experiments 3&4, using the same design we found that the infants failedto discriminate both 2 vs 3 and 2 vs. 4 (a 2:1 ratio). We propose that infants possess two systems for representing number: asmall 'number' system (the object-file model) which keeps track of up to 3or 4 objects, and a large number system which represents approximatenumerosity for sets of objects larger than 4. The small numberrepresentation is non-unitary, i.e., an array consisting of two elements isrepresented as 'an object and another object' but not 'two objects.' Incontrast, the large number representation is unitary and approximate. Wespeculate that the process of learning to count verbally is the process ofconjoining these two representations.
paper
Research conducted over the past 20 years has shown a range of numericalabilities in infants. Infants can discriminate between displays containingdifferent numbers of items, and they are able to compute the correctresults of additions and subtractions involving small numbers of objects.Recently, some researchers have proposed that these abilities may not bespecifically numerical. Clearfield & Mix (1999) claim that in studies inwhich infants discriminated between different numbers of items, number hastypically been confounded with total area and summed contour length of theitems; they suggest that infants are evidencing a sensitivity to one ofthese attributes rather than to number. Simon (1997) proposes that infants'numerical discrimination abilities are founded, not upon cognitiveprocesses dedicated to number, but upon processes of object-tracking thatexist within the visual cognition system. Leslie (1998), Simon (1997), andUller et al. (in press) appeal to these same object-tracking processes toaccount for infants' ability to compute the outcomes of additions andsubtractions. Feigenson (1999) suggests this ability stems from computationover continuous amount of substance (or possibly area) rather than overnumber of objects. These views are represented by other speakers in thissymposium.In this talk, I will argue against these proposals. First, a review ofrecent evidence shows that infants can successfully discriminate betweennumbers of even when number is not confounded with contour length, area,volume, etc., weighing against the first proposal. Moreover, severalexperiments show that infants can enumerate items that are not objects,weighing against the second proposal. Finally, there exist empirical andtheoretical reasons for favoring the view that infants' numericalcomputation ability stems from a number-specific mechanism that generatesrepresentations of discrete numerical values, rather than fromobject-tracking mechanisms within visual cognition or from computationsover continuous variables of amount of substance or area.