WITH his intelligent brown eyes fixed on the small screen, the young male prepares to confront a vexing mathematical problem: what does three plus one equal?
The eyes belong to a rhesus macaque monkey called G32 who lives wild on the island of Cayo Santiago in Puerto Rico. Although his answer probably won’t win him any prizes, it will tell cognitive psychologists something about how animals, including humans, get their heads around numbers. Macaque monkeys, it turns out, are surprisingly good at simple maths-better than one-year-old human babies. Surprisingly, because they do it all without human language, and it has long been believed that language is at the heart of innate mathematical talent.
The thing about counting is, it’s not really “as easy as one, two, three”. To count things, you have to recognise individual objects, group them, and tag them with some sort of value that allows you to keep track of the count. That said, some trained animals have, to a degree, mastered the necessary skills. One star performer, a female chimpanzee named Sheba, can count to nine. Sally Boysen of Ohio State University in Columbus spent years teaching Sheba that the symbol “1” corresponds to one unit, “2” to two, and so on up to nine. The chimp even understands that “7” is more than “6”.
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Sheba’s performance is impressive, but it reveals little about the mathematical capabilities of other animals or, indeed, a chimp’s innate ability to count-it took heroic efforts on the part of her trainer to get to nine. To find out whether animals really do have inborn numerical know-how, researchers needed a way to ascertain whether they can spontaneously do maths without prior training, as well as a consistent means of measuring these abilities. The technique they settled on was one that had proved successful in testing the counting skills of another type of beast that lacks language-the human baby.
In 1992, Karen Wynn, a developmental psychologist at the University of Arizona in Tucson, came up with the idea of using a technique called Preferential Looking Time (PLT) to study the mathematical abilities of babies. Wynn took advantage of the fact that infants will gape, eyes wide with surprise, at things they don’t expect to see, to show that babies as young as five to ten months old can add and subtract small numbers.
The setup involves a handful of Mickey Mouse dolls and a movable screen. The researcher shows the baby a Mickey Mouse doll and then places it behind a screen. As the baby continues to watch, the researcher places a second Mickey behind the same screen. In half the trials, the researcher then uses a hidden trapdoor to remove one of the dolls. When the screen is lowered, what do the babies expect to see?
Wynn found that babies stared much longer when only one Mickey doll is revealed-they had apparently expected to see two. The fact that the infants had to somehow keep a tally of the dolls as each in turn disappeared behind the screen suggests that they were counting rather than doing what cognitive psychologists call “subitising”-a quick means of intuiting a small number of things in a visual field, say the number of dots on a die, without counting the individual items. But were the babies really calculating “one plus one equals two”? Or did they simply realise that one plus one had to equal something more than one? To answer that question, Wynn tried the babies on “one plus one equals three”. The babies were appropriately surprised when three dolls appeared from behind the screen rather than two. Even babies too young to use language can tell when things just don’t add up.
Number cruncher
Some see these startling results as a challenge to the assumption that language is needed for number crunching. But babies’ brains are set up to produce language eventually, notes Marc Hauser of Harvard University. If researchers really want to find out whether numerical knowledge can be organised and stored in the brain in the absence of language, they need to look at animals.
So researchers like Hauser, Claudia Uller at Rutgers University in New Jersey, and Susan Carey at New York University set out to adapt PLT to investigate how primates process numbers. “The idea was to take advantage of methods used to study prelinguistic humans and use [them] with nonlinguistic creatures,” says Uller.
Although Hauser’s monkeys on Cayo Santiago were unimpressed by the appearance or disappearance of Mickey Mouse dolls, they were interested in the fate of bright purple aubergines (Americans call them eggplants). By measuring the amount of time the monkeys spent gaping at aubergines, the researchers found that the monkeys have no trouble grasping that “one plus one equals two”. Even more eye-opening was the finding that the monkeys, like babies, knew that “one plus one” did not equal three.
Doll stuff
It all added up to the conclusion that rhesus macaques can count a small number of objects. Or did it? Perhaps the monkeys, and the babies for that matter, were really paying attention to the total amount of “stuff”, rather than the exact number of objects-only staring quizzically when confronted with an unexpected change in the amount of “doll stuff” or “aubergine stuff”. To test this hypothesis, Hauser and his students collected some very large aubergines-fruits twice the size of the normal ones. Would the monkeys be surprised when presented with “one aubergine plus one aubergine equals one big aubergine”, even though the total amount of “aubergine stuff” was correct? The answer is yes. If two aubergines were placed behind the screen, then the monkeys fully expected to see two sitting there when the screen was raised.
Meanwhile, Uller was running modified PLT tests on members of a Harvard-based colony of cotton-top tamarin monkeys. Tamarins responded best to Froot Loops, a breakfast cereal. And like the rhesus monkeys and the babies, they too could figure out that “one plus one equals two”-no more, no less. Tamarins are New World monkeys, and even more distantly related to humans than rhesus monkeys, revealing just how deep the evolutionary roots of numeracy run.
But monkey mathematicians have their limits. Using PLT, Hauser and his colleagues have found that rhesus monkeys can count two things, three things, and maybe four, but they falter when they’re asked to distinguish between four things and five, or to add two plus two. “The rhesus are doing better than [human] one-year-olds,” says Hauser. “But how much higher they can get is an open question.” In his experience, tamarins don’t do as well as rhesus macaques. “Only one of the tamarins can do three versus two,” says Hauser. “The others only make it to two versus one.”
Yet the limitations of the primates’ performance are providing clues to the mental processing that underlies counting to low numbers in other primate species-including humans. To get an idea of the current thinking about counting, says Hauser, imagine playing a game of Monopoly. When the die is rolled, you know how many dots there are without counting them. That’s subitising. Now, when you go to move your game piece five squares, you need to count. Counting involves assigning numerical values to similar objects-in this case, squares one, two, three, four and five. And it requires understanding order-that five comes after four, which follows three and so on. This type of enumeration involves running along a mental “number line”, assigning the words “one, two and three” to the objects until you’ve named them all.
But how do babies-and in some cases, monkeys-numerically tag a group of objects, store that value in their memory, and then keep the numbers in order without the benefit of language? Psychologists have three main theories to explain the mental processing that goes into counting and the manipulation of small numbers in the absence of language.
The first, the numeron model, is similar to number-line counting, the only difference being that the objects are represented by an ordered list of mental symbols called “numerons”-say a “#” or a “@” or a colour or a shape-rather than the actual words for the numbers. To count three squares, each square gets assigned a successive numeron in the list, and the final numeron assigned represents the total number of objects counted.
According to the second theory, the accumulator model, it’s size that matters. It’s as if neurons involved in counting generate an electrical impulse of a certain size when one object is present. When two objects are present, the signal is stronger, with three objects, stronger still. The observer counts by assessing the total size of the mental signal.
Finally, there’s the object-file model which doesn’t involve symbolic representation at all. Instead, observers create a mental image of each counted object and store it as a “file” in their memory. In the PLT tests, for example, a monkey watching an aubergine being placed behind a screen creates a file containing one aubergine. Add another aubergine and she creates another file. When the screen is removed, the monkey compares in her “mind’s eye” the image she sees in front of her with the images she has filed away, to see if they match. If they don’t, she knows that something is amiss. She’s not capable of sitting and counting in the abstract way a human can, but, by comparing the mental images, she can keep track of small numbers of objects.
Uller, Carey and their colleagues argue that neither the numeron nor the accumulator model of counting would predict the dramatic cutoffs they’ve seen in infants’ and monkeys’ performances. Hauser agrees. “There’s no question,” he says. “Given the limitations that the monkeys and kids show, the object-file model is hands down the best of the three.” Why? Because storing and comparing object files takes up more memory and requires more mental agility than manipulating numerical information the numeron or accumulator way.
Imagine a tamarin monkey watching two Froot Loops being placed, one by one, into an opaque box. He generates two mental files. Now imagine three Froot Loops being dropped into a second box, again one at a time, and then the monkey being allowed to choose one of the boxes. To select the box that holds the larger number of treats, the monkey has to fish back in his memory to find the first two files and then directly compare them with the three new files he’s made for box number two. Too many Froot Loops, argue the researchers, and the monkey (or baby) runs into memory overload.
Hauser is using the Froot Loop technique to test monkeys’ ability to remember numbers. To date, he says the rhesus monkeys go for the larger quantity (it’s food, don’t forget) in trials of up to three versus four items. But they fail to distinguish four versus five. “I don’t think an adult [human] could do much better if he’s not specifically instructed to count the items before they’re put into the boxes,” he says.
But just because monkeys and young babies can add, subtract and count small numbers without a hitch, it doesn’t mean that language has no role to play in older humans’ mathematical abilities. One theory is that language helps, even with smaller numbers, because it allows you to identify and categorise objects so you can keep track of what it is you’re supposed to be counting. In fact, this is just where young babies run into trouble. Last year, Carey and her former student Fei Xu, now at Northeastern University in Boston, reported that 10-month-old babies failed arithmetical tests that required them to distinguish one type of object from another.
Ducks and trucks
The experiment went like this. A researcher held a toy duck and a toy truck behind a screen. As a baby watched the screen, the toy duck appeared from around the left-hand side and then returned behind the screen the same way. Then the truck appeared from behind the right-hand side of the screen and returned. In half the trials, the researchers then secretly concealed one of the objects. When the screen was lowered, 10-month-old babies were not at all surprised to see only one object-a duck or a truck- sitting behind the screen. Only children 12 to 13 months old, who were beginning to learn the words for different objects, gaped with surprise. The researchers concluded that the older babies had used language to tag the objects, and this helped them to understand that the duck and the truck were two distinct, individual items. Uller says that, like the younger babies, tamarins have so far flunked experiments in which they had to distinguish between items-in this case marshmallows and bits of monkey food.
But to everyone’s surprise, rhesus monkeys, like the one-year-old children, pass similar tests with flying colours. Using bright orange carrots and yellow squashes in place of ducks and trucks, Uller, Carey, and Hauser last year tested rhesus monkeys and found that they could grasp that “one squash plus one carrot equals two things”.
If the finding is confirmed, the implication is profound. Rhesus monkeys, which lack the human capacity for language, are happily engaging in what psychologists have always considered sophisticated processing as they count small numbers of objects. Perhaps in the absence of spoken language, rhesus monkeys have some way to tag the objects with non-verbal symbols, ventures Hauser. “Or maybe they’re using a whole other system to distinguish objects.”
Explaining exactly how language in humans might enhance the basic, spontaneous numeracy common to all primates is even more speculative. Carey and Uller argue that perhaps all primates, young and old, use the object-file model as their basic mechanism for counting low numbers. Then as the human baby gets older, perhaps she transfers her mental files into an ordered numeron-type list, which she eventually infuses with language. “It’s a very controversial area,” says Uller.
For his part, Hauser speculates that as babies get older, they gain access to every available mode of counting, from simple subitising to linguistic symbolising, and it is this that allows them to leave even the most numerate monkey in the dust.