Matching of numerical symbols with number of responses by pigeons

Pigeons were trained to peck a certain number of times on a key that displayed one of several possible numerical symbols. The particular symbol displayed indicated the number of times that the key had to be pecked. The pigeons signalled the completion of the requirement by operating a separate key. They received a food reward for correct response sequences and time-out penalties for incorrect response sequences. In the first experiment nine pigeons learned to allocate 1, 2, 3 or 4 pecks to the corresponding numerosity symbols s ₁, s ₂, s ₃ and s ₄ with levels of accuracy well above chance. The second experiment explored the maximum set of numerosities that the pigeons were capable of handling concurrently. Six of the pigeons coped with an s ₁–s ₅ task and four pigeons even managed an s ₁–s ₆ task with performances that were significantly above chance. Analysis of response times suggested that the pigeons were mainly relying on a number-based rather than on a time-based strategy. Received: 11 October 1999 / Accepted after revision: 27 January 2000     

Continuous visual properties of number influence the formation of novel symbolic representations

Numerical symbols are thought to be mapped onto preexisting nonsymbolic representations of number. A growing body of evidence suggests that nonsymbolic numerical processing is significantly influenced by the associated visual properties of continuous quantity (e.g., surface area, density), but their role in the acquisition of novel symbols is unknown. Forty undergraduate students were trained to associate novel abstract symbols with numerical magnitudes. Half of the symbols were associated with nonsymbolic arrays in which total surface area and numerosity were correlated (“congruent”), and the other symbols were associated with arrays in which total surface area was equated across numerosities (“incongruent”). As numbers are represented in multiple formats (words, digits, nonsymbolic arrays), we also tested whether providing auditory nonword labels facilitated symbol learning. Following training, participants engaged in speeded comparisons of the newly learnt symbols. Comparisons were affected by the ratio between the numerosities associated with each symbol, a characteristic marker of numerical processing. Furthermore, comparisons were hardest for large-ratio comparisons of symbols associated with incongruent area and numerosity pairing during learning. In turn, these findings call for the further investigation of visual parameters on the development of numerical cognition.     

The differentiation of response numerosities in the pigeon

Two experiments examined how pigeons differentiate response patterns along the dimension of number. In Experiment 1, 5 pigeons received food after pecking the left key at least N times and then switching to the right key (Mechner's Fixed Consecutive Number schedule). Parameter N varied across conditions from 4 to 32. Results showed that run length on the left key followed a normal distribution whose mean and standard deviation increased linearly with N; the coefficient of variation approached a constant value (the scalar property). In Experiment 2, 4 pigeons received food with probability p for pecking the left key exactly four times and then switching. If that did not happen, the pigeons still could receive food by returning to the left key and pecking it for a total of at least 16 times and then switching. Parameter p varied across conditions from 1.0 to .25. Results showed that when p= 1.0 or p=.5, pigeons learned two response numerosities within the same condition. When p=.25, each pigeon adapted to the schedule differently. Two of them emitted first runs well described by a mixture of two normal distributions, one with mean close to 4 and the other with mean close to 16 pecks. A mathematical model for the differentiation of response numerosity in Fixed Consecutive Number schedules is proposed.     

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.     

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.     

Summation of symbols by pigeons (Columba livia): the importance of number and mass of reward items

In each of 3 experiments, different sets of 4 pigeons (Columba livia) were trained to discriminate between 2 visual symbols that covered wells containing food items that varied in number, mass, or both. In Experiment 1, the symbols were associated with 0, 1, 3, 5, 7, or 9 pieces of grain reward. The pigeons learned to choose the symbol corresponding to the larger reward, and on summation tests, they chose the pair of symbols that summed to the larger total reward. When number of food pellets was varied but mass of reward was held constant in Experiment 2, preference for the larger number symbols failed to appear. When number was held constant and mass was varied in Experiment 3, the pigeons showed a clear preference for the larger mass symbols on single-symbol and summation tests. These findings show that pigeons summate the value of symbols and are more likely to represent symbols by mass of food reward than by number of food items.     

Pigeons (Columba livia) associate time intervals with symbols in a touch screen task: evidence for ordinality but not summation

The present experiments examined whether pigeons can sum symbols that are associated with various temporal consequences in a touch screen apparatus. Pigeons were trained to discriminate between two visual symbols that were associated with 0, 1, 2, 3, 4, or 5 s either of delay to 4 s of hopper access (delay group) or duration of hopper access (reward group). In Experiment 1, the pigeons in both groups learned to select the symbol associated with the more favorable outcome, and they successfully transferred this discrimination to novel symbol pairs. However, when tested with 2 pairs of symbols associated with different summed durations, they responded on the basis of a simple response rule rather than the sum of the symbol pair. In Experiment 2, the reward group was presented with four symbols at once and was allowed to successively choose one symbol at a time. All pigeons chose the symbols in order from largest to smallest. This indicates that pigeons formed an ordered representation of symbols associated with different time intervals, even though they did not sum the symbols.     

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.     

Origins of number sense. Large-number discrimination in human infants

Four experiments investigated infants' sensitivity to large, approximate numerosities in auditory sequences. Prior studies provided evidence that 6-month-old infants discriminate large numerosities that differ by a ratio of 2.0, but not 1.5, when presented with arrays of visual forms in which many continuous variables are controlled. The present studies used a head-turn preference procedure to test for infants' numerosity discrimination with auditory sequences designed to control for element duration, sequence duration, interelement interval, and amount of acoustic energy. Six-month-old infants discriminated 16 from 8 sounds but failed to discriminate 12 from 8 sounds, providing evidence that the same 2.0 ratio limits numerosity discrimination in auditory-temporal sequences and visual-spatial arrays. Nine-month-old infants, in contrast, successfully discriminated 12 from 8 sounds, but not 10 from 8 sounds, providing evidence that numerosity discrimination increases in precision over development, prior to the emergence of language or symbolic counting.     

Six does not just mean a lot: preschoolers see number words as specific

This paper examines what children believe about unmapped number words - those number words whose exact meanings children have not yet learned. In Study 1, 31 children (ages 2-10 to 4-2) judged that the application of five and six changes when numerosity changes, although they did not know that equal sets must have the same number word. In Study 2, 15 children (ages 2-5 to 3-6) judged that six plus more is no longer six, but that a lot plus more is still a lot. Findings support the hypothesis that children treat number words as referring to specific, unique numerosities even before they know exactly which numerosity each word refers to.     

Local rather than global processing of visual arrays in numerosity discrimination by pigeons (Columba livia)

A series of experiments investigated which stimulus properties pigeons use when they discriminate pairs of visual arrays that differ in numerosity. Transfer tests with novel stimuli confirmed that the birds’ choices were based on relative differences in numerosity. However, pigeons differed from other species in the non-numerical cues that affected their choices. In human and non-human primates, numerical discrimination is often influenced by continuous variables such as surface area or overall stimulus brightness. Pigeons showed little evidence of using those cues, even when summed area and brightness had been correlated with numerosity differences and reward outcome. But when array-element sizes were asymmetrically distributed across numerosities, the birds readily utilized information about item sizes as an additional discriminative cue. These novel results are discussed in relation to pigeons’ tendency to focus on local, rather than global dimensions when they process other non-numerical complex visual stimuli. The findings suggest there may be inter-specific differences in the type of perceptual information that provides the input stage for mechanisms underlying numerical processing. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Subitizing and similarity: Toward a pattern-matching theory of enumeration

Pattern-matching theories of subitizing claim that subjects enumerate displays with small numerosities by retrieving numerical responses associated with similar displays experienced in the past. Such retrieval implies that displays with small numerosities are similar to other displays of the same numerosity and dissimilar to other displays of different numerosities. These hypotheses were tested by having subjects rate the similarities of displays of dot patterns with numerosities in the range of 1-10. One group of subjects rated patterns of the same numerosity. Their ratings were higher for patterns in the subitizing range (numerosities of 1-3) than for patterns beyond the subitizing range (numerosities of 4-10). Another group rated patterns of different numerosities. Their ratings were lower in the subitizing range than beyond the subitizing range. An analysis based on multidimensional scaling suggested that numerosity could be retrieved accurately for displays of 1-3 dots, but not for displays of 4-10 dots.     

Response summation to a compound stimulus in a context of choice

Key pecks by two groups of pigeons were reinforced on concurrent schedules. For group Ē, pecks were reinforced during either a visual or an auditory stimulus; for group E, an additional, extinction component was available, during which both visual and auditory stimuli were absent. After training, both groups were given a compound test to measure preference among four stimuli, the three used in training plus a compound of the visual and auditory stimulus. Group E showed preference for the compound, emitting more pecks and spending more time in this stimulus than in other stimuli. Group Ē showed no preference between the compound and visual stimulus, nor between the auditory stimulus and the absence of both stimuli, but preferred the former pair over the latter pair of stimuli.     

Children acquire the later‐greater principle after the cardinal principle

Many have proposed that the acquisition of the cardinal principle (CP) is a result of the discovery of the numerical significance of the order of the number words in the count list. However, this need not be the case. Indeed, the CP does not state anything about the numerical significance of the order of the number words. It only states that the last word of a correct count denotes the numerosity of the counted set. Here, we test whether the acquisition of the CP involves the discovery of the later‐greater principle – that is, that the order of the number words corresponds to the relative size of the numerosities they denote. Specifically, we tested knowledge of verbal numerical comparisons (e.g., Is ‘ten’ more than ‘six’?) in children who had recently learned the CP. We find that these children can compare number words between ‘six’ and ‘ten’ only if they have mapped them onto non‐verbal representations of numerosity. We suggest that this means that the acquisition of the CP does not involve the discovery of the correspondence between the order of the number words and the relative size of the numerosities they denote.     

Increasing the variability of response sequences in pigeons by adjusting the frequency of switching between two keys.

Three experiments compared the amounts of behavioral variability generated with two reinforcement rules. In Experiments 1 and 2 pigeons received food whenever they generated a sequence of eight pecks, distributed over two keys, provided that the sequence contained a certain number of change-overs between the keys. Although no variability was required-the birds could obtain all reinforcers by repeating the same sequence-the pigeons emitted a large number of different sequences. In Experiment 3 pigeons received food whenever they generated a sequence that had not occurred during the last 25 trials. After prolonged training, the birds showed more sequence variability than in the first two experiments. The analysis of the internal structure of the response sequences revealed that, in general, (a) the location of the first peck was highly stereotyped; (b) as the trial advanced, the probability of switching to the initially preferred key decreased whereas the probability of switching to the other key increased; and (c) a first-order Markov chain model with transition probabilities given by a logistic function accounted well for the internal structure of the birds' response sequences. These findings suggest that, to a large extent, the variability of response sequences is an indirect effect of adjustments in changeover frequency.     

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.     

Mechanisms underlying the effects of unsignaled delayed reinforcement on key pecking of pigeons under variable-interval schedules.

Three experiments were conducted to test an interpretation of the response-rate-reducing effects of unsignaled nonresetting delays to reinforcement in pigeons. According to this interpretation, rates of key pecking decrease under these conditions because key pecks alternate with hopper-observing behavior. In Experiment 1, 4 pigeons pecked a food key that raised the hopper provided that pecks on a different variable-interval-schedule key met the requirements of a variable-interval 60-s schedule. The stimuli associated with the availability of the hopper (i.e., houselight and keylight off, food key illuminated, feedback following food-key pecks) were gradually removed across phases while the dependent relation between hopper availability and variable-interval-schedule key pecks was maintained. Rates of pecking the variable-interval-schedule key decreased to low levels and rates of food-key pecks increased when variable-interval-schedule key pecks did not produce hopper-correlated stimuli. In Experiment 2, pigeons initially pecked a single key under a variable-interval 60-s schedule. Then the dependent relation between hopper presentation and key pecks was eliminated by arranging a variable-time 60-s schedule. When rates of pecking had decreased to low levels, conditions were changed so that pecks during the final 5 s of each interval changed the keylight color from green to amber. When pecking produced these hopper-correlated stimuli, pecking occurred at high rates, despite the absence of a peck-food dependency. When peck-produced changes in keylight color were uncorrelated with food, rates of pecking fell to low levels. In Experiment 3, details (obtained delays, interresponse-time distributions, eating times) of the transition from high to low response rates produced by the introduction of a 3-s unsignaled delay were tracked from session to session in 3 pigeons that had been initially trained to peck under a conventional variable-interval 60-s schedule. Decreases in response rates soon after the transition to delayed reinforcement were accompanied by decreases in eating times and alterations in interresponse-time distributions. As response rates decreased and became stable, eating times increased and their variability decreased. These findings support an interpretation of the effects of delayed reinforcement that emphasizes the importance of hopper-observing behavior.     

Imprinted numbers: newborn chicks’ sensitivity to number vs. continuous extent of objects they have been reared with

Newborn chicks were tested for their sensitivity to number vs. continuous physical extent of artificial objects they had been reared with soon after hatching. Because of the imprinting process, such objects were treated by chicks as social companions. We found that when the objects were similar, chicks faced with choices between 1 vs. 2 or 2 vs. 3 objects chose the set of objects of larger numerosity, irrespective of the number of objects they had been reared with. Moreover, when volume, surface or contour length were controlled for using sets of 1 vs. 4, 1 vs. 6 or 1 vs. 3 objects, chicks resorted to choosing the larger object, rather than the familiar numerosity. When, however, chicks were reared with objects differing in their aspect (colour, size, and shape) and then tested with completely novel objects (of different colour and shape but controlled for continuous extent), they chose to associate with the same number of objects they had been reared with. These results suggest that identification of objects as different and separate individuals is crucial for the computation of number rather than continuous extent in numerical representation of small numerosities and provide a striking parallel with results obtained in human infants. Early availability of small numerosity discrimination by chicks strongly suggests that these abilities are in place at birth.     

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.     

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.