Easy as 1, 2, 3 People come into the world ready to count its wonders

The baby is just one day old and has not yet left hospital. She is quiet but alert. Twenty centimetres from her face researchers have placed a white card with two black spots on it. She stares at it intently.

A researcher removes the card and replaces it by another, this time with the spots differently spaced. As the cards alternate, her gaze starts to wander—until a third, with three black spots, is presented

Her gaze returns: she looks at it for twice as long as she did at the previous card. Can she tell that the number two is different from three, just 24 hours after coming into the world?

Or do newborns simply prefer more to fewer? The same experiment, but with three spots preceding two, shows the same revival of interest when the number of spots changes.

Perhaps it is just the newness? When slightly older babies were shown cards with pictures of household objects instead of dots (a comb, a key, an orange and so on), changing the number of items had an effect separate from changing the items themselves.

Could it be the pattern that two things make, as opposed to three? No again. Babies paid more attention to rectangles moving randomly on a screen when their number changed from two to three, or vice versa.

The effect even crosses between senses. Babies who were repeatedly shown two spots perked up more when they then heard three drumbeats than when they heard just two; likewise when the researchers started with drumbeats and moved to spots.

"One great blooming, buzzing confusion' was how William James, a 19th-century psychologist, described the way he thought the world looked to a newborn baby.

 But these experiments, and many others like them over the past few decades, have convinced researchers that, on the contrary, babies are born with many ways of making sense of what they see and hear.

The trick is to use their love of novelty to work out what is happening inside their brains: when shown the same things repeatedly, babies' eyes wander; when the scene changes, their gaze returns. That makes visible what to them constitutes a change in the world around them worthy of notice.

Dot and carry one
One of those ways of understanding the world is by number. People are born with an innate sense of how many items there are in small collections. Experiments in which older children and adults are shown randomly arranged dots and asked to say quickly how many there are show this sense is retained throughout life.

Up to three or four items, and the number is immediately visible without counting. Within a limited range, humans are born arithmeticians, too.

When babies a few months old were shown dolls placed and removed from behind a screen they had correct expectations of the number of dolls they would see when the curtain was drawn aside, and were surprised when trickery meant those expectations were violated.

 In fact, they were more surprised to see the wrong number of dolls than the right number, but different-looking ones.

Some animals also seem able to perceive and understand small numbers. From the 1930s Otto Köhler, a German zoologist, trained ravens to open boxes with the same number of dots on the lid as a card held by a researcher.

One raven learnt to distinguish two, three, four, five and six dots. Rats can learn to ignore a certain number of doors in a maze before choosing which one to enter.

 Chimpanzees have been taught to match the numerals 1 to 6 to the number of objects in a display and to find oranges hidden in two different places and point to the numeral that indicates their total number.

Even more strikingly, some wild animals appear to understand and use numerical facts without training.

Karen McComb of the University of Sussex, in England, played a variety of recordings of lions roaring at night in the Serengeti National Park—different numbers of lions; their roars in sequence and overlapping; and so on.

She wanted to test the theory that, since fights between lions are very costly, when lions heard large numbers of intruders" roars they would withdraw unless they were in superior numbers.

The best explanation of what she observed was that lions estimated the number of intruders from the number of different-sounding roars, compared that number to the number in their own group and then decided whether to attack or slink away.

That humans (and perhaps other animals) come ready-supplied with numbers contradicts two popular rival theories: the Platonic and the constructivist. Plato thought numbers (and geometric objects such as circles) existed in some abstract, eternal and perfect realm, of which mortals were granted only an occasional glimpse.

Constructivists follow Jean Piaget, a Swiss child psychologist, in thinking that by moving things in the real world around and observing the results people 'construct' an understanding of number in the first few years of their lives.

The distinction, though abstract, has practical relevance too. Could 'maths-phobes' be born, rather than made? Can they be cured? And could mathematics be taught better to all?

Numbers on the brain
Brian Butterworth, a cognitive neuroscientist at University College London, has spent much of his career teasing out which bits of humans'understanding of numbers are innate—and which learnt, and how.

 He thinks people are born with brain circuits that are dedicated to recognising and understanding the number of items in small collections.

On this foundation an entire 'number sense' is built, as children realise that bigger and bigger numbers can be reached by adding 'one more' and learn by experience how these bigger numbers behave.

His most recent work has confirmed that to develop a better understanding of numbers than that of a newborn baby, it is not necessary to be able to count with words.

He collaborated with some Australian researchers to test aboriginal children in the country's Northern Territory who were monolingual speakers of one of two languages, Warlpiri and Anindilyakwa, in which the only number words are one, two, few and many.

(Words for numbers have generally arisen when and where people grow crops or keep herds; hunter-gatherer bands, who have no herds or other stores of wealth, need not keep track of surpluses, or balances of trade.)

Since the children were too old for the baby-staring trick, but unable to answer the question: 'How many?', researchers laid out counters, then put them away and asked the children to 'do as I did'

To check that they were using the number of the counters, rather than mimicking their pattern, the researchers banged sticks together and asked them to 'make the counters like the noises'.

 The children performed about as well as English-speaking aboriginal children living in Melbourne.

Historically, one common method of counting has been to use body parts to keep track of a running total. The base-ten system used in modern arithmetic originates with the fingers, and linguistic traces of that fact remain in the similarity of 'five', 'finger' and 'fist', and the dual meaning of 'digit'.

Some think that the original inhabitants of Europe were 20-counters who used fingers and toes—the use of 'score' for both 20 and keeping count may be a remnant.

 And there remain tribal peoples who have elaborate methods using eyes, nostrils, elbows and so on.