The study confirmed for Butterworth that developmental forms of dyscalculia are the result of basic problems in comprehending numbers and not in other cognitive faculties. But determining exactly what those problems are would prove challenging.
Like nearly all human cognitive abilities, number sense is evolutionarily ancient — tens if not hundreds of millions of years old. Studies of chimpanzees, monkeys, newborn chicks, salamanders and even honeybees point to two parallel systems for representing quantities. One, called the approximate number sense, distinguishes larger quantities from smaller ones, be they dots flashing on a screen or fruits in a tree. Studies on monkeys reveal that certain neurons in a specific fold of the parietal lobe fire more vigorously in response to increasingly higher numbers. A second ancient number system allows humans and many other animals to instantly and precisely recognize small quantities, up to four. Primate studies show that individual neurons within the same fold, called the intraparietal sulcus, seem tuned to particular quantities, such that when a monkey is performing a task that involves numbers, one neuron will fire for the number 1, a different one will fire for 2 and so on.
People who are poor at distinguishing approximate quantities do badly in maths, suggesting that the approximate-number system is crucial. And some work shows that dyscalculics are poor at recognizing small numbers, suggesting that this ability is also fundamental to numeracy. Moreover, scans of people with dyscalculia suggest that their intraparietal sulci are less active when processing numbers and less connected with the rest of the brain compared with numerate children and adults.
Yet Butterworth views such results as consequences, not causes, of the poor numerical abilities that characterize dyscalculia. He argues that another cognitive capacity is even more fundamental to number sense. He calls this 'numerosity coding': the understanding that things have a precise quantity associated with them, and that adding or taking things away alters that quantity.
But Stanislas Dehaene, a cognitive neuroscientist who studies numerical cognition at INSERM, France's national institute for research on medicine and health, near Paris, sees number sense as being supported by a broader set of cognitive features. Approximation and a sense of small numbers, while critical, are not enough for humans to precisely grasp large numbers, he says. Language, he argues, empowers humans to integrate the two number systems — giving them the ability to intuitively distinguish, say, 11,437 from 11,436. Butterworth's concept of numerosity coding may be an important part of number sense, says Dehaene, but there is still much to learn about it — for instance, whether it is present in other animals or in children from a very early age.
One of Butterworth's favorite papers is titled 'Six does not just mean a lot: preschoolers see number words as specific'. In it, the developmental psychologist Barbara Sarnecka, now at the University of California, Irvine, and Susan Gelman, at the University of Michigan in Ann Arbor, showed that young children who could not yet count past two nonetheless understood that adding pennies to a bowl containing six somehow altered its number, even if the children couldn't say exactly how. If numerosity coding is fundamental, it predicts that dyscalculics such as Moorcraft or Christopher struggle to enumerate and manipulate all numbers, large and small. Butterworth hopes that, by honing this ability, the Number Sense games will help support his research ideas.