Alex’s Adventures in Numberland (3 page)

The lesson of Clever Hans was that when teaching animals to count, supreme care must be taken to eliminate involuntary human prompting. For the maths education of Ai, a chimpanzee brought to Japan from West Africa in the late 1970s, the chances of human cues were eliminated because she learned using a touch-screen computer.

Ai is now 31 and lives at the Primate Research Institute in Inuyama, a small tourist town in central Japan. Her forehead is high and balding, the hair on her chin is white and she has the dark sunken eyes of ape middle age. She is known there as a ‘student’, never a ‘research subject’. Every day Ai attends classes where she is given tasks. She turns up at 9 a.m. on the dot after spending the night outdoors with a group of other chimps on a giant tree-like construction of wood, metal and rope. On the day I saw her she sat with her head close to a computer, tapping sequences of digits on the screen when they appeared. When she completed a task correctly an 8mm cube of apple whizzed down a tube to her right. Ai caught it in her hand and scoffed it instantly. Her mindless gaze, the nonchalant tapping of a flashing, beeping computer and the mundanity of continual reward reminded me of an old lady doing the slots.

When Ai was a child she became a great ape in both senses of the word by becoming the first non-human to count with Arabic numerals. (These are the symbols 1, 2, 3 and so on, that are used in almost all countries except, ironically, in parts of the Arab world.) In order for her to do this satisfactorily, Tetsuro Matsuzawa, director of the Primate Research Institute, needed to teach her the two elements that comprise human understanding of number: quantity and order.

Numbers express an amount, and they also express a position. These two concepts are linked, but different. For example, when I refer to ‘five carrots’ I mean that the quantity of carrots in the group is five. Mathematicians call this aspect of number ‘cardinality’. On the other hand, when I count from 1 to 20 I am using the convenient feature that numbers can be ordered in succession. I am not referring to 20 objects, I am simply reciting a sequence. Mathematicians call this aspect of number ‘ordinality’. At school we are taught notions of cardinality and ordinality together and we slip effortlessly between them. To chimpanzees, however, the interconnection is not obvious at all.

Matsuzawa first taught Ai that one red pencil refers to the symbol ‘1’ and two red pencils to ‘2’. After 1 and 2, she learned 3 and then all the other digits up to 9. When shown, say, the number 5 she could tap a square with five objects, and when shown a square with five objects she could tap the digit 5. Her education was reward-driven: whenever she got a computer task correct, a tube by the computer dispensd a piece of food.

Once Ai had mastered the cardinality of the digits from 1 to 9, Matsuzawa introduced tasks to teach her how they were ordered. His tests flashed digits up on the screen and Ai had to tap them in ascending order. If the screen showed 4 and 2, she had to touch 2 and then 4 to win her cube of apple. She grasped this pretty quickly. Ai’s competence in both the cardinality and ordinality tasks meant that Matsuzawa could reasonably say that his student had learned to count. The achievement made her a national hero in Japan and a global icon for her species.

Matsuzawa then introduced the concept of zero. Ai picked up the cardinality of the symbol 0 easily. Whenever a square appeared on the screen with nothing in it, she would tap the digit. Then Matsuzawa wanted to see if she was able to infer an understanding of the ordinality of zero. Ai was shown a random sequence of screens with two digits, just like when she was learning the ordinality of 1 to 9, although now sometimes one of the digits was a 0. Where did she think zero’s place was in the ordering of numbers?

In the first session Ai placed 0 between 6 and 7. Matsuzawa calculated this by averaging out which numbers she thought 0 came after and which numbers she thought it came before. In subsequent sessions Ai’s positioning of 0 went under 6, then under 5, 4 and after a few hundred trials 0 was down to around 1. She remained confused, however, if 0 was more or less than 1. Even though Ai had learned to manipulate numbers perfectly well, she lacked the depth of human numerical understanding.

A habit she did learn, however, was showmanship. She is now a total pro, tending to perform better at her computer tasks in front of visitors, especially camera crews.

Investigating animals’ mastery of numbers is an active academic pursuit. Experiments have revealed an unexpected capacity for ‘quantity discrimination’ in animals as varied as salamanders, rats and dolphins. Even though horses may still be incapable of calculating square roots, scientists now believe that the numerical capacities of animals are much more sophisticated than previously thought. All creatures seem to be born with brains that have a predisposition for maths.

After all, numerical competence is crucial to survival in the wild. A chimpanzee is less likely to go hungry if he can look up a tree and quantify the amount of ripe fruit he will have for his lunch. Karen McComb at the University of Sussex monitored a pride of lions in the Serengeti in order to show that lions use a sense of number when deciding whether to attack other lions. In one experiment a solitary lioness was walking back to the pride at dusk. McComb had installed a loudspeaker hidden in the bushes and played a recording of a single roar. The lioness heard it and continued walking home. In a second experiment five lionesses were together. McComb played the roars of three lionesses through her hidden loudspeaker. The group of five heard the roars of three and peered in the direction of the noise. One lioness started to roar and soon all five were charging into the bushes to attack.

McComb’s conclusion was that the lionesses were comparing quantities in their heads. One vs one meant it was too risky to attack, but with five-to-three advantage, the attack was on.

Not all animal number research is as glamorous as camping in the Serengeti or bonding with a celebrity chimpanzee. At the University of Ulm, in Germany, academics put some Saharan desert ants at the end of a tunnel and sent them down it foragng for food. Once they reached the food, however, some of the ants had the bottom of their legs clipped off and other ants were given stilts made from pig bristles. (Apparently this is not as cruel as it sounds, since the legs of desert ants are routinely frazzled off in the Saharan sun.) The ants with shorter legs undershot the journey home, while the ones with longer legs overshot it, suggesting that instead of using their eyes, the ants judged distance with an internal pedometer. Ants’ great skill in wandering for hours and then always navigating their way back to the nest may just be due to a proficiency at counting strides.

Research into the numerical competence of animals has taken some unexpected turns. Chimpanzees may have limits to their mathematical proficiency, yet, while studying this, Matsuzawa discovered that they have other cognitive abilities that are vastly superior to ours.

In 2000 Ai gave birth to a son, Ayumu. On the day I visited the Primate Research Institute, Ayumu was in class right next to his mum. He is smaller, with pinker skin on his face and hands and blacker hair. Ayumu was sitting in front of his own computer, tapping away at the screen when numbers flashed up and avidly scoffing the apple cubes when he won them. He is a self-confident lad, living up to his privileged status as son and heir of the dominant female in the group.

In this task Ayumu is flashed the digits 1 to 7, which then become white squares. He must remember the positions of the numbers so that he can then tap the squares in order to win the food reward.

Ayumu was never taught how to use the touch-screen displays, although as a baby he would sit by his mother as she attended class every day. One day Matsuzawa opened the classroom door only halfway, just enough for Ayumu to come in but too narrow for Ai to join him. Ayumu went straight up to the computer monitor. The staff watched him eagerly to see what he had learned. He pressed the screen to start, and the digits 1 and 2 appeared. This was a simple ordering task. Ayumu clicked on 2. Wrong. He kept on pressing 2. Wrong again. Then he tried to press 1 and 2 at the same time. Wrong. Eventually he got it right: he pressed 1, then 2 and an apple cube shot down into his palm. Before long, Ayumu was better at all the computer tasks than his mum.

A couple of years ago Matsuzawa introduced a new type of number task. On pressing the start button, the numbers 1 to 5 were displayed in a random pattern on the screen. After 0.65 seconds the numbers turned into small white squares. The task was to tap the white squares in the correct order, remembering which square had been which number.

Ayumu completed this task correctly about 80 percent of the time, which was about the same amount as a sample group of Japanese children. Matsuzawa then reduced the time that the numbers were visible, to 0.43 seconds, and while Ayumu barely noticed the difference, the children’s performances dropped significantly, to a success rate of about 60 percent. When Matsuzawa reduced the time that the numbers were visible again – to only 0.21 seconds, Ayumu was still registering 80 percent, but the kids dropped to 40.

This experiment revealed that Ayumu had an extraordinary photographic memory, as do the other chimps in Inuyama, although none is as good as he is. Matsuzawa has increased the number of digits in further experiments and now Ayumu can remember the positioning of eight digits made visible for only 0.21 seconds. Matsuzawao reduced the time interval and Ayumu can now remember the positioning of five digits visible for only 0.09 seconds – which is barely enough time for a human to register the numbers, let alone remember them. This astonishing talent for instant memorization may well be because making snap decisions, for example, about numbers of foes, is vital in the wild.

Studies into the limits of animals’ numerical capabilities bring us naturally to the question of innate human abilities. Scientists wanting to investigate minds as uncontaminated as possible by acquired knowledge require subjects who are as young as possible. As a result, infants only a few months old are now routinely tested on their maths skills. Since at this age babies cannot talk or properly control their limbs, testing them for signs of numerical prowess relies on their eyes. The theory is that they will stare for longer at pictures they find interesting. In 1980 Prentice Starkey at the University of Pennsylvania showed babies between 16 and 30 weeks old a screen with two dots, and then showed another screen with two dots. The babies looked at the second screen for 1.9 seconds. But when Starkey repeated the test, showing a screen with three dots after the screen with two dots, the babies looked at it for 2.5 seconds: almost a third longer. Starkey argued that this extra stare-time meant the babies had noticed something different about three dots compared to two dots, and therefore had a rudimentary understanding of number. This method of judging numerical cognition through the length of attention span is now standard. Elizabeth Spelke at Harvard showed in 2000 that six-month-old babies can tell the difference between 8 and 16 dots, and in 2005 that they can distinguish between 16 and 32.

A related experiment showed that babies had a grasp of arithmetic. In 1992, Karen Wynn, at the University of Arizona, sat a five-month-old baby in front of a small stage. An adult placed a Mickey Mouse doll on the stage and then put up a screen to hide it. The adult then placed a second Mickey Mouse doll behind the screen, and the screen was then pulled away to reveal two dolls. Wynn then repeated the experiment, this time with the screen pulling away to reveal a wrong number of dolls: just one doll or three of them. When there were one or three dolls, the baby stared at the stage for longer than when the answer was two, indicating that the infant was surprised when the arithmetic was wrong. Babies understood, argued Wynn, that one doll plus one doll equals two dolls.

The Mickey experiment was later performed with the
Sesame Street
puppets Elmo and Ernie. Elmo was placed on the stage. The screen came down. Then another Elmo was placed behind the screen. The screen was taken away. Sometimes two Elmos were revealed, sometimes an Elmo and an Ernie together and sometimes only one Elmo or only one Ernie. The babies stared for longer when just one puppet was revealed, rather than when two of the
wrong
puppets were revealed. In other words, the arithmetical impossibility of 1 + 1 = 1 was much more disturbing than the metamorphosis of Elmos into Ernies. Babies’ knowledge of mathematical laws seems much more deeply rooted than their knowledge of physical ones.