Think about your daily activities and just how many involve 'number'. You wake up and look at the clock (base 60), you decide how many pieces of toast you'll eat (exact number) or how much milk you will add to your cereal (magnitude). You search for enough coins to pay the fare (addition) and jump on the No. 293 bus (number recognition). You arrive at the nursery and work out how many children are still to arrive (subtraction). You buy coffee for four of your colleagues and calculate how much cash you need (multiplication) and then you check your change. In the evening, you watch football and become aware of how many numerical computations the players constantly engage in.
As adults, number surrounds us at all times; in fact, it is so pervasive that we barely notice how critical it is to our daily lives. So how does our number knowledge begin, and how does it change over the initial course of development?
NUMERICAL PROCESSING IN EARLY INFANCY
It may surprise you to learn that in the very early months of life, infants are already sensitive to some numerical aspects of visual and auditory displays. Studies have shown that even in the first post-natal weeks, infants can discriminate between displays of two items and displays of three. Thus, if babies are repeatedly shown on a screen a series of two-dot displays in different sizes, different colours and different configurations, they become bored (technically speaking, they 'habituate'). Then, if the experimenter changes the display to three dots, they suddenly renew their interest by looking longer (they dishabituate).
In other words, despite interesting changes in colour and size, babies also seem sensitive very early on to differences in number - that is, they notice the difference between two and three. But this only works for displays of one, two or three objects. If young babies are habituated to four objects, they will not dishabituate when shown five objects. In other words, they will not show sensitivity to that change in number. The same applies to auditory sounds; they discriminate between two and three drumbeats but not between four and five drumbeats.
ADDITION AND SUBTRACTION DURING INFANCY
Imagine the following scenario. A little bear puppet comes on to a toy stage with one pot of honey in his hand. He places the pot in the centre of the stage and as he leaves with empty hands, a screen comes up and hides the pot. He then comes back with another pot in his hand, goes behind the screen, and exits with nothing in his hand. What do you expect to see when the screen is lowered? Two pots, of course.
You would be surprised if you saw only one, or if there were three. Well, babies as young as five months are surprised too. Using the 'violation of expectation' paradigm, developmental psychologists have shown that babies can add 1+1 and 1+2, build mental images of what they expect the outcome to be, and show surprise if the outcome is different.
And they can do the same for simple subtractions: leave two objects on a stage, hide them with a screen and then overtly take one away; infants expect one to be left and will show surprise (longer looking) if there are two left on the stage, or if the stage is completely empty.
So what do these findings mean? Can infants count? Or is this more about an ability to track objects as they move about in space, provided the number of objects is small (three or fewer)? The latter explanation would seem more reasonable, given the upper limit of four. But what about large quantities? Are infants sensitive to differences in displays, which adults can readily discriminate just by looking, even though they cannot count them accurately if they are asked to judge the differences very rapidly?
APPROXIMATE NUMBER: MAGNITUDE DISCRIMINATION
Experiments have shown that, indeed, by six months infants show sensitivity to differences in large approximate numbers - that is, magnitude - as long as the ratio is 1:2. Thus, if shown a series of displays of eight items until they habituate, and then shown a display of 16 items, babies will suddenly renew their interest and look longer.
However, at seven months, they do not do so if displays of eight are followed by a display of 12 items - that is, a ratio of 2:3. They are not yet sensitive to these smaller differences in magnitude. It is only by about nine months that they successfully discriminate ratios of 2:3.
Sensitivity to such large numerosities obviously cannot be explained by their ability to track individual objects, so it is highly likely to have something to do with the ability to roughly process different quantities quite early in the first year of life.
PREDICTIONS FOR LATER NUMERICAL ABILITIES
So is small exact or large approximate number discrimination predictive of later numerical abilities? To find out, sensitivity to small and large numerical differences was compared in two developmental disorders with a similar overall developmental delay.
A cross-syndrome study of infants with Down syndrome (with an extra third copy of chromosome 21) and infants with Williams syndrome (much rarer, due to a deletion of some 28 genes on one copy of chromosome 7) tested their ability to discriminate small exact number and large approximate magnitudes.
Interestingly, the infants with Williams syndrome looked just like typically developing control children on small-number discrimination, yet they failed to notice obvious differences in large magnitudes of the ratio 1:2. The infants with Down syndrome showed the opposite pattern: they discriminated the large numerosities but failed on the small exact numbers.
Yet when studied as adolescents and adults, it was those with Down syndrome who significantly outstripped those with Williams syndrome on a broad battery of number tasks, although both syndromes display clear number deficits compared to healthy controls. This suggests that one of the early foundations of later numerical abilities is, in fact, being able to discriminate largenumber magnitudes in infancy.
These approximate-number discrimination capacities emerge before language and well before the ability to count. Yet surely counting is what numbers are really about? Or is it?
WHAT DOES IT MEAN TO COUNT?
From the moment they start to speak, toddlers quickly learn something about the words in the count sequence. Parents tend to count as they climb stairs, count objects on a play table, count as they spoonfeed their children, count the children lined up for nursery, and count anything they can when they interact with their little ones. Toddlers rapidly pick up on this interesting adult behaviour.
But, initially, counting for toddlers is little more than a poem with count words. At first, 'one, two, three' has little to do with the cardinality of the set '3'. To represent number, the child has to understand that each item can only be counted once and no item can be skipped (one-to-one mapping), that items can be counted in any order ('three' is not the name of the third item), and that the final number of the count represents the cardinality of the set.
How often have we asked, 'How many?' and the child goes, 'One, two, three, four, five, six, seven' and stops? And you repeat, 'So, how many are there?', and this question sets them off counting to seven again and again and again. Their proficient counting doesn't yet mean that they understand cardinality.
Rhythm matters too. If you spread out objects unevenly, very young children find it more difficult to count them because, as a mere poem, the count list has to be produced with an even rhythm.
In a clever yet simple experiment, two-and three-year-olds were asked to count a line of objects. They all succeeded. They were then asked to fetch for the experimenter four marbles from a large bowl of marbles. Although they could easily count well beyond four, many of them had absolutely no idea that counting was even relevant to the marble task. They merely grabbed a handful of marbles and gave them to the experimenter.
None of the two-year-olds and only some of the three-year-olds actually used counting to pick out four marbles one by one. It turned out to be quite some time after they had learnt the count sequence by rote that they really understood the meaning of counting.
EVOLUTION AND THE NUMERICAL BRAIN
The human capacity for mathe- matics seems to be rooted in a basic brain system for the mental manipulation of approximate numbers, very ancient in evolution. Indeed, we share this system with many other species (chimpanzees, birds - even fish), and it seems that it emerges very early in human development, independent of language.
It is, of course, a primitive system, capable only of basic computations, such as estimation, comparison, addition (there's more) and subtraction (there's less) of approximate magnitudes. Yet it is on the basis of this simple evolutionary foundation of approximate number that human cultures have gone on to invent increasingly complex cultural tools such as Arabic symbols, counting routines, algorithms for exact addition, multiplication and so forth.
What do we know about the way in which the brain processes number? According to several authors, the adult brain comprises two different circuits involved in exact-number calculations (left angular gyrus area and other left-hemispheric perisylvian areas) and approximate large-magnitude computations (intraparietal sulcus). But research still has to establish whether the infant brain starts out with two different circuits for numerically relevant computations, or whether, as in other domains, it starts out with much bilateral processing across various brain regions and only gradually specialises and fine-tunes to specific number-relevant areas.
FURTHER READING
- Brannon EM, Abbott S, & Lutz D (2004) Number bias for the discrimination of large visual sets in infancy. Cognition, 93:B59-B68.
- Butterworth B (2000) The Mathematical Brain. MacMillan.
- Dehaene S (1997) The number sense: how the mind creates mathematics. Oxford University Press.
- Karmiloff, K & Karmiloff-Smith, A (2010) Getting to Know Your Baby. London: Carroll & Brown.
- Karmiloff-Smith, A (1992) Beyond modularity: A developmental perspective on cognitive science. Cambridge, MA: MIT Press.
- Lipton JS, Spelke ES (2003) 'Origins of number sense: large-number discrimination in human infants'. Psychological Science, 14:396-401.
- Paterson SJ, Brown JH, Gsodl MK, Johnson MH & Karmiloff-Smith A (1999) 'Cognitive Modularity and Genetic Disorders'. Science, 286, 2355-2358.
- Paterson SJ, Girelli L, Butterworth B & Karmiloff-Smith A (2006) 'Are numerical impairments syndrome specific? Evidence from Williams syndrome and Down's Syndrome'. Journal of Child Psychology and Psychiatry, 47(2), 190-204.
- Van Herwegen J, Ansari D, Xu F & Karmiloff-Smith A (2008) 'Small and large number processing in infants and toddlers with Williams syndrome'. Developmental Science, 11(5), 637-643.