In Oliver Sacks’ book ‘The Man Who Mistook His Wife for a Hat’ he describes the behaviour of two twins, John and Michael, who had been previously diagnosed as “autistic, psychotic and severely retarded”. What he observed in them, on the other hand, was closer to genius. Their mathematical abilities were so outstanding and unusual that they still provoke questions about our own innate mathematical ability.
Completely incapable of performing simple addition or subtraction, and lacking any concept of multiplication or division, these twins had an extraordinary ability to ‘feel’ or ‘see’ numbers. For example, Sacks recalls an occasion in which both twins said the number 111 aloud when a box of matches fell to the floor. On counting, it was realised that this number had been present in the box. The twins even repeated ‘37’ three times, showing recognition of this as a factor, despite their inability to perform any calculations.
The abilities of the twins also extended to other areas of maths, for example, listing prime numbers of twenty figures or telling the week day of any given date!
This is clearly unusual. For most people, using maths in everyday life is somewhat of a bugbear. Even figuring out the date of next Wednesday can cause a mild headache, let alone a date years into the future. However, we do have some mathematical capability that is seemingly not learnt through school. Children as young as five months have been shown to have an understanding of addition (study by Karen Wynn, University of Arizona). This experiment also relied on attention-grabbing Mickey Mouse dolls, as babies are not exactly known for their mathematical interest. The innate ability to see numbers is known as subitization, but, for most of us, applies only to embarrassingly small quantities.
A further way of learning about mathematical understanding comes from cultural studies. I can’t remember the day on which the chanting of times tables began. Having been drilled into my head so early, since so far back in childhood, I can’t imagine a life without these concepts. For all I know, they could have always been sitting there in my mind, a natural thinking process.
Through studying societies where education and language differ, we can reveal surprising truths about our own abilities and how they arise:
Do we rely on language for mathematical thinking?
Do we have the ability to see numbers?
What about numerical patterns?
The Mundurukú are an Amazonian tribe, living an isolated life in the forest with a rich culture, full of traditions. Their terminology for numbers is extremely limited, only reaching up to 5, with even some of these words appearing to translate to rough estimates, such as four-ish. So how does this affect their lifestyle and understanding? It’s easy to think of language and arithmetic as separate capabilities and people often consider themselves talented at one but not the other, however we rarely think about the way in which they interact. We now understand that human babies and other animals use this rough estimation of numbers, named ‘analogue representation’, rather than exact counting, which is only demonstrated by older humans. For example, when Pica showed Mundurukú individuals different numbers of dots on a screen, their answers were wildly different from those given in the west, for example the number 5 was identified only 28% of the time. It is the innate number recognition ability, differing from our counting system, that is reflected in the Mundurukú language.
Pica himself also provides a fascinating case study of the way in which numerical language can influence culture. On returning from his studies of the tribe he experienced extreme culture shock, having lost track of numerical concepts, causing great problems with timekeeping, among other things.
A further study of this same tribe has revealed findings related to geometry; a concept completely absent from their language. The study, however, found that their ability matched that of US children, suggesting that understanding images such as right angles and equilateral triangles was unaffected by language. Geometric understanding therefore seems to extend more widely across cultures, being less dependent on language. Other studies have also found that young children and tribes have a greater ability to recognise ratios and plot logarithmic scales, than carry out stepwise counting in the way we are accustomed to.
These results may be interpreted from an evolutionary perspective. In the natural environment, we perhaps rely on recognising ratios and rough estimations of numbers. This allows us to, at a flash, recognise which patch of berries it might be more profitable for us to collect etc. An understanding of geometry can be highly beneficial in navigation and other activities. Our style of numeracy, on the other hand, does not have a place in this lifestyle. It is only in modern routines, where exact counting of minutes, pennies or G&Ts has become a necessity.
The Man Who Mistook His Wife for a Hat
Alex’s Adventures in Numberland