The explanation of these effects of stimulus density may be that the birds "scan" each array. They were always meant to choose the stimulus with fewer items in it. Suppose that a bird looked at the S- array with more dots in it, but these were spaced out. If it did not completely scan the whole pecking key, there was a greater probability that it missed one or more dots than if the dots were all close together. So with a low density S- there was a greater likelihood that a bird would falsely choose that stimulus, as if it actually contained fewer dots. The pigeon would be less likely to overlook an item if the dots were close together, so this type of ‘false alarm’ error was less likely to occur with a high density S- array. The basic suggestion here is that birds scan across multi-item stimuli, processing items one after the other. If this is the case, then the way they deal with "simultaneous" arrays is essentially to process the items in them sequentially. This would provide an important link to the other types of experiments that have been done to investigate birds’ ability to differentiate numbers of items.

Discrimination of Sequential Stimuli

In these other studies, birds’ numerosity judgments have been tested with sequences of visual stimuli, rather than with spatial arrays. Although Koehler’s earlier work demonstrated that pigeons can learn to keep track of a given number of pea-seeds, presented to them one by one down a chute, there was no precise control over temporal parameters. Time factors were certainly varied, both intentionally by rolling the peas down the chute at slightly irregular intervals, and unintentionally due to the birds’ occasional difficulty in grasping a moving pea. More recent experiments on pigeons have used sequences of light flashes as the stimuli. With these stimuli, the duration of a light flash as well as the timing of the intervals between successive flashes can be controlled with much greater precision.

Alsop and Honig’s (1991) approach was to flash a series of colored lights onto a center key that pigeons had to peck. If there were more red than blue flashes in a series, then choice of one side key was correct. If instead blue flashes were in the majority, the correct response was to choose the other side key. The birds were tested with up to 9 flashes in a series but it was the relative rather than the absolute number of light flashes in one versus the other color that was important in determining which side key the birds chose. Alsop and Honig also discovered that later flashes in a sequence had more influence on the birds’ choices than did light stimuli that were near the beginning of a series. So in a mixed sequence of red-blue-blue-blue-blue lights, the birds were more likely to peck the side key indicating that there were more blue flashes than if the sequence was blue-blue-blue-blue-red. The experimenters varied either the total number of light flashes in a sequence or the duration of the dark-interval between flashes. This revealed that not only the ordinal position of a flash in a sequence affected the accuracy of a bird’s choices but also the time that went by between its seeing a flash of a particular color and having to peck a side key. The pigeons showed a "recency" effect in their memory for which flashes had occurred in the sequence. The duration of each light flash also had an effect on their accuracy. Extending the duration of each light flash probably made it more salient so that the birds’ performance was slightly better when flashes were longer than when they were shorter. This experiment showed that, although birds can discriminate the relative numerosity of items in a sequence, temporal factors influence their memory for how many items they have seen.

A recent experiment by Keen and Machado (1999) has also looked at the way pigeons discriminate the relative frequencies (or numerosities) of two series of events – a sequence of red lights shown on one key versus a series of green lights on another key. In this experiment (unlike Alsop and Honig’s), lights of the same color were always shown consecutively. Which series (red or green, and larger or smaller number) appeared first or last was counterbalanced across trials. After the stimulus sequences, two keys were lit red and green. The birds had to choose the color that had appeared least frequently. The total number of stimulus events and the difference in frequencies of red vs. green lights were systematically varied across trials. Both these parameters affected the birds’ accuracy in indicating which series included fewer events. Their key-choices were related to the relative frequencies of red and green lights -- the bigger the difference in number, the more accurate their choices, but their discrimination also depended on the total number of stimuli. The total number (that ranged between 4 and 28) had other effects on their choices too. The pigeons showed "recency" effects, similar to those reported by Alsop and Honig (1991), when there were more than 8 colored lights altogether, but they showed "primacy" effects with fewer stimuli. A "recency" effect means that the last stimulus series more strongly influenced the birds’ choices about the relative number of events, whereas a "primacy" effect means that the first series of stimulus events had more effect on the animals’ decisions. Although the birds’ choices were related to stimulus numerosities, Keen and Machado explained their results not in terms of numerical abstraction by the birds but rather with a mathematical model based on cumulative stimulus effects and decay functions (see section VI).

Roberts, Macuda and Brodbeck (1995) found a similar "recency" effect in pigeons’ memory for the number of light flashes. In their experiment, stimuli always consisted of series of red-light flashes. Initially, some birds were trained to choose one color of side-key if there was a small number of flashes (2) in the series and another color of side-key if there was a larger number (8). The overall time for the series was held constant at 4 sec. When there were only 2 flashes, one occurred at the start of the 4 sec period and the other at the end. With 8 flashes, these events were spaced evenly throughout the 4 sec. For other birds, time varied rather than number. Four flashes were spread out over 8 sec or else over 2 sec. Different side keys had to be chosen depending on the duration ("long" vs. "short") of the sequence. In the initial training, choices to the side-keys could be made immediately after the end of the stimulus sequence (0 delay). But in subsequent tests, the delay varied between the end of the light-flash series and the opportunity to choose a side-key.

The result of varying this delay indicated two things that were confirmed in a subsequent experiment. Firstly, the birds discriminated the number of events (light flashes), even when their training had been designed as a time-discrimination task. Secondly, they appeared to remember more accurately those events that had occurred in the most recent window of time. When the delay increased, birds in the number-discrimination group still chose the "small-number" side-key after 2 events. Given the choice between "small-number" and "large-number" this would be the appropriate response even if they remembered just the light flash at the end of the fixed 4 sec stimulus period. But as the delay increased these birds were more likely to choose the "small-number" rather than the "large-number" key after 8 flashes. So with longer delays they remembered only the events towards the end of the stimulus period and reported fewer events than there had actually been. Pigeons in the time-discrimination group also behaved differently when delays were introduced. As delays increased, there was little change in their accurate choice of the key for a "long" (8 sec) series of flashes. However, following a "short" 2 sec series they shifted their choices to the key for a "long" series instead. In a "long" sequence, the light flashes occurred at a low rate, so if the birds only remembered the events at the end of a stimulus period, they treated this stimulus sequence as if it contained only a small number of events. In a "short" stimulus series, there was the same number of events but they occurred in a briefer time period. At short delays, the birds responded as if they remembered "many" light  flashes. But as delays increased, they remembered only the most recent stimulus flashes and so responded as if there  were a small number of them. Essentially then the "long" key meant "small-number" for them and the "short" key meant "larger-number".

Other experiments (Roberts & Mitchell, 1994; Roberts & Boisvert, 1998) showed that pigeons can concurrently process both the time interval and number of events for series of light flashes. In the first of these experiments (described in the chapter by Sutton and Roberts, this volume), the stimulus series and the training procedure were similar to those used by Roberts et al. (1995), except that no delay interval was imposed between the end of a stimulus sequence and the opportunity to choose a side-key. In the second experiment, the stimuli and procedure were slightly different. Roberts and Boisvert (1998) presented light stimuli on a pecking key. A stimulus series began when the key turned green. When the key briefly flashed red instead of green, this was an ‘event’ in the series. In their initial training, time and number were confounded: the birds’ key-pecking was rewarded when 20 red flashes had occurred so that the stimulus series lasted for 20 sec. The rate at which red flashes appeared was then manipulated and the birds were tested using the peak-procedure. On test trials, no food reward was given but the stimulus series continued for 100 sec so that changes in pecking rate could be measured. These test trials indicated that either the number of events or else the time elapsed influenced the birds’ peak rate of responding, which was indicative of their expectation of a food reward. But the pigeons could also be trained that reward depended specifically on the number of events, for some birds, or the duration of a series, for others. These two experiments suggested that time is a more salient dimension than number for pigeons, which is slightly different from the conclusion reached by Roberts et al. (1995). However, the birds’ bias to base their decisions on time rather than number when either dimension could be used was easily altered by differential training. The close relation between timing and counting that these experiments revealed has been used by Roberts and his colleagues to develop a model to explain how birds process these two aspects of sequences of events (see section VI).

IV. Cardinal Numbers and Counting

In the studies described so far, the birds clearly showed that they could judge relative differences in the numbers of items, but they were not required to choose stimuli according to the absolute number of items they contained. Can birds in fact respond differentially on the basis of absolute number if required to do so? Can they actually count? Although the term "counting" is often applied loosely to any behavior that involves numerical discriminations, a number of criteria have to be met before an animal can really be said to count (Gelman & Gallistel, 1978;). These criteria include assigning in a one-to-one way a tag to each item in a set. Tagging might be evident if an animal responds by pecking, or touching, each item in turn. But a physical response might not be obvious. More important is the idea that the animal cognitively keeps track of each item in a set and assigns each one a code. These codes are called numerons by Gelman and Gallistel (1978). This is similar to Koehler’s suggestion that an animal uses a series of ‘inner marks’ to represent each item it has seen. The order in which the various items themselves are tagged is irrelevant. Also the animal should be able to tag any type of item so that the number in a set is an abstract quantity that is not tied to any specific characteristics (such as size or shape) of the items to be counted.

The mental tags, or symbols that represent each tagged item, must be applied in a particular order. For instance the tag for 4 things always follows the tag for 3 things which in turn follows the tag for 2 items. Then, when the final item in the set is tagged, the mental code or symbol applied to that item is the cardinal number – the total amount in the set. So counting involves the ability to judge absolute, or cardinal, numerical amounts or numbers of responses. The best evidence for counting comes from studies of non-avian species (e.g. rats: Capaldi & Miller, 1988; chimpanzees: Boysen & Berntson, 1989). Most of the work with birds demonstrates their ability to estimate relative differences in number rather than to count or judge the absolute number of things.

A notable exception is Pepperberg’s (1987, 1994) investigations of the abilities of Alex, an African Grey parrot. Alex had had extensive training to vocalize English words as verbal labels for different objects, shapes and colors. In addition, he learned to respond to the verbal question "How many?", spoken by his trainer, with a numerical label for the quantity of items he was shown. Alex was initially trained to use numerical labels to describe classes of shapes as "3-corner" or "4-corner" (Pepperberg, 1983). Then, over a period of several years, intermixed with training on other tasks, he was gradually taught the labels for between 2 and 6 objects (Pepperberg, 1987). In response to the question "How many?" he was at first required to vocalize not just the quantity but also the name for the type of objects. He accurately gave answers like "4 key" (the use of plurals was not required;  To see a video of Alex performing click here, this video is 58 seconds in duration, so download times will be longer then most of the videos in this book). His previous experience with shapes was extended to items such as "6-corner paper". He also learned to tell the number of objects in a subset within a heterogeneous array, e.g. when asked "How many cork?" he could answer "2" when shown a random mixture of 2 corks and 3 keys. His ability to label cardinal sets accurately transferred to other objects for which he knew the names but which had not been used in numerical training. So Alex used "spoken" numbers as abstract labels. Interestingly, when he was questioned about the number of unfamiliar objects in mixed arrays of known and novel items, his initial tendency was to respond with the total number in the array.

Later (Pepperberg, 1994), the range of Alex’s vocal numerical labels was extended to include "one". Also, his ability to give the number for a subset with a particular conjunction of properties was tested. In each test set there were four different numbers for the subset combinations. Test items could be in one of two colors (e.g. green or blue) or in one of two forms (e.g. truck or key). Alex was then asked a question about the quantity of a specific subset such as "How many green truck?". He was able to answer with a high level of accuracy. The aim of this study was to investigate whether Alex might have relied on subitizing or counting to judge quantity. Subitizing was considered to be a perceptual and preattentive mechanism by which numbers of items up to 4 are recognized holistically. Counting, on the other hand, would require abstraction. It was assumed to be necessary for quantities beyond the subitizing range and to require more spatial attention. Alex was equally accurate at judging subsets in the range of 1 to 6 items. He also had to process items defined by the conjunction of their properties (e.g. green and truck) when these items were scattered randomly amongst other similarly complex objects (e.g. blue trucks, green keys, blue keys) that would have acted as perceptual distractors. Although Pepperberg could not totally dismiss subitizing as a potential process contributing to Alex’s performance, she thought it was unlikely that he could easily segregate the target subset from all the other items and clump the relevant items together. The even distribution of errors that Alex made with both smaller and larger numbers, together with his ability to deal with conjunctive subsets, was probably more compatible with counting. On the other hand there was no evidence that Alex applied numerical labels in an ordinal fashion, for instance. Although all the principles of counting (Gelman & Gallistel, 1978) were not directly tested in this experiment, this parrot has obviously demonstrated an ability to respond in a flexible way to the absolute or cardinal number of things that make up a group, or part of a group. The extent to which parrots rely on a process like subitizing versus one like counting is still being investigated (I. Pepperberg, pers. comm.).

"Counting", in the sense of assessing cardinal numbers, can be expressed in quite different ways but roughly one can distinguish two categories of behavior: counting external items, as Alex probably did, and counting one’s own responses. Only a few studies have investigated the latter type of behavior. Zeier (1966), for example, explored the upper limit of pecking responses that pigeons could deliver in a precise manner. Pigeons had to peck the first of two response keys a specific number of times. By pecking the second key the pigeon signaled that it had completed the pecking sequence on the first key. Zeier successively increased the number of required pecks and found that the upper limit that individual pigeons could produce amounted to 8, but for the majority of the birds the upper limit was clearly below 8.

However, Zeier’s pigeons never had to cope with different response numbers at the same time or learn, in the way that Alex did, to link things to be counted with symbols representing their number. To find out whether pigeons would be able to produce specific numbers of pecks in response to given symbols, Xia, Siemann and Delius (2000) used 6 symbols (A, N, T, 4, U, and 5, some of them rotated) that were presented on one of the pecking keys (the "symbol" key) in a conditioning chamber. Each symbol was associated with a certain required number of pecks. A second key (the "enter" key) had to be pecked to indicate that the response requirement on the first key had been fulfilled. If this second key was pecked before the response requirement on the first key was completed, or if too many pecks had been delivered on the first key, a timeout followed (during which the houselight was turned off). Pigeons only received food reward for the exact production of the required number of pecks on the first key and a final single peck on the second key. (Click here for an explanation of the procedure, and click here for animated illustration of the procedure).

After prolonged training, 6 out of 9 pigeons reached a choice performance well above chance level for the first 4 symbols presented in a random order. Six of these birds managed to deal with 5, and 4 animals also dealt reliably with all 6 symbols. The mean choice performance was 79%, 84%, 74%, 68%, 53%, and  50% correct response  sequences for numbers 1 to 6 respectively. These results are shown in Figure 8. Note, that the chance level decreases with the response number as the number of wrong alternatives increases. For example with a response requirement of 3 there is one correct response sequence, but 4 wrong sequences, thus chance level is 20%; with a response requirement of 6, there is one correct, but 7 incorrect sequences, therefore chance level amounts to 12.5%. Response distributions to each given symbol revealed that errors were more frequent with an adjacent numerosity than with more distant numerosities (e.g. 4 responses were more frequent than 2 when the correct number was 5).

The task took several thousand trials for the pigeons to acquire but nevertheless shows that these birds are capable of associating cardinal numbers of responses with different visual symbols for each of these numbers. The birds’ behavior was also quite flexible since they accurately produced the required number of responses to whichever symbol was randomly presented to them.

V. Judging Proportion with Continuous or Discrete Quantities

Judging the number of things is one way of estimating quantity. However, differences in quantity also depend on other dimensions. For instance, food quantities differ not just as a result of the number of edible items available but also because of their sizes, volumes, weights, etc. Most of the experiments described so far have controlled for other dimensions in order to see if birds can choose on the basis of number alone. They obviously can do so but seem to be more adept at judging relative differences in number than absolute amounts. Another way of looking at relative quantities is to consider the proportions of items in heterogeneous sets. In the experiments described so far, proportion has been equated with the relative numerosity of two types of items in mixed stimulus arrays (Honig & Stewart, 1989). In another study (Emmerton, 2001) pigeons were trained to discriminate differences in color proportions within horizontal bars composed of continuous blocks of color. They were then tested under a variety of conditions to see if they still responded accurately to differences in the relative quantity of color when the stimulus displays were altered.

Initially, the pigeons learned to discriminate between a red and a green horizontal bar when these two stimuli were presented simultaneously on a computer monitor. The ‘correct’ color was counterbalanced across birds. Responses were sensed by a touchscreen, and a peck to the correct color led to food reward. Then the proportion of the two colors was varied in both stimuli. If a bird chose the bar with the greater proportion of the ‘correct’ color (e.g. red in the following stimulus examples), it was rewarded. If it chose the bar with the complementary lesser proportion of ‘correct’ color a timeout period followed. If the proportion of red to green was equal in both bars, one of the stimuli was arbitrarily programmed to be the correct one but the bird had to guess which one it was. For examples of the  stimuli the birds saw across a series of trials, click here. The accuracy with which the birds discriminated the paired bars was correlated with the color proportions in the bars (see Figure 9). When there was no difference in proportion, their performance was at chance level. Since a non-linear discrimination function was obtained when behavioral scores were measured on a scale of percentage of correct choices, these behavioral measures were linearized by converting the scores to a logit scale (defined as ln(% correct/(100 - % correct)); Macmillan & Creelman, 1991).

The stimuli presented in this initial condition were deliberately different from those used by Honig and Stewart (1989, 1993). Whereas they had used arrays of discrete items, this new experiment used stimuli consisting of continuous areas of color. In this case, proportion was no longer equated with the relative numerosity of component items but the results of both experiments were very similar: as the difference in the proportion of colored areas or discrete items decreased so did the accuracy of the birds’ discrimination. One of the questions in this recent study was whether the pigeons would still respond to changes in proportion when the stimuli were changed to consist of arrays that were similar to those used by Honig and Stewart. So in one part of the study, the birds were presented not with paired horizontal bars but with arrays of small red and green rectangles. There were two types of arrays. In one type the rectangles were configured in regular matrices. In another type the arrangement of the rectangles within the arrays was irregular to break up any potential texture effects. To see examples of the stimuli as they appeared across trials, click here. The birds transferred to these altered stimuli without any difficulty and their accuracy in choosing the array with the greater  proportion of ‘correct’ color in it still depended on the difference in color proportion between the paired arrays (see Figure 10). In this case, as in Honig and Stewart’s experiments, color proportion is synonymous with the relative numerosity of red and green rectangles. So proportion, irrespective of whether it refers to the relative number of discrete items in a mixture, or to a continuous dimension such as the relative length or area of different constituents, is another abstract property on which judgments about quantity are based.

Other experiments on numerical discrimination suggest that animals discriminate relative rather than absolute differences in numerosity. For instance, Boysen, Berntson, Hannan and Cacioppo (1996) found high correlations between chimpanzees’ discrimination accuracy and the disparity ratio of numerosity with real objects (candies or pebbles). In that context, the disparity ratio was defined as the difference in number between paired stimuli divided by the sum of their numbers. In the experiment on proportion discrimination with pigeons, the stimulus conditions described so far utilized stimuli in which the proportions of ‘correct’ color in the paired positive and negative stimuli were complementary. Disparity ratio in this experiment could be defined as the difference in the proportions of ‘correct’ color in paired stimuli divided by the sum of those proportions. However, with complementary values, this would be the same as calculating the absolute difference in color proportion within a stimulus pair. So another stimulus condition was devised to test whether the birds’ discrimination performance was really related to the relative difference in proportion, or disparity ratio.

The stimuli again consisted of horizontal bars but there were three sets of stimuli intermixed within sessions. Click here for examples of the stimuli. One set (the S+/S- set) was the same as in the initial condition for examining proportion discrimination – color proportions varied in a complementary fashion in positive and negative stimuli. Choice of the S+ bar with the greater proportion of ‘correct’ color in it was rewarded. In another (S+/0.5) set, the proportion of ‘correct’ color in one bar varied as it did in the S+ bars on rewarded trials. The other bar always contained equal proportions of red and green. For this set, a bird’s response was registered as correct if it chose the "S+" bar with the greater, variable proportion of ‘correct’ color (or guessed the arbitrarily ‘correct’ stimulus in the case of both bars containing equal proportions of color). In the third (0.5/S- ) set, one bar again contained equal amounts of red and green. The proportion of the two colors varied in the other one so that on average this other bar contained a lesser proportion of the ‘correct’ color as in the S- bars of rewarded trials. So for this third stimulus set the correct response was to avoid the bar that had the lesser proportion of ‘correct’ color and choose the one with the greater proportion (the 0.5 bar). Whenever a pair of stimuli from the second or third set appeared, the birds’ choices were recorded as correct or incorrect ones but they were given neither reward nor timeout, so these were all test trials. For these two test sets, the disparity ratio of color proportion was no longer the same as the absolute difference in proportion values. The accuracy of the birds’ choices for all three sets of stimuli was closely correlated with changes in the disparity ratio of color proportion (see Figure 11). 

VI. Models and Mechanisms for Numerical Processing

Various models and mechanisms have been proposed to account for animals' (and presumably humans') ability to assess numerical quantities. Pacemaker-accumulator models, and a stimulus control model were designed to explain how different numbers of sequential events are processed. Subitizing, and a neuronal filtering model on the other hand were primarily intended to account for the numerical processing of visual arrays.

Pacemaker-Accumulator Models

This model has been developed by Roberts and Mitchell (1994) as an extension of the pacemaker-accumulator model previously proposed by Meck and Church (1983). According to this model, the same, or similar, mechanisms are used for timing and counting, since, in both pigeons and rats, behavioral results for time and number are similar when the animals could have learned about either dimension. The basic mechanism in this model is a pacemaker, or oscillator, that emits pulses at a fixed rate. These pulses are then switched into other mechanisms. In Roberts’ version of the model, the pulses can be routed simultaneously into two different accumulators. One type of switch closes at the start of a sequence of events and continues to allow pulses to be accumulated until the sequence finishes. This channel is used for timing. The other type of switch closes for a brief, fixed period when each event in the series occurs. So the number of pulses that collect in a separate accumulator – the current value of its ‘contents’ – is proportional to the number of sequential events that have occurred. This channel is used for "counting". Information about the  total numbers of pulses in each type of accumulator is then transmitted to different compartments of working memory. By comparing the current information in working memory with previously stored information in reference memory, an animal decides how to behave, e.g. whether to peck a left or a right key. The main features of the ‘counting’ channel in this type of model are illustrated schematically in Figure 12. 

The model was designed to account for counting and timing with sequential events – series of light flashes, sequences of tones, or a run of behavioral responses emitted by the animal itself. At first, it seems to have little to do with a pigeon’s ability to discriminate the number of items in visual arrays since these items are presented simultaneously. However, if birds in fact scan arrays, as the effects of changes in dot density suggest (Emmerton, 1998), then the pacemaker-accumulator model could also apply to numerosity discrimination with those types of stimuli.

Mathematical Model of Stimulus Control Effects

An entirely different model has been proposed by Keen and Machado (1999). This model shows how birds make different choices depending on the relative numbers of events of two types, e.g. red vs. green lights presented sequentially. In contrast to all the other models of number-related behavior, however, this model does not assume that birds in any sense ‘count’, store numerical representations, remember the actual number of stimuli they have seen, or have an internal processing mechanism for numerosity itself.

Essentially, the model assumes that when a series of similar events occurs (like 5 successive red-light stimuli), the probability of the bird’s performing one behavior (e.g. pecking one key) increases linearly by a constant amount for each of the stimuli shown. When a series of different events occurs, these other events increase the likelihood of the animal’s making an alternative response (e.g. peck the other key). The amount of stimulus control (influence on responding by these stimuli) depends on the number of events of each type, but not because the animal counts or performs any numerical calculations. When each series of stimuli is no longer present, the memory of the cumulative stimulus effects for that series decays exponentially. There are two parameters in the model. One is the relative incremental effect on stimulus control of each stimulus in the first versus the second series. The effects of the first stimuli are assumed to interfere in part with the effects of the second set (proactive interference). The other parameter is decay following stimulus events. Decay refers to the degrading effects of the passage of time on a memory trace for a particular type of stimulus, the retroactive interference effects of presenting one type of stimulus after another type, or both. The bird’s choice of response depends on the net effectiveness of one series compared with the other when the bird has to decide what to do.

The model can account for the effect of relative numbers of sequential stimuli or events on birds’ choices. For instance, it provided a good fit to behavioral data that conformed to Weber’s law. The mathematical model also successfully predicted recency and primacy effects – whether later or earlier stimulus events have more influence on behavioral choices. It remains to be seen, though, how it would explain birds’ discrimination of numerosity with simultaneously presented arrays of items (even if the items themselves are scanned). When birds have to choose between arrays, it is not clear what role the two parameters of the mathematical model would play. Similarly, it is unclear how the model would explain birds’ conditional discrimination when only a single array is presented but the animals’ choices are related to the array’s numerosity value. With the latter type of stimulus displays, discrimination performance is remarkably robust, even with arrays of inhomogeneous items, or when the types of items are altered, or novel stimuli are introduced. Furthermore, the model is designed to deal with discriminations of the ‘more’ vs. ‘less’ type and so would not be applicable to judgments about the absolute number of things or events (see section IV).

Subitizing

Other types of mechanisms have been proposed to explain how animals make judgments about the numerosity of items in arrays. One of the earliest proposals was that animals assess small numbers by a process of subitizing. Subitizing refers to a type of pattern recognition by which quantities of about 1 to 4 items are rapidly assessed. For example, 2 dots form the end points of a straight line. Three dots either fall in a straight line or else define the corners of a triangle. Subitizing was originally thought to be a mechanism, separate from enumeration, by which humans make quick and accurate estimates of small numbers of things (Kaufman, Lord, Reese & Volkmann, 1949). People were reported to show very short reaction times while accurately judging up to 4 things whereas reaction times increased progressively when they had to assess greater numbers of items. However, these differences in reaction times have been called into question (Balakrishnan & Ashby, 1992). In animals, the necessary information about reaction times is lacking (Miller, 1993). Also, pigeons can make relative numerosity judgments even when the arrays consist of entirely novel numbers of dots, as Emmerton et al. (1997) showed in the study of serial ordering effects. In that case, the birds had no opportunity to learn about the patterns that 3, 4 or 5 dots can form. So subitizing, if it exists as a separate process, is neither necessary nor sufficient to account for birds’ numerosity judgments. On the other hand, the extent to which pattern recognition enables birds to make choices about familiar array configurations is unclear at the moment.

Neuronal Filtering Model

Dehaene and Changeux (1993) proposed a different model based on neuronal filtering mechanisms. In their model, an array of objects or elements is first coded as different clusters of activity in a network, such as the retina. Subsequently each cluster of activity from the input level is filtered so that only item location within the array is encoded, but other features, such as the size of each object, become irrelevant. The activity from each unit at this stage is then summated  and transferred to the next neuronal level. There, the summed activity is gated by other units which set different threshold levels. Activation that exceeds a particular threshold is then transmitted to a final neuronal level. At this level, units respond to only a selected range of activity values so that these units function as the numerosity detectors (see Figure 13). 

This model was primarily designed to explain how judgments are made about the numbers of items seen simultaneously in arrays. So it is easiest to see how it would explain the parallel processing of numbers of objects, rather than the serial processing of events. However, Dehaene and Changeux suggest that, with the addition of some form of sensory store, it could also be modified to deal with sequentially presented stimuli such as light flashes. With this type of modification it would still apply if pigeons serially scan the items in arrays, as suggested earlier.

Variance Assumptions

At present, it is not clear whether a pacemaker-accumulator model or a neuronal filtering model best accounts for animals’ numerical abilities (if indeed either one applies). Both models assume some inherent variability at the different  processing levels. This means that, although an animal’s average estimate about number might tally well with the actual number of items it encounters, its repeated estimates will show some variance about a mean value. For both models it is assumed that this variance will become greater as the number of items or events to be assessed increases. This idea is illustrated in Figure 14. This link between variance and number can explain why pigeons found it easier to discriminate arrays of 1, 2, 3 and 4 items in the experiment on serial ordering (Emmerton et al., 1997), but arrays of 5, 6 or 7 dots were less easily differentiated. The assumption of variability made by both processing models also corresponds to evidence that Weber’s law applies to numerosity judgments. Note, however, that Keen and Machado (1999) report data that by and large conform to Weber’s law, although their mathematical model includes no variance assumptions.

Weber’s Law and Judgments of Numerical Difference

Weber's law basically says that the ability to discriminate two stimuli is a ratio function of the stimuli, rather than a function of the absolute difference between them. Weber’s law has often been applied in measuring sensory abilities. For instance, if we (or any other vertebrate species) have to judge whether one light is brighter than another, or one sound is louder than another, our ability to choose the brighter or the louder stimulus depends on the value of the reference stimulus. The brighter the reference light (or the louder the reference sound) the greater must be the difference between it and a comparison stimulus for us to accurately tell the two apart. In addition, Weber’s law operates in a variety of situations that require discriminations of abstract cognitive dimensions. This law underlies discrimination of time as well as number when birds have to judge how many pecks they have made or how long a response series lasts (Fetterman, 1993). It also applies to temporal and numerical judgments made by rats when they are presented with tone series (Gibbon, 1977; Meck & Church, 1983). The results of the experiments reported here on the discrimination of color proportion (see Figure 11 in section V) conform to Weber's law. Transformation of the data from a recent experiment on numerosity discrimination indicates that Weber's law applies to birds' discrimination of visual arrays too.

In that experiment some pigeons were trained to choose consistently an array containing ‘more’ dots while others always had to choose the array with ‘less’ dots in it. (These opposite training conditions were designed to see if birds are biased to choose greater numbers, as will be discussed in section VII.) If numerosity discrimination obeys Weber’s law we would expect that as the number of items in the ‘correct’ array increases, so the difference in number that can be discriminated from this reference amount will increase proportionally. To see whether Weber’s law applies to the results in this experiment, numerical disparity ratios were calculated for all the array pairs that were used. As in the experiments by Boysen et al. (1996), a disparity ratio is defined as the difference in numbers of dots between paired arrays divided by the sum of all the dots in a stimulus pair. The scores of discrimination accuracy were again transformed to a logit scale. Linear regression analyses were then applied to these transformed data. From the start, the results from the birds trained to choose ‘more’ followed this linear relationship. At the start of training, the birds that had to choose ‘less’ did not show this pattern of results.  These initial data are shown in Figure 15. However, the results of pigeons in the ‘less’ group were due to the birds' bias to choose the more numerous array. A linear relationship between logit performance scores and numerosity disparity ratios did emerge after extended training with familiar stimuli had enabled the birds to overcome their bias. So long as their predisposition to choose the greater number did not interfere with the birds’ choice accuracy, the pigeons discriminated the arrays as Weber’s law would predict and differentiated the arrays on the basis of the relative difference in the numbers of dots they contained.

Various results now support the idea that Weber’s law, which is incorporated in the current models of "counting", applies to numerical discriminations. Whether or not any of these models prove to be valid, this seems to be a feature that any future models must also include.

VII. Do Birds’ Numerical Abilities Have Any Functional Significance?

Dating back to the earliest anecdotes about animals’ potential counting ability, there has been the implicit assumption that being able to keep track of the number of things must have survival value for any animal that can do so. A behavioral trait need not necessarily have functional significance (see Gould & Lewontin, 1979). For instance, being able to estimate numerosity may be just a spin-off of very precise timing abilities. However, there are some situations in which it might be advantageous to be able to differentiate numerosity. One situation that comes to mind is foraging. When an animal has to make feeding choices it may be useful for it to estimate the number of food items available in different patches. Another situation is in keeping track of what other animals in a group are doing. Some observations indicate that birds use numerosity discrimination to assess how many other members of a flock are currently engaged in watching out for predators, rather than in attending to the business of gathering food. 

In its natural environment, an animal is constantly faced with the problem of optimizing its energy intake, and being able to choose the larger of alternative food patches may contribute to survival. However, being able to assess the number of items in a food-patch is no guarantee of that patch being the best one to choose. Optimal foraging theory (Krebs, 1978) instead suggests that several factors determine how an animal maximizes its energy budget in the wild. The ability to discriminate and choose the larger number of food items in seed-patches or clumps of berries most likely serves as a "rule of thumb" (Krebs & McCleery, 1984; Shettleworth, 1984). This means that choosing the greater number of food items often, but not always, leads to a better payoff.

These ideas suggested that a bird, if given a choice between two arrays, should more readily choose the one containing the greater number of items. Such a bias had already been mentioned by Koehler (1941). When he trained a pigeon to choose one of two seed patches, placed at opposite ends of a strip of cardboard, he noticed a spontaneous tendency by the bird to approach the larger amount, even if the ‘correct’ one for the bird to eat was the less numerous patch. So in an experiment (mentioned in the previous section on Weber's law) pigeons were trained to discriminate between paired arrays of seed-like dots. Choice of the correct array led to a constant amount of food reward. The numbers of dots in the array pairs  varied across trials. But some birds always had to choose the more numerous array to obtain reward whereas the other birds had to choose the less numerous array. Otherwise training conditions were identical. As expected, the group that was trained to choose ‘more’ performed better in this task during the early stages of training than the group that was trained to choose ‘less’ (see Figure 16). With extended training, both groups reached similar levels of discrimination accuracy. Then a transfer phase started.

In the transfer sessions, some non-rewarded trials included new stimulus arrays. Not only were the patterns of dots in the test arrays novel to the birds, but also pairs of arrays were matched in brightness. Although both groups of birds were still equally accurate at choosing correctly on familiar training trials, differences between the groups re-emerged  on the test trials. The birds that were trained to choose ‘more’ were quite accurate in choosing the more numerous test arrays. However, the group that was trained to choose ‘less’ was poorer at choosing the less numerous novel arrays. The results are shown in Figure 17.  When trained to choose an artificial ‘patch’ that was associated with food reward, the birds seemed to be biased from the start of the experiment to choose the array with the greater number of dots in it. Even though the birds trained to choose ‘less’ could learn to overcome this bias with familiar stimuli, the predisposition to choose the more numerous array became apparent once more in the test trials.

These results suggest that birds might indeed use their numerical abilities in foraging. However, in this particular experiment the birds did not eat the arrays, and the reward they obtained was a constant amount of food. Other experiments with both rats and birds support the idea that numerosity discrimination might operate as a ‘rule of thumb’ in deciding between foraging opportunities, even when the weight of reward is equated. For instance, chickens that were trained to seek food at the end of a runway ran faster when the reward was presented as 4 pieces of a single kernel of corn rather than as 1 whole grain. They also were more accurate at choosing the correct arm of a T-maze when the reward was presented in multiple pieces rather than a single one (Wolfe & Kaplon, 1941). Contrary to these results, Olthof and Roberts (2000) found that pigeons rely on the overall mass, rather than the number of items, to assess the relative quantity of food rewards. However, the birds did not visually discriminate between food rewards directly. Rather, they had to choose between symbols (colored shapes) that represented specific amounts of reward.

Numerical discrimination may also be of functional significance to birds that forage in flocks and switch between searching for food and scanning the environment for predators. Observations of a variety of species have shown that birds generally spend a greater amount of time eating rather than being vigilant for predators as flock size increases (Elgar, 1989). However, several other factors also influence vigilance behavior. These include flock geometry – whether birds are aligned with each other or feeding in a circle (Bekoff, 1995). Another factor is whether other members of the flock are visible or are obscured by obstacles such as rocks or trees (see Elgar, 1989). A model of vigilance behavior (Bahr & Bekoff, 1999) incorporates the ability of birds to categorize their conspecifics’ behavior as either feeding or scanning and to discriminate the numbers of individuals showing each type of behavior. The model assumes that a bird monitors up to 8 of its nearest neighbors in a flock. Dependent on the relative number of birds that it can see feeding or scanning it will decide whether to forage or to be vigilant. The small number range seems to be important since observations show that, once flock size exceeds about 10 birds, further increases in the number of birds do not affect the proportion of time that an individual spends scanning (Elgar, Burren & Posen, 1984). The ability to assess the number of conspecifics is not the only factor that influences vigilant behavior, but, as with other aspects of foraging, it appears to be an important contributory factor.

VIII. Conclusions

This chapter has concentrated on the ability of birds to deal with different aspects of number and quantity when some form of visual stimulus is involved. This focus tends to conceal at least two points. One is that birds also differentiate number in contexts in which visual stimuli are unimportant, for instance, when they discriminate the relative numbers of responses they emit on different fixed-ratio schedules (Fetterman, 1993). The other point concerns the striking similarities between avian abilities and those of other species that have been tested on numerical tasks. For example, chimpanzees that have to choose between two arrays of real objects (candies or pebbles) are biased, as pigeons are, to choose the array containing ‘more’ items (Boysen & Berntson, 1995; Boysen, Berntson, Hannan & Cacioppo, 1996). In chimpanzees this bias is overcome, by the way, when they have to choose between Arabic numerals that stand for different numbers of candies as reward, a possibility that has not yet been tested with birds. Both parrots and pigeons can learn to associate arbitrary symbols with precise numbers of objects or events (but see Olthof & Roberts, 2000). These types of associations have been demonstrated in primates (monkeys: Olthof, Iden & Roberts, 1997, chimpanzees: Boysen, 1993, and of course humans: Gelman & Gallistel, 1978). Pigeons seem to order arrays containing different numbers of dots in a serial fashion (Emmerton et al., 1997). Serial ordering of visual arrays that vary in their numerosities has been demonstrated more explicitly in rhesus monkeys (Brannon & Terrace, 1998, 2000).

Even the earliest anecdotal speculations about animals’ "counting" behavior implied that the ability to make numerical discriminations is functionally significant. Choice biases seen in the artificial foraging situation of experiments that use food-rewards are one reflection of this. As long as there are no other factors (such as predators or aggressive competitors) to complicate the picture, it is often advantageous to choose the items that are greater in number. It may even be adaptive for an animal to be able to judge relative differences in number or quantity more easily than absolute amounts. This conjecture is based on the amounts of training required for birds to master different tasks, and considerations of how numerical abilities might be used in an animal's natural habitat. Most studies so far have tested birds’ performance in assessing numerosity differences rather than absolute or cardinal numbers. Birds learn quite readily, especially with visual stimuli, to differentiate ‘more’ from ‘less’, ‘many’ from ‘few’, or a mixture containing a greater proportion of one feature from a mixture containing a smaller proportion. In the few studies dealing with cardinal numbers of objects or responses the birds have required extensive training. However, it is not yet clear whether this is because the animals have to assess the exact numbers of items they see or pecks they should emit, or whether the additional requirement of associating a precise amount with a numerical symbol is what really taxes them. The ability to make relative judgments of number, as of other dimensions, may be of more general utility though. Being able to assess where there is more food, where there are fewer competitors, which tree contains more ripe fruit, etc. may often be more important than knowing the exact number of each of these things. But as the crow anecdote suggested, there may nevertheless be situations in which being able to keep track of exact small numbers is also essential for survival!

Unresolved Issues

In spite of the progress achieved in our understanding of different facets of birds’ numerical abilities, several issues still have to be resolved. One is the extent to which they integrate number with other aspects of quantity when they discriminate visual features of their environment. Another issue is whether birds deal with small compared with larger numbers of items or events in a similar fashion or whether there are any differences in their behavior across the number range. Can they, for instance, really count small numbers of things (according to the principles outlined by Gelman and Gallistel, 1978) but make only relative judgments of larger amounts? Perhaps related to this is still the question of whether there is just one common mechanism underlying numerical behavior (and, if so, which one), or are there several mechanisms that are employed to a greater or lesser extent in different situations? And how do birds, and other animals, utilize their ability to discriminate number and quantity in their natural environment? Perhaps some of the ongoing research in this area of animal cognition will soon enable us to understand this remarkable avian capability even better.

IX. References

    Alsop, B. & Honig, W. K. (1991). Sequential stimuli and relative numerosity discriminations in pigeons. Journal of Experimental Psychology: Animal Behavior Processes, 17, 386-395.

     Bahr, D. B. & Bekoff, M. (1999). Predicting flock vigilance from simple passerine interactions: modeling with cellular automata. Animal Behaviour, 58, 831-839.

     Balakrishnan, J. D. & Ashby, F. G. (1992). Subitizing: Magical numbers or mere superstition? Psychological Research, 54, 80-90.

     Bekoff, M. (1995). Vigilance, flock size, and flock geometry: information gathering by western evening grosbeaks (Aves, Fringillidae). Ethology, 99, 150-161.

     Boysen, S. T. (1993). Counting in chimpanzees: Nonhuman principles and emergent properties of number. In S. T. Boysen & E. J. Capaldi (Eds.) The development of numerical competence: Animal and human models. (pp. 39-59). Hillsdale, NJ: Erlbaum. 

     Boysen, S. T. & Berntson, G. G. (1989). Numerical competence in a chimpanzee (Pan troglodytes). Journal of Comparative Psychology, 103, 23-31.

      Boysen, S.T. & Berntson, G.G. (1995). Responses to quantity: perceptual versus cognitive mechanisms in chimpanzees (Pan troglodytes). Journal of Experimental Psychology: Animal Behavior Processes, 21, 82-6.

     Boysen, S. T., Berntson, G. G., Hannan, M. B., & Cacioppo, J. T. (1996). Quantity-based interference and symbolic representations in chimpanzees (Pan troglodytes). Journal of Experimental Psychology: Animal Behavior Processes, 22, 76-86.

     Boysen, S. T. & Capaldi, E. J. (1993). The development of numerical competence: Animal and human models. (pp. 39-59). Hillsdale, NJ: Erlbaum. 

     Brannon, E. M. & Terrace, H. S. (1998). Ordering of the numerosities 1 to 9 by monkeys. Science, 282, 746-749.

     Brannon, E.M. & Terrace, H.S. (2000). Representation of the numerosities 1-9 by rhesus macaques (Macaca mulatta). Journal of Experimental Psychology: Animal Behavior Processes, 26, 31-49.

     Candland, D. K. (1993). Feral children and clever animals: Reflections on human nature. New York, NY: Oxford University Press.

     Capaldi, E J. & Miller, D. J. (1988). Counting in rats: Its functional significance and the independent cognitive processes that constitute it. Journal of Experimental Psychology: Animal Behavior Processes, 14, 3-17.

     Davis, H. & Memmott, J. (1982). Counting behavior in animals: A critical evaluation. Psychological Bulletin, 92, 547-571.

     Davis, H. & Pérusse, R. (1988). Numerical competence in animals: Definitional issues, current evidence, and a new research agenda. Behavioral and Brain Sciences, 11, 561-615.

     Dehaene, S. & Changeux, J.-P. (1993). Development of elementary numerical abilities: A neuronal model. Journal of Cognitive Neuroscience, 5, 390-407.

     Elgar, M A., Burren, P. J. & Posen, M. (1984). Vigilance and perception of flock size in foraging house sparrows. Passer domesticus L.,  Behaviour, 90, 215-223.

     Emmerton, J. (1998). Numerosity differences and effects of stimulus density on pigeons’ discrimination performance. Animal Learning and Behavior, 26, 243-256.

     Emmerton, J. (2001). Pigeons’ discrimination of color proportion in computer-generated visual displays. Animal Learning and Behavior, 29, 21-35.

     Emmerton, J. & Delius, J. D. (1993). Beyond sensation: Visual cognition in pigeons. In H. P. Zeigler & H.-J. Bischof (Eds.) Vision, brain, and behavior in birds (pp. 377-390). Cambridge, MA: MIT Press.

     Emmerton, J., Lohmann, A., & Niemann, J. (1997). Pigeons’ serial ordering of numerosity with visual arrays. Animal Learning & Behavior, 25, 234-244.

     Fetterman, J.G. (1993). Numerosity discrimination: Both time and number matter. Journal of Experimental Psychology: Animal Behavior Processes, 19, 149-164.

     Gelman, R. & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge, MA: Harvard University Press.

     Gibbon, J. (1977). Scalar expectancy theory and Weber’s law in animal timing. Psychological Review, 84, 279-325.

     Gould, S. J. & Lewontin, R. C. (1979). The spandrels of San Marco and the Panglossian paradigm: a critique of the adaptationist programme. Proceedings of the Royal Society of London – Series B: Biological Sciences, 205, 581-598.

     Honig, W. K. & Stewart, K. E. (1989). Discrimination of relative numerosity by pigeons. Animal Learning & Behavior, 17, 134-146.

     Honig, W. K. & Stewart, K. E. (1993). Relative numerosity as a dimension of stimulus control: The peak shift. Animal Learning & Behavior, 21, 346-354.

     Kaufman, E. L., Lord, M. W., Reese, T. W. & Volkmann, J. (1949). The discrimination of visual number. American Journal of Psychology, 62, 489-525.

     Keen, R. & Machado, A. (1999). How pigeons discriminate the relative frequency of events. Journal of the Experimental Analysis of Behavior, 72, 151-175.

     Knorn, H.-J. (1987). Visuelle Diskriminationsleistung von Tauben und Menschen beim simultanen Punktmengenvergleich. [Visual discrimination performance of pigeons and humans when simultaneously comparing dot quantities.] Unpublished Diplomarbeit, Ruhr-Universität Bochum, Bochum, Germany.

     Koehler, O. (1941). Vom Erlernen unbenannter Anzahlen bei Vögeln. [On the learning of unnamed numerosities by birds.] Die Naturwissenschaften, 29, 201-218.

     Koehler, O. (1943). "Zähl"-Versuche an einem Kolkraben und Vergleichsversuche an Menschen. ["Counting" experiments on a raven and comparative experiments on humans.] Zeitschrift für Tierpsychologie, 5, 575-712.

     Krebs, J. R. (1978). Optimal foraging: Decision rules for predators. In J. R. Krebs & N. B. Davies (Eds.) Behavioural ecology: An evolutionary approach (pp. 23-63). Oxford, UK: Blackwell. 

     Krebs, J. R. & McCleery, R. H. (1984). Optimization in behavioural ecology. In J. R. Krebs & N. B. Davies (Eds.) Behavioural ecology: An evolutionary approach (2nd, ed., pp.91-121). Oxford, UK: Blackwell. 

     Macmillan, N. A. & Creelman, C. D. (1991). Detection theory: A user’s guide. Cambridge, MA: Cambridge University Press.

     Meck, W. H. & Church, R. M. (1983). A mode control model of counting and timing processes. Journal of Experimental Psychology: Animal Behavior Processes, 9, 320-334.

     Miller, D. J. (1993). Do animals subitize? In S. T. Boysen & E. J. Capaldi (Eds.) The development of numerical competence: Animal and human models. (pp. 149-169). Hillsdale, NJ: Erlbaum. 

     Olthof, A., Iden, C. M., & Roberts, W. A. (1997). Judgments of ordinality and summation of number symbols by squirrel monkeys (Saimiri sciureus). Journal of Experimental Psychology: Animal Behavior Processes, 23, 325-339.

     Pepperberg, I. M. (1983). Cognition in the African Grey parrot: Preliminary evidence for auditory/vocal comprehension of the class concept. Animal Learning & Behavior, 11, 179-185.

     Pepperberg, I. M. (1987). Evidence for conceptual quantitative abilities in the African Grey parrot: Labeling of cardinal sets. Ethology, 75, 37-61.

     Pepperberg, I. M. (1994). Numerical competence in an African Gray parrot (Psittacus erithacus). Journal of Comparative Psychology, 108, 36-44.

     Porter, J. P. (1904). A preliminary study of the psychology of the English sparrow. The American Journal of Psychology, 15, 313-346.

     Rilling, M. (1993). Invisible counting animals: A history of contributions from comparative psychology, ethology, and learning theory. In S. T. Boysen & E. J. Capaldi (Eds.) The development of numerical competence: Animal and human models. (pp. 3-37). Hillsdale, NJ: Erlbaum. 

     Roberts, W. A. & Boisvert, M. J. (1998). Using the peak procedure to measure timing and counting processes in pigeons. Journal of Experimental Psychology: Animal Behavior Processes, 24, 416-430.

     Roberts, W. A., Macuda, T. & Brodbeck, D. R. (1995). Memory for number of light flashes in the pigeon. Animal Learning & Behavior, 23, 182-188.

     Roberts, W. A. & Mitchell, S. (1994). Can a pigeon simultaneously process temporal and numerical information? Journal of Experimental Psychology: Animal Behavior Processes, 20, 66-78.

     Shettleworth, S. J. (1984). Learning and behavioural ecology. In J. R. Krebs & N. B. Davies (Eds.) Behavioural ecology: An evolutionary approach. (2^nd ed. pp. 170-194). Oxford: Blackwell.

     Thorpe, W. H. (1963). Learning and instinct in animals (2^nd ed.). London: Methuen.

     Wesley, F. (1961). The number concept: A phylogenetic review. Psychological Bulletin, 58, 420-428.

     Wolfe, J. B. & Kaplon, M. D. (1941). Effect of amount of reward and consummative activity on learning in chickens. Journal of Comparative Psychology, 31, 353-361.

     Xia, L., Siemann, M. & Delius, J. D. (2000). Matching of numerical symbols with number of responses by pigeons. Animal Cognition, 3, 35-43.

     Zeier, H. (1966). Über sequentielles Lernen bei Tauben, mit spezieller Berücksichtigung des "Zähl"-Verhaltens. [About sequential learning by pigeons, with special consideration of "counting" behavior.] Zeitschrift für Tierpsychologie, 23, 161-189.

Acknowledgement        I would like to thank Li Xia, Martina Siemann, and Juan Delius, Universität Konstanz, Germany, for giving me pre-publication information about their research. I am especially grateful to Martina Siemann for not only providing the summary of their experiment but also for creating the figures demonstrating conditional discrimination and cardinal responding.  I also much appreciate Irene Pepperberg's help in supplying the video clip of Alex.