Alex's Adventures in Numberland - Alex Bellos [14]
In a related experiment, also illustrated overleaf, Pica’s computer screen showed an animation of six dots falling into a can and then four dots falling out. The Munduruku were then asked to point at one of three choices for how many dots were left in the can. In other words, what is 6 minus 4? This test was designed to see if the Munduruku understood exact numbers for which they had no words. They could not do the task. When shown the animation of a subtraction that contained either 6, 7 or 8 dots, the solution always eluded them. ‘They could not calculate even in simple cases,’ said Pica.
The results of these dot experiments showed that the Munduruku were very proficient in dealing with rough amounts, but were abysmal in exact numbers above five. Pica was fascinated by the similarities this revealed between the Munduruku and Westerners: both had a fully functioning, exact system for tracking small numbers and an approximate system for large numbers. The significant difference was that the Munduruku had failed to combine these two independent systems together to reach numbers beyond five. Pica said that this must be because keeping the systems separate was more useful. He suggested that in the interests of cultural diversity it was important to try to protect the Munduruku way of counting, since it would surely become threatened by the inevitable increase in contact between the Indians and Brazilian settlers.
Approximate addition and comparison.
Exact subtraction.
The fact, however, that there were some Munduruku who had learned to count in Portuguese but still failed to grasp basic arithmetic was an indication of just how powerful their own mathematical system was and how well suited it was to their needs. It also showed how difficult the conceptual leap must be to having a proper understanding of exact numbers above five.
Could it be that humans need words for numbers above four in order to have an exact understanding of them? Professor Brian Butterworth, of University College London, believes that we don’t. He thinks that the brain contains a ready-built capacity to understand exact numbers, which he calls the ‘exact number module’. According to his interpretation, humans understand the exact number of items in small collections, and by adding to these collections one by one we can learn to understand how bigger numbers behave. He has been conducting research in the only place outside the Amazon where there are indigenous groups with almost no number words: the Australian Outback.
The Warlpiri aboriginal community live near Alice Springs and have words only for one, two and many, and the Anindilyakwa of Groote Eylande in the Gulf of Carpentaria have words only for one, two, three (which sometimes means four) and many. In one experiment with children of both groups, a block of wood was tapped with a stick up to seven times and counters were placed on a mat. Sometimes the number of taps matched the number of counters, sometimes not. The children were perfectly able to say when the numbers matched and when they didn’t. Butterworth argued that to get the answer right the children were producing a mental representation of exact number that was abstract enough to represent both auditory and visual enumeration. These children had no words for the numbers four, five, six and seven, yet were perfectly able to hold those amounts in their heads. Words were useful to understand exactness, Butterworth concluded, but not necessary.
Another important focus of Butterworth’s work – and of Stanislas Dehaene’s – is a condition called dyscalculia, or number blindness, in which one’s number sense is defective. It occurs in an estimated 3–6 percent of the population. Dyscalculics do not ‘get’ numbers the way most people do. For example, which of