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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.

     
    In Karen Wynn’s experiment, babies were tested on their ability to distinguish the correct number of dolls behind a screen.
     
    The Swiss psychologist Jean Piaget (1896–1980) argued that babies build up an understanding of numbers slowly, through experience, so there was no point in teaching aithmetic to children younger than six or seven. This influenced generations of educators, who often preferred to let primary-age pupils play around with blocks in lessons rather than introduce them to formal mathematics. Now Piaget’s views are considered outdated. Pupils come face to face with Arabic numerals and sums as soon as they get to school.
     
     
    Dot experiments are also the cornerstone of research into adult numerical cognition. A classic experiment is to show a person dots on a screen and ask how many dots he or she sees. When there are one, two or three dots, the response comes almost instantly. When there are four dots, the response is significantly slower, and with five slower still.
    So what! you might say. Well, this probably explains why in several cultures the numerals for 1, 2 and 3 have been one, two and three lines, while the number for 4 is not four lines. When there are three lines or fewer we can tell the number of lines straight away, but when there are four of them our brain has to work too hard and a different symbol is necessary. The Chinese characters for one to four are,,and. Ancient Indian numerals were,,and. (If you join the lines, you can see how they