THE GREAT nineteenth-century German mathematician Leopold Kronecker once wrote, “God made the integers, all else is the work of man.” (The integers are the positive and negative whole numbers.) His point was that, starting with the integers, it was possible to develop all of mathematics. Given that many contemporary branches of mathematics have little to do with numbers, his observation can be misleading when applied today. Nevertheless, the integers do play a fundamental role in mathematics. And, of course, they represent most people’s first introduction to mathematics.

As I indicated in the previous chapter, our ability to handle numbers—to count collections and to do arithmetic—rests largely on three mental capacities: number sense, numerical ability, and algorithmic ability. How common are these abilities? To what degree do other species have the same or similar abilities? How and when did our ancestors acquire them, and what survival benefits did they confer?

In this chapter, we’ll take a look at the number sense. In the following chapter, we’ll move on to examine arithmetical ability.

Number sense holds a number of surprises. One is that, no matter how mathematically inept we think we are, every one of us has a built-in number sense and a rudimentary arithmetical ability. Another surprise is that babies exhibit these basic abilities when they’re just a few days old. Still another surprise is that many animals, from the pigeon to the chimpanzee, possess similar number sense and arithmetical ability.

Unlike our ability to perform mathematical reasoning, our basic number sense seems quite independent of language. Thus the discussion of number in this chapter and the next is not a part of my overall argument about language and mathematics, which will come later. Yet there is a connection. As I shall describe in the pages that follow, we use our facility with language to extend our innate number sense and make numbers perform useful work for us.

THE NUMBER SENSE

So you think you don’t have a head for figures? Okay, answer the following questions as fast as you can:

As soon as you have done that, pick a number between 12 and 5; any one, the first one that comes into your head.

Done that?

You picked the number 7, didn’t you? How did I know? Because I knew you would follow your innate number sense. (If, despite my instructions, you did not pick the first number that came into your head, you might not have chosen 7; but the chances are that, even then, 7 was your choice.) Why 7?

Here is the explanation offered by the cognitive psychologists. The first four questions were all subtractions. Although they were all very easy, answering them got your mind into “subtraction mode.” Then, when presented with the numbers 12 and 5, you subconsciously computed—or at least estimated—the distance between them, namely 12—5 = 7, making the number 7 salient in your mind. (If you are one of the rare people who, despite following my instructions, did not give 7 as your answer, the chances are very high that you chose a number close to 7—namely, 6 or 8.)

Notice that 7 is not roughly halfway between 5 and 12; if that were the principle subconsciously guiding your choice, you would have picked 8 or 9. The number 7 lies toward one end of the given range. Why did you not pick 10, which is placed at the other end of the range? After all, 10 is much more common in daily life than 7, given that 10 is the base for our method of arithmetic.

When you think about it, there are several rational choices. But when you are first presented with the challenge, 7 is the number that spontaneously pops into your mind, the distance between 5 and 12.

Quick, what is 8 times 7? It’s 54, right?

Or is it 64? Or 56, perhaps? If you are like most people, each of these answers seems “reasonable.” Why is it that, despite hours of drill in elementary school, most of us have so much trouble with our multiplication table? Especially when each of the numbers being multiplied is 6, 7, 8, or 9?

Okay, one more. Here’s a number comparison test. For each of the following pairs of numbers, say which is greater:

1 and 50
5 and 4
25 and 24

You may not have been aware of it, but if someone had been timing your response (say by asking you to press one of two buttons depending on which of the pair was the greater), you would discover that it took you longer to answer the second question than the first, and longer still to answer the third. Why?

You might explain that you answered the first more quickly because “the two numbers are so far apart, it’s obvious that 50 is bigger than i.” But what’s that got to do with it? I didn’t ask you how far apart they were. I asked which was the greater.

How about the second and the third examples? Arithmetically, these are the same: each is a pair of successive numbers, with the greater written first. Indeed, if you simply ignore the first digits in the third example (which you may, since it’s the same digit in both numbers), the two examples are identical. And yet it has been shown on many occasions that everyone takes measurably longer to decide between 25 and 24 than between 5 and 4. A difference of 1 is somehow more easily recognized for pairs of small numbers than for larger ones.

The fact is, you have an innate sense of number. You have had it since you were a few days old, and quite possibly you were born with it. And we humans share this number sense with chimpanzees, rats, lions, and pigeons.

I am not saying you are “good at sums” or that you “know your multiplication table.” But regardless of your prowess in the mathematics class, your mind does have a sense for number. Numbers—at least small ones—have meaning for you, just as do words and music, and that meaning is not something you had to work at to learn. You were either born with it or born with a natural ability to acquire it effortlessly and unavoidably at a very early age.

The term “number sense” was introduced by Tobias Dantzig in his 1954 book Number: The Language of Science. He wrote:

Man, even in the lower stages of development, possesses a faculty which, for want of a better name, I shall call number sense. This faculty permits him to recognize that something has changed in a small collection when, without his direct knowledge, an object has been removed or added to a collection.

Stanislas Dehaene took the term “number sense” as the title of his recent book. It’s a good book. It contains most of what you will find in this chapter, and a great deal more. I recommend that you read it. It does, however, have one flaw: its subtitle, “How the Mind Creates Mathematics.” Not only does the kind of basic number sense that Dehaene focuses on have little to do with mathematics, but at no point does he even begin to tell us how the mind creates mathematics. Indeed, he barely scratches the surface of how the mind performs arithmetic, which is just one branch of mathematics, and an atypical one at that. This book, in contrast, is very definitely about all of mathematics. My intention is to understand how the human mind acquired the ability to perform mathematical reasoning.

Since mathematical thinking seems to be unique to humans, we may gain some insight into the key factors that led to it by comparing our mathematical ability with that of other species. In particular, to demonstrate that mathematical ability is nothing other than linguistic ability used in a slightly different way, it should be helpful to see what mathematical abilities are possessed by creatures that do not have language.

There is in fact a fairly extensive literature on laboratory studies of number sense in animals. (Dehaene lists some good sources in his book.) One of the first to realize that animals have a sense of number was the German psychologist Otto Koehler. During the 1940s and 1950s, Koehler suggested that two important prerequisites for arithmetic were the ability to compare the sizes of two collections presented simultaneously and the ability to remember numbers of objects presented successively in time. Both of these abilities form part of what I am here calling number sense. Koehler showed that birds have both abilities.

In one case, a raven called Jakob was repeatedly presented with two boxes, one of which contained food. The lids of the boxes had different numbers of spots arranged randomly. A card placed alongside the two boxes bore the same number of spots—although arranged differently—as the lid of the box with the food. Through many repetitions, the raven learned that to obtain the food, it had to open the box whose lid bore the same number of spots as the card. In this way, it was eventually able to distinguish two, three, four, five, and six spots.

In another experiment, Koehler trained jackdaws to open the lids of a row of boxes to obtain food until they had taken a given number of pieces, say four or five. Each box contained one, two, or no pieces of food, distributed randomly on each repetition, so there was no possibility of the birds basing their actions on a geographic feature such as the length of the row of boxes they opened. Rather, they had to keep an inner tally of how many pieces they had taken; in our terms, they had to count.

Another illustration of birds’ numerical abilities comes from Irene Pepperberg, who trained her African Gray parrot Alex to say how many objects it saw on a tray, a task that required that the bird not only distinguish numerosities but also associate an appropriate vocal response with each number.

Many birds also exhibit a sense of numerosity in the number of times they repeat a particular note in their song. Members of the same species born and reared in different regions acquire a local “dialect,” with the number of repetitions of a particular note varying from one location to another. Thus, although many aspects of a bird’s characteristic song may be genetically determined, the number of repetitions of a particular note seems to be acquired by a young bird imitating the older birds around it, most likely its parents. For example, a canary raised in one area may repeat a particular note six times, whereas one raised elsewhere will repeat the same note seven times. Since the number of repetitions is constant for each bird, this means that the bird can “recognize” the number of repetitions in its song.

One obvious survival advantage to being able to compare numbers of objects in collections is that it helps a group of animals to know whether to defend their territory against an attack or to retreat. If the defenders outnumber the attackers, it might make sense to stay and fight; if there are more attackers, the wisest strategy might be to make a bolt for it. This suggestion was put to the test a few years ago by Karen McComb and her colleagues. They played tape recordings of roaring lions to small groups of female lions in the Serengeti National Park in Tanzania. When the number of different roars exceeded the number of lions in the group, the females retreated; but when there were more females, they stood their ground and prepared to attack the intruders. They seemed able to compare number across two different senses: the number of roars they heard versus the number of lionesses they observed, a task that seems to require a fairly abstract number sense.

THE HORSE THAT DIDN’T KNOW 2 + 2 = 4 AND THE RAT THAT DID

Claims that various animals have been shown to possess a number sense are sometimes dismissed by the experts. Much of the blame for this reaction can be laid at the feet of a horse living in Germany at the start of the twentieth century. After more than ten years’ effort, a certain Wilhelm von Osten claimed to have taught arithmetic to his horse, Hans. Both horse and master became celebrities, and the German newspapers carried stories about “Clever Hans.”

A typical demonstration would see von Osten and his horse surrounded by an eager audience. “Ask him what is three plus five,” someone would call out. Von Osten would write the sum on chalkboard and show it to the horse, who would then carefully tap his hoof on the ground exactly eight times. Other times, von Osten would show Hans two piles of objects, say four in one pile and five in the other. Hans would tap his hoof nine times.

Even more impressive, Hans could apparently add fractions. If von Osten wrote the two fractions ½ and ⅓ on the board, Hans would tap his hoof five times, then pause, then tap six times, to give the correct answer ⅚.

Of course, there were suspicions of a trick. In 1904 a committee of experts gathered together to investigate the matter, among them the eminent German psychologist Carl Stumpf. After carefully observing a performance, the committee concluded that it was genuine—Hans really could do arithmetic.

One person, however, was not convinced by the committee’s findings. Stumpf’s student Oskar Pfungst insisted on further testing. This time, Pfungst wrote the questions onto the board himself, and he did so in such a way that von Osten could not see what was written. This enabled him to do something Stumpf had not. On some occasions, Pfungst wrote down the question that had been given to him. Other times, he changed it. Whenever Pfungst wrote down the question as given to him, Hans got it right. But when he changed the question, Hans gave the wrong answer —in fact, he answered the question von Osten thought had been given to the horse.

The conclusion was inescapable: von Osten had been doing the arithmetic. Through some subtle cue, perhaps a raised eyebrow or a slight shrug, he had been instructing Hans when to stop tapping his hoof. As Pfungst acknowledged, von Osten could well have been oblivious to this. Having worked so hard to train him, von Osten very much wanted his four-legged protégé to succeed. Doubtless, he became very tense as Hans’s tapping got to the crucial number, and presumably Hans was able to detect some external manifestation of that tension. Thus, while Pfungst’s investigation showed that Hans’s performance did not require unusual arithmetical powers, it did show that humans could communicate with horses by means of the subtlest actions.

The case of Clever Hans showed the importance of proper design for any psychological experiment, to eliminate any possibility of subtle communication of clues. Unfortunately, the affair made subsequent claims of arithmetical abilities in animals extremely difficult to get taken seriously. And yet nothing Pfungst did showed that animals could not have number sense. He simply showed that, in Hans’s case, it was von Osten who had performed the calculations, not the horse.

In fact, a number of carefully conducted studies have shown that some animals can indeed perform some kind of arithmetic. One convincing series of demonstrations was carried out in the 1950s and 1960s by the American animal psychologists Francis Mechner and Laurence Guevrekian.

The idea was to deprive a rat of food for a short period and then put it into a closed box with two levers, A and B. Lever B was connected to a mechanism that delivered a small amount of food. But to activate lever B, the rat first had to depress lever A a fixed number (n) of times. If the rat depressed lever A fewer than n times and then pressed lever B, it received a mild electric shock and no food. Thus, to eat, the rat had to learn to press lever A n times and then press lever B.

At first, by trial and error, the rats discovered that, in order to get food, they had to press lever A a certain number of times and then press B. With repeated trials, they gradually learned to estimate the number of times they had to press lever A. If the apparatus was set up so that four presses of lever A were required to activate lever B, then, over time, the rats learned to press lever A about four times before pressing lever B.

The rats never learned to press lever A exactly four times on every occasion. In fact, they tended to overestimate, pressing it four, five, even six times. Given that they received an unpleasant shock if they pressed lever A fewer than four times, this “play safe” strategy makes sense. In any event, it did seem that the rats were able to estimate four presses. Likewise, in an apparatus set up so that lever A had to be pressed eight times, they learned to press it about eight times. In fact, they could learn to press lever A as many as sixteen times.

To avoid any possibility that the rats were judging time rather than the number of presses, the experimenters subsequently varied the degree of food deprivation. The more hungry the rats were, the more rapidly they pressed the lever. Nevertheless, rats trained to press level A four times continued to do so, and there was a similar effect with the rats trained to another number. Time was not the factor; they were estimating the number.

Notice that I have not said that the rats counted. What the experiment showed is that, through training, rats are able to adjust their behavior to press a lever about a certain number of times. They may have been counting, albeit badly. But there is no evidence for this. I think it’s far more probable that they were simply judging or estimating the number of presses, and moreover doing so as well as we ourselves could if we did not count.

Other experiments performed on rats—some of which are described by Dehaene—point to the same conclusion: rats have a sense of number.

What evolutionary advantage led to selection for a number sense in rats? One possibility is the need to remember navigational information, such as its hole was the fourth one along after the third tree. It’s also useful in keeping track of other animals in the vicinity, be they friends or predators.

As it happens, an experiment was performed that could be said to have addressed this very issue, but as things turned out, the result was not at all what was expected.

The experimenters presented their subjects with a corridor of doors, each of which led to food, and all but one of which were locked. For example, in a row of ten doors, only door number 7 might be unlocked. The experimenters wanted to see if the rats would learn to ignore the first six doors and go straight to door 7. The experiment seemed to be a huge success. After a number of initial learning cycles in which they tried one door after another, the rats seemed to figure out what was going on. Soon, each rat would race along the corridor at top speed until it reached door 7, and then push open the door to get the food.

When the experimenters watched a videotape of the rats in slow motion, however, they discovered that the rats were not counting at all. As each rat dashed along the corridor, it gave each door a light push with its hind leg until it encountered one that yielded. Whereupon it screeched to a halt and shot into the room. This did not show that rats cannot count, nor that they could not have used a counting strategy. But that is not what they did. The lesson the experimenters learned was to be careful in ascribing an explanation to what has been observed. Things may not always be as they first appear.

WHAT ABOUT THE CHIMPANZEES?

So much for rats. What about chimpanzees? Given their similarity to humans, we might expect them to exhibit the best-developed number sense. Do chimpanzees in fact have any arithmetical ability? Guy Woodruff and David Premack of the University of Pennsylvania set out to investigate this in the late 1970s and early 1980s.

Woodruff and Premack started out by aiming high. In their first experiment, the two investigators showed that chimpanzees can understand fractions. For instance, they showed the chimp a glass half-filled with a colored liquid and then got the animal to choose between two further glasses, one half-filled, the other three-quarters full. The subjects had no difficulty mastering this task. But was the chimp basing its choice on the volume of the water in the glass or on the fraction by which it was full? The answer was obtained by making the task more abstract. This time, after the chimp was shown a half-full glass of liquid, for example, it would be presented with half an apple adjacent to three-quarters of an apple. The chimp consistently picked the half-apple over the three-quarter apple. The same thing happened when the chimp was shown half a pie against one-quarter of a pie. In fact, when presented with any choice between one-quarter, onehalf, and three-quarters, the chimp was able to spot the correct fraction. It knew, for instance, that one-quarter of a glass of milk is the same fraction of a whole glass that one-quarter of a pie is of a complete pie.

Going a step further, Woodruff and Premack first presented two initial stimuli, say a quarter of an apple together with half a glass of liquid, and then asked the chimp to choose between a full pie and three-quarters of a pie. The chimp tended to choose the three-quarters pie—not always, but too often for its choices to be random. Apparently the chimp could perform the computation ¼ + ½ = ¾, at least in an approximate, intuitive fashion.

Many other experiments show that chimps possess an ability in basic arithmetic. For example, a chimp is presented with two alternative choices of a treat. On one tray are placed two piles of chocolates, one with three chocolates, the other with four. The alternative tray has one pile of five chocolates together with one additional chocolate by itself. The chimp can only choose one tray. Which one does it choose? If it bases its choice on the largest pile it sees, it should pick the tray with the pile of five chocolates. But if it can add up the total number of chocolates on each tray, it will realize that the first tray has seven chocolates whereas the other has only six. Most of the time, without any special training, the chimp will select the tray with seven chocolates. The chimp can estimate that 3 + 4 = 7 and 5 + 1 = 6, and moreover can tell that 6 is less than 7.

The numerical approximating ability shown by rats and chimpanzees resembles the innate numerical estimating ability in humans. But humans can do more. We can count precisely and perform exact arithmetic. One of the keys to these skills is that we can use symbols to denote numbers. Arithmetic can then be performed in an essentially linguistic fashion, by manipulating symbols according to precise rules. Can chimpanzees be taught symbolic notation?

The answer is yes—up to a point. In the 1980s, Tetsuro Matsuzawa, a Japanese researcher, taught a chimpanzee named Ali how to use correctly the Arabic numerals 1 to 9. Ali was able to use these numerals to give the number of objects in a collection with up to 95 percent accuracy. Based on his response times, it appears that Ali could recognize at a glance the number of objects when they were three or fewer, but resorted to counting for larger collections. Ali could also order the numerals according to their magnitude.

A number of subsequent investigations have produced similar results. In one of the most impressive, Sarah Boyson provides her chimp Sheba with a collection of cards, on each of which is printed a single digit between 1 and 9. Sheba can correctly match each of the printed digits with a collection of between one and nine objects presented to her. Sheba can also do simple addition using symbols. For instance, if Boyson holds up the numerals 2 and 3, Sheba can successfully pick out the card bearing the numeral 5. More recently, in 1998, Elizabeth Brannon and Herbert Terrace have demonstrated a similar ability of rhesus monkeys to learn to distinguish the numbers 1 to 9 and to associate symbols with those numbers.

I should stress that it took many years of slow and painstaking training to achieve the kind of performance exhibited by Sheba and the various other chimpanzees, monkeys, dolphins, and what-have-you in such experiments. Teaching animals the link between the abstract symbols and collections of objects is a long and arduous process. Performance is never perfect and is limited to very small collections. Young children, by contrast, take just a few months to catch on to numbers. And once they do, they do so in a big—and accurate—way. When it comes to numbers, humans are very different from all other animals, and the difference shows up at a very early age.

THE RISE AND FALL OF PIAGET

Much of our current popular wisdom about small children’s mental abilities originates in the work of the cognitive psychologist Jean Piaget fifty years ago. Piaget’s influence can be found not only in many of our current beliefs about the way children learn, but also in our educational systems. Unfortunately, as often happens with ground-breaking research, subsequent investigations have shown that many of Piaget’s conclusions were almost certainly wrong. (I say “almost certainly” because some psychologists still maintain that Piaget was right, and that the experimental results I shall describe admit alternative conclusions.)

In the 1940s and 1950s, Piaget developed a “constructivist” view of child development. According to this view, a newborn baby enters the world with a cognitive clean slate and, by observing the world around it, gradually pieces together a coherent and steadily increasing understanding of that world. In other words, the child constructs a mental model or conceptualization of the world.

Piaget did not arrive at his conclusions by armchair speculation. He was an experimentalist, and his experiments are one reason why his work was so influential. It took great ingenuity—and equipment not available in Piaget’s time—for subsequent generations to devise more reliable experiments. When they did so, they reached very different conclusions.

For example, according to Piaget, children younger than ten months old have no proper sense of physical objects as things that endure in the world. Piaget based this conclusion on his observation that, when an object such as a toy is hidden under a cloth, a baby ten months old or younger will fail to reach for it. According to Piaget, “object permanency,” as he called it, is not innate but is acquired sometime after ten months of age.

Similarly, Piaget believed that children do not have a number sense until they acquire it at around four or five years of age. In one of Piaget’s experiments, repeated many times by different groups, a psychologist would show a four-year-old child two equally spaced rows of six glasses and six bottles and ask whether there were more glasses or more bottles. The child invariably answered that there were the same number. Presumably the child observed a one-to-one correspondence between the rows. The experimenter then spread out the glasses to form a longer row and asked the child again whether there were more glasses or more bottles. Now the child would answer that there were more glasses, apparently misled by the longer length of that row. “Obviously,” Piaget concluded, “this shows that the child does not have a properly developed number sense.” In particular, Piaget claimed, four- and five-year-old children have not yet grasped the idea of number conservation—the notion that rearranging the objects in a collection does not change their number.

At the time, Piaget’s experiments were held up as triumphs of experimental science in psychology. As a pioneer, Piaget was blazing a trail for future generations. And that is good science. Unfortunately, his methods had serious flaws. He relied on the motor actions of the babies in the object permanency test and on a dialogue between the experimenter and the subject for the various number tests performed on older children.

In the case of object permanency, a baby’s failure to reach for an object hidden under a blanket does not support the rather dramatic conclusion that the baby thinks the object has ceased to exist. Perhaps he simply does not yet have sufficient hand-arm coordination to reach for a hidden object. In fact, we now know that this explanation is correct. Recent experiments, more sophisticated than Piaget’s, indicate that even very young babies have a well-developed sense of object permanency.

Likewise, dialogue with a small child is highly unreliable. Communication via language is never 100 percent objective and free of the influences of context, emotion, social factors, and possibly several other things. Just how unreliable dialogue can be was demonstrated by Jacques Mehler and Tom Bever at MIT during the late 1960s.

In one experiment, Mehler and Bever carried out the original Piaget experiment to test for number conservation, but with two- and three-year-old children instead of Piaget’s four- and five-year-olds. The children succeeded perfectly. Consequently, unless we believe that children temporarily lose their sense of number conservation between the ages of four and six, we clearly need some alternative explanation for Piaget’s results. One is readily available.

Around five years of age, children begin to develop the ability to reason about another person’s thought process (“What Daddy means by this is ... ”). This provides the most likely explanation of Piaget’s observations. Remember the way the experiment was performed. First the experimenter arranges the glasses and bottles in two equally spaced rows and asks the child which row has more objects. Then the experimenter rearranges one of the rows, making it longer, and again asks the child, “Which row has more objects?”

Now, by four or five years of age, a young child knows that adults are powerful and are knowledgeable. Moreover, she has probably observed the respect her parent showed the experimenter when they arrived at the laboratory. How is this child likely to react when she sees the experimenter rearrange the objects in one of the two rows and then ask the very same question as a moment earlier, “Which row has more objects?” She may well reason, “Hmm. That’s the same question she just asked me. Adults are not dumb, and this is a special kind of adult who knows a lot. We can both see that the number of objects hasn’t changed. So I must have misunderstood the question the last time. I thought she was asking me about the number of objects in the row, but obviously she was really asking me about the length, since that’s what she just changed.” And so the child gives the answer she thinks is expected of her.

Of course, we can’t know for sure. Attempts to find out by interrogating the child are unlikely to yield conclusive evidence, for the same reason that the original Piaget experiment is suspect! This is where the Mehler and Bever experiment came into its own. The kind of “what-doesshe-really-want?” reasoning just described is beyond two- or three-year-olds. Mehler and Bever’s younger subjects took the experimenters’ questions literally, and counted correctly.

What Piaget’s original experiment really showed is that four- and five-year-old children can reason rationally about the motivations and expectations of another person. That’s an important and useful discovery. But it’s not the one Piaget thought he had made!

To confirm that children from age two upward have a good sense of number, Mehler and Bever redesigned the Piaget test to avoid the reliance on language. Their idea was breathtakingly simple. Instead of glasses and bottles they presented the child with two rows of M&Ms. One row contained six M&Ms, the other had four. Sometimes the rows were the same length; sometimes the row of six M&Ms was longer; other times the row of four M&Ms was longer. Instead of being asked to indicate which row had more candies, the child was simply told he could pick one row and eat them. The outcome was precisely what any parent would predict. The child invariably plumped for the row of six candies, regardless of its length. He knew full well which row had more members, and moreover realized that the number was not dependent on the arrangement. The result was just as conclusive with two-year-old children as with four-year-olds.

Another ingenious variation of the original Piaget experiment reached the same conclusion. This time, James McGarrigle and Margaret Donaldson of the University of Edinburgh carried out their experiment in a small puppet theater. Like Piaget, they started by aligning two rows of the same number of objects and asking the child which row had more objects. After the child responded correctly, the experimenter pretended to look away while a teddy bear puppet lengthened one of the rows. Turning back, the experimenter exclaimed, “Oh dear, that silly teddy has mixed up the rows. Can you tell me which row has more objects again?” Children from two to five invariably gave the correct answer. Since the teddy bear had rearranged one of the rows, unseen by the experimenter, the child presumably found it reasonable for the adult to ask the same question again. Yet when the experimenter repeated the process with the same children but rearranged the objects him- or herself, the four- and five-year-old children responded exactly as they had for Piaget, basing their answer on length.

CAN BABIES COUNT?

Many studies have shown that Piaget’s conclusions about number sense were wrong. Children as young as two have a well-developed sense of number and of number conservation. What about the underlying constructivist hypothesis, that such numerical abilities are acquired through observation? Perhaps the child learns such abilities in its first two years.

A number of remarkable experiments carried out during the 1980s and 1990s show that’s not true either. The first of these experiments was performed in 1980 by Prentice Starkey and his colleagues at the University of Pennsylvania. Starkey’s subjects were babies aged between sixteen and thirty weeks. The challenge was to find a way to detect what such young subjects were thinking.

The experimenters relied on measuring the baby’s attention span. As any parent knows, any novelty, such as a new toy, will capture a young child’s attention for a while. Then that attention starts to wane, or some other novelty takes over. Using video technology, Starkey and his colleagues were able to track the child’s attention by recording its gaze. The baby was generally held by its mother on her lap, facing the experimental apparatus.

In one experiment, the baby was shown slides projected onto a screen. A slide was displayed showing two dots side by side. When this first appeared, it caught the baby’s attention, and the baby gazed at it for some time. When the baby’s attention began to fade and its eyes started to wander, the slide was replaced with a new one that differed slightly from the first. The baby glanced back briefly. The slide was changed again. Each new slide was a slight variation of the previous one. With each repetition, the baby’s rearoused gaze—as timed from the videotape—became more and more brief. Then, without warning, the slide showed three dots, not two. Immediately, the baby’s interest was aroused, and it gazed longer at the picture (from 1.9 seconds to 2.5 seconds in one run of the experiment). The child clearly detected the change from two dots to three. Another group of babies shown the slides in reverse order noticed the change from three dots to two.

Subsequently, Sue Ellen Antell and Daniel Keating of the University of Maryland used the same method to show that babies just a few days after birth can also discriminate between two and three.

Incidentally, by showing the dots in different arrangements or replacing them with pictures of objects in different configurations, both groups of experimenters eliminated any possibility that some feature other than number was catching the babies’ attention. Regardless of what objects were shown or their configuration, the change in number was what caught the babies’ interest. The conclusion seems definite: babies as young as a few days old can distinguish between collections of two and three objects.

In another experiment, performed by Ranka Bijeljac and her colleagues at the Laboratory for Cognitive Science and Psycholinguistics in Paris, babies just four days old were subjected to auditory stimuli instead of visual ones. Since the stimuli were not visual, the babies’ attention could not be measured by timing their gaze. Instead, the experimenters made use of the babies’ sucking reflexes to measure their attention. Each baby was given a nipple, connected to a pressure transducer and thence to a computer. When the baby sucked on the nipple, the computer delivered a nonsense word of a fixed number of syllables. Typical three-syllable words might be “akiba” or “bugaloo.”

At first, each baby showed great interest in the fact that its sucking produced sounds, and it sucked vigorously. After a while, however, interest waned and the sucking rate dropped. The computer, detecting this drop, at once shifted to delivering words with only two syllables. The baby at once started to suck vigorously again.

Changes from one three-syllable word to another or from one twosyllable word to another did not produce the same increased interest in the baby. Nor did changes in the speed of delivery of each word. It had to be the number of syllables that generated the response.

The Paris experiment shows more than that four-day-old babies can distinguish two from three. It also shows that, even at that age, babies can recognize a syllabic structure in a stream of sound and, in an innate and subconscious way, can detect differences in the number of syllables.

Given that even very young babies can discriminate between two objects and three, does this number sense extend to a sense of (abstract) twoness and threeness? To test this, one would have to show that a baby can see a similarity between collections of, say, two apples and two dots, or three marbles and three bell rings. But how can this be done with a young baby that has no power of speech? Remarkably, ten years ago Starkey and his colleagues found a way to perform just such a test.

The researchers sat babies aged six to eight months in front of two slide projectors. The projector on the left showed a picture of three objects, randomly arranged; the projector on the right showed a picture of two objects, also arranged randomly. At the same time, a loudspeaker situated between the two screens played a sequence of drumbeats. A hidden video camera recorded the baby’s eyes to measure its attention.

Initially, each baby examined both pictures, spending a little more time on the picture with three objects, because it was more complex. After the first few trials, however, a remarkable pattern emerged. The baby spent more time looking at the picture whose number matched that of the sequence of drumbeats it heard. When two drumbeats were played, the baby spent more time looking at the picture with two objects, but when three drumbeats were played, it paid more attention to the picture showing three objects.

Starkey and his fellow researchers did not suggest that their subjects were at all conscious of numerosity. The babies’ behavior was probably an in-built neuronal response, in which activation of a certain pattern of neuronal firing caused by hearing two sounds caused preferred receptivity to scenes showing two objects. That does not amount to a sophisticated conception of number. But it is most definitely a number concept. And it tells us that however we may rate our mathematical ability, we all have a built-in number sense. We were born with it.

Perhaps we were also born with the ability to add—at least to the limited extent that chimpanzees can add.

BUT CAN THEY ADD?

Can babies add? Does a baby know that 1 + 1 = 2 or that 2 + 1 = 3 In 1972 the American psychologist Karen Wynn stunned the world when she announced her findings that babies as young as four months were capable of performing (in an innate, non-conscious fashion) simple additions and subtractions. As with all experiments performed on babies, the main challenge was to find a way to discover what was going on in the mind of so young a subject.

Wynn made use of babies’ sense of “the way the world is.” Even very young babies are troubled when they encounter something that runs counter to the laws of physics. For example, an object apparently suspended in mid-air with nothing visible to support it will elicit an intent stare.

Wynn sat her young subjects in front of a small puppet theater. The experimenter’s hand came out from one side and placed a Mickey Mouse puppet onto the stage. Then a screen came up, hiding the puppet. The experimenter’s hand appeared again, holding a second Mickey Mouse, which it placed behind the screen. Then the screen was lowered. Sometimes the baby would see two Mickey Mouses on the stage. On other occasions, the lowering of the screen would reveal just one Mickey Mouse, the other having been removed unseen through a trapdoor. By videotaping the baby’s responses, Wynn was able to measure the time the baby gazed at the stage. On average, when the lowering of the screen revealed one puppet rather than two, the babies looked for a full second longer.

The most obvious conclusion is that, having seen two puppets placed on the stage, one after the other, and none removed, they expected to see two puppets at the end. They “knew” that 1 + 1 = 2 and were surprised to see a manifestation of the incorrect rule 1 + 1 = 1.

The subjects’ attention times were also longer when the lowering of the screen revealed three puppets. Apparently, they “knew” that 1 + 1 was equal to neither 1 nor 3, but exactly 2.

The babies also seemed to know subtraction. To test this, the demonstration started with two puppets on the stage, then the screen went up, and then the baby saw one puppet removed. When the lowering of the screen revealed two puppets (illustrating 2—1 = 2), the babies gazed at the stage by as much as three seconds longer than when only one was left (illustrating 2—1 = 1).

To eliminate the possibility that the babies in Wynn’s experiment were relying on a visual memory rather than a numerical one, the French psychologist Etienne Koechlin repeated the procedure, but placed the objects on a slowly revolving turntable. The constant movement on the stage prevented the babies from forming a fixed mental image of what they saw; they could not predict what the scene would look like when the screen was lowered. Nevertheless, they showed far greater attention when the number of objects on the turntable was not what it should have been.

Interestingly, of all the features of the physical world that babies are born with, number seems to rank among the most significant—far more so than physical form or appearance. This was demonstrated by the American psychologist Tony Simon and his colleagues in the early 1990s, using yet another variation of Wynn’s experiment. If a baby sees two puppets disappear behind the screen, it shows no surprise when the screen is lowered to reveal, say, two red balls; but it appears troubled when just one ball is revealed. Apparently the idea that one object can transform into another is less troubling than is a change in number. This provides a new and totally unexpected twist to Piaget’s concept of object conservation.

Further confirmation of this odd view of object conservation is provided by the outcome of another experiment, in which a baby less than twelve months old is placed in front of a screen from behind which, say, a red ball and a blue rattle alternately appear. Provided the baby does not see the two objects at the same time, it will not show surprise to see just one object (either a ball or a rattle) when the screen is lowered. It appears happy to accept that there is a single object that sometimes looks like a red ball and at other times like a blue rattle. Only when it is a year old or more do the two different appearances begin to imply the existence of two objects. In a baby’s first year of life, number is apparently a more important “invariable” than color, shape, or appearance.

Incidentally, these remarkable arithmetical abilities of young babies are strictly limited to simple additions and subtractions involving the numbers 1, 2, and 3. Babies younger than one year seem unable to distinguish four objects from five or six.

In the next chapter, we shall turn to the uniquely adult-human numerical world beyond 3. But before then, spare a thought for the very small number of individuals who have to live in a number-dependent society without having any sense of what numbers are.

BUTTERWORTH’S CASEBOOK

The British psychologist Brian Butterworth, who studies brain abnormalities, has collected a number of fascinating cases of individuals who have lost or never had any sense of number. He describes some of them in his 1999 book The Mathematical Brain (published in the United States as What Counts).

For instance, one of Butterworth’s examples, Signora Gaddi, suffered a stroke that left both her language and reasoning abilities intact, but completely destroyed her numerical capacity. She could not determine or estimate the number of objects in any collection. Moreover, she could only recite the number words up to four and, as a result, could only count the members of a collection of four or fewer objects.

Or consider Frau Huber, who had an operation to remove a tumor in her left parietal lobe. After the surgery, her general intelligence and language ability seemed just fine, but numbers literally had no meaning for her. She could not even be taught finger addition. She could recite the multiplication table, but it was just a “nonsense poem” to her. Although she was able to learn new arithmetical facts verbally, they had no meaning for her and she could not make any use of them. She was unable to work out any arithmetical fact.

Prior to her surgery, Frau Huber did have a number sense; but some individuals are born without it and are never able to acquire it. Another of Butterworth’s examples, whom he refers to as Charles, is a highly intelligent young man who earned a degree in psychology. But Charles has virtually no sense of numbers. Faced with simple arithmetic, his only recourse is to use his fingers. To perform any kind of calculation, he needs a calculator. The answer he gets means nothing to him. He is unable to tell which of two numbers is the larger. Forced to make such a comparison, he has to resort to counting to see which number he reaches first. If Charles is shown a collection of, say, three objects, he is unable to say how many there are, but must count them. When asked to add or subtract two numbers, he does so by counting on, but his performance is slow and erratic. In one test, he took eight seconds to add 8 and 6, and twelve seconds to subtract 2 from 6. He failed on 7 + 5 and on 9 + 4. Not surprisingly, it took Charles longer than usual to earn his degree.

Another highly intelligent individual having virtually no number sense is Julia. When Butterworth first examined her, she had completed a first university degree and was embarking on a postgraduate program. She could perform simple arithmetic only by using her fingers and was helpless when faced with numbers beyond the range of her hands. She had no sense of fractions or of the basic arithmetical properties—for example, she did not know that 3 × (2 + 5) is the same as (3 × 2) + (3 x 5). She was able to count 1, 2, 3, etc., but could not count in threes except by counting in ones and stressing every third number: one, two, three, four, five, six, seven,... Unlike Charles, however, she was able to compare two numbers to see which was the larger.

People like Charles and Julia suggest that number sense is not something that can be learned. Despite having the intelligence to earn college degrees, they were unable to acquire a sense of number. They could learn facts about numbers, but those facts had no meaning for them.