Mathematical Brain
1. Thinking By Numbers
2. Everybody Counts
3. Born To Count
4. Numbers In The Brain
5. Hand, Space, & Brain
6. Bigger & Smaller
7. Good & Bad
      At Numbers
8. Home, Street, &
      School Mathematics
9. Hard Numbers &
      Easy Numbers
    Notes & References
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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?)

Chinese Edition)

Mathematical Brain
Brian Butterworth

I am not a mathematician. In fact I am not particularly good at maths or calculation. But, like everyone else, I do use numbers every waking hour of every day. Numbers invade my dreams and fantasies, my hopes and anxieties. Perhaps because we see the world through numerical spectacles which we never take off, even in our sleep, it is hard for us to realize how utterly dependent we are on numbers. I am just looking at the front page of this morning's newspaper. It is fairly typical, I suspect.

The paper costs 45p, it is published on 12 June 1998; Sport is on page 28; the Chancellor of the Exchequer will sell £12 billion of public assets a £4 billion a year, local authorities expect to raise £2.75 billion, public sector salaries will rise by 2.25%, Government investment has fallen by 0.8% of GDP, net investment of infrastructure to 2002 will be £14 billion, public sector spending will rise by 2.75%, the Government will raise £l billion a year from selling unused assets, leader comment is on page 19, financial analysis on page 21; Catherine Cookson dies at 91 years, 13 days before her 92nd birthday, she had 85 best-selling books at the rate of 2 a year since 1950, they had sold 100 million copies, earning her £14 million, she was the 17th richest woman in Britain, her husband Tom is 87, she gave £100,000 to a charity in 1996, her neighbour Gertrude Roberts is 78, the obituary is on page 20, and a notice of the death of writer Hammond Innes is on page 4; Stephen Lawrence, aged 18 was murdered in 1993, his mother is 45, 5 men summonsed to give evidence are aged 21, 22, 21, 20, and 22; the maximum charge for Cable & Wireless telephone calls on Saturdays is fixed at 50p until the end of September, you can call 0800 056 8182 to find out more; there's a horrifying racist murder reported on page 3; the other sections in the paper run from Home on page 5, through pages 15, 17, 18, 20, 23, 24, and 29 to Radio schedules on page 31, and the newspaper's bar code is 9770261307354.

There are 51 separate numbers on just one page. It took me less than five minutes to read over breakfast. I was keen to get to the sporting pages, with World Cup results and cricket scores numbers by the bucketful. So in the half-hour it takes me to get through the paper, I probably see, and at least half attend to, about 300 visual numbers. Radio 4, 93.5 MHZ, was on at the same time, and other numbers were going into my ears, occasionally making it all the way into my consciousness (which doesn't mean that the others were not processed at all). I had to check the numbers on my watch - two out of the twelve - to make sure I wasn't late cooking daughter Anna the approved number of bacon rashers - three. The two digital clocks on the kitchen appliances advanced by 35 minutes while I was reading and cooking. Anna's elder sister, Amy, needed £70 for a school trip. As I walked Anna to school, we passed 73 houses, each with a number, and many cars, also with numbers. All this before work, which of course involves lots more numbers.

At a very, very rough guess, I would say that I process about 1,000 numbers an hour, about 16,000 numbers per waking day, nearly 6 million a year. People whose job entails working with numbers, in supermarkets, banks, betting shops, schools, dealing rooms, will process many more than this.

Behind these numbers are vast systems of other numbers. The time now is set in relation to the 1,440 minutes of a 24-hour day. Today's date is set in relation to the number of days that have passed since 1 January, AD 1; people's ages at death depend on this numerical system. The Chancellor's statistics rest on other statistics of public accounts, economic growth, and so on, which rest on the daily, weekly, and monthly transactions of an enormous number of public bodies, private companies and individuals, including Mrs Cookson, her publisher, its printer, and her accountant and PR company.

Not all numbers are the same. On the front page of that newspaper there are whole numbers and decimal fractions. There are numbers to denote how many things there are in a collection; real visible things such as 5 men, invisible things like 91 years, or things that are potentially visible, I guess, like 50 pence, 100,000 pounds, or 14,000,000,000 pounds. There are also numbers used solely for ordering things in a sequence, for example, Cookson's 92nd year; the date 12 June falls into this category, as does the page number 15. And there are also numbers for which both their value and their position in a sequence are irrelevant: the telephone number and the bar code. These are no more than numerical labels.

I confess that I didn't focus on our dependence on numbers until, as part of my work as a neuropsychologist, I started to test people who couldn't use them. One of the most remarkable was an Italian hotelier, who kept the hotel's accounts until she suffered a stroke, after which she was blind and deaf to numbers above 4. This meant she couldn't shop or make phone calls or do innumerable other things she had previously taken for granted.

Then there was an intelligent young man with a degree and professional qualifications. He was even good at statistics, provided he could use a computer, yet he was unable to do even the simplest numerical things in a normal way. Arithmetic was a disaster area for him, but it went further than that. Most of us can tell how many things there are at a glance, without counting, as long there aren't more than about five. This young man had to count when there were just two things! This wasn't a failure of education - it was something of a very different order.

In the 1980s reports began to appear in the learned journals of experiments showing that newborn babies, who certainly hadn't learned to count, were able to do what this young man could not: recognize the number of things they saw at a glance. Naturally, I tried a version of one of these experiments on our first daughter, when she was just four weeks old. She was sat inside a large cardboard box in which industrial quantities of nappies were delivered, to watch one, two, three, or four light green rectangles appear and disappear on a dark green computer screen. Her response to what she saw was measured by how often she sucked on a rubber teat attached to a pressure transducer connected to the computer. The more she sucked, the more interested she was. We were getting very promising results until the subject decided that she wasn't going to suck the teat any more. She has remained averse to doing things she can't see the point of ever since.

It crossed my mind at the time, though without really registering, that if you are born with the ability to recognize the number of things you see, then maybe you could also be born with a handicap that prevented this ability from developing normally. I remember thinking that perhaps there is a numerical equivalent of colour blindness. Ten years later, I began to wonder whether this was what was wrong with people like the young man who needed to count two items. From this it was a relatively short conceptual step to wondering whether the human genome normally contained instructions for building circuits in the brain that are specialized for numbers. But were these circuits built for just this purpose, or had they evolved for some other purpose and been coopted by the need to cope with numbers?

Colour vision is universal. Everyone, save those with particular identifiable genetic abnormalities, sees the world in colour. But does everyone see, or think about, the world in terms of numbers, save those with some genetic abnormality? If thinking in numbers is something that has to be taught, then there should be people who had not been taught and couldn't do it.

We, in our advanced technological, trading society, need to be able to use numbers, so numeracy has emerged as a key ingredient of our educational system. But what about "stone-age" societies with little technology and little trade? Do they use numbers? Do they count? Is number ability really universal?

Finding out is by no means as easy at might at first seem. Let me give an example. One way to tell whether a culture uses numbers is to see whether they have symbolic representations for numbers, either written or spoken. English has special words for numbers, and a syntax that allows us to name numbers as large as we like, but most indigenous Australian languages have words only for "one", "two", and "many". Users of these languages - particularly the Aborigines of the Central Desert, who are traditional hunter-gatherers - have little to trade. Their technology, although exquisitely adapted to their way of life, is limited to a few implements such as boomerangs, and bark shields and containers. If anyone is likely not to use numbers, not to think about the world in terms of numbers, it is them. The problem is how to tell. Nowadays, all Aborigines have encountered Western money culture, and the English language with its number words and written numerals. So the question turns into a historical one: before contact, did they use numbers? If they didn't, this would be one strike against the universality of a specialized number ability.

An obvious objection to this idea is that, even within a society, some people are really good with numbers while others approach numbers with fear and loathing. Surely, if we are all born with much the same brain circuitry for numbers then our abilities should be much the same, just as almost all of us are born with nearly identical abilities to see colour, or to use language (also thought to depend on special genes, as yet unidentified). But perhaps that's like requiring everyone to show the same sense of colour in the colour coordination of their clothes, or the way they decorate and furnish their homes, or maintaining that we should all be equally good at putting words together to create narratives or poetry. It may turn out that there are basic capabilities that are indeed innate and universal, and that the differences in the level of adult performance will depend on experience and education.

Thinking along these lines, I began to wonder what these basic capabilities could be. The things that babies can do without instruction seemed to me a good place to start. How do babies think of the world? Do they see it in terms of numbers of things, just as they see it in terms of colour? Another line of attack was to see which numerical ideas seemed natural and easy to grasp. For example, I noticed in my own children that what I was taught to call "proper fractions" (1/2, 1/4, 7/8) were easy, while "improper fractions" (3/2, 5/4, 8/7) were hard. Most people find probabilities obscure. Can calculus be made easy? Are the ideas that seem natural and easy those we are born with, or are they just learned earlier, or taught better?

Certainly there is much anxiety about mathematics education. Children are distressed when they fail, as are their parents. Governments worry that their working population is not properly equipped to compete in a highly technological, and therefore highly numerical, world. Could we improve how well we understand mathematical ideas if the education system were to base its teaching more firmly on the mathematical toolkit we are born with?

These are some of the questions that led to this book - They have led me into all kinds of fascinating subjects completely new to me: thermoluminescent dating of rock faces and the intricacies of Venetian house numbering, Aboriginal sign language and New Guinea bodycounting, Ethiopian farming practices and ancient Indus Valley poetry, the origins of number words and the Venerable Bede's system of fingercounting. I have also had to rethink much of what I thought I knew about numbers and about the brain.

On the way I have had help from many people. I have been struck by how willing busy experts have been to answer naive and often stupid questions from a complete stranger. Among them are Jean Clottes, Gordon Conway, Josephine Flood, Les Hiatt, Rhys Jones, Deborah Howard, Alexander Marshack, Karen McComb, Bert Roberts, Robert Sharer, Stephen Shennan, and David Wilkins.

Over the years I have had the enormous benefit of working closely with brilliant and knowledgeable scientists on how the brain deals with numbers: Bob Audley, Lisa Cipolotti, Margarete Delazer, Franco Denes, Marcus Giaquinto, Luisa Girelli, Jonckeere, Carlo Semenza, Elizabeth Warrington, and Marco Zorzi. This work was generously supported by the Commission of the European Union and by the Wellcome Trust.

I have also benefited enormously from discussions with Mark Ashcraft, Peter Bryant, Jamie Campbell, Marinella Cappelletti, Alfonso Caramazza, Laurent Cohen, Richard Cowan, Stanislas Dehaene, Ann Dowker, Karen Fuson, Randy Gallistel, Rochel Gelman, Alessia Granà, Patrick Haggard, Thom Heyd, Jo-Anne LeFevre, Giuseppe Longo, Daniela Lucangeli, George Mandler, Ference Marton, Mike McCloskey, Marie-Pascale Noël, Terezinha Nunes, Mauro Pesenti, Manuela Piazza, Lauren Resnick, Sonia Sciama, Xavier Seron, Tim Shallice, David Skuse, Faraneh Varga-Khadem, John Whalen, Karen Wynn, and the late Neil O'Connor. Some of these discussions were held at a workshop on "The Concept of Number and Simple Arithmetic", sponsored by the Scuola Internazionale Superiore di Studi Avvanzati in Trieste, organized by Tim Shallice, the Director of Cognitive Science. Sean Hawkins and Martin Hill provided invaluable library research. Being Editor of the academic journal "Mathematical Cognition", published by Psychology Press, has helped me keep in touch with the latest developments.

The philosopher Marcus Giaquinto inspired my approach, and he kindly read all the chapters and offered penetrating but helpful comments. Designer and film-maker Storm Thorgerson worked with me on adapting some of the ideas in the book for other media, and tried to ensure that the written material was as clear and gripping to the reader as it was to me. My daughters Amy and Anna were major inspirations, not just because they have been handy, and often unwitting, sources of data on how children construct numerical ideas, but also because their witting insights into their own mental processes have been invaluable. My partner, Diana Laurillard, contributed so much and in so many ways that it is now hard to identify her contributions, except negatively: the least sensible ideas are not hers.

Lisa Cipolotti, Margarete Delazer, and Norah Frederickson expertly scrutinized some of the chapters. My original editor at Macmillan, Clare Alexander, commissioned the book and provided perceptive comments on the first two chapters. Georgina Morley at Macmillan and Dr Michael Rodgers offered detailed advice on many aspects of the text. Stephen Morrow at my US publisher, The Free Press, offered strategic suggestions on the organization of the whole book, as well as detailed comments. John Woodruff scrupulously edited the whole text.

Probably none of this would have happened without Peter Robinson, my agent, whose faith that people would want to read a book with the word "mathematical" in the title reassured me, and persuaded my publishers, that it was a viable project.

For me, this has been a wonderful adventure into the history, anthropology, psychology, and the neuroscience of the ideas that shaped how we all think about the world. I hope you find it so too.

(October, 1998)

Brian Butterworth


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