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Understanding of the Concept of Numerically Less by BottlenoseDolphins (Tursiops truncatus)

Kelly Jaakkola, Wendi Fellner, Linda Erb, Mandy Rodriguez, and Emily GuarinoDolphin Research Center

In 2 experiments, bottlenose dolphins (Tursiops truncatus) judged the ordinal relationship between novel

numerosities. The dolphins were first trained to choose the exemplar with the fewer number of items

when presented with just a few specific comparisons (e.g., 2 vs. 6, 1 vs. 3, and 3 vs. 7). Generalization

of this rule was then tested by presenting the dolphins with all possible pairwise comparisons between

1 and 8. The dolphins chose the exemplar with the fewer number of items at levels far above chance,

showing that they could recognize and represent numerosities on an ordinal scale. Their pattern of errors

was consistent with the idea of an underlying analog magnitude representation.

The idea that nonhuman animals evidence numerical abilities is

no longer controversial. Data from numerous laboratories haveshown that many nonhuman animals have at least some numerical

capacities (for reviews, see Davis & Perusse, 1988; Dehaene,

1997). The debate in comparative cognition has moved toward two

issues: (a) to what extent different animal species understand

particular numerical properties, and (b) what representational sys-

tems underlie these capacities.

Ordering Numerosities

The current article is specifically concerned with the property of

relative numerositythat is, the understanding that numerosities

(e.g., oneness, twoness, etc.) can be judged according to their order

in an inherent series. Even if one can distinguish three objects from

two objects, this distinction could theoretically be recognized as asimple property of a set, like distinguishing redness from green-

ness. However, the concept of number necessarily entails more

than that. Inherent in the number concept is the idea that numbers

form an ordered seriesnot only is three different from two and

four, but it is more than two and less than four.

Relative numerosity has been studied in the animal kingdom

using a variety of methods, including presenting an animal with

sets of objects or series of events (e.g., light flashes) and requiring

it to pick the one that has more or less objects/events (e.g.,

Alsop & Honig, 1991; Beran, 2001; Boysen & Berntson, 1995;

Call, 2000; Dooley & Gill, 1977; Hauser, Carey, & Hauser, 2000;Killian, Yaman, von Fersen, & Gunturkun, 2003; Machado &

Keen, 2002; Shumaker, Palkovich, Beck, Guagnano, & Morowitz,

2001; Thomas & Chase, 1980); presenting an animal with symbols

or objects associated with particular numbers of food pieces and

assessing whether it chooses the symbol/object associated with the

greater number of food pieces (e.g., Mitchell, Yao, Sherman, &

ORegan, 1985; Washburn & Rumbaugh, 1991); training an ani-

mal to respond to numerosities in a particular numerical order

(e.g., ascending 1 32 33 34) and then testing its generalization

of the rule with new numerosities (e.g., 5, 6, 7, 8; Brannon &

Terrace, 1998, 2000); and training an animal to make a differential

response to two numerosities and then testing its response to

intermediate numerosities (e.g., Breukelaar & Dalrymple-Alford,

1998; Emmerton, Lohmann, & Niemann, 1997).

The results of these studies have led to claims of understanding

relative numerosity for many animal species, including chimpan-

zees (Beran, 2001; Boysen & Berntson, 1995; Dooley & Gill,

1977), orangutans (Call, 2000; Shumaker et al., 2001), rhesus

monkeys (Brannon & Terrace, 1998, 2000; Hauser et al., 2000;

Washburn & Rumbaugh, 1991), squirrel monkeys (Thomas &

Chase, 1980), dolphins (Killian et al., 2003; Mitchell et al., 1985),

rats (Breukelaar & Dalrymple-Alford, 1998), and pigeons (Alsop

& Honig, 1991; Emmerton et al., 1997; Machado & Keen, 2002).

Note, however, that to qualify as explicit knowledge of relative

numerosity, four criteria must be met: (a) Nonnumerical cues such

as stimulus area, density, or duration must be controlled for

otherwise, the animals may be judging the relative magnitude ofthese continuous dimensions rather than relative numerosity per

se;1 (b) particular numerosities must be judged as greater than

some numerosities and less than others (rather than simply desig-

nating some numerosities as large and others as small); (c) the

ability must be demonstrated with numerosities on which the

animals were not originally trained; and (d) this generalization to

1 This is true even in cases in which stimulus items are presented

sequentially rather than simultaneously. Even if an animal is shown one

M&M at a time, for example, it is not possible to determine whether its

responses are based on the total number of M&Ms or the total amount

(volume) of candy (Beran, 2001; Call, 2000).

Kelly Jaakkola, Wendi Fellner, Linda Erb, Mandy Rodriguez, and Emily

Guarino, Dolphin Research Center (DRC), Grassy Key, Florida.

We thank Adrian Dahood, Shelly Samm, Peter Sugarman, Tammie

Anderson, Margaret Thomas, Cheryl Sullivan, and numerous research

interns at DRC for help with data collection, as well as Tammie Anderson,

Susan Carey, Pat Clough, Kathy Roberts, Elizabeth Spelke, and Marie

Trone for helpful comments on earlier versions of this article. We are also

grateful to Peter Sugarman for help with design and construction of the

testing apparatus and Tommi Jaakkola for technical assistance. Finally, a

special thanks to the staff and dolphins of DRC, under the leadership of

President Jayne Shannon-Rodriguez, for their cooperation and patience

during this project.

Correspondence concerning this article should be addressed to Kelly

Jaakkola, Dolphin Research Center, 58901 Overseas Highway, Grassy

Key, FL 33050. E-mail: [email protected]

Journal of Comparative Psychology Copyright 2005 by the American Psychological Association2005, Vol. 119, No. 3, 296 303 0735-7036/05/$12.00 DOI: 10.1037/0735-7036.119.3.296

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novel numerosities must occur in such a manner that inadvertent

cuing from trainers or experimenters is impossible. Using these

criteria, researchers have thus far convincingly demonstrated

knowledge of relative numerosity only in primates (Brannon &

Terrace, 1998, 2000; Hauser et al., 2000).

The current study explored the understanding of relative numer-

osity in the bottlenose dolphin. To our knowledge, there have onlybeen two prior studies of numerical understanding in dolphins. In

the first (Mitchell et al., 1985), a dolphin was asked to choose

between different objects, each of which was associated with a

particular number of fish (e.g., an ice-cube tray was worth one fish;

a mixing bowl was worth four fish). Whatever choice the dolphin

made, the associated number of fish were thrown to her one at a

time. The dolphin learned to choose the object with the greatest

fish value. Unfortunately, it is not possible to determine whether

the dolphin based her response on number, per se, because both the

amount of fish (mass, volume), and the duration of the rewarding

process also varied systematically with number. In the second

study (Killian et al., 2003), a dolphin was trained to choose which

of two stimulus arrays contained the fewer number of elements.

However, during the crucial generalization phase in which the

dolphin was tested on novel discriminations, the stimuli were

placed in the water by an assistant who kept observing the animal

(Killian et al., 2003, p. 138). It is unclear from the description

whether the dolphin could see this person. Without this piece of

information, it is unfortunately not possible to rule out inadvertent

cuing. Thus, to a large extent, the level of dolphins numerical

understanding remains an open question.

Numerical Representation

To fully understand numerical competence in humans and non-

human animals, a second issue that must be addressed concerns the

type of representational system that underlies any demonstrated

numerical capacities. The two types of representational systems

most often proposed in the current literature are depicted in Fig-

ure 1.

The first is an analog magnitude system (see Figure 1B), in

which the numerosity of a set is mentally encoded as a single

continuous magnitude, like a number line, such that a greater

magnitude is indicative of a larger numerosity (e.g., Gallistel &

Gelman, 1992, 2000). Comparisons between sets are performed by

comparing the extent of the magnitudes representing each set.

Because of the nature of this representation, judgments using this

system are subject to Webers law, in which the difficulty of

discriminating between two magnitudes is a function of their ratio.

For example, discriminating seven from eight should be moredifficult than discriminating two from three.

The second type of representational system proposed to account

for nonlinguistic numerical competence is the object-file model

(e.g., Hauser et al., 2000; Simon, 1997; Uller, Carey, Huntley-

Fenner, & Klatt, 1999; see Figure 1C). In this model, individuals

in a set are represented as separate symbols (object files), with no

single mental entity representing the numerosity of a set. Rather,

comparisons between sets are performed by one-to-one matching

between the object files for each set. Because there is a limit to the

number of objects that can be held in mind simultaneously, nu-

merical judgments are subject to a magnitude limit. That is, accu-

rate judgments should not be possible for numerosities that exceed

that attentional limit, typically taken to be four or fewer objects

(Feigenson, Carey, & Hauser, 2002).

The Current Study

The purpose of the current study was to look for evidence of

understanding relative numerosity in the bottlenose dolphin, and if

successful, to explore the underlying representational system. Dol-

phins were presented with two arrays of dots that varied with

respect to dot sizes and positions (see Figure 1A). The dolphins

were trained to choose the exemplar with the fewer number of

items, using just a few specific training pairs (e.g., 2 vs. 6, 1 vs. 3,

and 3 vs. 7). Generalization of this rule was then tested by

presenting the dolphins with all possible pairwise comparisonsbetween 1 and 8. If the dolphins performance is consistent with

Webers law, this would suggest an underlying analog magnitude

representation. If their performance instead evidences magnitude

limits, this would suggest an underlying object-file representation.

Experiment 1

Method

Subject and testing environment. The subject was a male Atlantic

bottlenose dolphin (Tursiops truncatus) named Talon, who was born at the

Dolphin Research Center in Grassy Key, Florida, and was 10 years old at

the time training was initiated. Talon was selected because he was male (to

Figure 1. Examples of stimuli (A) and corresponding analog magnitude

(B) and object-file (C) representations.

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avoid a potential baby-induced hiatus), relatively young, and had time

available in his training and public program schedule. He resided primarily

with another 12-year-old male dolphin in one of three natural seawater

lagoons situated on the Gulf of Mexico. Lagoon A measured 36.6

27.4 up to 6.1 m deep, depending on tide state; Lagoon B measured

36.6 13.7 up to 3.7 m deep; and Lagoon C measured 41.1 15.2

up to 4.9 m deep. Training took place in any of the three lagoons, and

testing took place exclusively in Lagoon A. All sessions were videotapedusing a camera located across the lagoon from the testing dock. At various

times throughout training and testing, the group was joined by any of three

other male dolphins. Talon was fed between 5.5 and 7.7 kg (M 6.7 kg)

of capelin, smelt, and herring daily, approximately 33% of which he

received during experimental sessions. During nonexperimental sessions,

he continued to participate in other training sessions, including public

programs and in-water interactions with trainers and guests.

Stimuli and apparatus. On each trial, the dolphin was presented with

two stimulus arrays of dots. Each array was presented on a 61 61-cm

black plywood board divided into 16 evenly spaced locations (arrayed in a

4 4 matrix). Four 1.3-cm-wide strips of black hook-and-loop fastener

were attached horizontally to the front of the plywood, running through the

center of each row of the matrix. White plywood dots of three different

diameters (2.5, 5, and 10 cm) served as the stimuli and were attached to the

boards with hook-and-loop fastener. Across trials, the stimuli varied with

respect to both dot sizes and positions, to ensure that the dolphin could not

respond on the basis of nonnumerical cues such as stimulus area or pattern.

Dot sizes were assigned randomly, subject to surface area constraints. Dot

locations varied randomly, subject to the constraint that each dot must be

adjacent to at least one other dot in a single continuous configuration (see

Figure 1A). This prevented a situation in which it could appear that there

were two separate groups of dots on the same board.

The two stimulus arrays were attached to two arms of a presentation

apparatus constructed of PVC pipe, at a distance of 1.3 m apart (see Figure

2). The apparatus was designed so that the stimulus boards could be

simultaneously rotated from the loading position at 1.0 m back from the

edge of the dock, facing up, down to the presentation position of 0.2 m

above the water, facing the dolphin. The area between the two arms of the

apparatus was blocked by a research screen constructed of PVC pipe andopaque canvas. An experimenter stood behind the screen out of the

dolphins field of view and used a periscope-like device for one-way

viewing of the dolphin. A trainer sat in front of the research screen facing

the dolphin. The trainer could not see the front of the stimulus arrays and

was blind as to which array was correct. Between trials, canvas side flaps

were raised to prevent the dolphin from viewing preparations for the next

trial.

Procedure. Trials ran as follows: Once both boards were loaded onto

the apparatus, the side flaps were lowered and the experimenter called out,

Ready. The trainer asked the dolphin to station with his head out of the

water, then called back, Ready. The apparatus was rotated into position

and then the trainer gave a hand signal while saying Less. The experi-

menter observed the dolphin through the periscope and blew a whistle if he

touched the correct board or simply removed the stimuli if he was incor-

rect. On correct trials, the trainer provided positive reinforcements of fish

and social interaction. On incorrect trials, the trainers response remained

neutral. During the training phase, training decisions such as whether to

repeat a trial, how much to reward, and whether to end a session early were

left to the discretion of the trainer. During the testing phase, all such

decisions were fixed.

Normally, one session was run per day, 5 days per week. Sessions

typically lasted 20 to 30 min. During sessions, any other dolphins present

in the lagoon were kept busy at a separate dock by another trainer.

Training. Three pairs of numerosities were designated as training

pairs: 2 versus 6, 1 versus 3, and 3 versus 7. All other combinations were

presented for the first time during the testing phase. Initial training made

use of only a single numerosity pair: 2 versus 6.2 For the earliest stages of

training, the discrimination was simplified as much as possible by present-

ing only medium-sized dots at fixed locations. In addition, the dots on the

correct stimulus array were affixed to the board with springs and the board

was jostled, providing a wiggle cue. Errorless trials in which only the

two-dot array was presented were used to train the dolphin to approach the

2 On a single occasion, Talon was accidentally presented with a

2-versus-5 comparison when a dot fell off of the six-dot board.

Figure 2. Experimental set-up. A: Front view. Note that the trainer

cannot see the stimulus arrays, and is blind as to which side is correct. The

experimenter watched the dolphin through the periscope located on top of

the research screen. B: Side view, showing apparatus at loading position,

then rotated into presentation position.

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wiggling board and dots.3 Training progressed in several steps: First,

introduce choice (i.e., nonerrorless) trials; second, eliminate the wiggle

cue; third, vary dot locations; fourth, introduce small- and large-sized dots;

and fifth, introduce reversed trials (i.e., trials in which the array with the

fewer number of dots covered more surface area than the array with the

greater number of dotse.g., the two-dot board may have had two large

dots that covered 162 cm2 of surface area, whereas the six-dot board may

have had two small and four medium dots that covered only 91 cm2 ofsurface area).

Training sessions typically consisted of 20 trials and included three types

of trials: (a) standard trials at a difficulty level that the dolphin had

demonstrated he was capable of solving successfully, (b) errorless trials

that previewed the next training step, and (c) probe trials consisting of

choice trials at the next training step. For example, consider the point at

which the wiggle cue had just been eliminated and varying dot locations

were being introduced. At this point, standard trials consisted of medium

dots in fixed locations (which the dolphin had already mastered), errorless

trials consisted of one board with two medium dots in any location

(showing the dolphin the correct answer for the next step), and probe trials

consisted of a board of two versus a board of six medium dots in any

location (i.e., the next step). In a typical session, 25% of the trials were

errorless. The number of standard and probe trials changed dynamically in

such a way that the number of probes increased each time the dolphin

demonstrated competency, until the previous probes became the new

standard at the next training step.

This procedure worked well until the last step, in which we introduced

reversed trials. It became apparent that Talon had been discriminating

based on the amount of white space (i.e., surface area of the stimuli) on

each board rather than on the number of dots. We tried two strategies to get

him past this conceptual difficulty. First, we asked Talon to carry the dots

to the boards, and also to try the discrimination using common objects in

place of the dots (i.e., we attached objects such as soda cans and paint

brushes to the boards). The objective was to try to get him to view the dots

as objects rather than as amounts of stuff. However, Talon reacted to this

by losing his motivation to participate in the research. We thus selected a

new combination of dots that resulted in nearly equal amounts of white

space on both boards: two large dots (162 cm2

of surface area) versus onesmall, four medium, and one large dot (167 cm2 of surface area). We then

repeated Training Steps 2 (eliminating the wiggle) and 3 (varying dot

locations).

After Talon had reached criterion on the comparison of 2 versus 6 with

matched surface area, a second pair of numerosities1 versus 3was

introduced in probe trials, also with nearly equal amounts of white space on

both boards. The number of probe trials was gradually increased until the

session was divided evenly between both training pairs (2 vs. 6 and 1 vs.

3). The final step was to gradually increase the difference in the amount of

white space between the boards, with half of the trials being reversed, until

the difference in white space became noticeable. This time, performance on

reversed trials was equal to performance on consistent trials (i.e., trials in

which the array with the fewer number of dots also covered less surface

area than the array with the greater number of dots), indicating that Talon

had come to disregard the amount of white space and was now discrimi-

nating based on the number of dots.

Finally, a third numerosity pair3 versus 7was introduced in probe

trials. Talon was immediately correct on 100% of these trials over two

sessions, including both consistent and reversed trials. An additional ses-

sion was run consisting of an even mix of all three known numerosity pairs.

This marked the end of the training phase.

In total, by the end of training, Talon had seen 572 errorless trials (i.e.,

a single array of two dots), 2,764 trials of 2 versus 6, 332 trials of 1 versus

3, and 15 trials of 3 versus 7.

Testing design. Once training was completed, generalization of the less

rule was tested by presenting the dolphin with all possible pairwise com-

parisons between 1 and 8. As in training, stimuli were varied with respect

to dot sizes and positions. In addition, testing exemplars for each compar-

ison were counterbalanced such that half the trials were reversed with

respect to surface area, and half were consistent. Each new pair of numer-

osities was tested eight times, distributed over 19 sessions. Each session

consisted of 20 trials, with familiar trials (i.e., 2 vs. 6, 1 vs. 3, and 3 vs. 7)

and test trials intermixed. For the first 7 sessions, there were 12 familiar

trials and 8 test trials; for the remaining sessions, there were 8 familiar

trials and 12 test trials. The specific numerosity pairs tested in each session

were chosen randomly, with the constraints that no more than two of thesame numerosity pair could occur in any given session, and no consecutive

numerosity pairs (e.g., 3 vs. 4) occurred during the first 3 sessions. During

each session, half of the trials were reversed with respect to surface area

and half were consistent; order of trials was randomized, with the con-

straint that there were no more than 3 test trials in a row (in the final session

there were 5 consecutive test trials); and correct side was assigned ran-

domly, with the constraint that there were never more than three consec-

utive correct answers on either side.

Coding. The dolphin was coded as making a choice when his rostrum

contacted one of the stimulus boards. Accuracy of his choices was coded

live during the sessions and later checked by a second experimenter from

the videotapes. Initial orientation was defined as the first direction the

dolphin faced after the start of the signal and was coded from the video-

tapes. Because 12 test trials were not videotaped, analyses of initial

orientation were conducted using the remaining 188 test trials. A secondexperimenter independently coded initial orientation for 20% of the gen-

eralization trials. Reliability between the two coders was 99% for final

accuracy and 100% for initial orientation.

Results

All analyses were conducted on the data from novel numerosity

pairs only. The data from trained pairs are included in tables for

completeness.

Overall accuracy. Overall, the dolphin chose the exemplar

with the fewer number of dots on 83% of the generalization trials

(binomial test, p .001), showing that he could recognize and

represent numerosities on an ordinal scale. The accuracy of thesejudgments was not significantly related to whether surface area

was consistent or reversed, 2 corrected for continuity (1, N

200) 1.25, ns.

Because every trial presented a new combination of dot sizes

and positions, the only way to succeed was to recognize the

numerosity of the arrays. It is theoretically possible, however, that

the dolphin could have recognized these numerosities as nominal

categories and then learned their ordinal relations extremely rap-

idly after the first reinforcement. However, first-trial data showed

that the dolphin correctly judged the relative numerosities even the

first time he was presented with each comparison (76% correct,

binomial test, p .01).

Error patterns. To assess underlying numerical representa-

tion, we next analyzed the dolphins error patterns. The proportionof correct responses for each numerosity pair is presented in the

top panel of Table 1. A regression analysis showed a significant

linear effect of the ratio of large to small numerosity on the

proportion of correct responses for each numerosity pair (

3 Errorless trials were those in which only a single, correct board was

presented. Technically speaking, there is no correct answer regarding

which board has fewer dots when only one board is presented. However,

the only comparison presented in the early trials was 2 versus 6, in which

the two-dot board was always the correct answer. The errorless trials were

included to help the dolphin learn that rule. Errorless trials were faded out

entirely when reversed trials were introduced (Training Step 5).


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.488, p .02). In other words, the dolphin tended to make more

errors on trials in which the ratio between the two presented

numerosities was small, as predicted by Webers law. This is

consistent with the predictions of the analog magnitude model.

We next looked at the effect of set size. The object-file model

would predict perfect or near-perfect performance for set sizesunder a particular threshold, coupled with chance performance

above that threshold. We tested each small numerosity for such a

magnitude limit. Binomial tests showed greater than chance per-

formance (ps .05) both above and below each small numerosity

from 1 through 6. It was only above a possible threshold of 7

where performance was at chance. This is much higher than the

magnitude limit of four or fewer items typically reported for

humans and nonhuman primates (e.g., Feigenson et al., 2002;

Hauser et al., 2000). Moreover, the failure at 7 versus 8 (the only

numerosity pair above this threshold) is also predicted by the

analog magnitude model. Thus, we found no clear evidence to

support the predictions of the object-file model.

First look. As a further check on the dolphins possible nu-

merical representations, we next examined his initial orientation tothe stimulus boards. On all trials, the dolphin oriented toward one

of the boards immediately as the signal was being given. This

initial orientation was correct on 78% of the generalization trials

(binomial test, p .001), showing that he had made an initial

choice prior to the signal. On some trials, he then made an

orientation reversal, in which he turned back toward the other

stimulus array (sometimes choosing that second array, and other

times coming back to his original choice). One possibility, there-

fore, is that he might have initially used an object-file representa-

tion but then reverted to an analog magnitude representation when

the set sizes were not sufficiently small for object files to handle.

If so, one would expect that the error pattern for initial orientation

would show the magnitude limit signature of the object-file model

rather than the Webers law signature of an analog magnitude

system.

The proportion of correct initial orientations for each numeros-

ity pair is presented in the bottom panel of Table 1. A regression

analysis showed a significant linear effect of the ratio of large to

small numerosity on the proportion of correct initial orientationsfor each numerosity pair ( .643, p .01), consistent with

Webers law. Further, binomial tests showed above chance per-

formance (ps .05) for each small numerosity from 1 through 6,

with chance performance only appearing above a threshold of 7

(i.e., for the comparison of 7 vs. 8). As was the case with final

accuracy, the results of initial orientation are more consistent with

the predictions of the analog magnitude model than with those of

the object-file model.

Discussion

When presented with displays of dots that controlled for non-

numerical cues, a bottlenose dolphin was able to consistently

choose the array with the fewer number of dots. This was true even

for first-trial data on novel numerosity pairs. His performance,

measured by both initial orientation and final choice, was predicted

by the ratio of the numerosities presented, consistent with Webers

law. The only evidence of a magnitude limit came at a possible

threshold of 7. However, as the only comparison in that set was 7

versus 8, this result is also predicted by Webers law. Thus, the

dolphins performance seems better explained by an underlying

analog magnitude model than by the object-file model.

One puzzling result from Experiment 1 was that the dolphin

performed poorly on the comparison of 1 versus 2 dots. This

should be an easy problem on any theory of numerical comparison.

It is important to note, however, that this difficulty could have

arisen as an artifact of the training procedure. Because the vastmajority of his training was performed on the 2-versus-6 discrim-

ination (in which 2 was the correct answer), it could be that the

dolphin simply developed a strong bias to pick 2 whenever it was

present.

In Experiment 2, we replicate Experiment 1 with another dol-

phin, with two differences in the training procedure. First, the

majority of training used the numerosity pair 1 versus 8, rather

than 2 versus 6. If Talons difficulty with 1 versus 2 was a training

artifact, then a second dolphin with a different training history

should not show this same difficulty. Second, we equalized the

white space between stimulus boards earlier in the training process

to avoid the difficulties that Talon encountered when he initially

based his choices on stimulus area rather than on numerosity.

Experiment 2

Method

Subject and testing environment. The subject was a male Atlantic

bottlenose dolphin named Rainbow, who was collected from the Gulf of

Mexico at approximately 4 years of age and was approximately 24 years

old at the time training was initiated. He resided in one of two natural

seawater lagoons situated on the Gulf of Mexico. Lagoon D measured

28.6 18.3 up to 4.9 m deep, depending on tide state; and Lagoon E

measured 27.4 13.1 up to 2.4 m deep. Training took place in either of

the two lagoons, and testing took place exclusively in Lagoon D. All

sessions were videotaped using a camera located across the lagoon from

Table 1

Experiment 1: Proportion of Talons Correct Responses for

Each Numerosity Pair

Smallnumerosity

Large numerosity

2 3 4 5 6 7 8

Final choice

1 .38 (.85*) 1.00* 1.00* 1.00* 1.00* 1.00*2 .88* 1.00* .88* (1.00*) 1.00* 1.00*3 .63 .63 .88* (.88*) 1.00*4 .38 1.00* .75 .88*5 .50 1.00* 1.00*6 .75 .88*7 .25

Initial orientation

1 .50 (.79) 1.00* 1.00* 1.00* 1.00* 1.00*2 .88* .88* .83 (.96*) 1.00* 1.00*3 .63 .29 .88* (.78) .714 .50 .67 .75 .675 .75 .75 .756 .86 .507 .75

Note. Talon is the bottlenose dolphin who was the subject in Experiment1. Parentheses indicate trained discriminations.*p .05.


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the testing dock. At various times throughout training and testing, Rainbow

was joined by any of five other dolphins. Rainbow was fed between 7.3 and

10.4 kg (M 8.5 kg) of capelin, smelt, sardines, and herring daily,

approximately 33% of which he received during experimental sessions.

During nonexperimental sessions, he continued to participate in other

training sessions, including public programs and in-water interactions with

trainers and guests.

Stimuli and apparatus. The stimuli and apparatus were identical to

those used in Experiment 1.

Procedure. The procedure for individual trials was identical to that of

Experiment 1.

Training. Four pairs of numerosities were used as training pairs: 1

versus 8, 3 versus 7, 2 versus 4, and 4 versus 7. All other combinations

were presented for the first time during the testing phase. Initial training

made use of only a single numerosity pair: 1 versus 8. For the earliest

stages of training, the discrimination was simplified as much as possible by

presenting only medium-sized dots at fixed locations. In addition, the dot

on the correct stimulus array was affixed to the board with a spring and the

board was jostled, providing a wiggle cue. Errorless trials in which only the

one-dot array was presented were used to train the dolphin to approach the

wiggling board and dot. Training progressed in several steps: First, intro-

duce choice (i.e., nonerrorless) trials; second, eliminate the wiggle cue;third, vary dot locations; and fourth, equalize white space on the boards.

We selected a new combination of dots that resulted in nearly equal

amounts of white space on both boards: one large dot (81 cm2 of surface

area) versus five small and three medium dots (86 cm2 of surface area).

As in Experiment 1, training sessions typically consisted of 20 trials and

included three types of trials: (a) standard trials at a difficulty level that the

dolphin had demonstrated he was capable of solving successfully, (b)

errorless trials that previewed the next training step, and (c) probe trials

consisting of choice trials at the next training step. The number of standard

and probe trials changed dynamically in such a way that the number of

probes increased each time the dolphin demonstrated competency, until the

previous probes became the new standard at the next training step. Error-

less trials were faded out entirely at the point where we introduced

equalized white space between boards (Training Step 4).

After Rainbow had reached criterion on the comparison of 1 versus 8

with similar surface areas, a second pair of numerosities3 versus 7was

introduced in probe trials, also with nearly equal amounts of white space on

both boards. The number of probe trials was gradually increased until the

session was divided evenly between both training pairs (1 vs. 8 and 3 vs.

7). The final step was to gradually increase the difference in the amount of

white space between the boards, until the difference in white space became

noticeable. Then a third numerosity pair2 versus 4 was introduced in

probe trials. Rainbow was correct on seven out of eight (88%) of these

trials over two sessions, including both consistent and reversed trials.

At this point, Rainbow unfortunately began to decline to participate in

research sessions, because of the distraction of a female dolphin who had

been moved into his lagoon. Over the next 3 months, he chose to partic-

ipate on only 4 days. When he regained interest in research in December,

we decided to delay testing until January, to accommodate staff vacations.In the meantime, we continued to run research sessions a few times a week,

using all three training pairs to date. In January, we introduced a fourth

numerosity pair4 versus 7in probe trials. Rainbow was correct on 12

out of 14 (86%) of these trials over three sessions, including both consistent

and reversed trials. This marked the end of the training phase.

In total, by the end of training, Rainbow had seen 301 errorless trials

(i.e., a single array of one dot), 1,861 trials of 1 versus 8, 524 trials of 3

versus 7, 118 trials of 2 versus 4, and 14 trials of 4 versus 7.

Testing design. The testing design was the same as in Experiment 1,

with the following differences: Each new pair of numerosities was tested

eight times, distributed over 18 sessions. Each session consisted of 20

trials, with familiar trials (i.e., 1 vs. 8, 3 vs. 7, 2 vs. 4, and 4 vs. 7) and test

trials intermixed. For the first 6 sessions, there were 12 familiar trials and

8 test trials; for the remaining sessions, there were 8 familiar trials and 12

test trials.

Coding. Coding was identical to that of Experiment 1. Because 8 test

trials were not videotaped, initial orientation analyses were conducted

using the remaining 184 test trials. Reliability between the two coders was

100% for both final accuracy and initial orientation.

Results

All analyses were conducted on the data from novel numerosity

pairs only. The data from trained pairs are included in tables and

figures for completeness.

Overall accuracy. Overall, the dolphin chose the exemplar

with the fewer number of dots on 82% of the generalization trials

(binomial test, p .001), showing that he could recognize and

represent numerosities on an ordinal scale. The accuracy of these

judgments was not significantly related to whether surface area

was consistent or reversed, 2 corrected for continuity (1, N

184) 2.42, ns. As in Experiment 1, first-trial data showed that

the dolphin correctly judged the relative numerosities even the first

time he was presented with each comparison (88% correct, bino-

mial test, p .01).

Error patterns. To assess underlying numerical representa-

tion, we next analyzed the dolphins error patterns. The proportion

of correct responses for each numerosity pair is presented in the

top panel of Table 2. As in Experiment 1, a regression analysis

showed a significant linear effect of the ratio of large to small

numerosity on the proportion of correct responses for each numer-

osity pair ( .507,p .02). That is, the dolphin tended to make

more errors on trials in which the ratio between the two presented

numerosities was small, as predicted by Webers law.

We next looked at the effect of set size. The object-file model

would predict perfect or near-perfect performance for set sizes

Table 2

Experiment 2: Proportion of Rainbows Correct Responses for

Each Numerosity Pair

Smallnumerosity

Large numerosity

2 3 4 5 6 7 8

Final choice

1 .88* .75 1.00* 1.00* 1.00* 1.00* (1.00*)2 .63 (.81*) .63 .88* 1.00* .88*3 1.00* .25 .88* (.93*) .88*4 .75 .88* (.88*) .88*5 .75 .75 .88*

6 .50 .88*7 .75

Initial orientation

1 .63 .63 .88* 1.00* .88* 1.00* (.98*)2 .67 (.71*) .50 .86 1.00* .753 .88* .38 1.00* (.82*) .88*4 .63 .75 (.80*) .635 .88* .71 .88*6 .38 1.00*7 .63

Note. Rainbow is the bottlenose dolphin who was the subject in Exper-iment 2. Parentheses indicate trained discriminations.p .07. *p .05.


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under a particular threshold, coupled with chance performance

above that threshold. We tested each small numerosity for such a

magnitude limit. As in Experiment 1, binomial tests showed above

chance performance (ps .05) both above and below each small

numerosity from 1 through 6, with chance performance only

appearing above a threshold of 7 (i.e., for the comparison of 7 vs.

8). Thus, the dolphins performance was more consistent with thepredictions of the analog magnitude model than with the object-

file model.

Note that in contrast to Talons performance in Experiment 1,

Rainbow performed above chance with the comparison of 1 versus

2. It thus seems that Talons difficulty with this comparison was

likely an artifact of training history.

First look. As a further check on the dolphins possible nu-

merical representations, we also examined his initial orientation to

the stimulus boards. On all trials, the dolphin oriented toward one

of the arrays immediately as the signal was being given. This

initial orientation was correct on 78% of the generalization trials

(binomial test, p .001), showing that he had made an initial

choice prior to the signal. The proportion of correct initial orien-

tations for each numerosity pair is presented in the bottom panel ofTable 2. As in Experiment 1, a regression analysis showed a

significant linear effect of the ratio of large to small numerosity on

the proportion of correct initial orientations for each numerosity

pair ( .445, p .03), consistent with Webers law. Further,

binomial tests showed above chance performance (ps .05) for

each small numerosity from 1 through 6, with chance performance

only appearing above a threshold of 7 (i.e., for the comparison of

7 vs. 8). Thus, as was the case with final accuracy, the results of

initial orientation are more consistent with the predictions of the

analog magnitude model than with those of the object-file model.

Discussion

The dolphin in Experiment 2 showed essentially the same pat-

tern of results as the dolphin in Experiment 1. He consistently

chose the array with the fewer number of dots, even the first time

he was presented with novel numerosity pairs. His pattern of errors

was predicted by the ratio of large to small numerosity, consistent

with Webers law. The only evidence of a possible magnitude limit

came at a threshold of 7 (i.e., for the comparison of 7 vs. 8), which

is also predicted by Webers law. Thus, the dolphins performance

seems better explained by an underlying analog magnitude model

than by the object-file model.

General Discussion

This study adds two new findings to the literature on animalconcepts of number. First, when presented with displays of dots

that controlled for nonnumerical cues, dolphins were able to con-

sistently choose the array with the fewer number of dots. This was

true even for first-trial data on novel numerosity pairs. Thus,

bottlenose dolphins are able to discriminate numerosities and to

reason about them with respect to an ordinal scale.

Second, the pattern of errors in the dolphins initial orientation

and final choice conformed to Webers law. This suggests an

underlying analog magnitude representational system as has been

proposed to account for other human and nonhuman animal results

(e.g., Dehaene, 1997; Gallistel & Gelman, 1992; Whalen, Gallistel,

& Gelman, 1999). In contrast, we found no evidence of the type of

magnitude limit that would suggest that the dolphins were using an

object-file model in this task.

In their review of the literature, Davis and Perusse (1988)

proposed thatsubitizingwhich they characterized as a perceptual

process that rapidly assesses the numerosity of a small quantity of

itemscan account for much of the data on animals numerical

abilities. Although the experiments presented here were not de-signed to speak to this issue, there are several aspects of our results

that suggest that subitizing was not behind the performance of the

dolphins in this task.

First, subitizing is generally agreed to operate only on numer-

osities up to 3 or 4 in humans (e.g., Dehaene, 1992; Mandler &

Shebo, 1982). The dolphins, however, showed success at compar-

isons as large as 6 versus 8. Indeed, if subitizing were at work, the

pattern we would expect to see would be the same sort of magni-

tude limit signature predicted by the object-file model, most likely

with respect to the dolphins first response (i.e., initial orientation).

Recall, however, that we found no such evidence of a magnitude

limit for either of our dolphins.

Second, the model of subitizing advocated by Davis and Perusse

(1988) holds that each numerosity is recognized as a nominalcategory, similar to triangle and square, and as such is not a

numerical process at all. On that model, there is no such mecha-

nism that can account for the fact that the dolphins error patterns

conformed to Webers law. Indeed, that model of subitizing cannot

account for ordinal results at all (Brannon & Terrace, 2000). If

dolphins were to recognize 1, 2, and 3 in the same way that they

might recognize cat, horse, and dog, then they would have

to learn individually that 3 is greater than 2, which is in turn greater

than 1. This study demonstrated that they can make exactly these

kinds of judgments without needing to learn them individually.

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Received January 24, 2004Revision received December 8, 2004

Accepted December 11, 2004


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