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 1500 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 (3×83+7×82+2×81+0×1), 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 sing-song 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 less 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.