Lisa takes away the blocks and shows Mrs Gaddi a card with three dots on it, and asks her to say how many there are. She looks carefully at each dot in turn, mentally counting them, and says "Tre". Lisa asks her to do it without counting, but she cannot. She then shows her a card with two dots on it. Again Mrs Gaddi counts in her head: one, two. She clearly cannot do what any of us can do - see at a glance how many dots there are, at least up to four. This process, called "subitising", can be carried out by infants and many species of animal. But Mrs Gaddi cannot subitise, thanks to a stroke she suffered 15 months ago.
We test other number abilities. Can she tell which of two numbers is larger? This is basic indeed. The whole of arithmetic depends on being able to put numbers in order of size. She is shown a card with the numerals 2 and 3, and correctly picks the 3. She is shown 2 and 4 and she correctly picks the 4. Then she is shown 5 and 10. Now she confesses to being completely at a loss, and guesses that 5 is the larger.
Perhaps, we think, there is something amiss with her ability to read and understand the numerals. So the same test was tried with dots instead of numerals. We got exactly the same results. If the numbers were 4 or below, she was accurate, if very slow; if they were above 4, she was completely confused, and resorted to guessing.
Then we asked her to compare one number that was 4 or less with one that was larger than 4. Now she was really quite good, scoring 70 per cent with numerals and almost 100 per cent with dots. So she had a sense that there were numbers beyond the reach of her counting, which had to be bigger than the numbers she could count. It wasn't that all numbers finished at 4, just that her numbers finished at 4.
The stroke Mrs Gaddi had suffered damaged the left parietal lobe of her brain. This is the classical site for "acalculia" - an inability to use numbers - but we had never seen a patient, or even heard of a patient, so severely affected by a stroke that they could not even subitise two dots.
Since her stroke, her life had been full of frustration and embarrassment. She was unable to do things that previously had been second nature. She could not give the right money in shops; she had no idea how much she spent or how much change she received. At the check-out, all she could do was open her purse and ask the assistant to take the right amount. She could not use the phone, tell the time or catch the right bus - or to convey ordinary facts if they involved numbers. Despite her best efforts, her numerical abilities did not improve in the year and half we were testing her.
What makes victims of stroke, like Mrs Gaddi, so interesting and important to science is that the damage is often localised, which can mean that just one function of the brain is disturbed, so we can see how the brain divides up the work it has to do.
For this method to work in her case, we needed to be sure that Mrs Gaddi's problem really was numerical, and not the result of some other cognitive deficit. We were encouraged by the old findings of Salomon Henschen, a neurologist who worked in the Karolinska Institute in Stockholm until the late Twenties, and had introduced the term "acalculia". He had collected data on 260 neurological patients with some disturbance to their numerical abilities, out of a total collection of more than 1,300 cases. He concluded that there "exists in the brain an independent system subserving arithmetical processes".
However, when we were testing Mrs Gaddi it was still by no means accepted that core numerical abilities were an independent system. Some researchers, following the linguist Noam Chomsky, have argued that numeracy is just a special aspect of language: bigger numbers are like longer sentences. Others, following the Swiss developmental psychologist Jean Piaget, have claimed that numerical abilities are constructed from more primitive logical concepts, without a life of their own.
When we tested Mrs Gaddi's spoken language, it was unimpaired. As for logic and reasoning, she scored 100 per cent on all our tests. She could solve unerringly problems such as: Giorgio is taller than Carlo; Pietro is shorter than Carlo: Who is taller, Pietro or Giorgio? Nor was her memory affected. She could still remember geographical and historical facts from her school days, provided they didn't involve numbers.
We concluded that there was indeed a specialised brain circuit for numbers in the left parietal lobe, and that damage to this circuit had caused Mrs Gaddi's acalculia. If we were right, there should be other patients in whom brain damage has spared this area, leaving mathematical abilities intact, but has affected brain regions that support language, memory and reasoning. Our idea has been vindicated by the discovery of a remarkable patient by a team at the Catholic University of Louvain, Belgium.
Mr Van is an 86-year-old manual worker whose schooling finished after primary level. A neurodegenerative disease, probably Alzheimer's, has affected large parts of his brain, leaving him confused and forgetful of many facts about his own life. His most remarkable deficit lies in his inability to reason. For example, when given 27 playing cards from 1 to 7 in each of the suits, he was asked to find the missing card. "He proved totally unable to organise the search," the investigators remarked.
Extraordinarily, he was also quite unable to carry out the tasks Piaget had claimed to be a prerequisite for having the idea of numbers. When a row of 10 wooden cubes was lined up, one-to-one, with a row of 10 wooden discs, he agreed that there were the same number of cubes as discs. When three discs were removed he correctly said that there were fewer discs than cubes, indeed that there were three fewer discs.
Then the two rows of 10 cubes and 10 discs were set out again. This time the experimenter spread out the cubes to make a longer line, but without adding or subtracting any cubes or discs. Mr Van now said that there were more cubes than discs. On a second trial, the discs were squeezed together; again he said there were more cubes than discs. Surely he would understand, as even a child of four does, that to change the number of things in a collection a thing must be taken away or a new thing added? Mr Van was unable to do it.
Now, if this kind of reasoning ability underpins our skill at using numbers, then Mr Van should be very bad indeed. Was he?
Quite the opposite. He was as good as normal people half his age on standard tests of simple arithmetic, but what made his performance exceptional was the way he tackled difficult calculations. For example, he was able to tell which of a set of three- and four-digit numbers, such as 839, 841, 4,096, 4,099, were squares and which were not. In another task, he had to pick the square root for a four-digit number. In one task he was given 3,844, and had to pick one of the following: 42, 61, 62, or 68. It is not easy. (The answer is 62. I had to use a calculator.) He did it quickly. Mr Van, despite his profound dementia, was able to calculate at levels better than most of us, showing that autobiographical memory and good reasoning skills are not needed even for difficult calculations.
Patients like Mrs Gaddi and Mr Van (not their real names) are helping us understand how our brains contain circuits specialised for one of the most important of all human abilities: the use of numbers.
The writer is professor of neuropsychology at University College London. His new book, `The Mathematical Brain', is published by Macmillan, price pounds 20Reuse content