Naze sugaku ga tokui na hito to nigate na hito ga irunoka?
(Why are some people good, but others bad at maths?)
 

 
 
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Naze sugaku ga tokui na hito to nigate na hito ga irunoka?
(Why are some people good, but others bad at maths?)
 
Brian Butterworth
 
Brian Butterworth
 
 

Reviews
 

Confidence Counts
The Mathematical Brain by Brian Butterworth
Macmillan, 446pp, £20: Rating: 4/5
 

Was Einstein's brain physically different, or did he just exercise it well? One man who has thought about it more than most is Brian Butterworth, who claims in a fascinating new book that there is indeed a physical sector of the brain, the number module, which is part of our genetic inheritance. Butterworth argues that the evidence for some sort of awareness of numbers, some sort of ability to answer the question: "how many?" goes back in human prehistory to well before the first art or the first written representations of language. The evidence also exists in the responses of newborn babies. Thrust a white card with two black dots on it in front of a day-old baby. Replace the card with one on which the two dots are more widely spaced and the baby shows some interest, but soon gets bored. When a card appears with three dots, the baby perks up. Not because it's a different card, but because the most important factor in renewing the baby's interest is the change in the number of dots.

Nor is the phenomenon limited to humans: a pride of lionesses in the Serengeti National Park show an ability to count the number of intruders - to calculate whether their own group is larger or smaller, and plan accordingly whether to fight or run. Butterworth has also looked at people incapacitated by strokes or by degenerative brain diseases. People like Signor Strozzi, a market trader in northern Italy who, after a stroke, was unable to add two and two. Or Frau Huber, a farmer from Austria, who recovered from an operation to remove a brain tumour from her left parietal lobe - where Butterworth locates the number module - capable of talking and doing her times tables, but unable to add up. She was unable to make sense of numbers.

So is this yet another attempt to explain human difference in terms of the genetic coding which we receive? Not at all. "Everyone counts" is the central, democratic and enthusiastically-delivered message. There is good evidence that almost all of us are able, from birth, to take in at a glance if there are one, two, three or four in a collection of objects. Even for adults, it takes longer to establish the number of objects when there are more than four. This awareness of number is the foundation-stone from which subsequent mathematical ability is built. So why are some people more gifted than others? Why are some countries better at fostering numeracy than others? The answer, Butterworth argues, lies not in nature but in nurture. Just as Einstein's brain has been found to have unusually densely packed cells in the left parietal lobe, so musicians have been shown to have a bigger and more elaborately connected motor cortex - the region of the brain that controls hand and finger movements - than non-musicians. This is not a sign that these people are predisposed towards music, but simply that practice makes perfect.

The more we find out about the brain, the more we discover its flexibility. It directs more cells towards those parts that are in most use. When pianists stop practising, their motor cortices go on a diet. So does this mean we should be force-feeding maths to our children? Butterworth gives short shrift to the modern fad for rote learning, arguing that what matters is not whether children learn their tables, or whether they use calculators instead, but whether they understand, and enjoy, what they are doing. On the one hand, he says, there is a vicious circle: "Lack of understanding leads to confusion, confusion to anxiety, avoidance, and no further learning." On the other, a virtuous circle, where increased confidence leads to the inclination to practise more. Einstein starts here.

 

John Yandell

 

© Guardian Media Group plc. 1999

 

 

Numbers, their human psychology and cultural history. Restores to numerals - those Indo-Islamic signs the medieval west adopted to record transactions computed on the abacus - a depth of character and long back-stories. Four, eight and nine have dark meaningful pasts, 60 acquired in Babylon its power over the division of time and angles of space and thus over the dimensions of the known universe; the total sum of the Yupno of Papua New Guinea, who figure by naming body parts in sequence, is 33, signifying the penis (Yupno woman don't count). Mega.

 

Vera Rule

 

© Guardian Media Group plc. 2000

 

 


 
 
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What Counts:
How Every Brain Is Hardwired for Math
(Free Press, $26)
The Washington Post: Sunday August 29, 1999
  

In What Counts: How Every Brain Is Hardwired for Math, cognitive psychologist Brian Butterworth argues that we are born with brain circuits specialized for answering the question "How many?" While all of us possess this Number Module, as he refers to it, Butterworth deftly slips in the question of whether a collection of experimentally confirmed number-crunching chimpanzees, ravens and at least one parrot possess a "predecessor" of our Number Module. Nor do we know if these savant-like animals use the same brain areas to carry out their numerical tasks.

We do know that adults with brain damage can lose the ability to perform numerical operations that would provide little challenge to the average primary grader. As examples Butterworth introduces us to a cast of fascinating patients including: Signora Gaddi, who despite otherwise normal cognition cannot count above the number four; Mr. Bell, whose understanding of speech or written language is almost nonexistent but who nevertheless retains a serviceable ability in arithmetic; Mr. Morris, who after hearing a series of numbers cannot repeat back more than two of them yet can carry out accurate mental calculations involving two three-digit numbers. On the basis of these examples Butterworth concludes that "arithmetical facts and arithmetical procedures occupy different circuits in the brain." Even more intriguing, "writing words and writing numerals,reading words and reading numbers all involve distinct brain circuits, despite having common input pathways from the eyes and common output pathways to the hands."

Given this emphasis on the brain as an explanation for mathematical abilities, Butterworth's conclusion that "anybody can be a math prodigy" comes as asurprise. To support this contention, he refers to a famous study by the French psychologist Alfred Binet showing that experienced cashiers at the Bon Marche department store in Paris could calculate more rapidly than two math prodigies competing against them. Butterworth favors the explanation that "in those days a cashier was recognized as highly skilled" rather than the more reasonable one that "self-selection played a part: those who couldn't do [math] or didn't enjoy doing it moved on to other positions within the store." At another point, using the English shorthand for "mathematics," he concedes what we mathophobics have always known: "Having good natural abilities for maths may be exactly the reason for choosing maths in the first place."

  
Richard Restak
© Washington Post,1999.
 


 
 
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What Counts: How Every Brain Is Hardwired for Math
Brian Butterworth
Free Press, New York, 1999 ($25).
Scientific American: September, 1999
 

Butterworth is a neuropsychologist (professor of cognitive neuropsychology at University College London) rather than a mathematician, but he has thought and read extensively about how people deal with math and has concluded that a basic mathematical ability is inborn. He notes that "everyone can count or tally up small collections of objects, and can carry out simple arithmetical operations, whether they are Cambridge graduates or tribesmen in the remote fastnesses of the New Guinea highlands." Why, then, do so many people have a hard time with more advanced forms of mathematics? Because "maths more than any other subject is sensitive to earlier failures to understand." And how well children understand "depends on how well they learn at each stage, and this in turn depends on how well the curriculum is designed and the teaching is carried out." Butterworth writes engagingly about the hardwiring of the brain for mathematical fundamentals and about the amazing quantity of numbers that each of us confronts every day.

The Editors Recommend
© Scientific American, 1999.
 


 
 
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Counting on our brains: Stanislas Dehaene
The Mathematical Brain by Brian Butterworth
Macmillan: 1999. 480 pp. £20
Nature 401, 114 : 9 September, 1999
  

"One, two three, four. My mathematics finishes here." Those are the words of Signora Gaddi, an alert, 59-year-old Italian woman whose puzzling impairment has helped neuroscientists understand how the brain does arithmetic. Signora Gaddi suffered a stroke that damaged the left parietal lobe of her brain. Since then, she has become largely hopeless with arithmetic. She cannot read, write, compare or calculate with any numbers other than one, two, three and four. Even with numbers below four, she is definitely not performing normally. For instance, when shown two wooden blocks, she has to laboriously count on her fingers in order to establish their numerosity. Because Signora Gaddi performs normally on many other tests that do not involve numbers, her affliction can be described as a selective loss of arithmetic.

The detailed study of Signora Gaddi is just one of many fascinating pieces of evidence gathered by Brian Butterworth in his effort to illuminate the relations between brain, mind and mathematics in his book The Mathematical Brain. The title itself is something of a misnomer, for one would search this volume in vain for investigations of the cognitive bases of higher mathematics, or even of simple geometry, algebra or topology. The book focuses on a single mathematical object, but one that is rightly seen by Butterworth as a fundamental cornerstone of the mathematical edifice: the concept of number.

Butterworth's central hypothesis is that our brain is "born to count". Our genes contain instructions that specify how to build a number module, a set of neural circuits specialized for processing numbers. Those circuits, which are associated in part with the left inferior parietal lobe, make us sensitive to numerosities in our environment and allow us to understand and to manipulate numbers mentally. Loss of those circuits, as in Signora Gaddi's case, results in a selective inability to grasp the meaning of numbers. The number module is not unique to humans: behavioural experiments reveal that many animals can also attend to numerosity. What makes the human numerical ability unique, however, is that it can be extended through the invention and spreading of cultural tools, such as number symbols and arithmetic algorithms.

In recent years, the cognitive neuroscience of numeracy, or 'numerical cognition', has emerged as an important area where the interaction between brain architecture and human culture can be studied empirically. The hypothesis of a modular architecture underlying number processing has been fruitful in many areas of research, from developmental psychology to brain imaging, animal behaviour or behavioural genetics. Several previous reviews of these findings are available, some aimed at specialists (for example, The Nature and Origins of Mathematical Skills by J. I. D. Campbell; Elsevier, 1992), others at a wider audience (for example, The Number Sense by S. Dehaene; Oxford University Press, 1997). The Mathematical Brain falls into the second category: it is a skilful overview of the area for the non-specialist, with remarkable depth and breadth in many cases, but with occasional oversights that may frustrate the expert.

Butterworth's review of prehistory is particularly original and commendable. He convincingly pulls together little-known evidence from cave-paintings and bone-carvings to suggest that the dawn of arithmetic in stone-age populations dates back at least as far as 30,000 years. More puzzling, however, is the almost complete omission of brain-imaging evidence in the discussion of the neural bases of the number module. Although the modern tools of positron emission tomography, functional magnetic resonance imaging and electro- and magneto-encephalography have been applied only recently to mathematical cognition, a review of the available evidence would have been welcome, especially since it confirms the presence of numerical circuits in a localized brain region: the left inferior parietal region.

Specialists will be delighted, however, by Girelli and Butterworth's latest evidence on developmental dyscalculia, some of which is published here for the first time. If there is a genetic plan for a number module, then one might expect to find an occasional child who is born without it, either due to a genetic defect or to pre- or perinatal cerebral damage. Butterworth claims to have identified one such patient, Charles, who is "born blind to numerosities". Although Charles is now a very bright adult, with a university degree in psychology, he has experienced profound, lifelong difficulties in mathematics, to the point of still having to count on his fingers in order to solve single-digit addition problems.

Chronometric tests reveal at least two major impairments. First, Charles cannot "subitize": he cannot decide how many items are presented on a computer screen, even if there are only two or three, unless he painstakingly counts them one by one. Second, he has an abnormal intuition of number size, which is reflected in an inverse distance effect in a number-comparison task: whereas we normally take less time to decide which of two numbers is larger as the distance between them gets larger, Charles takes more time for more distant numbers, presumably because he is using a very indirect counting strategy.

Charles has not been subjected to brain imaging, but another case of developmental dyscalculia, recently scanned with the novel technology of magnetic resonance spectroscopy, shows a small, isolated area of damage exactly where number circuits are postulated to lie -- the left inferior parietal cortex.

The finding that early focal brain damage can have such a permanent and restricted effect on mathematical competence is perhaps the best evidence to date in favour of the number-module hypothesis. Such evidence imposes strong limits on brain plasticity and clearly speaks against purely constructivist theories that view mathematical competence as the result of a general learning device.

In the end, I suspect that Butterworth's hypothesis of a direct link between genes, number circuits and higher mathematical competence may be too simple. Still, the cogent arguments of The Mathematical Brain should be required reading for anyone interested in the modularity of higher cognitive functions.

Stanislas Dehaene is at Unicog, INSERM 562,
Service Hospitalier Frédéric Joliot,
4 Place du Général Leclerc,
F91401 Orsay, France.
   
© Nature Macmillan Publishers Ltd 1999
 



 
 
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Try holding up your fingers to represent the number three. The result is more informative than you might expect, as Brian Butterworth demonstrates inThe Mathematical Brain (Papermac £12, 446 pages: published in US as What Counts,Free Press, 26 Dollars). If you flourished your index, middle and ring fingers, he states, you are from Northern Europe or one of its former colonial outposts; if instead you raised your thumb, index and middle fingers, then you hail from the Mediterranean.

This simple test shows that cultural difference plays an important role in the ways people formulate numbers. So is numeracy solely a learnt skill? Certainly not, argues Butterworth, who is a neuropsychologist: his invigorating book fleshes out an idea that number knowledge is innate and universal, a basic capability to be ranked alongside seeing and feeling.

According to his hypothesis, we are born with a mathematical toolkit genetically hardwired into our brains, which Butterworth calls the Number Module. Its operation is automatic, enabling us "to categorize the world in terms of numerosities - the number of things in a collection." A baby, for instance, will have no idea of what pineapples are, but should be able to identify that there are two of them. More advanced abilities, from puzzling over restaurant bills to solving Fermat's last theorem, are built onto this starting-point, and the scale of their development depends mainly on cultural factors.

Butterworth sets out his case by first exploring the origins of number perception. It is a dazzling tour, stretching back to prehistoric times, in which he elucidates the etymology of number words, the emergence of Arabic numerals and the use of body-parts as counting aids. Non-linguistic methods of keeping score are embedded in the terms many people use to describe numbers: indeed, "score" is one such example, as is "digit".

In subsequent steps, he attempts to isolate the part of the brain in which the Number Module might reside. This involves work with individuals who suffer from forms of number-blindness: a stroke sufferer, for example, who could only count to four. Similar examples of localised brain damage allow him to isolate a particular lobe, which in turn leads him to discuss why some people loathe maths and others love it. Forcing blank-eyed children to learn multiplication tables by rote, it turns out, is not the best way to stimulate further interest in the world of numbers. Butterworth's book is itself the perfect panacea for anyone to whom maths is a distant or unpleasant memory. Despite its complexities, he handles his subject matter with great deftness and good humour, while his argument sweeps in epic style from mathematical habits around the globe to the inner working of the brain's hemispheres.

 

Ludovic Hunter-Tilney

 

© The Financial Times, 2000

 

 
 
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Commendations
 
Brian Butterworth is one of the most distinguished exponents of cognitive neuroscience in Great Britain. He has had the courage. and foresight to tackle a difficult and somewhat neglected field of human capability. In what is clearly a ground-breaking work on the psychology of mathematics, he has written a lucid and entertaining book with an enormous range of reference. It is a significant contribution to the history of ideas and a fascinating account of experimental work in this perplexing subject.
 
 

Brian Butterworth shows that mathematics is as natural as breathing. In his fascinating book he tells its where it comes from, how it can go wrong and where it is going. "The Mathematical Brain" is a book that counts.
 
Professor Steve Jones.
 

This is a rich, fascinating and important book,which explores every sort of evidence bearing on the innateness and development of our mathematical powers. Brian Butterworth presents his theme vividly, challengingly, but always courteously and fairly, and seems to solicit the readers own thoughts at every point. I found "The Mathematical Brain" a delightful read.
 
Dr. Oliver Sacks.
 



 
 
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Reviews
Everything counts: Margaret Wertheim
The Mathematical Brain by Brian Butterworth
446pp. Macmillan. £20.
0333735377
Times Literary Supplement
December 24th 1999: Page 24
  

How is it that Homo sapiens - evolved to meet the raw contingencies of life on the African savannas - came to have brains capable of handling calculus and trigonometry, or the arcane abstractions of topology and group theory? The mystery of humankind's mathematical ability has become the focus of intense research and speculation in recent years, as few of our abilities seem so extravagantly unnecessary from a Darwinian point of view. Brian Butterworth, Professor of Cognitive Neuropsychology at University College London, is the latest to weigh in on this fascinating subject with his new book, The Mathematical Brain.

Until recently, Butterworth notes, significant mathematical ability was regarded as a talent vouchsafed to relatively few individuals. One either had it or one didn't, was the prevailing view - and most believed they didn't. Butterworth wants to release us from this myth. At the heart of this deeply humanist work is the message that the ability to do mathematics is a primary human trait that every one of us possesses in abundance. From his own research, and that of a growing group of scientists studying both human and non-human facility with numbers, Butterworth has concluded that "we are born with brain circuits specialized for identifying small numerosities" - that is, the number of things in a group (such as the number of deer at a water hole). He calls this set of circuits "the number module", and it is, he writes, "the start-up kit for all our mathematical abilities". As evidence for the innateness of at least some mathematical ability, Butterworth surveys a wide swath of cultures and finds none in which this ability is totally absent. The Aranda people of central Australia have no words for numbers above three, but they answer the question "How many?" by drawing lines in the sand. Likewise, the Yupno people of highland Papua New Guinea have no specific number-words, but they count using their bodyparts, including fingers, toes, facial features, and testicles. No culture we know of is without a sense of number, Butterworth says.

There is even evidence that our Homo erectus ancestors were able to count. Recent research has also revealed that babies only a few months old are able to distinguish between one and two objects, and infants of eighteen months can do simple addition and subtraction. Moreover, the basics of a mathematical brain are not confined to humans; birds have been taught to match numerosities by choosing a box with the same number of dots as on a card, lions can count the number of foes in an enemy pride, and a chimp named Sheba has learnt both to add and to use the first three Arabic numerals.

But if nature provides us with the rudiments of a neuronal mathematical machinery, that still leaves open the question of higher mathematical function. For Butterworth, the key is not in neuronal structure but in culture. Thus, New Guinea highlanders are no less mathematically able than Westerners, they are just less practised, largely because, Butterworth says, the contingencies of their environment have produced little pressure for the development of this skill.

Likewise, Butterworth believes that mathematical geniuses of any culture do not have different brains from the rest of us. The secret to being a Gauss or a Ramanujan is practice, he says. As with great musicians, so, too, great mathematicians spend vast amounts of ti playing with numbers, learning their mathematical scales, as it were.

"Prodigies work very hard to learn the tricks of their trade." If all of us - except certain stroke victims and those with rare neuron aberrations (who form an invaluable research cohort) - have an innate mathematical a ability then why are so many of us so bad at basic maths? According to Butterworth, the answer lies in our educational system; most children are taught mathematics in a way almost guaranteed to stamp out enthusiasm for the subject. is particularly incensed by the practice of forcing ch children to learn tables and rules by rote. What Butterworth would like to see is revolution in mathematics education where focus would shift to making maths fun and to reinforcing children's already sophisticated though often idiosyncratic, mathematical intuitions and "street" learning.

Much of Butterworth's book covers similar territory to Stanislas Dehaene's wonderful The Number Sense (reviewed in the TLS on September 11, 1998). Both conclude with call to arms for the revitalization of maths education by building on, rather than denying children's innate number instincts. Both should be read by anyone interested in the future of education.

 
© Times Literary Supplement, 1999.
 



 
 
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Counting Body Parts: John Allen Paulos
The Mathematical Brain by Brian Butterworth
Macmillan: 1999. 446 pp. £20
0333 735277
The London Review of Books:
20th January 2000: Pages 27-28
  

Most people nowadays who claim to lack a 'mathematical brain' can easily sit down to multiply 231 by 34 or divide 2119 by 138 and come up with the answers. Yet in the 15th century Northern European merchants had to send their mathematically gifted sons to Italy to learn how to accomplish these feats. Arabic numerals were not yet in wide use, and German universities weren't the place to find out about the arcane arts of multiplication and division. Before smiling indulgently, however, try multiplying the Roman numerals DCL and MLXXXI or dividing MDCCCVII by CCLXIV without first translating them into our own system of numerals. So who has more number smarts, the present-day self-styled innumerate or the mathematically gifted German student from five hundred years ago?

Brian Butterworth, a cognitive psychologist who has done much work on the neural and cognitive bases of mathematical thinking, says it's a tie. The thesis of The Mathematical Brain is quite simple. The left parietal lobe of the brain contains certain specialised circuits, which Butterworth terms the Number Module, that enable all of us to recognise small numbers automatically, to match up the elements of small collections of objects, and to tell which of them is the larger. We unthinkingly perform these tasks and others depending on them in something like the way we take note of colours without consciously trying to do so.

Any numerical achievements beyond this (multiplying and dividing, for instance) are a result of our slowly mastering various conceptual representations of numbers as determined by the surrounding culture. These include body parts - fingers primarily, but, as Butterworth notes, for the Yupno people of New Guinea, other appendages as well, ranging from nipples to penises - specialised counting words, external aids such as tallies and calculators, and written symbols such as Roman or Arabic numerals. Other cultural tools enable us to master more advanced mathematical notions such as probabilities, differential equations and infinite sets.

Nevertheless, Butterworth claims that we all start with the same basic equipment, the basic mathematical brain. To establish this claim and defend it against rival accounts, he conducts a fascinating, if at times maddeningly repetitive, tour of the relevant research in neural and cognitive psychology, digressing occasionally into general psychology, ethnography, ethology, history, mathematical pedagogy and, near the end of the book, some real mathematics. The whole thing is reminiscent of Stanislaus Dehaene's The Number Sense, but without that book's reductionist claims that numbers are somehow present in our brains and virtually a social construction. I was reminded also of Oliver Sacks's The Man who Mistook His Wife for a Hat, since many of the stories of stroke patients included here have a 'man who mistook his 5 for an 8' flavour.

Butterworth begins by describing at length various concrete means of indicating numbers: markings on bones and rocks, pebbles of different sizes, Egyptian hieroglyphics, the ubiquitous abacus and counting board, and, finally, the most personal of personal computers - human hands. Bede devised methods for counting up to a million using the hands, but counting on our fingers is an almost universal phenomenon, ultimately giving rise to the most commonly used written bases. Our base-10 system derives from it, while the French words for 20, 80 and 90 - vingt, quatre-vingt, quatre-vingt-dix - imply an older base-20 system (most likely the result of counting on fingers and toes). The Maya, one of several peoples to invent the principle of positional notation, also used a base-20 System 1,500 years ago to create calendars more accurate than the Gregorian one we use today. Even the ancient Babylonian-Sumerian base-60 system, which survives in our measurement of time, angles and geographic position, was probably derived from finger counting.

We are also given a historical sketch of methods of numerical representation: abstract symbolisation; the tally, collection and composition principles that led on to the idea of a multiplicative base for numerical systems (e.g. base-10); positional notation (826 is very different from 628 or 682); and the holy grail, the invention of zero (allowing us to distinguish easily between 36, 306, 360 and 3006) - all of which are an essential though almost invisible part of our cultural heritage. It is one of the virtues of The Mathematical Brain that it makes this heritage not only visible but vivid. just how much we take it for granted is underscored by the hoopla surrounding the year 2000. To deflate this numerologically inspired enthusiasm, I note that had we adopted a base-8 system of numeration, the year 2000 would be indicated by the numeral 3720 (3 x 83 + 7x82 + 2 x 81+ 0 x 1), a representation much less likely to induce celebration.

Butterworth takes great pains to establish that number notions, words and representations are not, as our alphabet certainly is, an invention that spread from a single source. Rather, he argues, they are part of our neural hardware. To this end he cites evidence drawn from prehistoric cave paintings and markings, the seemingly innate number sense of infants, as well as animals' abilities in a mathematical direction.

What do we know, in fact, of the relationship between very small people and very small numbers? The book describes,experiments in which babies are presented with a series of white cards on which two black dots have been placed. Each card in turn is placed a few inches from the babies' eyes and the length of time they stare at it is noted. The babies soon lose interest but extend the length of their stares once again when the cards are changed for ones carrying three dots. After a while they lose interest in these, but regain it when again shown a card with only two. The babies appear to be responding to the change in number and to be disregarding the colour, size and brightness of the dots.

On the strength of such evidence, Butterworth claims that the babies have a rudimentary sense of arithmetic. (The other researchers he cites agree with him.) In another experiment, two dolls are placed behind a screen in front of them, but only one remains when the screen is removed. The babies are surprised, as they are when one doll placed behind the screen somehow becomes two on removal of the screen. They are not surprised, on the other hand, if two dolls turn into two balls or a single doll into a single ball. The conclusion he draws is that the babies are aware that one and one make two, and that violations of arithmetic are more disturbing to them than changes of identity.

This seems dubious. There is ample room for an experimenter's expectations to skew the statistics, for example. And if a baby looks at an object for three seconds, looks away for two, and then looks back for two more, is this counted as looking at it for three seconds or for seven? It's possible, too, to find alternative theoretical explanations for the phenomena in question, something that's even more true for the research being done into the number notions and arithmetic skills of chimps and other animals.

In attributing a well-developed sense of number to small children, Butterworth is taking issue with Piaget, who believed that a child's numerical understanding is based on a long developmental process: children must first master principles of transitivity, conservation and so on before they can be said, at the age of five or so, truly to understand numbers. In one of Piaget's classic experiments younger children are shown two identical sets of objects. After the experimenter has spread one of the sets out, the children commonly say that it has more objects in it than the other. Butterworth's criteria for ascribing a number sense to children are looser than Piaget's, but it does now appear that young children know more about numbers than Piaget thought, although perhaps not as much as Butterworth claims.

Butterworth's general position is more compatible with that of Chomsky, who has argued for decades that the logic of grammar is hard-wired into our brains and forms an innate cognitive structure. But he parts company with Chomsky when it comes to the origins of the concept of number. Chomsky conceives of it as a special aspect of language, whereas Butterworth believes our numerical notions originate in the Number Module, or those specialised circuits in the left parietal lobe, a claim that finds support in what occurs in victims of brain disorders, with their resulting cognitive deficits and coping strategies. An Italian woman has a stroke, for example, which damages her left parietal lobe and, although her language abilities are unaffected, she can no longer tell without counting whether there are two or three dots on a sheet of paper. An Englishman with Pick's disease can barely speak but retains his ability to calculate. An Austrian woman with a tumour in the left parietal lobe cannot connect the arithmetic facts she recites in a singsong way to any real-world application of them. One patient understands arithmetic procedures but can't recall any arithmetic facts, while another has the opposite condition.

Particularly intriguing is Gerstmann's syndrome, two of whose salient characteristics are finger agnosia (an inability to name one's own fingers or point to them on request) and acalculia (an inability to calculate or do arithmetic). Butterworth's theory here is that during a child's development the large area of the brain controlling finger movements becomes linked to the circuits of the Number Module, and the fingers come to represent numbers. (It's interesting, too, that the 'reading finger' of a Braille reader is associated with considerably more brain cells than are the other fingers.) I am interested in using narrative - stories, vignettes, scenarios - to impart mathematical ideas to the young, and although Butterworth doesn't devote much time to this Chomskyian issue, one must assume that the large areas of the brain involved in the development of language also become linked to the Number Module.

Since the Module must be similar in everyone in whom it develops normally, Butterworth argues that the primary reasons (sometimes he appears to be saying the only reasons) for disparities in mathematical achievement are environmental - the quality of teaching, the amount of exposure to mathematical tools, motivation. He cites the burden imposed on students by the cumulative nature of mathematical ideas and points to the self-perpetuating nature of different attitudes to the subject, contrasting in particular the virtuous circle of encouragement, enjoyment, understanding and good performance leading to more encouragement, with the vicious circle of discouragement, anxiety, avoidance and poor performance leading to more discouragement.

Butterworth reports on the huge disparities between the performance of students from different countries - the score of the average Iranian student is higher than that of only 5 per cent of students from Singapore, for example - to bolster his fairly obvious contention that local curriculum and standard of teaching are highly influential. His pedagogical prescriptions near the end of the book are more or less on the mark (although they don't follow from the neural and cognitive findings in the earlier parts): more emphasis should be put on applications that interest students; mathematics should be made a more engaging subject, via the use of puzzles and games, for example, there should be emphasis on drill and rote memorisation, although some drill is needed, so long as it's not mind-numbing long division; there should be greater freedom for students to discover mathematical notions, or at least play around with them, on their own. What Butterworth doesn't cite, however, is the evidence we have that discovery learning of this sort isn't very effective in the less elementary areas. Very few students are going to come up with the Poisson distribution or the fundamental theorem of calculus on their own.

The best thing about The Mathematical Brain, its scope and variety, is related to its main weakness, its bagginess. The superiority of Chinese number names, the Indian mathematical genius Ramanujan, this patient or that with an obscure neurological deficit, Pascal's triangle, Indo-European number words, and dialects in finger counting - all these find a place. And for no apparent reason, an appendix even contains an outline of Gödel's incompleteness theorem. Anyone interested in the development of numeracy has plenty to go on in this engaging book.

 
John Allen Paulos is the author of Once upon a Number,
among other books.
 
© London Review of Books, 2000.
 



 
 
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Author Argues that Everyone Is Born with a Head for Numbers:
Malcolm J. Sherman
The Mathematical Brain by Brian Butterworth
Free Press: 1999. 320 pp. $26
American Scientist: Scientists' Bookshelf:
September - October 1999

0333 735277
  

Brian Butterworth, a British cognitive neuropsychologist and founding editor of the journal Mathematical Cognition, has summarized several lines of evidence pointing to the conclusion that the normal human brain contains a "number module" - a highly specialized set of neural circuits that enable us to categorize small collections of objects in terms of their so-called numerosities. When we see three brown cows our brains immediately tell us both that there are three of them and that they are brown. Just as we see colors automatically and involuntarily and without being taught the concept of color, so we immediately recognize and distinguish small numerosities without being taught the meaning of number. In order to communicate, we need to learn the words "brown" and "three," but our perception of small numerosities is as innate and as automatic as is our perception of color.

Of course, through instruction and practice we can greatly extend the capacities of our number module, just as we can similarly improve our ability to read, type or play the piano. But reading, typing and piano playing are not based on hard wired, specialized, genetically created neural circuits ("cognitive modules"); they depend instead on the slow, purposeful development of general-purpose brain circuits ("central processes").

Butterworth disagrees with the influential Swiss psychologist Jean Piaget, who denied that "any a priori or innate cognitive structures exist in man." For Piaget, a child's understanding of number was founded on years of sensorimotor interactions with physical and temporal realities. Before children can understand number, they must master, for example, transitive inference (if a <b and b <c, then a <c) and must be capable of separating number from the sensory properties of objects. Piaget concluded that the concept of number cannot be understood by children below the age of four or five.

Some of the disagreement between Piaget and Butterworth stems from Piaget's more stringent criterion for what it means to understand number. Piaget did not deny that toddlers recognize the difference between two and three. But Piaget did not regard the ability to make this distinction as proof of an understanding of number. In Piaget's famous experiments, children were shown two identical collections, and then the objects in one collection were moved farther apart. The children were then prone to say that the more spread-out collection had more objects. Piaget regarded the failure to realize that number is conserved when objects are moved as a failure to understand number.

More recent experiments, described in Stanislas Dehaene's 1997 book The Number Sense (cited by Butterworth), seem to show that Piaget's subjects knew perfectly well that number is conserved when objects are moved; the problem was that they did not understand the questions they were being asked. According to Dehaene, if children are asked to choose between four pieces of candy spread apart and five pieces close together, they are unlikely to be fooled. Piaget may have underestimated children's early understanding of number, but he is probably still correct in claiming, for example, that children below the age of five or six cannot count two sets of objects and compare them unless the collections are simultaneously present. Such capacities for abstract or symbolic representation may plausibly depend on more than the number module. Butterworth's view of the origins of our mathematical abilities is analogous to linguist Noam Chomsky's thesis that the logic of grammar is built into our brain -that spoken language depends on an innate cognitive structure. (Piaget, of course, denies that such structures exist.) Books and conferences have been devoted to attempts to reconcile Piaget's and Chomsky's views of the foundations of cognition.

Although Chomsky and Butterworth have similar theoretical perspectives about knowledge, they disagree about math. Chomsky sees the number concept as a special aspect of language, whereas Butterworth argues (citing, for example, studies of the cognitive consequences of injuries to various parts of the brain) that math and language use different regions of the brain. Butterworth also disagrees with those who explain math as a combination of language, general intelligence and spatial ability.

Modern cognitive science and physical investigations of brain structure may someday resolve or clarify an ancient philosophical issue: Does knowledge have a large innate component (as Kant, Chomsky and various religious philosophers would argue), or is the mind a tabula rasa whose contents are determined by the social and physical environment (Locke and Piaget)?

What are the implications for math education of various cognitive perspectives? Because Piaget believed that number itself was dependent on abstract and logical thought, Piagetians are prone to deduce that premature exposure to mathematics will lead to rote learning without understanding and to disabling confusions and anxieties. "Developmentally appropriate practice" has become a shibboleth in U.S. schools of education--largely reflecting the Piagetian belief in fixed stages of cognitive development. France severely de-emphasized the early teaching of numerals and counting words, believing that such instruction was useless or harmful. Even if Piaget's theories are right, it is an empirical issue whether it is helpful or harmful to teach children to memorize counting words before they can abstractly link these words to collections of objects.

Although Butterworth sensibly rejects Piagetian-based pessimism about what children are capable of learning at various ages, he is implausibly optimistic about our mathematical potential. Although his central thesis, the number module, is genetic, he argues that the main sources of individual differences in developed math ability are environmental: "provided [that] the basic Number Module has developed normally ... differences in mathematical ability ... 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." Butterworth denies that there is any "essential and innate difference between children ... who find maths [the British usage for math ] really easy and those who find it a struggle. There may have been differences in their capacity for concentrated work or in what they found interesting ... but there was no difference in their innate capacity specifically for maths."

Butterworth cites international comparisons that show large differences in performance (for instance, a test on which the average score of Iranian children is equal to that of the lowest 5 percent of children in Singapore). Cultural resources and pedagogy clearly matter. But it does not follow that all individual differences in developed math ability are due to temperamental and environmental factors (ability to concentrate, ambition, interest and time devoted to math). Intelligence, verbal aptitude and spatial ability are also likely to be important for math.

According to Piaget, children must discover or construct for themselves certain regularities about the world (that objects continue to exist even when we can't see them, for example). Some constructivists go beyond Piaget, claiming that all genuine knowledge must be gained through a process of discovery. Although infants and toddlers do need to learn basic facts and distinctions (hard versus soft, solid versus liquid) through their experiences with external objects, it does not follow that more advanced material must be learned by recapitulating the original process of discovery. If so, the potential for human progress in science and other areas would be severely limited.

Although Butterworth rejects Piaget's theoretical framework, he agrees with most Piagetians in advocating discovery learning. Butterworth argues that schools limit children's potential for growth when they insist that there is a preferred way to do math problems, then drill students in approved methods. He reasons that since we all have a number module, we all have the capacity to work out our own approach. Butterworth approvingly quotes educational researcher Lauren Resnick:

The failure of much of our present teaching to make a cognitive connection between children's own math-related knowledge and the school's version of math feeds a view held by many children that what they know does not count as mathematics. This devaluing of their own knowledge is especially exaggerated among children from families that are traditionally alienated from schools, ones in which parents did not fare well in school and do not expect - however much they desire - their children to do well, either. In the eyes of these children, math is what is taught in school.

But a large body of empirical evidence (not cited by Butterworth) shows that discovery learning is ineffective with all but the most basic material. Few children will discover for themselves efficient ways of multiplying three digit numbers, and virtually none will discover Archimedes' law by experimenting with floating bodies. To be sure, children may work out ways of doing simple arithmetic problems. Butterworth cites as an example an untutored Brazilian coconut seller, who calculated the price of 10 coconuts without understanding decimal place notation. But the ad hoc methods children discover for themselves are most unlikely to be suitable building blocks for more advanced knowledge. Even when successful, discovery learning is inefficient, taking time that could be better devoted to practice. Butterworth correctly relates that great mathematicians all steeped themselves in mathematics. Yet he disparages school practice and somehow regards it as antithetical to understanding.

Butterworth sees the international comparisons he cites as proof that children can learn more math than they typically do. But the best countries (such as Singapore) are the ones that emphasize direct instruction and drill, not the student centered discovery methods he advocates. Butterworth's findings and views of mathematical cognition may well be sound. But the existence of a number module does not in and of itself establish the relative soundness of various educational methods. Doing so would require an evaluation of empirical research in educational settings, and this Butterworth has not done. Cognitive theorists are too prone to jump from models of cognition to classroom practice without empirical testing under realistic classroom circumstances.

  
Malcolm J. Sherman is a professor in the Department of Mathematics and
Statistics at the University at Albany, State University of New York,
where his primary interests include statistics
and mathematics education.
  
© The American Scientist, 1999.
 



 
 
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Journal for Research in Mathematics Education

Neuropsychologist Brian Butterworth (University College, London) weaves scholarly analysis with down-to-earth humor and practical examples as he argues that the ability to do mathematics is inborn, not learned. Butterworth proposes the existence of a "number module" in the brain, an area devoted to the ability to count and to understand numbers. The evidence for this proposal is drawn from history, animal studies, infant learning, and a range of other disciplines. The abilities to distinguish between quantities and to perform primitive calculation seem inherent in infants and even in those animals and birds that have been tested. Studies of stroke patients who have lost their mathematics ability indicate that key mathematical functions reside in the left parietal lobe of the brain. A chapter on the history of various methods of counting on fingers (or other body parts) shows a similar relationship between another specific brain region and mathematics ability. In other chapters, Butterworth explores the question of why some people are particularly good or bad at mathematics and the ways that children learn mathematics at home, on the streets, and in school.

Telegraphic Reviews: November 1999: Page 590
 
© JRME, 1999.
 

 
 
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Two books about mathematical thinking
Helen Joyce: "Plus" Magazine: April, 2002.
 
The Mathematical Brain: Brian Butterworth
The Number Sense: Stanislas Dehaene
 

Over the last decade, the discipline of neuropsychology has shed light on many aspects of human thought. Brain scans, carefully structured behavioural experiments, and the study of individuals who have suffered brain damage, have taught us much about which abilities are native to humans and which learned; which abilities can be lost and what happens when they are.

These two books describe what is currently known about the foundations of human mathematical ability, and speculate a little further than is known. Both are fascinating, full of the sort of information you feel impelled to pass on. They discuss careful and ingenious experiments on children, including newborn babies (don't let your imagination run away with you - these experiments involve nothing more sinister than observing where babies look and how hard they suck a dummy!), which show conclusively that some numerical abilities do not have to be learnt, but are present from birth, hardwired into our brains.

It seems that human beings are born with a few core numerical abilities - for example, we are innately able to tell without counting how many objects are in a small collection, and to predict correctly the results of adding to and subtracting from these small collections. It may well be that the whole abstract edifice of modern mathematics is built on these biologically innate foundations. Studies of patients suffering brain damage make it clear that these abilities are quite separate from general reasoning and language skills - there are unfortunate individuals who are literally unable to count to 2, although their IQ's appear to be normal, and others whose maths is still reasonable, despite their almost total lack of language.

"The Mathematical Brain" and "The Number Sense" cover much of the same ground. Both describe the experimental evidence - some behavioural, some from brain scans - for the existence of this core numerical ability, and its location within our brains (left parietal lobe, apparently). Brian Butterworth and Stanislas Dehaene both discuss the implications of this research for mathematics education, and both describe extraordinary case studies, casting light on our entire understanding of mathematics. In places both books are reminiscent of Oliver Sacks ("The Man Who Mistook His Wife for A Hat"; "Awakenings"), with their meticulous and enlightening descriptions of bizarre and baffling deficits in stroke and accident victims.

However, the models of innate mathematical ability put forward by the two researchers don't fully agree. Butterworth's "number module" is an ability to recognise small cardinalities (that is, to see without counting when groups of objects consist of 1, 2 or 3 things) and to make arithmetic predictions about such small groups. Dehaene's "number sense" is based on an "accumulator" - an analogue procedure which allows us to keep track of quantities of various sizes, although accurately only for small quantities. Such disagreement is hardly surprising when you consider how new and active this area of research is, and no doubt the next few years will clarify the situation further.

The authors' differing backgrounds also show in the two books. Butterworth is a neuropsychologist who came to studying mathematical ability via his work on natural languages. He was intrigued by strange cases of brain-damaged patients - usually stroke victims - who had lost almost all language and reasoning abilities - except mathematical ones. Dehaene, on the other hand, started off as a mathematician, but became fascinated by the abstractness of his subject. He began to wonder where mathematical ability came from, and why some people are so bad at it, and others so good. He now works on the neuropsychology of maths, studying the physical basis for the mathematical abilities he earlier used for research.

Butterworth is a gifted writer, and his understated sense of humour makes "The Mathematical Brain" a pleasure to read. Clearly a man with a mission - to improve mathematical teaching and learning - he is closely associated with efforts to tackle the problem of dyscalculia (the number equivalent of dyslexia) and has advised the DofES on supporting dyscalculic children through the national numeracy strategy.

"The Number Sense", on the other hand, is a translation from the French (by the author), and it shows. The English is idiosyncratic - but soon you stop noticing, because the content is so enjoyable. No doubt as a result of his background as a mathematical researcher, Dehaene is clearly interested in the philosophy of mathematics, and allows himself to wander off in his last chapter into (highly interesting) speculation about the provenance of advanced mathematical ability, and mathematical inspiration.

Why should you read these books? Firstly, because they are interesting. I rate both highly on the most meaningful scale for a factual book - the number of times I was inspired to say "did you know?" to friends and colleagues, all of whom were intrigued (or are implausibly good actors!). Secondly, because their subject really matters. Children are not blank slates, to be inscribed with mathematics according to whichever scheme is currently fashionable. Rather, cognitive science tells us that it is possible to teach mathematics in a way that fits with our psyche, a way that minimises maths-induced fear and boredom. Teachers, parents, politicians and voters (education ranks high on the list of public concerns, according to polls) need to hear what these authors have to tell us about maths education. And thirdly, because to anyone who cares about mathematics - which surely includes readers of "Plus" - the question "What is a number, that a man may know it, and a man, that he may know a number?" (Warren McCullough, quoted by Dehaene in The number sense) must surely resonate.

 

The Mathematical Brain: Brian Butterworth
Paperback - 448 pages (2000): Papermac
ISBN: 0-333-76610-5
 
The number sense: Stanislas Dehaene
Paperback - 288 pages (1999): Penguin
ISBN: 0-14-026134-6
 

Plus Magazine: Issue 19: April, 2002
 
 

 
 
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Amazon.com

At first glance, neuropsychologist Brian Butterworth's "What Counts: How Every Brain Is Hardwired for Math" might infuriate mathphobes who insist that they just can't get a handle on numbers. Could it be true that natural selection produced brains preprogrammed with multiplication tables? Read a few pages, though, and you'll see that Professor Butterworth has more than a little sympathy for the arithmetically challenged, and indeed confesses that he too has a hard time with figures. His thesis isn't that we are born doing math, but that we are born with a faculty for learning math, much like our ability to learn language. He goes on to argue that unique individual differences in this faculty combine with our educational experiences to make us either lightning calculators or klutzes who can't figure tips.

Butterworth's style is perfect for his subject, seamlessly weaving scholarly analysis with down-to-earth humor and practical examples that will satisfy the researcher and the lay reader alike. Drawing on archaeology, anthropology, linguistics, and his own neuropsychology, he makes his case like a masterful attorney while remaining careful to leave room for scientific falsification. The history of counting is engrossing and will be new to many readers, as it has been a rather arcane field until recently - but it's just one of the many new vistas opened for the readers of "What Counts."

Rob Lightner

 



From Kirkus Reviews

A neuropsychologist (University College, London) argues that the ability to do math is inborn, not learned. Butterworth proposes a "number module" in the brain, containing the ability to count and to understand numbers. The evidence for this is drawn from history, animal studies, infant learning, and an impressive range of other disciplines. While few of us are professional mathematicians, numbers are an inescapable feature of everyone's life: grocery prices, phone numbers, children's ages, sports scores, speed limits, interest rates, and many other examples. The ability to use these numbers on some basic level appears to be as widespread as the ability to use language; yet the two appear not to be directly related. Number systems were developed independently in several parts of the world, and there are marked differences between them; the Babylonians used a base of 60, the Mayans one of 20, as counterexamples to the 10-based math Western cultures use. This argues against some single prehistoric genius having come up with an idea that then diffused to other cultures. In fact, the ability to distinguish between quantities and to perform primitive calculation seems inherent in infants and even in those animals and birds that have been tested. Studies of stroke patients who have lost their math ability indicates that key mathematical functions reside in the left parietal lobe of the brain. A fascinating chapter on the history of various methods of counting on fingers (or other body parts) shows a similar relationship between another specific brain region and math ability. Other chapters explore the question of why some of us are particularly good or bad at math and the ways that children learn math at home, on the streets, and in school. Butterworth writes clearly and entertainingly, with plenty of examples drawn from everyday life and flashes of humor that belie the notion that math is a dry subject. A pioneering study of a fascinating area of the human mind.

© 1999, Kirkus Associates, LP. All rights reserved.

 

 
 
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Summary
 

The ability to use numbers has been the key to raising us from makers of stone tools living in caves to creators of modern science living in great cities. But where does this ability come from? Is it like reading, something that depends crucially on learning a system that someone else has invented, or is it as natural as talking? In a new and provocative study, Brian Butterworth argues that our genes contain a set of instructions for building a "mathematical brain", and this is why, without benefit of teaching, human beings are born to count.

But can babies really use numbers? Can remote tribes count., even though they have no number words? Why are some people so good, and others so bad, with numbers? Is there number blindness just as there is colour blindness? Why are some, types of numbers so hard to understand? What connects our hands to our sense of number? Why does schooling leave us so muddled and discouraged that we close the door on our mathematical brain? And what can we, do to open the door again?

Brian Butterworth answers these and other intriguing questions as he takes us on a riveting biological and historical journey from tallymarks on the walls of Ice Age caves and the body-counting of New Guinea to his own cutting-edge research on mathematical brains, normal and abnormal. He tells us about the book-keeper who could no longer count above four, and about the science graduate who has to solve the simplest problems by counting on his fingers, as well as about the, calculating prodigy who can find powers and roots in seconds.

Fascinating, challenging and completely engrossing, "The Mathematical Brain" throws remarkable new light on our understanding of the extraordinary world of numbers.

 


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