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
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:
Paperback - 448 pages (2000): Papermac
The number sense:
Paperback - 288 pages (1999):