Avian Visual Cognition

Birds’ Judgments of Number and Quantity

Jacky Emmerton
Department of Psychology, Purdue University







In the last two decades there has been renewed interest in animals’ abilities to abstract information about number and quantity. Number refers to "how many" discrete things or events are perceived. Quantity is a more general term for amount and may involve an assessment of number or of other dimensions. The broader question underlying this research is how animals, that are by definition non-verbal organisms, can encode information about number or quantity when they are presented with stimuli that are variable in other respects. Often they also have to retain this abstracted information at least long enough to be able to use it as a basis for making behavioral decisions. The question of animals’ cognitive abilities is interesting in its own right. But some of the research to be described in this chapter may also give insights into the abilities of pre-verbal infants to process number or quantity, or indeed into the adult capability of assessing these abstract dimensions without having to enumerate or measure the things they perceive.

This chapter looks mainly at how birds perform on a variety of number-related tasks in which visual stimuli have been used. Some of the pioneering studies that were done on this topic have often been overlooked. They provide an important foundation for more recent research and will be reviewed briefly. Then newer research is discussed which has examined how birds judge relative differences in the numbers of items they see. Besides making relative judgments, some birds can assess absolute amounts within the small number range. Birds can also discriminate relative differences in proportion when the stimulus quantities are either continuous areas of color or are mixtures of numerous colored items. Various models have proposed how non-verbal organisms might process such an abstract property as number. This chapter also considers ways in which the ability to assess the number of things birds encounter in their general environment may be advantageous to them in an evolutionary sense.

Chapter Outline and Navigation

I.   Delving into History
        Anecdotal beginnings
         The need for controls
         Ethological experimentation
         Developments post-Koehler

II.  Number as an Abstract Property

III. Relative Judgments and Numerosity

        Discrimination of visual arrays
        Discrimination of sequential stimuli

IV. Cardinal Numbers and Counting

V.  Judging Proportion with Continuous or Discrete Quantities

VI. Models and Mechanisms for Numerical Processing
        Pacemaker-accumulator models
        Mathematical model of stimulus control effects
        Neuronal filtering model
        Variance assumptions
      Weber’s law and judgments of numerical difference

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

VIII. Conclusions

        Unresolved issues

IX. References

I. Delving into History

Anecdotal Beginnings

For at least two hundred years, people have speculated about whether animals can in some sense "count", or make choices based on number. This possibility was first raised in a popular anecdote about crows. The story goes that whenever a hunter tried to approach a crow and shoot it, the bird kept its distance. So the man enlisted the help of others in a ruse that was designed to trick the bird into deciding it was safe to return to its nest. A group of hunters entered a hiding place in sight of the bird, then after a while all but one left again, leaving the last person concealed. When there were up to 5 people in the group, the crow stayed out of harm’s way until the remaining hunter had also given up and left. The bird was fooled into returning prematurely, and met its demise, only when there was a total of 6 hunters. No doubt this story is apocryphal, but it points to two ideas. The first is that birds can accurately keep track of the number of things (such as hunters in a group) up to some limit (in this example 5 but not 6); the second is that being able to do so might have survival value, and so be of evolutionary significance.

The Need for Controls

The earliest attempts to find experimental evidence of numerical abilities in birds (and in other animals) were not very successful though. For instance, in an experiment with house-sparrows, Porter (1904) tested their response to number by hiding food under, say, the third container in a row, then recording which container a bird flew to first on repeated trials. Then he changed the number of the baited pot and tested the birds’ ability to redirect their choices. After extensive testing, Porter concluded that the sparrows based their choices mainly on the location of the food container -- the relative distance of the baited one from the end of the row, rather than its numerical order within the row. Although this early work failed to provide evidence for a response to number, it pointed to the importance of the conditions for testing an animal’s potential numerical abilities, and the need for controls. In Porter’s study, there were at least two procedural defects. As Porter himself recognized, the serial number of the baited container was confounded with a relative distance measure, since the containers were placed equidistant from each other in all the tests. Also, the birds were always allowed to take a bit of food, whether the correct container was chosen or not. So in modern parlance, the birds were non-differentially reinforced. Apparently, Porter tried to "make them do without" food when they chose the wrong container, but this made their behavior even more erratic.

The other early lessons about the importance of controlled procedures stem from work that was done with a horse, nicknamed Clever Hans. Its owner, von Osten, trained the animal to perform a variety of feats, including responding to numbers that were written on a chalkboard, and performing arithmetic calculations. The horse responded by tapping its hoof on the ground the correct number of times. Initial investigations of its behavior convinced a panel of scientists that the animal could indeed answer questions that involved counting, apparently without its owner’s intervention. However, one scientist, Pfungst, eventually discovered that the horse answered correctly only when its owner knew the answer, and when it could see its owner while it responded. Clever Hans was actually able to detect minute changes in von Osten’s behavior that he made unwittingly when the horse reached the correct count with its hoof-tapping (see Candland, 1993, for a full account). Although von Osten had not intended to deceive others about his horse’s abilities, most studies since then have tried to preclude the possibility of cueing an animal to give the correct response. One way of doing this is to test animals in a setting in which interactions with the experimenter are minimized. This is more easily achieved now than was possible at the beginning of the last century. Many of the more recent experiments use standardized procedures with the animals in training chambers in which stimuli are presented, and the animals’ responses are recorded with the help of computers.

Ethological Experimentation

Although there were several other early studies on animals’ numerical abilities (see Thorpe, 1963; Rilling, 1993), the first real progress was made in this field by Otto Koehler and his collaborators. Their experiments were performed in quasi-naturalistic settings on a variety of animals, including several avian species – pigeons, jackdaws, a raven, an African Grey parrot, and budgerigars (see Koehler, 1941). Koehler concluded that birds have two basic numerical abilities. One, based on a visuo-spatial sense, enables them to assess the number of items presented simultaneously in a group, while the other allows them to assess the number of events that occur successively, or spread out in time. Koehler devised various training procedures to investigate these abilities.

Pigeons were trained, for instance, to approach a strip of cardboard on which there were two sets of grain that differed in number. A bird had to choose the set containing a particular amount (e.g., 4 grains) and was allowed to eat this set as a reward. (See Figure 1 for a reconstruction of this type of choice.)Click here to view Figure 1 To prevent it eating the other set (of say 3 grains), Koehler shooed the bird away if it reached toward the incorrect group. At different stages of training, the correct set sometimes contained the larger and sometimes the smaller number of grains. Every effort was made to avoid cueing of the type that Clever Hans had utilized. The experimenter hid behind a screen, out of sight of the bird. The punishment of shooing a bird away was delivered in a standardized fashion by a mechanical device. Punishment for incorrect choices was withheld on some test trials. Also, the reactions of the birds were filmed to provide an objective record of their behavior.

One jackdaw in particular was successful in a matching to sample task. After looking at an array of blobs on a "sample" card, the bird had to remove the lid from one of two pots in order to find a hidden food reward. On each lid were other arrays. A correct choice, which led to reward, was to remove the lid with the same number of marks as on the sample card. An incorrect (and thus unrewarded) choice was to remove the lid in which the number of marks differed from the sample number. The bird that performed best on this task could match the numbers of items on the sample card and the comparison lid even when the configuration of blobs and their sizes differed (both between and within trials), so that the only common feature was their number. Another jackdaw was correct in its matching behavior when the patterns of the blobs differed between the card and the numerically matching lid, but its choices were not statistically reliable when the blob sizes varied. In this case, as Koehler recognized, the bird’s performance did not guarantee that it was discriminating solely on the basis of equality of number. Instead, it could have been comparing the overall areas of the stimulus marks on the card with those on the lids.

Other studies looked at birds’ ability to "act on number", as Koehler put it, i.e. to respond sequentially until a specific number of items had been obtained or events had been completed. For instance, pigeons and budgerigars were trained to eat only a given number of seeds from a much larger number they saw. So if they were required to take exactly 4 seeds their behavior was scored as correct if they walked away after eating the fourth food item, but they were automatically shooed away if they tried to eat a fifth item. The accuracy of their performance was tested on trials in which this mild punishment was withheld. In another experiment with pigeons, peas were delivered one at a time down a chute into a large dish. In this experiment, the time interval between deliveries was randomly varied to prevent the birds estimating the total time that had expired, rather than the number of items taken. Koehler also argued that it was the number of peas, rather than the number of pecks, or the pecking rhythm, that was important since the pigeons often had to peck several times at a rolling pea before they could grasp it.

With jackdaws, the task consisted of taking the lids off pots until a specific number of hidden food items had been retrieved, after which the remaining pots in the row would be empty. The important feature of this experiment was that on successive trials, the same number of food items was differently distributed. Suppose that the number of food items available was 2, but their distribution varied across trials. The bird had to open just one pot if both items were in the first pot, 5 pots if there was one piece of food in the first and in the fifth pots, 2 pots if 1 item was hidden in each of the first two pots in the row, etc.

From these and other studies, Koehler concluded that birds have at least a limited ability to discriminate objects or events on the basis of their numerosity and inferred that animals have some way of internally tagging the items they have seen or responded to. Koehler was careful to say that animals do not seem to count in the way that an adult human might by precisely enumerating items with a fixed series of symbolic labels (e.g. one, two, three, four, or eins, zwei, drei, vier, etc., depending on linguistic culture). Rather, he argued that animals learn what he called "unnamed numbers", so that four items might be represented by a series of inner marks or tags. He also noted that different species showed remarkable similarities in the limits of their ability to discriminate numerosity. Mostly the accuracy of their performance broke down when the number of items or events they had to respond to was between 5 and 6, or 6 and 7 (Koehler, 1943).

Developments Post-Koehler

Despite his efforts to eliminate extraneous cues, Koehler’s work has still been criticized for lack of control. Wesley (1961) pointed out, for instance, that there was little control for olfactory cues in the experiments involving hidden food, and that seeds were arranged by hand on cards, which could have led to some subjective bias. In spite of these remaining criticisms, subsequent experiments have done more to corroborate than to refute Koehler’s basic findings. As one example (Knorn, 1987; see Emmerton & Delius, 1993), 8 pigeons were trained using standard operant techniques to discriminate between simultaneously presented arrays of white dots upon the dark background of two pecking keys. All the events in the training chamber (presenting the stimuli, recording the birds’ pecks, giving them a food reward, etc.) were controlled by a computer. The number and the size of dots in the arrays varied across training trials, but pecks at the array containing the greater number of dots always led to food reward whereas pecks to the less numerous array led to timeout (a short period of darkness). After they hadClick here to view figure 2 reached a fairly stable level of choosing the greater numerosity on training trials, transfer trials with novel stimulus arrays were later added. Some of these novel test stimuli are shown in Figure 2.  As an additional control for brightness differences, luminance was equated in each pair of test arrays. Finally, neither food reward nor timeout Click here to view Figure 3 was given on the transfer trials. When the test data for various numerical combinations were averaged across birds and test sessions, the results resembled those found by Koehler. In this case, the pigeons could still discriminate arrays of 6 vs. 7 dots at an above chance level, but performance dropped to chance on test trials in which 7 vs. 8 dots were shown (see Figure 3). 

This experiment confirms Koehler’s findings when there is a difference of one item between paired stimulus displays. However, more recent research, described later in this chapter, has shown that the question of the limits of discrimination has wider implications than was previously supposed. Koehler’s work provided an innovative foundation for subsequent research into animals’ numerical abilities. He and his colleagues devised a remarkable variety of procedures for training and testing a wide range of species. The published reports of these experiments also contain a wealth of detailed observations about the animals’ behavior during the experiments. Since the '30’s and '40’s when most of Koehler’s work was done, there has been renewed interest and considerable progress in investigating how animals discriminate things in terms of their number. Some of this work has been reviewed in articles by Davis and Memmott (1982), by Davis and Pérusse (1988), and in a book edited by Boysen and Capaldi (1993). Further research has been completed since those reviews appeared.

II. Number as an Abstract Property

Number represents one of the abstract properties of items or events in the environment, or of actions that organisms can perform. When we refer to number we mean that this aspect of quantity – "how many" discrete elements -- remains the same even when other characteristics change. For instance, "five" might be our verbal label to describe a group of large oranges, a line of small pebbles, a series of whistles, or how many times we knock on a door. Animals presumably also have a way of internally representing (or encoding and remembering) these different amounts. However, as Koehler pointed out, animals obviously rely on non-verbal coding mechanisms to discriminate and store information about number. Only a few studies have shown that animals can learn symbols, similar to our digits, to represent the number of things, but they do this only after prolonged training.

Davis and Pérusse (1988) have discussed in detail the different types of competence that animals may show in numerical discrimination tasks. Most of the experiments described in this chapter correspond to two types of numerical competence. Birds have been tested mainly for their ability to discriminate numerosity. Numerosity discrimination requires only relative or ordinal judgments. For instance, a bird must show that it perceives one stimulus as having "more" items compared with another that has "less", or one consists of "many" elements while another has "few". However, with these relative judgments the animal does not necessarily have to recognize the precise number of elements in a stimulus. To demonstrate proper counting ability, several criteria must be met (Gelman & Gallistel, 1978). These include tagging items one by one, irrespective of their type, so that the final tag represents the precise or cardinal number of items in a stimulus. Essentially, the animal must be able to recognize that a stimulus consists of exactly three elements, for example. Some birds have learned to respond to small numbers of elements quite accurately. An African Grey parrot has learned to "speak" the correct number (Pepperberg, 1987). Pigeons have been trained to emit specific numbers of pecks when shown one of several digit-like symbols (Xia, Siemann & Delius, 2000). These studies will be described later (see section IV on cardinal numbers and counting).

III. Relative Judgments and Numerosity

When animals are tested for their ability to make numerosity judgments, they are presented with the task of deciding whether 5 items are more than 3, or whether there is a greater number of sound pulses in a series of 8 versus 4 tones. Click here to view Figure 4 Although the experimenter devises stimuli with knowledge of the exact numbers of items, this does not mean that the animal has to know anything about the precise numbers in order to make a correct choice. To get an idea of why this should be so, look at Figure 4. Without having to enumerate each grain, it is still possible to say that there are more seeds in one patch than in another.

Experiments on numerosity discrimination examine, on the one hand, how birds judge the relative numbers of items that are presented simultaneously in visual arrays, and, on the other hand, how they judge differences in the numbers of items shown sequentially.

Discrimination of Visual Arrays

Visual arrays were used by Honig to test pigeons with a go/no-go procedure. In the initial experiments (Honig & Stewart, 1989), the birds were trained that food reward would follow when they pecked a key on which they saw a uniform array of "positive" colored dots (e.g. red) for 20 sec. If an array with the same number of dots but all in a "negative" color (e.g. blue) appeared for the same 20 sec, pecks were not rewarded. Once they had learned to respond to the positive array but not to the negative one, they were tested in extinction (without reward) with various arrays in which the total number of dots stayed constant, but the relative numbers of red to blue ones varied. The relative amount of pecking to these arrays diminished as the proportion of positive-colored dots in the array decreased from 100% to 50%. By itself, this does not prove that the pigeons could discriminate numerosity since they might instead have responded to a particular summed area of, say, the red color. But other tests did show that they were relying on the relative number of different types of items in the arrays. Changing the total number of dots per array did not alter their discrimination performance. In other tests the dots differed in size rather than color so that the birds had to discriminate how many large versus small dots there were. The stimuli were devised so that a simple discrimination of area was unlikely. In other parts of the experiment, black and white shapes (Xs and Os) were used instead of colored dots. Finally, tests were conducted with arrays made up of complex and rather variable shapes that fell into two categories – stylized pictures of birds versus flowers. In spite of this variability in the color, size or form of the items in the arrays, the pigeons’ amount of responding changed with the relative numbers of items within a given array.

With similar procedures Honig and Stewart (1993) have shown that pigeons respond to the "abstract" dimension of numerosity in ways that resemble their reactions to the "perceptual" dimensions of much simpler stimuli. An example of this is the "peak shift" effect which is shown when there is more responding to a test value than to the rewarded value of the positive or S+ training stimulus. The direction of this shift is away from the negative or S- value. It is seen with dimensions such as hue (or spectral wavelength) or stimulus size, and it shows that animals often assess relative differences amongst stimulus values. Honig and Stewart also demonstrated the peak shift when pigeons were trained to peck arrays consisting of equal numbers of two types of elements (e.g. red and blue dots), and not to respond when the arrays consisted of relatively more of the negative color (e.g. more blue than red dots). The birds were then tested with arrays in which the relative numbers of dots in the two colors varied. In these tests they responded more to arrays containing a greater proportion of positive elements (e.g. red dots) than they did to the originally rewarded S+ array with its equal numbers of red to blue items. This peak shift effect still occurred when the total number of dots in the test arrays was increased (while the proportions of red to blue were maintained). The effect was also seen when arrays consisted of horizontal and vertical rectangles rather than colored elements. So once more, the important dimension was the relative number of items. Honig and Stewart gave examples of how the peak shift phenomenon might be of functional significance for birds, and other animals. If a bird has to select amongst food patches where berries are ripening, they can learn that it is better to choose the patch in which half the berries are ripe, rather than one in which only a small proportion has ripened. But it is better still to find a patch in which more than half the berries are already ripe.

Instead of testing pigeons with a go/no-go procedure, Emmerton, Lohmann and Niemann (1997) used a conditional discrimination procedure. First, a bird had to peck at a visual array that was shown on a center key. If the array contained "many" items (6 or 7) then the pigeon had to peck at one of the side keys (e.g. the right-hand red-lit one) to obtain a food reward. If instead the center array contained "few"Click here to view the animated illustration items (1 or 2) then the correct response was to choose the other side key (e.g. the left-hand green-lit one). Incorrect choices led to a timeout period of waiting several seconds in the dark. (Click here for animated illustration.) A variety of these arrays containing 1, 2, 6 or 7 items were shown until the birds had learned to discriminate accurately between "many" and "few" items. Then they were tested not only with new versions of the "many" and "few"  stimuli, but also with arrays consisting of the intermediate numbers 3, 4 and 5. These numbers were completely novel for them. Also, on test trials there was Click here to view Figure 5 no reward or timeout and so no feedback about correct or incorrect choices. The test choices were plotted as the percentage of choices made to the side key that denoted "many" (see Figure 5). Most of the "many" choices were made when the center test-array contained 6 or 7 elements, and the least choices when the array consisted of 1 or 2 items. In this respect the birds treated these novel arrays as they had the familiar training stimuli. When the intermediate numbers were shown, the birds’ choices were distributed in an orderly fashion. Compared to their responses with arrays of 6 or 7 items, they made slightly fewer choices of the "many" key when the test-array contained 5 items, fewer when it had 4 items in it, and fewer still when there were 3 items. This distribution of choices indicates that pigeons can serially order numerical quantities so that 7/6 > 5 > 4 > 3 > 2/1.

Several tests were conducted in this experiment to make sure that the birds really discriminated the numbers of items in the arrays, rather than detecting some other confounds. To begin with, all the items in a given array were the same size and shape (e.g. all white dots on a dark background). Although item sizes differed between arrays, it was likely that, as the number of items changed, so on average did the summed area of these elements, as well as their overall brightness. So in other tests, the brightness was equated across stimuli with filters, or the total area of the dots was held equal by altering their sizes. This did not change the birds’ performance with the various test numerosities. In further tests, shapes otherClick here to view Figure 6 than dots were used, dots of mixed sizes were presented, or the arrays consisted of mixtures of outline and filled-in shapes (to mimic mixtures of light and dark seeds). (See Figure 6 for examples of various test arrays.) In these and other tests, none of the alterations to the specific stimulus characteristics had any consistent effect on the birds’ distributions of choices to the "many" and "few" side-keys. In all the tests, they behaved as if they were responding in an orderly way to the number of items they saw. In spite of this pattern of behavior, the experiment still does not show that the birds recognized there were exactly 3 or else 6 items in an array. Although the distribution of choices was orderly, it was not exactly linear. Instead, it seemed to be easier for the birds to discriminate amongst the smaller numbers (1 to 4) than amongst slightly larger numbers (5 to 6). This non-linearity gives some clues about the processing of relative number, as will be discussed later.

A parameter that was not systematically altered in the last experiment was the density of the arrays, or the spacing between the dots. The effect of changes in density was examined separately (Emmerton, 1998). This time, pigeons were trained to discriminate pairs of arrays that were shown simultaneously on two pecking keys. The birds’ task was to choose the array with fewer dots in it in order to obtain reward. Different combinations of small numbers of dots were used. Mostly, the birds were more accurate at choosing the smaller numerosity when the difference between the S+ (smaller-number array) and the S- (larger-number array) was greater rather than smaller. For instance, the birds were better at discriminating 3 from 7 dots than they were 3 from 5 dots. But in addition to the relative difference in numerosity, the inter-dot spacing in the various arrays affected the accuracy of the pigeons’ discrimination. In some training sessions, the correct, S+ array consisted of only 1 dot, and the incorrect, S- array contained either 2 or 3 dots. Discrimination performance was better when the S- array had closely spaced dots than when they were far apart. When both arrays in a pair contained multiple dotsClick here to view Figure 7 there were four combinations of high density ("near" spacing) and low density ("far" spacing) for each numerosity pair. In different series of training sessions, the stimulus pairs consisted of 2 vs. 3 or 4 dots, 3 vs. 5 or 7, or else 5 vs. 6 or 7. (See Figure 7 for examples of the stimuli.) In each case, the best discrimination scores on average were obtained when the small numerosity, S+ array had widely spaced dots and the larger numerosity, S- array had closely spaced dots. The poorest performance was obtained with the opposite density combinations, i.e. "near" spaced S+ array vs. "far" spaced S- array. These rankings of performance level are summarized in Table 1. 

Table 1. Array density and discriminablity of numerosity

Numerosity Combinations

1/2 - 1/3

2/3 - 2/4

3/5 - 3/7

5/6 - 5/7


Performance rankings

N-> F-

F+N- > N+N- > F+F- > N+F-

F+N- > F+F- > N+N- > N+F-

F+N- > N+N- > F+F- > N+F-


Array size: + = smaller number           Dot spacing: F = "far"    = low density
                   -  = larger number                                   N = "near" = high density

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 Click here to view the animated illustration (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 Click here to view Figure 8 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 Click here to view Figure 9 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 withinClick here to view examples of the stimuli 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 Click here to view Figure 10 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.

Click here to view examples of the stimuliThe 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 tClick here to view Figure 11hey 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 Click here to view Figure 12 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).


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 Click here to view Figure 13 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 Click here to view Figure 14 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. Click here to view Figure 15 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 Click here to view Figure 16 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 Click here to view Figure 17 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.

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       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.