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The innate number sense we share with much of the animal kingdom boils down to two capacities: We can represent small numbers exactly, and we can represent larger numbers approximately. (The following largely follows Feigenson, L. Dehaene, S. and Spelke, E. (2004))
Small Numbers, Exactly
A 1994 experiment with wild lions illustrates the first of these numerical capacities. In the experiment (McComb, Packer & Pusey 1994), researchers used hidden speakers in the African jungle to play between 1 and 5 lion calls. If a group of female lions hears fewer calls than the number in their group, they will go and explore. If they hear more calls than the number of their group, they will leave, presumably to avoid being outnumbered. To do this, lions must be able to represent and compare the number of sounds the hear with the number of lions in their group. Clearly it's important that they be exact; being off by even one could be a life-or-death mistake!
Consider another experiment (Feigenson, L. et al. 2002), demonstrating this ability in rhesus monkeys and human infants. In the experiment, 10- and 12-month-old infants watched an experimenter hide crackers, one-by-one, in two buckets. The infants were then allowed to choose a bucket. Infants spontaneously chose the bucket with more crackers as long as there are 3 or fewer crackers in each bucket. However, when the experimenter puts, for example, 2 crackers in one bucket and 4 in the other, the infants choose randomly. They also choose randomly for 3 vs. 4 crackers, 3 vs. 6 crackers, and even 1 vs. 4 crackers. The same experiment performed on (adult) rhesus monkeys yielded similar results. Like the lions, infants and monkeys were able to represent small numbers. This experiment also highlights the limit of this capacity.
We adults are also familiar with our limited ability to discriminate all but the smallest numbers. If we see a grouping of 2 or 3 things, we can quickly, confidently, and correctly tell you how many there are. This ability is called subitizing. When there are more than about 5 or 6 items, however, we begin to make mistakes, become less confident, and eventually must resort to explicit counting.
The experiments sketched here barely scratch the surface of existing research, and you can undoubtedly think of alternative interpretations for their results. Infant and animal research is notoriously difficult, and I would encourage the curious to pursue the links in the references section further for more detail about the experimental designs. However, for the time being, let's look at our second numerical capacity...
Larger Numbers, Approximately
Immediately intuiting small numbers is only one part of our innate number sense. Here's a quick demonstration of the other. Could 43 + 17 be 500? You don't need to think much to answer. Like our ability to subitize, humans can make judgments like this quickly, confidently, and correctly. However, as the numbers all become larger, or the magnitudes closer together, we take longer, become less certain, and make more mistakes. Could 43 + 17 be 59? This judgment is far less intuitive.
We already know that infants can't represent more than about 3 or 4 things. Yet six-month-old infants are able to tell the difference between much larger pairs of numbers. For example, they can discriminate between 8 vs. 16 dots and 16 vs. 32 dots. For these experiments (Xu, F. and Spelke, E.S., 2000), it isn't the exact numbers of dots that matter, but the ratio of dots to be distinguished--how "close" the two numbers are. While six-month-olds are able to tell the difference between numbers in a 1:2 ratio, they fail with a 2:3 ratio (as when, for example, they are presented with 8 vs. 12 or 16 vs. 24 dots). All of these experiments control for non-numerical cues that you might expect someone to use such as size of the display, density of items, and so forth. Adults, by the way, can discriminate ratios as small as 7:8. This suggests that infants' do have some way of thinking about larger numbers, but that it's in terms of their approximate magnitude.
Trained rats can also represent approximate numerical magnitudes. In one experiment (reviewed in Dehaene, S., 1997), rats were rewarded with food whenever they pressed a lever in their cage a specific number of times, and then switched to press a different lever. For larger numbers, such as 41 presses, the actual number of times the rats pressed before switching formed a perfect bell curve around 41. Since individuals don't press at the same speeds, the experimenters were able to show that they could not have been using elapsed time to help them. It seems that they had to have some notion of the number 41, but that it was only an approximate one.
Getting from "fish-fish-fish-fish-fish-fish" to "6 fishes"
Strogatz' post includes an excellent clip from Sesame Street which he describes as "the clearest and funniest explanation of what they [numbers] are and why we need them." In it, one of the characters learns that the number 6 is a more reliable way to communicate a dinner order than saying "fish" six times. But how do children make the conceptual leap illustrated by the clip? How do they go beyond the numerical abilities they share with other animals to harness numbers in a more useful way?
No one yet has a full answer to these questions, but language and writing both appear to play an important role. Unlike the character in Sesame street, children aren't suddenly able to use the number system as adults do: they reliably go through several intermediate learning stages (detailed in B.W. Sarnecka, S. Carey, 2008).
Children spend countless hours learning and practicing counting. But children who can count to 10 aren't immediately able to use this skill to help them, for example, put six fish on a plate.
Initially, children who have learned to recite their numbers haven't actually learned the meaning of any of the number terms. They will count successfully, but when asked to "Give three cookies to the bear", they will grab any number of cookies at random. They will do this even when asked to count the cookies out. For this reason, children at this stage have been termed "grabbers". They can perceptually distinguish between 1, 2, 3, and 4 items; they know the word "three" comes between "two" and "four"; but they don't know what the word "three" actually means.
Over several months, children successfully learn to give 2 items, then 3, then 4. At no time do they use counting to help them--they still grab, but they grab the correct number. And at each stage, a request to give a larger number results in the random grab. At this stage, they are learning that the word "three" names their perceptual experience of 3 things. But they still don't know how counting works.
Then a revolution happens. Children suddenly become able to give any number of items up to their counting limit. When this happens, they graduate from "grabbers" to "counters". Indeed, whereas they used to grab to give 3 items, or 4 items, they now dutifully count the items out. At this stage, children have learned the key principal of counting: The number of items that a number names is determined by its position in the counting sequence.
Only this insight can get a child from "fish-fish-fish-fish-fish-fish" to "six fishes" with the aid of counting. Even if they can't perceptually distinguish 6 things from 5 things or 7 things, language has given them the tools to conceptually distinguish between them. They know that the numbers occur in a sequence, that each number-word names a specific amount, and that each successive word in the sequence adds one to that amount.
The Language and Culture of Numbers
As the last section suggested, number words provide a crucial cognitive scaffold for learning to count, and thereby to conceptualize larger numbers exactly. They act as anchors in a sea of thought to which meanings can be attached, organized, and preserved.
Though I can't seem to locate the references right now, a number of experiments have been performed with speakers from an isolated Amazon tribe whose language contains no exact number words. Their only number words translate roughly to "one-ish" and "two-ish", or perhaps better, "less" and "more". In these experiments, adult speakers performed equivalently on numerical tasks with children prior to learning language.
For me, the Amazonian culture brings out the question of what features of a culture create a need for counting and naming exact numbers. It's easy to imagine why hunter-gatherer cultures might not need to distinguish 57 fro 58, or even 19 from 20. Though it's harder, I can also imagine that, for smaller collections, naming the individual items could suffice, obviating the need for specific numbers like "three".
I know there is research on these topics, but unfortunately I haven't been able to find it (and, sadly, without a university connection I can't read online journal databases!). If anyone knows any general references, I would love to follow up on this strand.
References:
Feigenson, L. Dehaene, S. and Spelke, E. (2004) Core Systems of Number. Trends in Cognitive Science Vol. 8 No. 7. 308-314
B.W. Sarnecka, S. Carey (2008) How counting represents number: What children must learn and how they learn it. Cognition 108, 662–674
Dehaene, S. (1997), The number sense: How the mind creates mathematics, New York: Oxford University Press, ISBN 0195132408 (The precise is freely available online).
Feigenson, L. et al. (2002) The representations underlying infants’ choice of more: object-files versus analog magnitudes. Psychol. Sci. 13, 150–156
Xu, F. and Spelke, E.S. (2000) Large number discrimination in 6-month old infants. Cognition 74, B1–B11
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