Do fish count? Spontaneous discrimination of quantity in female mosquitofish

The spontaneous tendency to join the largest social group was used to investigate quantity discrimination in fish. Fish discriminated between shoals that differed by one element when the paired numbers were 1vs2, 2vs3 and 3vs4, but not when 4vs5 or larger. Using large numerosities (>4), the ability to discriminate between two numbers improved as the numerical distance between them increased and a significant discrimination was found only with ratios of 1:2 or smaller (4vs8, 8vs16 and 4vs10). Experiments to control for non-numerical variables evidenced the role played by the total area of stimuli with both large and small numerosities; the total quantity of movement of the fish within a shoal appeared also important but only when large numerosities were involved. Even though the pattern of discrimination exhibited by female mosquitofish is not fully consistent with any of the existing models of quantity representation, our results seem to suggest two distinct mechanisms in fish, one used to compare small numbers of objects and one used when larger numerosities are involved.     

Numerosity discrimination in preschool children

We used a numerical bisection procedure to examine preschool children's sensitivity to the numerical attributes of stimuli. In Experiment 1 children performed two tasks. In the Cups Task they earned coins for choosing a green cup after two drumbeats and a blue cup after eight drumbeats. In the Gloves Task they earned coins for raising a red glove on their left hand after two drumbeats and a yellow glove on their right hand after eight drumbeats. Then in each task a psychometric function was obtained by presenting intermediate numerosities and recording the percentage of trials in which children chose the "many" option. In Experiment 2 children's performance in a '2 vs. 8' discrimination was compared with their performance in a "4 vs. 16" discrimination. Results showed that the individual psychometric functions were of two types, one in which the percentage of "many" choices increased gradually with stimulus numerosity and another in which it increased abruptly, in a step-like manner. Although the average point of subjective equality was close to the geometric mean of the anchor numerosities and the average functions for "2 vs. 8" and "4 vs. 16" superimposed when plotted on a common scale (the scalar property), the individual data were highly variable both across tasks (Cups and Gloves) and numerosity ranges ('2 vs. 8' and '4 vs. 16'). It is suggested that between- and within-subjects variability in the psychometric function is related to children's verbalizations about the sample stimulus.     

Dissociation between small and large numerosities in newborn infants

In the first year of life, infants possess two cognitive systems encoding numerical information: one for processing the numerosity of sets of 4 or more items, and the second for tracking up to 3 objects in parallel. While a previous study showed the former system to be already present a few hours after birth, it is unknown whether the latter system is functional at this age. Here, we adapt the auditory‐visual matching paradigm that previously revealed sensitivity to large numerosities to test sensitivity to numerosities spanning the range from 2 to 12. Across studies, newborns discriminated pairs of large numerosities in a 3:1 ratio, even when the smaller numerosity was 3 (3 vs. 9). In contrast, newborn infants failed to discriminate pairs including the numerosity 2, even at the same ratio (2 vs. 6). These findings mirror the dissociation that has been reported with older infants, albeit with a discontinuity situated between numerosities 2 and 3. Two alternative explanations are compatible with our results: either newborn infants have a separate system for processing small sets, and the capacity of this system is limited to 2 objects; or newborn infants possess only one system to represent numerosities, and this system either is not functional or is extremely imprecise when it is applied to small numerosities.     

Core information processing deficits in developmental dyscalculia and low numeracy

There are two different conceptions of the innate basis for numerical abilities. On the one hand, it is claimed that infants possess a 'number module' that enables them to construct concepts of the exact numerosities of sets upon which arithmetic develops (e.g. Butterworth, 1999; Gelman & Gallistel, 1978). On the other hand, it has been proposed that infants are equipped only with a sense of approximate numerosities (e.g. Feigenson, Dehaene & Spelke, 2004), upon which the concepts of exact numerosities are constructed with the aid of language (Carey, 2004) and which forms the basis of arithmetic (Lemer, Dehaene, Spelke & Cohen, 2003). These competing proposals were tested by assessing whether performance on approximate numerosity tasks is related to performance on exact numerosity tasks. Moreover, performance on an analogue magnitude task was tested, since it has been claimed that approximate numerosities are represented as analogue magnitudes. In 8-9-year-olds, no relationship was found between exact tasks and either approximate or analogue tasks in normally achieving children, in children with low numeracy or in children with developmental dyscalculia. Low numeracy was related not to a poor grasp of exact numerosities, but to a poor understanding of symbolic numerals.     

Enumeration of small and large numerosities: the effect of element visibility

Precise enumeration is associated with small numerosities within the subitizing range (<4 items), while approximate enumeration is associated with large numerosities (>4 items). To date, there is still debate on whether a single continuous process or dual mutually exclusive processes mediate enumeration of small and large numerosities. Here, we evaluated a compromise between these two notions: that the precise representation of number is limited to small numerosities, but that the approximate representation of numerosity spans across both small and large numerosities. We assessed the independence of precise and approximate enumeration by looking at how luminance contrast affected enumeration of elements that differ by ones (1-8) or by tens (10-80). We found that enumeration functions of ones and tens have different characteristics, which is consistent with the presence of two number systems. Subitizing was preserved for small numerosities. However, simply decreasing element visibility changed the variability signatures of small numerosities to match those of large numerosities. Together, our results suggest that small numerosities are mediated by both precise and approximate representations of numerosity.     

Auditory-visual intermodal matching of small numerosities in 6-month-old infants

Recent studies have reported that preverbal infants are able to discriminate between numerosities of sets presented within a particular modality. There is still debate, however, over whether they are able to perform intermodal numerosity matching, i.e. to relate numerosities of sets presented with different sensory modalities. The present study investigated auditory-visual intermodal matching of small numerosities in infancy by using a violation-of-expectation paradigm. After being familiarized with events of a few objects impacting a surface successively, 6-month-old infants were alternatively presented with two and three tones while the movement of each object remained hidden behind an opaque screen. The screen was then removed to reveal either two or three objects. Results showed that the infants looked significantly longer at the numerically nonequivalent events (the three-tone/two-object and the two-tone/three-object events) than at the numerically equivalent events (the two-tone/two-object and the three-tone/three-object events) irrespective of the rate or duration of auditory tones presented. These findings suggest that infants are capable of performing intermodal matching of small numerosities and that they might possess abstract representations of numerosity beyond sensory modalities.     

Is approximate numerical judgment truly modality-independent? Visual, auditory, and cross-modal comparisons

The numerosity of any set of discrete elements can be depicted by a genuinely abstract number representation, irrespective of whether they are presented in the visual or auditory modality. The accumulator model predicts that no cost should apply for comparing numerosities within and across modalities. However, in behavioral studies, some inconsistencies have been apparent in the performance of number comparisons among different modalities. In this study, we tested whether and how numerical comparisons of visual, auditory, and cross-modal presentations would differ under adequate control of stimulus presentation. We measured the Weber fractions and points of subjective equality of numerical discrimination in visual, auditory, and cross-modal conditions. The results demonstrated differences between the performances in visual and auditory conditions, such that numerical discrimination of an auditory sequence was more precise than that of a visual sequence. The performance of cross-modal trials lay between performance levels in the visual and auditory conditions. Moreover, the number of visual stimuli was overestimated as compared to that of auditory stimuli. Our findings imply that the process of approximate numerical representation is complex and involves multiple stages, including accumulation and decision processes.     

Enumeration of small and large numerosities: The effect of element visibility

Precise enumeration is associated with small numerosities within the subitizing range (<4 items), while approximate enumeration is associated with large numerosities (>4 items). To date, there is still debate on whether a single continuous process or dual mutually exclusive processes mediate enumeration of small and large numerosities. Here, we evaluated a compromise between these two notions: that the precise representation of number is limited to small numerosities, but that the approximate representation of numerosity spans across both small and large numerosities. We assessed the independence of precise and approximate enumeration by looking at how luminance contrast affected enumeration of elements that differ by ones (1–8) or by tens (10–80). We found that enumeration functions of ones and tens have different characteristics, which is consistent with the presence of two number systems. Subitizing was preserved for small numerosities. However, simply decreasing element visibility changed the variability signatures of small numerosities to match those of large numerosities. Together, our results suggest that small numerosities are mediated by both precise and approximate representations of numerosity.     

Non-symbolic arithmetic in adults and young children

Five experiments investigated whether adults and preschool children can perform simple arithmetic calculations on non-symbolic numerosities. Previous research has demonstrated that human adults, human infants, and non-human animals can process numerical quantities through approximate representations of their magnitudes. Here we consider whether these non-symbolic numerical representations might serve as a building block of uniquely human, learned mathematics. Both adults and children with no training in arithmetic successfully performed approximate arithmetic on large sets of elements. Success at these tasks did not depend on non-numerical continuous quantities, modality-specific quantity information, the adoption of alternative non-arithmetic strategies, or learned symbolic arithmetic knowledge. Abstract numerical quantity representations therefore are computationally functional and may provide a foundation for formal mathematics.     

Discrimination of small numerosities in young chicks

Chicks were trained to discriminate small sets of identical elements. They were then tested for choices (unrewarded) between sets of similar numerosities, when continuous physical variables such as spatial distribution, contour length, and overall surface were equalized. In all conditions chicks discriminated one versus two and two versus three stimulus sets. Similar results were obtained when elements were presented under conditions of partial occlusion. In contrast, with sets of four versus five, four versus six, and three versus four elements chicks seemed unable to discriminate on the basis of number, although nonnumerical discrimination based on perceptual cues was observed. This adds to increasing evidence for discrimination of small numerosities of up to three elements in human infants and nonhuman animals.     

Magnitude comparison in preschoolers: what counts? Influence of perceptual variables

This study examined numerosity comparison in 3-year-old children. Predictions derived from the analog numerical model and the object-file model were contrasted by testing the effects of size and ratio between numerosities to be compared. Different perceptual controls were also introduced to evaluate the hypothesis that comparison by preschoolers is based on correlated perceptual variables rather than on number per se. Finally, the relation between comparison performance and verbal counting knowledge was investigated. Results showed no evidence that preschoolers use an analog number magnitude or an object-file mechanism to compare numerosities. Rather, their inability to compare sets controlled for surface area suggests that they rely on perceptual cues. Furthermore, the development of numerosity-based representations seems to be related to some understanding of the cardinality concept.     

Quantity matching by an orangutan (Pongo abelii)

An adult male orangutan (Pongo abelii) was presented with a series of delayed matching-to-sample (DMTS) tasks in which he was to match images based on (a) the number of individual animals depicted in the photograph (from 1 to 4), (b) the number of abstract shapes presented in the stimulus (from 1 to 4), or (c) the number of dots presented in the stimulus (from 1 to 4, 4–7, or 7–10). The spatial arrangement of the dots and the background color of the stimuli varied, and the size of the dots was manipulated to control for overall ratio of foreground to background. The subject’s performance was not affected by these perceptual features, but was affected by the absolute difference and ratio between number of elements in the comparison stimuli. However, the relationship between these variables and his performance was not always linear as predicted by the analog magnitude model. In addition, the subject showed a high degree of transfer to novel numerosities up to ten, indicating that orangutans are capable of estimating quantity for a greater number of items than can presumably be subtilized by humans.     

Do infants have a sense of numerosity? A p‐curve analysis of infant numerosity discrimination studies

Research demonstrating that infants discriminate between small (e.g., 1 vs. 3 dots) and large numerosities (e.g., 8 vs. 16 dots) is central to theories concerning the origins of human numerical abilities. To date, there has been no quantitative meta‐analysis of the infant numerical competency data. Here, we quantitatively synthesize the evidential value of the available literature on infant numerosity discrimination using a meta‐analytic tool called p‐curve. In p‐curve the distribution of available p‐values is analyzed to determine whether the published literature examining particular hypotheses contains evidential value. p‐curves demonstrated evidential value for the hypotheses that infants can discriminate between both small and large unimodal and cross‐modal numerosities. However, the analyses also revealed that the published data on infants’ ability to discriminate between large numerosities is less robust and statistically powered than the data on their ability to discriminate small numerosities. We argue there is a need for adequately powered replication studies to enable stronger inferences in order to use infant data to ground theories concerning the ontogenesis of numerical cognition.     

Conflict adaptation depends on task structure

The dependence of the Simon effect on the correspondence of the previous trial can be explained by the conflict-monitoring theory, which holds that a control system adjusts automatic activation from irrelevant stimulus information (conflict adaptation) on the basis of the congruency of the previous trial. The authors report on 4 experiments showing that conflict adaptation is mediated by task structure as well as previous congruency. Experiment 1 showed that for 2 completely segregated sets of stimulus-response pairs there was no conflict adaptation across sets. However, if the stimulus sets overlapped for 2 separate response sets, conflict adaptation could be observed across the response sets. Experiment 2 showed that this effect was not due to the use of stimulus-response sets lateralized to 1 hemisphere each. Experiment 3 showed that if response sets are common for 2 separate stimulus sets, then conflict adaptation can again be observed across sets. Finally, Experiment 4 showed local control effects in the absence of confounding feature-overlap effects. These results indicate that deployment of control as evidenced by conflict adaptation reflects task structure.     

Stimulus control over sex-typed play in cross-gender identified boys.

Discriminative stimulus control over masculine and feminine sex-typed play behaviors was investigated in five boys, aged 5–8 yr, with “childhood crossgender identification.” Reliable observational measures of play were obtained with two sets of toys: (a) “dress-up toys” (girls' apparel vs boys' apparel), and (b) “affect toys” (maternal-nurturance play vs masculine-aggression play). With an ABA reversal intrasubject design, certain stimulus conditions (e.g., presence of father, mother, male, or female stranger) were found to be discriminative for reliable changes in sex-typed play. Sex-typed play was found to vary as a function of the social stimulus situation and as a function of the type of play response required. All children played predominantly feminine while alone in the playroom. While no single environmental stimulus was consistently discriminative for masculine play across children, at least one stimulus condition was found for each subject under which he played predominantly masculine.     

How do people apprehend large numerosities?

People discriminate remarkably well among large numerosities. These discriminations, however, need not entail numerical representation of the quantities being compared. This research evaluated the role of both non-numerical and numerical information in adult judgments of relative numerosity for large-numerosity spatial arrays. Results of Experiment 1 indicated that judgments of relative numerosity were affected by the amount of open space in the arrays being compared. Further, the accuracy of verbal estimates of the numerosities of the arrays made upon completion of the comparison task bore little relation to performance on that task. Experiment 2, however, showed that numerical estimates for individually presented arrays were affected in much the same way by open space within or around the edges of the array as were the comparative judgments examined in Experiment 1. The findings suggest that adults heuristically utilize non-numerical cues as well as numerical information in apprehending large numerosities.     

Implicit Memory in Aging: Normal Transfer Across Semantic Decisions and Stimulus Format

To investigate the effect of aging on the fexibility of implicit memory, the current study compared the performance of elderly and young participants on semantic decision tasks in which the processing task or the stimulus format was varied across study and test. In Experiment 1, the required semantic decision (i.e., size judgments vs. corner judgments) either remained the same or was varied from study to test. In Experiment 2, the stimulus format of the item (i.e., pictures vs. words) either remained the same or was varied. Elderly participants demonstrated normal priming effects, and equivalent transfer across manipulations of processing task and across manipulations of stimulus format, despite showing a deficit in recognition memory compared with young participants. The results suggest that aging does not infuence the fexibility of implicit memory.     

Effects of Task Complexity and Goal Specificity on Change in Strategy and Performance Over Time

In this study we examined whether task complexity interacts with goal specific- ity over multiple performance periods. A 2 x 2 x 3 design manipulated goal condition (specific difficult vs. do-your-best) and task complexity (simple vs. complex) over performance on 3 separate days. Results across performance periods indicated that do-your-best goals led to higher quantity of performance than specific difficult goals on a complex task, whereas specific difficult goals led to higher quantity of performance than do-your-best goals on a simpler version of the task. Additionally, goal specificity and task complexity led to greater change in strategy over repeated performance periods, with those in the complex, specific difficult condition exhibiting the highest amount of change in strategy. Finally, there were no differences in quality of performance for indi- viduals working on simple versus complex tasks. Implications of these findings and directions for future research are discussed.     

Matching auditory and visual signals: is sensory modality just another feature?

In order to perceive the world coherently, we need to integrate features of objects and events that are presented to our senses. Here we investigated the temporal limit of integration in unimodal visual and auditory as well as crossmodal auditory-visual conditions. Participants were presented with alternating visual and auditory stimuli and were asked to match them either within or between modalities. At alternation rates of about 4 Hz and higher, participants were no longer able to match visual and auditory stimuli across modalities correctly, while matching within either modality showed higher temporal limits. Manipulating different temporal stimulus characteristics (stimulus offsets and/or auditory-visual SOAs) did not change performance. Interestingly, the difference in temporal limits between crossmodal and unimodal conditions appears strikingly similar to temporal limit differences between unimodal conditions when additional features have to be integrated. We suggest that adding a modality across which sensory input is integrated has the same effect as adding an extra feature to be integrated within a single modality.     

Number sense in human infants

Four experiments used a preferential looking method to investigate 6-month-old infants' capacity to represent numerosity in visual-spatial displays. Building on previous findings that such infants discriminate between arrays of eight versus 16 discs, but not eight versus 12 discs (Xu & Spelke, 2000), Experiments 1 and 2 investigated whether infants' numerosity discrimination depends on the ratio of the two set sizes with even larger numerosities. Infants successfully discriminated between arrays of 16 versus 32 discs, but not 16 versus 24 discs, providing evidence that their discrimination shows the set-size ratio signature of numerosity discrimination in human adults, children and many non-human animals. Experiments 3 and 4 addressed a controversy concerning infants' ability to discriminate large numerosities (observed under conditions that control for total filled area, array size and density, item size and correlated properties such as brightness: Brannon, 2002; Xu, 2003b; Xu & Spelke, 2000) versus small numerosities (not observed under conditions that control for total contour length: Clearfield & Mix, 1999). To investigate the sources of these differing findings, Experiment 3 tested infants' large-number discrimination with controls for contour length, and Experiment 4 tested small-number discrimination with controls for total filled area. Infants successfully discriminated the large-number displays but showed no evidence of discriminating the small-number displays. These findings provide evidence that infants have robust abilities to represent large numerosities. In contrast, infants may fail to represent small numerosities in visual-spatial arrays with continuous quantity controls, consistent with the thesis that separate systems serve to represent large versus small numerosities.