From What Counts: How Every Brain Is Hardwired for Math by Brain Butterworth.
© 1999 by Brian Butterworth. Reprinted by permission of the Free Press, a division of Simon & Schuster, Inc.
To many readers (not to mention a legion of long-suffering math phobics) nothing may appear less likely than a hardwired, innate capability for mathematics. If somehow that can be swallowed, the next hurdle is to consider the proposition that this capacity may be the same in the reader, in Albert Einstein, and in every human with a normal brain. The difference in actual achievement in math would have to be explained in terms of motivation and hard work—in a word favored by Francis Galton, the ninteenth century pioneer of intelligence studies, “zeal.”
These propositions are Prof. Brian Butterworth’s “bottom line” (another mathematics metaphor) in his book, What Counts: How Every Brain Is Hardwired for Math. Dr. Butterworth, professor of cognitive neuropsychology in the Institute of Cognitive Neuroscience at University College, London, founded Mathematical Cognition, a journal of studies on the psychology and neuropsychology of numbers, in 1993.
What Counts, published by The Free Press in August, draws upon a range of information from anthropology, psychology, neurophysiology, linguistics, psychometrics, and other ﬁelds to make the case for a “Number Module”—specialized circuits of the brain that categorize the world in terms of “numerosities,” the number of things in a collection. Setting down seven distinct tests that would have to be met by his thesis, Prof. Butterworth moves through comparative anthropology, examination of prehistoric man, the evidence of linguistics, brain lesions, split brain studies, and examination of aspects of memory. His concluding chapter examines the question that has plagued parents, teachers, and generations of anxious students: Why are some of us good, even prodigious, at mathematics, while others of us are complete “duffers”?
Following are selected excerpts from What Counts: How Every Brain Is Hard-wired for Math.
According to Butterworth, the brain has specialized circuits for categorizing the world in terms of the number of objects in a collection.
THE NUMBER MODULE
I shall argue that the human genome—the full set of genes that make us what we are—contains instructions for building specialized circuits of the brain, which I call the Number Module [above]. The job of the Number Module is to categorize the world in terms of numerosities— the number of things in a collection. It makes us, and other creatures who possess it, sensitive to the number of things in a collection. Compare seeing the world in terms of colour with seeing it in terms of number. Both processes operate automatically: we cannot help but see the cows in the field as brown or white, nor can we help seeing that there are three of them. Both processes fail to develop normally in some individuals. Just as there are people born colour-blind, there are also...people born with a kind of number-blindness.
What makes human numerical ability unique is the development and transmission of cultural tools for extending the capabilty of the Number Module.
What makes human numerical ability unique is the development and transmission of cultural tools for extending the capability of the Number Module. These tools include aids to counting, such as number words, finger-counting, and tallying, and also the accumulated inventions of mathematicians down the centuries— from numerals to calculating procedures, from counting-boards to theorems and their proofs.
Our Mathematical Brain, then, contains these two elements: a Number Module and our ability to use the mathematical tools supplied by our culture. This may seem obvious, but it is contentious...
Testing the Number Module
All scientific hypotheses are empirically testable. That, according to philosopher of science Karl Popper, is what distinguishes scientific hypotheses from the propositions of mysticism and religion. The skeptical reader has no doubt already formulated some tests that my hypothesis of a Number Module in everybody’s brain will need to pass before it is accepted as credible.
First, if the Number Module is built into all our brains when we are born, then everybody should show evidence of being able to carry out tasks that depend on categorizing the world in terms of numbers of things—what I call numerosities—whether or not they have had the opportunity for instruction, formal or informal, in numbers and arithmetic. Everybody should be able to match the numerosities of collections of things on the basis of one-to-one correspondence between the members of the collections, and they should be able to tell which of the collections is larger. Not much, you may think, but these two operations form the basis of everything else we know about numerosities...
Second, if specialized brain structures are in place when we are born, though of course they will mature and develop over time, then we should see evidence of capacities that depend on numerosity even in the infant.
Third, we should be able to locate these specialized structures in the brain. Certain localized brain damage could affect number skills only. Imaging of the brain’s activity should show the same localized hot spots whenever the brain is calculating.
Fourth, if we have a Number Module, we should be able to find evidence that will satisfy... criteria for modules: fast and automatic numerical operation, at least for basic operations such as identifying or comparing numerosities.
Fifth, if this numerical capacity is part of all our brains, then we need to explain why some people are good or very good at arithmetic, while others are bad or hopeless.
Sixth, if we inherit a Number Module, then it is encoded in our genome, which in turn we inherited, with mutations, from our ancestors. Does this mean that there is a set of genes that code for the building of the specialized brain circuits? Does it mean that our near ancestors, in the last ice age, counted as well? What about our more distant ancestors, in the last ice age, Homo erectus, or great apes, other mammals, birds, reptiles, worms?
Seventh, how does the way we talk about numbers and the way we write them affect the development of number skills that go beyond basic numerosity? Does the Mathematical Brain hypothesis have anything to say about how we can help our children learn more advanced arithmetic, or how we should design our education system?
Finally, there is something strange and emotive about numbers and counting. Many cultures from Africa to orthodox Jews forbid counting people. Some numbers are regarded as lucky or unlucky. Some people suffer anxiety at the thought, not to mention the practice, of arithmetic. Does this have anything to do with our Mathematical Brain?
The chapters that follow fully investigate all these issues.
When we look back through the prehistory of the Mathematical Brain, we see that some kind of tallying can be traced back through the Babylonians and Sumerians, deep into the ice-age cave-painters and bone-engravers of the Magdalenian era 15,000 years ago, before plants and animals had been domesticated.
THE PREHISTORIC MATHEMATICAL BRAIN
When we look back through the prehistory of the Mathematical Brain, we see that some kind of tallying can be traced back through the Babylonians and Sumerians, deep into the ice-age cave-painters and bone-engravers of the Magdalenian era 15,000 years ago, before plants and animals had been domesticated. That in itself is surprising enough, but 15,000 years before that we can find tally marks on bones and possibly on the walls of caves as well. We do not know what these people were counting, but physically they were people like us, as far as we can tell, and their brains were as our brains. So the capacity to count, the roots of the Mathematical Brain, go back to what has been called the Great Leap Forward or the Big Bang, when art was first produced and perhaps language, in something like its modern form, was first used. This certainly pushes the use of numbers back beyond an invention by an ancient Einstein in Ur 10,000 years ago. In itself, it does not rule out an invention, say, in France 32,000 years ago which spread quickly to other sites in France and eastwards to Moravia, and perhaps onward to Asia. But against this invention-diffusion account we may cite the Ishango Bone and the Koonalda Cave herringbone pattern from cultures thought to have had no contact with Europe, yet the methods of marking seem remarkably similar.
If people in the depths of the ice age already possessed a Mathematical Brain, and the urge to count, and especially the urge to count the phases of the moon, it is perhaps not wholly surprising that they devised similar solutions to keeping track of their counts. If the Mathematical Brain is encoded in genes of these fully human ancestors, we may still ask from whom they inherited it. Tantalizingly, there are glimpses beyond our own species to tallying in our cousins, the Neanderthals, and our ancestors Homo erectus perhaps a million years ago.
THE EVOLUTION OF THE NUMBER MODULE
Laboratory experiments show that animals have a latent capacity to do simple numerical tasks. As well as in the recently evolved apes, it is found in birds, which evolved after the dinosaurs. For this capacity to be preserved for millions of years, it must surely have offered some advantage to those species that possessed it. But what could this be...
Squirrels will have more nuts for winter if they consistently choose branches with three nuts rather than two, and in general foraging will be more effective if the animal can decide which patch has most food. It is certainly true that animals make more trips to patches where there is more food. If there are two patches, A and B, so that A has twice as much food as B, then animals tend to go twice as often to A as to B, and make the ratio of trips match the ratio of food. Why do they not spend all their time at the patch with the most food, which would seem to be the optimal strategy, is a problem we cannot go into here. The main point for us is this: animals can estimate and compare two quantities. Many researchers have suggested that this is precisely what constitutes the adaptive advantage of number capacity.1 But squirrels may not be counting nuts; they may be relying on a concept of quantity that is not numerousness—the amount of nut-stuff on the two branches. This is how we estimate the volumes of milk or granulated sugar, for example. Foraging success critically depends on making quantitative estimates without using numbers. Even invertebrates do this. Worker bees, for example, when they have found an abundant source of pollen or nectar, return to the hive and “dance” a message to the other bees indicating the direction and the distance of the food source. When the quantity of food decreases below a certain threshold, the bees may still visit the patch but will not dance on their return. They thus make estimates of relative quantity.2
Evolutionary explanation of behaviour can easily become Just-So stories, whose acceptability depends too often on the imagination of the teller and the gullibility of the audience. What is needed is documented examples of behaviours in the wild that cannot depend on quantity estimates, but must depend on numerosities. These examples are almost impossible to discover, since numerosity in nature almost always varies with quantity. This may of course be why a sense of numerosity is adaptive...
The best examples so far of number use in the wild do not come from foraging. In the Serengeti National Park in Tanzania, a lioness is returning at dusk to her pride. Eighteen females, one adult male, and seven cubs are waiting over a kilometre away. She hears a roar, one she does not recognize. It must be an intruder into her territory. Her cubs are safely with her sisters, and she is all alone. She hears the roar again, it’s the same lion. Should she try to drive off the intruder? It would be an even match—her against one intruder—and it could turn into a real fight. With lions, that could be fatal. Dusk turns to the inky black of night, and she returns silently to the pride.
The following week, again at dusk, she hears roaring 500 metres away. She hears one unfamiliar voice, then a chorus of roars, one overlapping the next, and none of them the familiar voices: three intruders. This time she is with four of her sisters from the pride. Three of them, five of us. Their ears prick. The roaring is coming from a stand of trees over to their left. They wait, they peer into the night and at each other. One of them, the leader, approaches the roaring cautiously at first, as the others join her, more quickly until they are charging headlong into the trees.
By the time they reach the trees, the roaring has stopped and there is no sign of the intruding lions. This is not surprising. The roaring was actually coming from a loudspeaker set up by Karen McComb and her colleagues at the University of Sussex.4 She was testing a theory about the way lions make numerical assessment of the threat from intruders and the strength of the defence forces. The theory predicts that lions (like many other animals, fighting among which would be costly) will contest resources only when they are very likely to win; otherwise they will withdraw. The one exception is when a lioness is with her cub: then she will always attack an intruder. The lions’ decision to attack depends on the number of intruders—in this experiment one or three—and how many adult defenders there are. The lioness leader identifies roaring as coming from the individuals who are not members of the pride; she will also represent the defenders as known individuals. The best explanation of her decision is that she enumerates the number of distinguishable roarers and the number of her sisters, and then compares the two numbers. Only when the number of defenders is greater than the number of intruders will she launch an attack. This is remarkable because the number of intruders comes from the sound they make (they are not visible), while the number of defenders comes from another sense or other sense, vision probably, and is stored in the lioness’s memory. Thus she has to abstract the numerosity of the two collections—intruders and defenders—away from the sense in which they were experienced and then compare these abstracted numerosities.
NUMBERS IN THE BRAIN
Signora Gaddi is sitting at her desk in the office of the family hotel in Friuli, northern Italy, where for many years she has kept the accounts. She is a small, neat woman of fifty-nine years, well organized and clearly very competent, with thirteen years of education behind her. She speaks clearly and fluently about her family and the problems of running a hotel. My colleague Lisa Cipolotti places five small wooden blocks in front of her, and asks her to count them. Uno, due, tre, quattro. La mia matematica finisce qui. “One, two, three, four. My mathematics finishes here.”
Lisa takes away the blocks and shows Signora 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 Signora Gaddi counts in her head: one, two. She clearly cannot do what the rest of us can do—see at a glance, without counting, how many dots there are, at least up to four. This process, called subitizing, can be carried out by small children, probably even by infants and by many species of animals... But Signora Gaddi cannot subitize.
Lisa tests other basic abilities. First, how accurately and quickly can she judge which of two numbers, two single-digit numbers, is the bigger? She is shown a card with the numerals 2 and 3, and she correctly picks 3. She is shown 2 and 4 and she correctly picks the 4. Then she is shown 5 and 10, which is a bigger absolute difference than in either of the two other trials, but she confesses she is 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. For up to four dots she was accurate if very slow; for four or more 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% with numerals and almost 100% with dots. So she had a sense that there were numbers beyond the reach of her counting, which had to be bigger than t he numbers she could count. It wasn’t numbers that finished at 4, just that her numbers finished at 4.
Counting objects, as children have taught us, is not as simple a task as it seems to us adults. We need to coordinate the sequence of words with tagging each of the objects counted or to be counted.
Counting objects, as children have taught us, is not as simple a task as it seems to us adults. We need to coordinate the sequence of words with tagging each of the objects counted or to be counted. There are lots of ways in which this can go wrong. A possible problem could be in remembering the sequence of number words. To test this, we asked her just to recite the number names, starting at “one”. But again, she could not get above “four”.
Even number facts, which seem to require just memory, were extraordinarily difficult for her. To retrieve the number of wheels of a car, she summoned up the image of a car, and counted aloud the wheels she could see in her mind’s eye. To answer Lisa’s question about the number of arms on a crucifix, she asked Lisa to hold out her arms and then counted them. If numbers up to 4 were hard, those above 4 were impossible. She could not recall the number of days in the week, her house number, her age, or her shoe size.
Fifteen months earlier, Signora Gaddi had suffered a stroke that damaged the left parietal lobe of her brain. This is the classic site for a condition called 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 subitize two dots. Since her stroke, her life had been one of frustration and embarrassment. She was unable to do things that previously had been second nature to her. She could not give the right money in shops; she had no idea how much she was spending or how much change she was getting. When she got to the check-out, all she could do was to open her purse and ask the assistant to take the right amount of money. She could not use the phone. There was no way to call her friends. In case of emergencies, the phone was specially adapted so that by pressing one button it rang pre-set numbers in sequence until one answered. She was unable to tell the time, or catch the right bus. She could not convey ordinary facts if they involved numbers. Despite her best efforts, her performance did not improve in the year and half we were testing her.
The Independence of Number Areas
We immediately realized that this could be an important case. The brain is organized into different specialist regions, each doing its own job. One of the best ways of finding out which bit of the human brain does which job is to find patients whose brain cannot do a particular job, and try to link the disability to the location of the brain damage...
We were encouraged by the old findings of Salomon Henschen, a neurologist who worked in the Karolinska Institute in Stockholm until the late 1920s, and 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 over 1,300 cases. On the basis of this extraordinary database, he concluded that there “exists in the brain an independent system subserving arithmetical processes and that it is independent, or nearly so, of the systems for speech and music.”6 However, at the time when we were testing Signora Gaddi, it was by no means generally accepted that core numerical abilities were an independent system. Many, following Jean Piaget, saw the number concept as emerging from more primitive logical concepts without, as it were, a life of its own, at least not in the development of the child.7 Perhaps by the time the child becomes a fully competent adult, the number concept could have freed itself from its logical past. Others, following Noam Chomsky, believed that number was just a special aspect of language.8
There was also a whole school of thought which claimed that the core mental representation of numbers, the representation of their size, was not numerosity but analogue. It was as if each number were represented in the brain by a rod of a particular length (plus or minus some margin of error).9 Neurologists too had tried to explain acalculia in terms of more general abilities. One review, published in the same year as our report on Signora Gaddi, argued that there was no one site in the brain for acalculia, since acalculic symptoms could appear following damage to many different brain areas. Yet other neurologists tried to argue that our number abilities are based on general intelligence and reasoning, or on our spatial abilities, or on linguistic abilities, or on some combination of the three, in various proportions; and that different types of symptoms arise from damage to the areas of the brain supporting reasoning (frontal lobes), spatial awareness (parietal lobes), memory for general knowledge (left temporal lobe), or linguistic ability (left temporal lobe).10
The Language of Numbers Is Not Language
It was critical to establish that Signora Gaddi’s problem with numbers could not be explained in other ways. We had already seen that her spoken language was unimpaired,11 but perhaps the brain damage had affected non-modular capacities—those we deploy in a wide range of processes and tasks—such as the ability to reason, the ability to deal with any kind of quantity, or just memory for any kind of fact, not just numerical facts. We developed a whole new battery of tests to check each of these functions. She passed all with flying colours. Not a hint of a problem. She scored a 100% on our reasoning tests. She could solve unerringly problems such as, “Giorgio is taller than Carlo, Pietro is shorter than Carlo: who is taller, Pietro or Giorgio?” She could still remember all kinds of geographical and historical facts she had learned at school, provided they didn’t involve numbers. She had no problems with remembering her friends and family, or her past history.
It was clear from these tests that language, memory, and reasoning were not sufﬁcient in themselves for good number performance. Something crucial was clearly missing. But was the number circuit really independent of these other systems?
So there was clearly nothing wrong with her memory. With respect to quantities, she knew which was heavier, a kilo or a gram, a kilo or a tonnellata; 12 which was longer, a metre, a centimetre, or a kilometre. She had no trouble ordering pictures of objects according to their size in real life. It was clear from these tests that language, memory, and reasoning were not sufficient in themselves for good number performance. Something crucial was clearly missing. But was the number circuit really independent of these other systems?
To be confident that there was an independent number region in the brain, we needed cases the converse of Signora Gaddi. Could there be people with good numerical abilities but minimal language, severely defective memory, and very poor reasoning? To begin with, perhaps, someone whose language was disastrous, but who could still calculate. Even Henschen had never seen such a patient; they were bound to be rare. Language processes take up a large chunk of the left hemisphere of the brain, so anything that damaged the language circuits could well affect neighbouring regions, including the inferior left parietal region, which we suspected were critical. These cases would be very different from stroke victims like Signora Gaddi. They would be the result of a disease which causes widespread damage, and moreover which leaves a kind of island of spared brain. These diseases are associated with the dementias. The best known is Alzheimer’s disease, which spreads slowly, or quickly, through the brain. These are truly tragic conditions that ultimately affect most brain functions. Unlike strokes, there is no recovery.13
Even today, only one or two such cases have been discovered in which widespread damage has spared numerical processing. Martin Rossor, Elizabeth Warrington, and Lisa Cipolotti from London’s National Hospital for Neurology and Neurosurgery have recently reported a patient, a Mr. Bell, suffering an awful degenerative disease of the brain called Pick’s disease, whose effects are similar to those of Alzheimer’s.14 Mr. Bell’s language had almost completely disappeared. He was left being able to utter just a few stereotyped phrases, such as “I don’t know” and curiously, “Millionaire bub”. His understanding of speech or of written language was almost nonexistent. Nevertheless he was still pretty good at calculation, and could accurately add and subtract, and though he seemed to have lost some multiplication facts he still showed a good understanding of what it was and how to do it...15 He could also select the larger of two- and three-digit numbers, showing that he still understood about numbers as being ordered by size, and the way the Arabic numeral system worked.
NUMBER MEMORY IS NOT JUST MEMORY
Neuroscientists and psychologists distinguish three separate memory systems, each with its own location in the brain. First, there is long-term autobiographical or “episodic” memory, for events we have experienced in our life, tagged for when and where these events happened. These are the memories lost in amnesia. Second, we have long-term “semantic” memory for general knowledge, including what we learn in school. Items in semantic memory are not tagged for where or when we learned them. Semantic memory is the repository for facts like the capital of Botswana or the meaning of “verisimilitude,” and also the verbal form of multiplication tables. In cases of classic retrograde amnesia, patients may have forgotten where they went to school or that they married, but can still recall the capitals of European states, the meanings of words, and tables. Third is “short-term memory.” For psychologists, this is the capacity, measured by “span,” for keeping a list of unrelated items, for example a list of digits, in the right order until needed. Without rehearsal the items are lost within a few seconds. The normal span is about seven digits or six words. Whether they are letters, words, or numerals, there is good evidence they are stored in verbal form. Looking up a telephone number and keeping it in mind to dial is the textbook use for short-term memory.
Now the question for us is this: which of these memory systems do we use to store our knowledge of numbers and arithmetic? One could make a plausible case for all three. There is that indelible memory of a sunlit classroom. You were waiting for the bell to ring so that you could run into the playground, when Miss Brown asked you for the answer to 8x7. Flustered, you ventured “fifty-four, Miss.” Humiliatingly, your worst enemy Lucy confidently piped, “that’s wrong, Miss, it’s fifty-six.” Maybe we do have a few of these memories. But what about those table facts when you weren’t the humiliated—or the humiliator? One way of finding out is to study the numerical abilities of severely amnesic patients. Have they lost their arithmetical facts as well?
Long-term Autobiographical Memory
Margarete Delazer, Thomas Benke, and I studied a group of amnesic patients,16 all of whom suffered both retrograde and anterograde problems. That is, they were very bad at recalling what happened before their illness began (retrograde), but also very bad at remembering things that had happened to them since (anterograde). They would usually be unable to remember whether they had been to the clinic before, though they all had, frequently. These patients were tested with standard arithmetical tests, and also with some special tests of our own devising. On tests of simple arithmetic most were just as accurate and just as fast as our control subjects with normal memories. This suggests that for number facts, autobiographical memory is not really very important.
My collaborators had the ingenious and original idea to use “priming” with these patients. Priming is a technique I had been using for quite other purposes with normal subjects. If you see the same multiplication problem twice, you are able to answer it faster the second time. The first occurrence of 7x9, which we call the “prime,” activates the fact 7x9=63 in memory. This activation dies away only slowly, so when you encounter 7x9 for the second time, which we call the “target,” that fact is still more active than it was the first time, which makes it quicker to retrieve. Eventually, after some minutes or hours, the activation is back to its previous level. We also used a slightly more sophisticated version, where the subject sees 7x9 but later has to solve its complement, 9x7. In the experiment subjects just had to answer a long string of problems presented one at a time on a computer screen. They were not told that they would sometimes see the same problem more than once. The amnesic patients have no conscious recollection of the problems they have just seen...
…arithmetical problems elicit the same responses in the amnesic brain as in the normal brain…except that the amnesics don’t remember having recently seen the problem. It fails to register in their autobiographical memory.
This shows that arithmetical facts can be well preserved in the long-term semantic memory of amnesics: they are not “amnesic for arithmetic.” What is more, arithmetical problems elicit the same responses in the amnesic brain as in the normal brain—the activation seems to decay in the same way—except that the amnesics don’t remember having recently seen the problem. It fails to register in their autobiographical memory. We can therefore rule out autobiographical memory as the locus for stored numerical knowledge.
Long-term Semantic Memory
What about semantic memory, where our general knowledge is stored? Signora Gaddi had well-preserved semantic memory. There are also the converse cases, where general knowledge can be severely affected but arithmetic is well preserved. These cases are rare because impairments to semantic memory are usually the result of progressive diseases of the brain, like Alzheimer’s, which cause widespread damage and in the end affect most brain areas. Recently, a French neurologist and a team of Belgian neuropsychologists have described a really extraordinary case: Monsieur Van, whose semantic memory was severely compromised, but whose calculation remained almost at prodigy level...17
When we do a mental arithmetic problem which involves more than two digits, carrying or borrowing, most of have the sensation of saying the intermediate results in our heads. This helps us keep track, or seems to. Try adding 87 and 56. Did you hear something like “carry 1” in your head after adding 7 and 6? If you did, this was your short-term memory working. This is the verbal code in which you store information for brief periods. Is it necessary for mental arithmetic, or is it a kind of epiphenomenon, something that pops up in consciousness while the real work, the real calculating, is being carried out elsewhere?
We know from studies of normal children and normal adults that short-term memory span—how many digits you can repeat back immediately after a single presentation—correlates highly with scores on arithmetic tests.18 Short-term memory professionals, and there are hundreds round the world who make a decent living out of studying our ability to recall lists of unrelated items, stress its importance.
Charles Hulme and Susie Mackenzie, in their book Working Memory and Severe Learning Difficulties (short-term memory is often called “working memory”) write that “Such temporary storage is obviously necessary for a wide variety of tasks....such as mental arithmetic.”19 However, as we all know, correlation is not cause. We cannot say whether short-term memory helps arithmetic, or arithmetic skills help short-term memory. The other problem is that span correlates with all sorts of cognitive abilities, and could therefore be used to diagnose mental ability generally. Oliver Wendell Holmes called it “a very simple mental dynamometer which may yet find its place in education.”20 Looked at the other way around, lots of other abilities may also be contributing to span performance.
There is one way to find out what role short-term memory plays in arithmetic: test someone whose short-term memory is very poor and see what effect this has on their calculation skills.
One day I was attending a case conference at the national Hospital for Neurology and Neurosurgery in Queen Square, London, the oldest neurology hospital in the world. This is a meeting where a patient is tested before an “audience” of neurologists and psychologists who together apply their expertise in an attempt to make sense of the symptoms, formulate a diagnosis, and recommend a treatment. The patient was Mr. Morris, a brisk, well-dressed, confident man in his fifties. We heard that he had been a pharmacist and still owned a chain of chemist shops. He had suffered a stroke to his left hemisphere that had left him with a halting speech and a mild form of Broca’s aphasia (a condition in which speaking is affected more than understanding speech). He was being presented by Professor Elizabeth Warrington, head of the neuropsychology department and one of the founders of modern neuropsychology.
Elizabeth showed that Mr. Morris had difficulty uttering grammatical sentences, and was also suffering a rather rare kind of dyslexia that was a result of his stroke. She then asked Mr. Morris to repeat back a list of numbers. She started with one number, then went on to two numbers. No problem. He could clearly hear the numbers properly and he could say them. But when she got to three numbers, e.g. 6,2,9, he started to make mistakes. He would miss out one of the numbers, or get them in the wrong order. The best he could manage reliably was two digits. Most people can manage at least six numbers, so this was really defective short-term memory. Here was an ideal opportunity to explore the contribution of short-term memory to arithmetic. So, what was his mental arithmetic like?
Elizabeth tested him with examples from her own Graded Difficulty Arithmetic test, a mental arithmetic test known to neuropsychologists around the world by its initials, GDA. It starts easy and gets harder. The first question is “fifteen add thirteen”, the fifth is “one hundred and twenty-three add twenty-nine,” and the twelfth “Two hundred and forty-four add one hundred and twenty-nine.” If you complete this question or if you fail three questions in a row, you continue with subtraction, starting with “nineteen take away seven” and ending with “two hundred and forty-six take away one hundred and seventy-nine.” Note that all these questions exceed Mr. Morris’s span of two digits.
Mr. Morris astonished us. He seemed to have no trouble at all with most of the questions. Even questions such as 128+149 and 119-35, which he would find impossible to repeat because they were 3 or 4 items greater than his span, he answered correctly. As Elizabeth fired questions at him, I was trying to answer the questions in my head, but he got his answer out quicker than I could, most of the time.
LEFT BRAIN, RIGHT BRAIN
The next problem is to locate the functional architecture in the actual architecture of the brain. How are the number circuits arranged in the brain? Are they all in the same hemisphere? Is there a single brain region dedicated exclusively to numerical processes—is the Mathematical Brain a well-defined neural structure? To begin to answer these questions, we have to understand the overall structure of the brain.
The brain is organized into many specialist areas which have quite definite tasks. In perception there are specialists for detecting edges of objects, colour, location in space, and so on. There are also specialists for controlling action. One curious feature of brain organization is that the specialists for the right side of the body are in the left hemisphere of the brain, and for the left side of the body in the right hemisphere. That is, the main perceptual and action circuits cross over: the left hand is controlled by the right hemisphere, sounds going into the left ear are analyzed primarily in the right auditory cortex, and the left visual field (what you see to the left of your centre line) is analyzed in the right hemisphere. The two hemispheres are linked by the corpus callosum, a massive tract of more than 200 million long neural fibres that carry information from one hemisphere to the other.
…one brain area emerges as a key area for numbers: the left parietal lobe….Patients with other cognitive abilities shot to pieces, but with preserved numerical skills, seem to have the left parietal lobe intact.
Number processes share the main perceptual and action systems with other processes. But one brain area emerges as a key area for numbers: the left parietal lobe. It is the area that is almost always demonstrably damaged in cases of acalculia. Patients with other cognitive abilities shot to pieces, but with preserved numerical skills, seem to have the left parietal lobe intact. This doesn’t mean that no other brain areas are specialized for some number abilities. Does the right hemisphere play any part in number tasks? We all know, or at least many people believe, that the left hemisphere is sequential, logical, and analytic, while the right hemisphere is parallel, global, and artistic. Perhaps the left brain does all the work while the right brain appreciates the beauty of numbers. How can we find out what the right side of the brain does? There are four ways.
What Are the Effects of Right-Hemisphere Damage?
First, we can study patients with lesions in the right hemisphere to see what effects this might have on numerical skills. The problem here is that we are not sure which part of the right hemisphere we should be looking at, and lesions in irrelevant areas will have no effect. There are two ways round this. We could survey a whole group of patients with right-hemisphere lesions and compare them with a group of patients with left-hemisphere lesions. For simple arithmetic, as measured by Jackson and Warrington’s Graded Difficulty Arithmetic test, lesions in the left hemisphere caused severe acalculia in 16% of patients (patients who scored worse than 99% of control subjects), while lesions in the right hemisphere resulted in no such severe problems.21 On the other hand, there is evidence that one of the most basic numerical abilities, the ability to subitize, may be represented in both the left and right hemispheres since damage to the right parietal lobe will, in some cases, lead to an impairent in this ability.22
Normal arithmetic seemed to have been abolished almost completely, though the patient could still identify the numerosity of a collection of objects, recognize numerals, and compare number magnitude.
A second approach is to find people who have had the whole of their left hemispheres removed, so that whatever they can still do must be the responsibility of the right hemisphere. One such patient was a Vietnam veteran studied by Jordan Grafman and his colleagues.23 Normal arithmetic seemed to have been abolished almost completely, though the patient could still identify the numerosity of a collection of objects, recognize numerals, and compare number magnitudes. Incidentally, almost all language ability, a left-hemisphere function, had been lost.
Third, we could look at “split-brain” patients in order to examine what happens to the specialist processes in each hemisphere...
This is what happened to a patient referred to Paris’s most famous neurological hospital, La Salpetriere (where Jean-Martin Charcot taught Sigmund Freud about advanced neurology, and hysteria). The posterior part was destroyed, which prevented communication between the visual areas. In a brilliant series of experiments, comparing the presentation of stimuli to one side of the brain or the other, Laurent Cohen and Stanislas Dehaene showed that the patient could recognize numerals in both hemispheres. We have known since the work of Jules Dejerine in the late nineteenth century that the right hemisphere could recognize and read numerals even when reading words was impossible.
The patient could even compare the magnitude of single-digit numbers in both hemispheres, though with a 13% error rate, and she was slower than when the numerals were presented to the right hemisphere. She was very bad at naming numerals presented to the right hemisphere, but errors showed an interesting and important pattern. When she was wrong, the number name she produced was similar in magnitude to the target (more similar than one would expect by chance). If she saw ‘5,’ she might say quatre or six. What she could not do was arithmetic of any sort. Cohen and Dehaene conclude that she was unable to transfer the identity of the numerals across the corpus callosum to the left hemisphere for naming, whereas she could transfer approximate size information. The reason for this is that the anterior fibres of the corpus callosum, the fibres that connect the two parietal lobes, were still intact; according to their theory, the parietal lobe in each hemisphere is able to process analogue magnitudes-sizes. Since the information is only approximate, she will make mistakes, but these won’t be outlandish mistakes. She will never say neuf when her right hemisphere reads ‘2.’ Although her right hemisphere was able to manage these very simple numerical tasks, though less efficiently than the left hemisphere, calculation was virtually impossible. She made errors on half the occasions she was asked to add 1 to a single digit. Even when the task did not require a spoken response—and hence the involvement of the left hemisphere—but a pointing response or just indicating whether a sum was right or wrong, she was still very bad. This supports the idea that calculation is exclusively the job of the left hemisphere.24
...when we compare numbers we use an analogue representation of their size, in much the same way we might compare the water levels in two glasses.
Is numerosity represented in both hemispheres? According to Cohen and Dehaene, when we compare numbers we use an analogue representation of their size, in much the same way we might compare the water levels in two glasses. They claim that this representation is used by both hemispheres, and is what can still be transferred across the anterior fibres in their patient’s corpus callosum. Even if the right hemisphere uses analogue magnitudes, which is why responses originating from it tend to be approximate, the left hemisphere could still be using numerosity representations, which is why responses originating there are almost always exactly right. Of course, the left hemisphere might be able to represent analogue magnitudes as well, but the crucial point is that only the left hemisphere can represent numerosity.
Imaging the Brain in Action
Fourth, it is now possible to image human brain activity while it is occurring, using one of several new methods that do not require the skull to be opened up (though there is that method as well). They all depend on the fact that when brain areas become active, the blood flow to them is increased, and with it comes an increase in the metabolism of glucose and in electrical activity...
Unfortunately, this exact investigation has yet to be done. Studies of numerical processes using functional brain imaging are still in their infancy, but they hold out the hope of being able to localize the circuits with much more accuracy than is possible from lesion studies, and, combined with EEG, we should be able to determine very precisely both when and where these processes are occurring.
The Left Parietal Lobe
We have seen a whole series of patients whose numerical abilities were affected when just one brain area was damaged: the parietal lobe in the left hemisphere. We have also seen patients whose brains have suffered damage, often quite widespread, that has affected other cognitive processes, but has spared the numerical ones.
The parietal lobe is actually quite a big area of the brain, stretching from the central sulcus all the way back to the occipital lobes. Not all of this area is devoted to numbers. The current best guess is that a relatively small part, the inferior lobule, is the core of our numerical abilities. It is difficult to be precise for two reasons. First, our best evidence at the moment comes from patients, and unfortunately the things that cause the brain damage—strokes and diseases—don’t target just this part of the brain. In every patient the damage will affect other regions, possibly unconnected with numbers, but we cannot tell which part of the lesion is which. Second, everyone’s brain is different in size, in shape, and in the pattern of folding. Although the major landmarks are easy to locate, a small area might be much harder to identify in different brains. In one brain it could be tucked into the side of a sulcus (groove), while in another it may be nearer the top of the gyrus (the bump).
The best hope of pinning down the crucial part of the left parietal lobe is through imaging the brain in action while it is doing very well-defined and well-understood numerical tasks. But imaging methods are still in their early stages of development, and we don’t yet know quite what we are seeing through the scanning cameras. We are in a position similar to Galileo using the first telescopes. This made possible unprecedented advances in astronomy. He knew that light from the sky was causing the images he could see, but he had no theory of optics. There was thus a big logical jump from the spots of light on the eyepiece to the moons of Jupiter. Crucially, when we image the brain in action, we need to identify just those bits of the activity that are devoted to numbers. Since the brain is always busy doing lots of tasks, from monitoring the environment to controlling breathing to wondering about dinner, we still need to find good ways of sorting out the activity we are interested in.
Nevertheless, it is now clear that our Mathematical Brain is located in the left parietal lobe.
GOOD AND BAD AT NUMBERS
My nature is subdued To what it works in, like the dyer’s hand. William Shakespeare, Sonnets, 110
Why are some people good at numbers and others bad? In the movie Good Will Hunting, our hero, played by Matt Damon, is a young man working as a janitor at MIT, the most prestigious scientific university in the world. As Will mops, the math professor sets his class an end-of-term test. They have the whole vacation to find the solution, and whoever can solve it will prove himself or herself to be the most outstanding mathematician in a class of students already outstanding by virtue simply of being good enough to get to MIT. After the lecture theatre has cleared, Will Hunting leaves his mop to write the solution on the blackboard. Next day, the professor, astonished, asks the solver to step forward, but of course none does. Eventually he discovers that it was Will, who turns out to be a mathematical prodigy comparable to Ramanujan, the greatest prodigy of them all. However, instead of studying to become a professional mathematician, Will prefers to go out drinking and getting into scrapes with his friends from the neighbourhood. The professor, despite being a Fields Medallist (like a Nobel prize for mathematics, but awarded only every four years instead of Nobel’s every year), is in awe. “I am nothing compared to this young man,” he confesses.
So where does the talent come from? Will tries to explain to his girlfriend. He compares himself to Mozart: “he looked at a piano... he could just play. I could always just play. That’s the best I can explain it.” There are two opposed ideas about mathematical abilities. Nature: it is some kind of biological gift, like the gift for music, perhaps. And nurture: it is all due to hard work and educational opportunities. There is also, of course, the “sweet pickle view”: mathematical ability contains both nature ingredients and nurture ingredients, in variable proportions. As with all human abilities, there are really two distinct questions. First, why am I a bit better at sums than Eric and a bit worse than Diana? That is, what explains the variation in skills among 90% of the population? The second issue concerns the extreme cases. What is it that makes some people real-life counterparts of Will Hunting, and why are other people absolute duffers?
A Biological Gift for Numbers?
A few years ago the papers carried the story of the (re)discovery of Einstein’s brain. His left parietal lobe had cells more densely packed than normal...This is the area of the brain crucially involved in numerical processes. Is being born with these extra cells in the key brain areas what made him a great mathematician? The theory of the biological gift goes something like this: our genes (and perhaps our early nutrition) will determine the number of parietal-lobe neurons we are born with; those with more will be better at numbers than those with fewer. This sounds plausible, but it cannot be provided simply by correlating the number of parietal neurons with numerical ability. Being good at numbers could be the cause of more neurons rather than the consequence—that is, the brain could assign more parietal neurons to number tasks, or hold on to more parietal neurons (since neurons start dying from the day we are born), precisely because that part of the brain is constantly “exercised.”
The important thing to remember is that what gets us beyond simple numerosity is acquiring what I call “cultural resources”: the words for numbers, the notations we use for recording and manipulating numbers, the myriad methods and inventions our predecessors have bequeathed to the great subject of mathematics, and so on. One of the things that makes Good Will Hunting so implausible is that Will does not seem to have spent time acquiring these resources. Imagine, if you can, asking Archimedes, the greatest mathematician of antiquity, to solve the equation 2a2+ 3ab-4b2=0 He would have less chance than an averagely educated fourteen-year-old, simply because he would not know what the strange symbols 0, 2, 3, and 4 mean because they weren’t invented till seven centuries after his murder; nor “+” and “-” , German inventions of the fifteenth century; not to mention “=”, which was invented by Englishman Robert Recorde in the sixteenth century. He would also have had a problem with the idea that equations can have negative roots. As to calculus, no chance at all. Of course, Archimedes could have learned readily enough, but he would still have had to spend time just mastering the notation and getting up to date on ideas that were not around in his time. How, then, could Will even understand the problems that the MIT professor was setting his class? However gifted, he would have to have spent less time in his cups, and more time in his books....
The Dyer’s Hand
Mathematicians like nothing better than doing maths, and they spend as much time as possible doing it. Even idiot savants, who are calculators rather than real mathematicians, spend an extraordinary amount of time playing around with numbers and working out problems. They need to, because there is so much to learn. Will Hunting seems to do none of that, and has none of that characteristic passion.
I will argue that differences in mathematical ability, provided the basic Number Module has developed normally in our Mathematical Brains, are due solely to acquiring the conceptual tools provided by our culture. Nature, courtesy of our genes, provides the piece of specialist equipment, the Number Module. All else is training. To become good at numbers, you must become steeped in them. This is the “dyer’s hand theory.”
Surely, you will be thinking, there was some essential and innate difference between the children in your class at school who seemed to find maths really easy and those who always found it a struggle. I am not denying that there may be. In particular, I would not deny that there may have been differences in their capacity for concentrated work or in what things they found interesting. My claim is that there was no difference in their innate capacity specifically for maths.
A GENE FOR NUMBERS
Why are otherwise intelligent children so bad at numbers? To say that they suffer from “developmental dyscalculia” is just to give a name to the condition. There’s no evidence that it runs in families, in the way dyslexia does, though there has been very little research.
One important clue was discovered by Marcel Kinsbourne and Elizabeth Warrington. They found that Gerstmann’s syndrome can occur in children. Sufferers from this syndrome have defective mental representations of the fingers—they cannot tell which finger is being touched, without looking, are confused about left and right, and have difficulty writing and drawing...
[U]sing fingers for counting is intimately linked with the development of normal concepts of numerosity, and the fact that representations of numerosity in the brain are right next to the representations of fingers. If the brain representations of fingers are defective in the child, could this lead to abnormalities in the basic concept of numerosity? Kinsbourne and Warrington found that the children with Gerstmann’s syndrome were certainly much poorer at simple arithmetic than matched control subjects were, and some even had difficulty in counting.25
Byron Rourke, a U.S. psychologist specializing in development disorders, looked at the issue from the opposite perspective. Starting with children whose academic abilities are poor only in arithmetic, he found that many of them also have the other symptoms of Gerstmann’s syndrome.26 Children with Gertsmann’s syndrome often have a history of difficult births or diseases of infancy which may have damaged the sensitive developing brain, but some do not.27 If brain injury is not responsible for all cases of developmental dyscalculia, then what is? Could it be in the genes? If, as the Mathematical Brain hypothesis claims, our genes code for building a specialized Number Module in the brain, then something could go wrong with this process. Of course, finding dyscalculics with no obvious cause of brain injury doesn’t mean that the cause is genetic. One would need to be able to establish a genetic abnormality quite independently of the dyscalculia.
One group with poor numerical skills are those born with Fragile X syndrome, an abnormality of a sex-determining chromosome. The abnormality is now known to be located on the long arm of the X chromosome, though it is still not known why. These children often develop Gertsmann’s syndrome, with severe dyscalculia. Another group are females with Turner’s syndrome. To oversimplify, these girls are missing one of the two X chromosomes that makes them girls rather than boys who have one X and one Y chromosome. This condition seems to have no effect on language skills, memory, or general intelligence, but arithmetical abilities are about two years behind the average for their age...28
The answer to the puzzle has to connect genes, perhaps in the X chromosome, with the development of the parietal lobes, and the parietal lobes with the capacities of the Number Module. And this is what we are currently working on in my lab. With luck, we should have an answer within the next five years.
NUMBERS AND THE DEVELOPING BRAIN
One of the most unexpected recent findings in neuroscience is that the brain turns out to be quite flexible—the technical term is “plastic”— in how jobs are assigned to different areas. These findings support the importance of practice, but they offer no comfort for those who believe in innate talent...
Musicians, pianists, and violinists develop manual skills requiring great accuracy and precise timing of hand movements. We now know that to program the intricate dance of fingers and thumbs, the brain assigns extra brain cells to the job, just as your computer assigns more memory for powerful software. The motor cortex—the region that controls hand and finger movements—of the brains of professional keyboard and string players has been measured using very accurate imaging techniques. This area turns out to be much bigger in the musicians than it is in the brains of otherwise similar non-musicians. And it is not only the motor cortices in each hemisphere that are bigger, but also the white matter that connects them together. The defenders of innate talent might say, “Aha, the people who ended up as professional musicians started out with brains that were bigger (better) in the relevant respects!”, but that would be to ignore one further finding: the size of the motor cortex is related to the age at which the musician began training. The earlier their training started, the bigger their motor cortex...
These long-term adaptations depend on short-term practice effects. Like the text you are now reading, Braille texts need to be proofread before they are published. And Braille proofreaders work about 6 hours a day. Alvaro Pascual-Leone and his colleagues mapped the brain regions controlling the fingers of the reading hand of a Braille proofreader after a working day, and also after a day off from work. The resulting maps were clearly bigger after 6 hours of hard work. The brain recruits more cells to do a job, but when the job is over the cells are re-assigned.29
In the long term, repeated practice at a skill will increase the number of neurons the brain assigns to that skill on a more or less permanent basis, though it seems that if you stop altogether the neurons that are no longer in employment will become available for other tasks. Long-lasting structural changes in the brain are dependent on practice.30 Use it or lose it!
A student acquiring math skills—a strongly “progressive and cumulative” process—can easily fall into “virtuous” or “vicious” patterns of reinforcement or discouragement, which set up increasingly powerful motivation to succeed or fail.
VICIOUS AND VIRTUOUS CIRCLES
If deliberate practice is critical, particularly practice directed to understanding the basic principles of numbers, then it is easy to see why some people get left behind by the educational system, and therefore end up both incompetent and anxious about their incompetence. They don’t do enough practice. But it’s important to understand how this comes about, and why it has particularly disastrous effects where mathematical skills are concerned.
Maths, like music and many other skills, is progressive and cumulative. If you slip behind you will get worse, incurring unwanted reactions from peers, teachers, and parents: disappointment or worse—even anger or ridicule. This will lower your own confidence and your enjoyment of maths tasks. A likely consequence is avoidance, as far as possible, of anxiety-inducing, self-esteem lowering maths tasks. And that means not practising. The gap will continually widen between understanding the ideas needed to keep up with the class and ideas that you have a confident grasp of. And as the “chasm of incomprehension” gets wider, class performance will get worse, anxiety will increase, and practice will be avoided even more: a vicious circle (see figure above).
On the other hand, it is also easy to see how maths abilities can be improved: more deliberate practice. When the grasp of current concepts is sure, and performance on current tasks is good, teachers and peers are encouraging, confidence rises, enjoyment of number increases, practice increases. New concepts are acquired, and you can keep up with or even exceed current conceptual demands. The result: a virtuous circle (see right figure). One can imagine the virtuous circle rotating fastest in prodigies. The degree to which they exceed current expectations becomes greater very quickly. The rewards of excelling at mental arithmetic are, in fact, considerable...
Sir Francis Galton knew that “zeal” and “very laborious work” were essential to eminence in any field, including mathematics. He also believed that differences in our innate intellectual abilities, which went beyond differences in innate capacity for hard work and innate interest and passions, put an upper bound on our potential achievements, on the likelihood of what he called “eminence”. We have seen good evidence—statistical, biographical, psychological, and historical—for the contribution of hard work. There is evidence for zeal (a plausible cause of hard work) from the biographies of prodigies. What is lacking is any evidence for differences in intellectual ability that precede zeal and hard work in mathematics. We have found no warrant for the Will Hunting story. A real-life case like Will, who really didn’t work abnormally hard at maths, would be a counterexample to my thesis.
There is no evidence for individual differences in upper bounds to achievement that cannot be explained in terms of training, the best predictor of achievement (in music, at least) being simply the number of hours of solitary deliberate practice. Of course, the type of practice, the programme of learning, and the inspiration of the teacher are all likely to be important.
It remains a theoretical possibility that some innate intellectual capacity for numbers will predispose a child to apply zeal and hard work to that field, rather than some other; but again there is no evidence that this is so. The intellectual capacity, if it existed, could not be something like general intelligence, since many prodigies score only average, or even well below average, on the relevant tests. This capacity would therefore have to be specific to numbers...
Most of us are born to count, but beyond that the established limits to mathematical achievement are, in Galton’s words, zeal and very laborious work.
I have also discussed people—like Julie and Charles—who never get the hang of numbers, despite hard work and good teaching. These are the developmental dyscalculics. I attribute this to the abnormal development of the Number Module in the brain. Without properly functioning brain circuits for numerosity, the upper bound to numerical abilities is very low: roughly what they count out on their fingers. Most of us are born to count, but beyond that the established limits to mathematical achievement are, in Galton’s words, zeal and very laborious work.