What kind of quantitative information do infants represent when they see a set of objects such as a box of toys or a handful of cheerios? Do infants track the number of cheerios, the overall amount of cereal, or the size of each O? Much research has been dedicated to these questions, primarily focusing on the more specific question of whether infants are capable of representing something as abstract as number. Results of a slew of studies have revealed that, despite significant changes in item size (e.g., Xu and Spelke, 2000), item identity (Starkey et al., 1983), or stimulus modality (Kobayashi et al., 2005; Jordan and Brannon, 2006; Izard et al., 2009) infants do represent number (although see Clearfield and Mix 1999, 2001; Cohen and Marks, 2002; Feigenson et al., 2002a). In fact, findings from studies stringently controlling for other array properties that tend to co-vary with number, such as surface area, density, and item size (in simultaneous visual displays) or duration (in sequential presentations) have revealed that infant discriminations of number are ratio-dependent, such that 6-month-olds are capable of detecting a twofold change in number (e.g., 8 vs. 16) but not a 1.5-fold change (8 vs. 12; e.g., Xu and Spelke, 2000; Lipton and Spelke, 2003; Wood and Spelke, 2005).

A prevailing belief, both in the developmental and animal cognition literature, has been that discriminating continuous quantities is trivial for both non-human animals (Davis and Memmott, 1983) and preverbal infants (Piaget, 1952; Mix et al., 2002a,b). In fact, the default assumption was that infants attend to continuous quantities instead of number. While number was thought to be too abstract for a preverbal child or non-verbal animal, representations of surface area were considered perceptual and thus cognitively effortless.

Despite these prevailing views, until recently there were few empirical tests that actually addressed whether infants were even capable of representing continuous dimensions, such as the size of items, duration, or the total amount of stuff. Results of these new studies have revealed that when required to quantify the continuous amount of a single entity (i.e., the duration of a single event and/or the surface area of a single item), infants discriminate continuous extent with an identical level of precision as that with which they discriminate number. That is, 6-month-olds notice a twofold change in the duration of an event (1 vs. 2 s) but not a 1.5-fold change (1 vs. 1.5 s; vanMarle and Wynn, 2006; Brannon et al., 2007). Similarly, they detect a twofold change in the size of a single element (an Elmo face), but fail to notice a 1.5-fold change (Brannon et al., 2006; see Feigenson, 2007 for review). Based on these findings it would appear that, contrary to Piaget and others, infants are no better at discriminating continuous extent than they are at discriminating number (Cordes and Brannon, 2008b).

An important difference however between the studies addressing numerical discrimination in infancy and those described above that addressed discrimination of duration or size is that stimuli from numerical discrimination tasks necessarily involved sets of multiple items (i.e., numbers greater than 1; e.g., Xu and Spelke, 2000) in contrast to the single elements (event or item) used in the duration or size discrimination studies.

Only two studies to date have made an explicit comparison between the ability to discriminate numerosity and the continuous variables that summarize a set. In Brannon et al. (2004) and Cordes and Brannon (2008a), infant abilities to discriminate changes in the cumulative (total) surface area of arrays of dots were investigated. In both studies, infants were habituated to arrays in which the number of items in the arrays changed across trials, however, the cumulative area of the arrays stayed the same and were then tested with arrays in which the cumulative area was the same or different from that of habituation. Results of these studies were surprising: despite previous findings that infants discriminate a twofold change in the area of a single item (Brannon et al., 2006), infants required as much as a fourfold change in the cumulative area of an array of items in order to detect a change. That is, infant discriminations of the continuous extent of an array were remarkably poor.

Although the precision with which infants represent the cumulative area of arrays is poor relative to that with which they represent number, is this indicative of how they represent all aspects of extent within an array? In particular, are infant representations of other continuous variables in an array, such as item size (individual element area as opposed to cumulative area) similarly imprecise? Whereas it is conceivable that the process by which infants represent cumulative area may involve a noisy (imprecise) summation computation (Cordes et al., 2007), tracking the size of individual items of a display is much less likely to involve extraneous computations that may compromise precision. Thus, in light of these findings, it is still possible that infants discriminate continuous extent within an array with equal or better precision than they represent number.

Here, we explore this question by asking how precisely infants track the size of individual items presented within an array. Seven-month-old infants were presented with homogeneous arrays in which the size of the individual items of the arrays was the same throughout habituation and then tested with arrays in which the size of the items was either the same or different from that of habituation. Controlling for changes in other quantitative aspects of the displays (e.g., number and cumulative area) by varying them throughout habituation and test, we evaluated how well infants keep track of the continuous extent of individual elements presented within the context of an array.

Note that demands in our task do not differ substantially from that of the single-item area discrimination task (Brannon et al., 2006). In order to succeed, infants need not attend to all elements of the display or perform any overt computation, but simply attend to the size of a single element in each of the homogeneous displays. So it would be reasonable to predict infants to reveal a similar level of discrimination as they did in the single-item study, discriminating a twofold change in the size of the individual elements of the array. However, given that all arrays in our task are homogeneous in item size and so novel test displays not only involve changes in the size of a single element but of multiple elements simultaneously, it is conceivable that infants may perform better than in previous studies. That is, the simultaneous change in the size of multiple items may provide redundant information to the infant which could serve to facilitate their discriminations, as has been found in other studies involving sequential and numerical ordering (e.g., Bahrick and Lickliter, 2000; Lewkowicz, 2004; Suanda et al., 2008). If so, data should support the claim that infants discriminate extent cues with better precision than that with which they discriminate number (i.e., they discriminate less than a twofold change in extent), as posited by Piaget and others (Piaget, 1952; Mix et al., 2002b; Soltesz et al., 2010).

Alternatively, these early “number is harder” claims may have simply been wrong and continuous extent may not be privileged over number – data revealing infants to be poor discriminators of the extent of an array may be the rule, not the exception (Brannon et al., 2004; Cordes and Brannon, 2008a). When presented with an array, extent representations may be hindered because number is a salient property of a set and infants preferentially attend to number at the expense of area (see Cordes and Brannon, 2009b). Under this scenario, infants have trouble attending to any continuous property in an array of elements because they are distracted by enumeration of the items. If so, infants should require a significantly greater change in the size of the individual items (more than a twofold change) in order to succeed in our task.

In addition to exploring these opposing predictions regarding infant discriminations of item size within an array, the current studies also explore the role of numerical information on representations of an item's size. In our first study we allow number to vary between arrays in an attempt to eliminate numerical cues as a basis for responding (i.e., Brannon et al., 2004; Cordes and Brannon, 2008a). The question is whether they extract the invariant item size and ignore changes in cumulative surface area and number. One possibility, however, is that providing inconsistent numerical information is detrimental to the formation of extent representations – that is, we may unduly bias the infants toward failure in our task by varying both number and cumulative area. Thus, in our second experiment, number, and cumulative area are held constant throughout habituation.

Despite strenuous controls for the intensive parameters of an array (i.e., item size) in most studies of numerical discrimination in infancy (e.g., Xu and Spelke, 2000; Xu, 2003; Brannon et al., 2004; Cordes and Brannon, 2008a), no studies have explicitly examined infants’ abilities to discriminate the size of items within a set. Thus, in two experiments, we explore infant discriminations of changes in the size of items presented within an array. These studies will serve as the first investigation into the reasoning behind the implementation of standard design controls in infant numerical cognition and also provide data to address the on-going debate regarding the relative precision with which infants discriminate number and extent of arrays.

Results of Experiment 1 reveal that infants require as much as a fourfold change in the size of individual items of a display in order to detect a change in the size of the items. This result is surprising in light of previous findings indicating that 6-month-olds can detect as little as a twofold change in the size of a single item (Brannon et al., 2006). While these findings suggest that the presence of more than one item detracts from the ability to represent surface area, there is another explanation that may account for the discrepancy in discrimination precision that must first be ruled out. In the Brannon et al. (2006) study that used a single Elmo face, the number of items did not change across trials as it did in Experiment 1 here. It may be the case that infants were distracted by the changing number of items in each display and thus were less likely to attend to continuous extent cues not because set size was greater than one but because set size varied from trial-to-trial. Although it is common practice in studies of infant quantitative cognition to vary irrelevant quantitative dimensions throughout habituation while keeping the relevant quantitative dimension constant (e.g., Xu and Spelke, 2000; Xu, 2003; Brannon et al., 2004; Cordes and Brannon, 2008a, 2009a), it is also possible that variability in number hampers the cognitive procedure used to represent continuous extent. Experiment 2 tests this hypothesis by examining infant abilities to detect three- or fourfold changes in the size of items in an array when the size of the array is kept constant throughout habituation.

Our results suggest that infants show coarse discrimination for the continuous variables of the individuals that make up a set. Specifically, in two experiments 7-month-old infants required as much as a fourfold change in the size of the individual items of a set in order to detect a change. In contrast to previous findings indicating that 6-month-olds detect as little as twofold changes in the number of items in a set (e.g., Xu and Spelke, 2000), our data reveal that when presented with similar arrays of items, infants represent surface area with remarkable imprecision. This is surprising given that the area representations required for successful discrimination in our task did not require any overt computations such as summing area over multiple elements. At a minimum, success in our task only required attention to a single item in each array whereas representations of number inherently require attention to all individuals in the set. Despite the seemingly simplified task demands in our area discrimination task, infants required twice as much of a change in element size as they did for number in order to succeed. Thus, contrary to common assumptions in the field of cognitive development (e.g., Piaget, 1952; Mix et al., 2002a,b), when presented with a set of items, infants are much more likely to detect a change in number than in continuous extent (Cordes and Brannon, 2008b, 2009a).

In addition to providing the first test of infant discriminations of the size of items in a set, results of the current study speak to the discrepancy in precision with which infants discriminate changes in the area of a single item (Brannon et al., 2006) and of the cumulative area of a set of items (Cordes and Brannon, 2008a). Despite detecting relatively small (twofold) changes in the size of an individual item, infants required a fourfold change in element size when elements were presented within an array of identical elements in order to successfully detect a change. This is somewhat surprising because, as noted above, success in our task simply required attention to a single element in each display – properties of the set as a whole were irrelevant to the discrimination. That is, infants encountered identical task demands to that of the single element area discrimination task in Brannon et al. (2006), and yet they apparently failed due to the presence of more than one item. In fact, one might have predicted that the homogeneous arrays in our experiments would facilitate area discrimination in that they provide multiple redundant cues or a “super-stimulus.” Abundant research reveals infant discriminations benefit from redundant information (e.g., Bahrick and Lickliter, 2000; Lewkowicz, 2004). This however was not the case. Instead, our data suggest that the presence of more than one item in a set may in fact detrimentally impact infant abilities to represent area. Of course a caveat is that infants may have been more interested in the red Elmo faces used by Brannon et al. (2006) compared to our red dot arrays⁴, however overall looking times in Brannon et al. (M = 6.6 s) and our study (M = 6.5 s) did not differ [t(111) = 0.1, p > 0.9]⁵ suggesting that this saliency explanation is unlikely.

One potential explanation for our results is that when infants are presented with arrays they automatically attempt to track number rather than area and that this numerical process interferes with tracking the continuous properties of the elements of the set. Prior work supports this view in that infants, children, and adults are more likely to detect changes in number than cumulative area when presented with large (>3 items) sets (e.g., Xu and Spelke, 2000; Brannon et al., 2004; Barth, 2008; Cordes and Brannon, 2008a, 2009b; Cantlon et al., 2010; see Feigenson et al., 2002a; Cordes and Brannon, 2009a; Xu et al., 2005 for discussion of small/large distinction). The claim that numerical representations interfere with continuous extent representations however is not entirely consistent with our pattern of results. Specifically, when the number of items in the set varied throughout habituation and test (Experiment 1), infants habituated to the displays at the same rate and revealed the exact same precision in discrimination of element size as when the number of items stayed the same (Experiment 2). Thus, the identical pattern of results obtained in Experiments 1 and 2 indicates that infants did not attend to changes in number at the expense of attending to changes in surface area. Instead our data suggest that these were seemingly independent processes. The presence of multiple items appeared to detract from infant abilities to represent surface area regardless of whether number was held constant or varying. This is consistent with previous findings revealing infants are poor discriminators of the continuous extent across sets (cumulative area – Cordes and Brannon, 2008a). Even though infants did not need to attend to more than one item in the set in our current study, when presented with a large set of items, infants spontaneously attended to all elements of the display and this spreading of attention detrimentally impacted the precision with which they represented continuous extent. Surprisingly, this spreading of attention does not appear to have a similar effect on their ability to represent number in other studies (e.g., Xu and Spelke, 2000), again supporting the claim that infants are better discriminators of number than of continuous extent.

Given that infants represent the size of items within a homogeneous set and the cumulative area of a set of items with the same level of precision, our data also suggest that their poor performance in representing cumulative area in previous studies was not solely due to a noisy summation process. Instead, infants are simply less sensitive to changes in element size when elements are presented within an array compared to when they are presented in isolation.

In conclusion, results of the current study provide further evidence that infants are poor discriminators of continuous extent. Despite the prevalent view in the field of cognitive development that representing number requires greater cognitive effort compared to representing size and cumulative area (e.g., Piaget, 1952; Mix et al., 2002a,b; Soltesz et al., 2010), our findings suggest otherwise. In fact, it appears that when presented with sets continuous variables are not automatically encoded in infancy and that instead infants may often fail to track seemingly large changes in the size of objects in their world.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.