The Origins of Small Number Representation

Earlier this week I wrote about the developmental and evolutionary origins of large number representation. A series of studies in human infants, monkeys, rats, and fish demonstrated that animals and humans spontaneously represent large (>4), abstract, approximate numerosities. Animals, human infants, and human adults, show the same ratio signatures (based on Weber's Law). Adult tamarins are on par with 9-month-old human infants. With age or training, discriminability becomes more precise, and the the critical ratio is reduced a bit. There is good evidence that the large number cognitive system is evolutionarily-ancient and non-verbal, and is innate.

ResearchBlogging.orgWhat about for small numbers? Small numbers do not need to be counted or estimated; instead, they are subitized. Upon seeing a scene with a small number of objects you have a sudden, immediate sense of how many objects there are. This happens in parallel rather than serial- you do not need to count the items individually. Therefore, judgments made about displays of 1, 2, 3, or 4 items are rapid, accurate, and confident. As the number of items in the scene increases, judgments are increasingly less accurate and made with less confidence. Response times also increase, with an extra 250-300 milliseconds added for each additional item to be counted beyond four.

Previous work has demonstrated that animals have some pretty sophisticated numerical abilities, but often these experiments depended on a small number of highly trained captive animals, in highly artificial testing situations. What about spontaneous number representation in a population of non-trained free-ranging monkeys? Or human infants?

Cayo Santiago is a small island off the coast of Puerto Rico that is home to approximately 1,000 free-ranging rhesus macaques living socially. The fact that these monkeys are able to thrive in social groups isolated from any natural predators makes them an ideal population to study, and since they are so used to humans on the island, it is relatively easy to approach and work with them.


Figure 1: A rhesus macaque on the beach at Cayo Santiago.

The monkey study had two goals: (1) establish at least one natural context in which number is encoded spontaneously, and (2) establish the limits of this capacity, as a point of comparison for other animal studies, and for studies of human infants.

Two hundred male and female rhesus macaques (Macaca mulatta) living on Cayo were tested. Two researchers placed themselves two meters apart from each other, and 5-10 meters away from the monkey. Each person had a bucket and each bucket was of a different color. The researchers indicated to the monkey that the buckets were empty by tipping them over and placing their hands inside. Then the buckets were placed on the ground in front of the researchers' feet. The first researcher placed one or more apple slices into the bucket, one at a time, making sure the monkey was watching, then stood up, and looked back down at the bucket. Then, the second researcher did the same thing with the second bucket. Both researchers then turned and walked away in separate directions at a natural pace. Since the experimenters no longer appeared to be watching the buckets, the monkeys were able to approach one of them to retrieve a snack.

First, they made sure that the monkeys preferred apples in the first place. One researcher puts a rock into his bucket, and the other puts an apple slice. 15 out of 15 monkeys approached the bucket that contained the apple slice.

After settling that, the researchers went on with the rest of the experiment: how proficient were the subjects at comparing quantities of apple slices? Everything was counterbalanced: sometimes the greater quantity was presented first, and sometimes the lesser quantity was presented first. Sometimes experimenter A presented the greater quantity and sometimes he presented the lesser quantity. Sometimes the greater quantity was on the right side, and sometimes it was on the left side. All quantities were presented sequentially, to prevent the monkey from making a judgment based on a correlate of number, such as volume or surface area.


Figure 2: Results

Condition A was the apple slice (1F) versus the rock (1NF). Conditions B-J are all varying quantities of apple slices. The subjects preferred the larger quantity for comparisons of 1:2 (B), 2:3 (C), 3:4 (D), and 3:5 (G). For comparisons of 4:5 (E), 5:6 (F), 4:6 (H), 4:8 (I), and even 3:8 (J), the monkey did not differentiate the quantities.

Since each monkey was only tested once, it could not have learned anything useful about the task. Instead, it must have spontaneously kept track of the quantities of apple slices in each bucket and represented those quantities in memory. The monkeys also must have been able to compare the relationship of the two numbers in memory by establishing an ordinal relationship, since they always chose the box with the larger number of apple slices.

This experiment suggests that monkeys spontaneously distinguish between sets of 1, 2, and 3 objects. and can further differentiate 3 from 4 or 5 objects.

Recall that the cotton-top tamarins were able to distinguish 8 from 16 items. If there was only one number system that depended on ratio, and success was found for 8:16, then you might expect that 4:8 would yield success as well, in this experiment since it is the same 1:2 ratio. But this was not the case, suggesting that there are two distinct systems for number representation: a large number system, and a small number system for quantities less than or equal to four.

This study was repeated with human infants. Instead of using apple slices, they used graham crackers. Does the small number system in infants depend on a ratio signature, as with the large number system? Or do they represent and encode individual objects in sets of about 4 or fewer items like rhesus macaques?

The experiment included 124 infants, half of whom were 10 months old, and half of whom were 12 months old. Approximately half of the group were males, and half were females. Graham cracker pieces (of identical size) were removed from a small plastic bucket and placed into two opaque containers, which were too tall for the infant to see inside. Each infant sat on the floor, about 1 meter away from the experimenter.

As with the monkey study, the researchers first ensured that the infants were able to approach one of the boxes. The infant watched a toy being placed into a bucket, and was encouraged to crawl to the bucket to retrieve the toy.

Once this had been established, the researcher showed the infant that both containers were empty. She placed them on the floor between herself and the infant, at equal distances from the infant. All crackers were placed sequentially, just as in the monkey study, with similar counterbalancing procedures. After the presentation of the crackers, the experimenter looked down to avoid providing subtle cues to the infant. If the infant did not approach within 10 seconds, the experimenter provided verbal encouragement, but did not look up or make eye-contact with the infant. A choice was recorded if the subject reached into one of the containers, or approached and sat in front of the container for at least 8 seconds without reaching in. Infants who examined both containers were excluded from analysis.


Figure 3: Data.

Infants in both age groups chose the container containing the larger number of graham crackers in the 1:2 condition and the 2:3 condition. For 3:4, neither age group showed a preference for the greater number, and they were also at chance for 2:4 and 3:6. These results indicate that infants recognize more/less relationships. Again, since each infant participated it only one trial, there was no chance to learn the task. They had to spontaneously track the graham crackers, establish the numerical relationship for both quantities, and compare them.

The set-size effect for the infants in this study was three: they discriminated 1:2, and 2:3, but not 2:4 or 3:4, or even 3:6. The monkeys were able to discriminate 3:4 and 3:5, though they were adult monkeys, and the sharpened ability to discriminate small numbers may have grown through experience.

Given the monkey and infant studies, there is reasonable evidence to carefully conclude that the small number system is probably innate, especially considering the common signatures.

But to really drive the point home, we should consider one more study, of infant chickens, in just the first few days of their lives. Chickens turn out to be a really good model organism for study in the laboratory, since they are precocial. Chickens do not need any parental care - that is, starting from the day they hatch, they can feed themselves, find water, and generally survive. Further, upon hatching, their motor systems are such that they can walk around and interact with the world, and their visual systems are well developed. Compare this to human infants who are severely dependent on parental care, can't walk around until one year of life, and don't have decent visual acuity for several months. By the time human infants can participate in a reaching or looking preference experiment, they've had at least several months of experience in the world. For an investigator interested in innate processes separate from experience, chickens are a perfect species. Further, with a cleverly designed lab set-up, the chicks' experiences can be carefully controlled.

Rosa Rugani and colleagues, from the University of Trento in Italy, used chickens to test whether or not the small number system previously seen in infants and monkeys could be found in chickens. The chicks were imprinted to a small ball. This is critical because when chicks are separated from their imprinted object, they have a strong drive to reunite with it. Moreover, it is known that given a choice between hanging out with one of their imprinted objects (one ball), and several imprinted objects (several balls), they will always choose to hang out with the larger number of imprinted objects, when all items are visible. When it came time for testing on the third day of life, the chicks were placed in a small holding chamber with a transparent door while they watched balls that were identical to their imprinted object disappear behind one of two opaque walls. Then they were released from the chamber and allowed to approach one of the walls.

i-c05484b303feaf533241475e3518d67b-chick arithmetic.jpg

Figure 4: Testing apparatus. The holding chamber can be seen on the left, and the two opaque walls on the right.

Each chick participated in two sessions of 20 trials each. Success on a trial required a simple comparison between 2 objects and 3 objects. In one condition, the balls disappeared behind the walls simultaneously, and in a second condition, they were placed behind the walls consecutively. The chicks chose to approach the screen hiding three balls more often than expected by chance in both the consecutive condition (CDT) as well as the simultaneous condition (SDT). There was no difference in success rate between the two conditions.

Could these results have been the product of learning? In an additional analysis, only the first five trials from each session was analyzed, and the statistical significance of the results was preserved. These results confirm that infant chicks spontaneously discriminated sets of 2 and 3 objects, preferring the larger set, even when they must do so on the basis of short-term memory.

i-3f1b15116541b5428671a15423b7274b-rugani 2.jpg

Figure 5: Results.

However, it is possible that the chicks were relying on a correlate of number instead of the actual number of items, in making their decision, such as surface area or contour length. A second experiment was therefore conducted with new chicks, that was identical to the first, except for one key change. Instead of balls, the chicks were imprinted to and tested with red squares. The experimenters were able to systematically control the properties of squares so that the set of three squares would have equivalent total surface area or contour length as the set of two. As before, each chick participated in 20 trials on day 3 of life. The results of this experiment confirmed the prediction that the chick was using the small number system, and not relying on some correlate of number: chicks preferred to approach the screen hiding the larger set in all conditions.

These studies, combined with the ones discussed on earlier this week, suggest that:

  • There is one system that approximates sets of large numbers, and distinguishes among two sets of large sets on the basis of a ratio determined by Weber's law.
  • There is a second system for determining the exact amount of object in a set of small numbers, when the number of objects is four* or less.
  • There is good evidence that each of these systems is innate, though they can be sharpened with age or experience.

The importance of these studies may not be immediately obvious or apparent. But I think these experiments are important in figuring out how the human mind evolved, which cognitive capacities are innate and evolutionarily-ancient, and which require some amount of learning or experience to emerge.

*Shouldn't this number be the magical 7? Early research determined that short-term memory has a capacity for 7 plus or minus 2 items. However, later research suggests that the true magic number is indeed four. That people can remember ~7 items is instead due to "chunking" of larger sets of items four or fewer smaller sets.

Hauser, M., Carey, S., & Hauser, L. (2000). Spontaneous number representation in semi-free-ranging rhesus monkeys Proceedings: Biological Sciences, 267 (1445), 829-833. DOI: 10.1098/rspb.2000.1078

Feigenson L, Carey S, & Hauser M (2002). The representations underlying infants' choice of more: object files versus analog magnitudes. Psychological science : a journal of the American Psychological Society / APS, 13 (2), 150-6. PMID: 11933999

Rugani R, Fontanari L, Simoni E, Regolin L, & Vallortigara G (2009). Arithmetic in newborn chicks. Proceedings. Biological sciences / The Royal Society, 276 (1666), 2451-60 PMID: 19364746


More like this

This post considering the evolutionary origins of numerical cognition, specifically in terms of the approximation of large numbers, is meant as a companion to this week's series on the developmental origins of numerical cognition and developmental dyscalculia, at Child's Play. What are the origins…
Behold! The second installment of the Science Online Lemur Cognition series. If you missed the first installment, you should check out the cyborg lemurs of the Duke Lemur Center. There's some pretty good evidence that numerical cognition emerged fairly early in the primate lineage, at least, if not…
There's a very well-known experiment in developmental psychology called the "A-not-B task." The experiment goes something like this: you, the experimenter, are seated opposite a human infant. Within the reach of both you and the child are two boxes: box "A," and box "B." You hide a toy in "A," in…
Over the last couple decades there's been a pretty heated debate about which, if any, nonhuman animals possess a "theory of mind," that is, the ability to think about what others are thinking. Much of the research bearing on this debate has used false belief tasks. There are many variants, but the…

Love this one!

Very nice post.

Haven't looked at this subject since reading about corvines and shotguns!

By LeonRover (not verified) on 20 Aug 2010 #permalink