Alex's Adventures in Numberland - Alex Bellos [11]
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 Now Piaget’s views are considered outdated.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 turned into the modern 1, 2, 3 and 4.)
In fact, there is some debate about whether the limit of the number of lines we can grasp instantly is three or four. The Romans actually had the alternatives IIII and IV for four. The IV is much more instantly recognizable, but clock faces – perhaps for aesthetic reasons – tended to use the IIII. Certainly, the number of lines, dots, or sabre-toothed tigers that we can enumerate rapidly, confidently and accurately is no more than four. While we have an exact sense of 1, 2 and 3, beyond 4 our exact sense wanes and our judgements about numbers become approximate. For example, try to guess quickly how many dots are at the top of the page opposite.
It’s impossible. (Unless you are an autistic savant, like the character played by Dustin Hoffman in Rain Man, who would be able to grunt in a split second ‘Seventy-five’.) Our only strategy is to estimate, and we’d probably be far off the mark.
Researchers have tested the extent of our intuition of amounts by showing volunteers images of different numbers of dots and asking which set is larger, and our proficiency at discriminating dots, it turns out, follows regular patterns. It is easier, for example, to tell the difference between