Learning & Development: Estimating: Part 1 - Roughly speaking

Most people associate mathematical skills with the ability to count or solve complex equations. Counting is an ability that is culturally dependent as it relies on language abilities and thus, the development of counting can only start once some basic linguistic milestones have been reached, generally around the age of 18 to 24 months. Yet we know that pre-verbal infants can discriminate between different amounts, even if they do not have language.

They are able to do this because humans possess two number estimation systems: one is precise and works only for small quantities (called 'subitising'); the other, meanwhile, is imprecise and works for larger quantities (called the approximate number sense).

SUBITISING

Subitising is the ability to estimate quickly the number of 'items' in a visual field at a glance, without counting. For example, in experiments where we show adults a set of dots on a screen for a very short time and ask them to say how many there are, they are very quick to identify up to four dots. For larger numbers, response times become slower and more errors are made.

So, the subitising system is fast and precise, but only when used with up to four items.

One theory suggests that the subitising system is similar to a memory, rather than a number, system - and that because our memory is limited, there is likewise an upper limit for identifying quickly the number of objects at a glance.

APPROXIMATE NUMBER SENSE

In other experiments, we show adults larger numbers of dots for a very short time and they have to say which number set is greater. Adults can determine between two sets of items (for example, eight versus 16 dots) without knowing the precise number of items in each set, using their approximate number sense (ANS).

Reactions are faster and more accurate where there is a big difference in the number of dots on each side (say, eight versus 16 dots) than when the numbers are closer together (eight versus ten dots).

The ease of distinguishing between the two displays relies on the ratio between the numbers (that is, how far apart numbers are from each other on a number line). This is known as Weber's fraction: as the ratio between the two sets of objects increases, the more easily the difference between the sets will be perceived. Thus, in contrast to the subitising system, the ANS deals with large numbers but relies on number ratios and is, therefore, imprecise.

IN OUR EVERYDAY LIVES

All of us use our ANS abilities in daily life. For example, when we take a taxi we need to know approximately whether we have enough money to pay the driver at the end of the journey. When we pay a bill, we use our ANS abilities to check whether that bill is roughly correct. When we need to get to an appointment, we need to estimate how long it will take us to get there, and so on. So, we use ANS a lot to estimate large or unknown quantities. However, the ANS system only allows an approximation of the amount.

We also use our subitising system in daily life. Subitising mainly works on numbers smaller than four or displays that we have seen often enough that we do not need to count the objects again. For example, we know that a hand has five fingers and that two hands have ten without counting them. Likewise, we do not have to count the actual dots on the dice when we play board games as we recognise the faces of the dice and instantly know the number displayed.

This system relies on memory and thus we need to have been exposed to the displays before we can estimate accurately a visual array at a glance without counting.

INFANTS

When we look at studies in infants, we find that infants can also distinguish between two sets of dots, for both small and large numbers. In these experiments we use a 'habituation-dishabituation' approach: while the infant is sitting in front of a screen (on a parent's lap or in a car seat), we present them with pictures of a certain number of dots until the infant gets bored and looks away from the screen. Infants generally look longer at something new.

After we have habituated the infant with a certain number of dots, we then show them a new number of dots and see how long the infant looks at these before looking away again. Infants look longer at three dots when they have been habituated with two dots, and thus can distinguish between small numbers. Infants can also discriminate between sounds and actions, but the upper limit is three. So, infants have a subitising system just like adults which is very precise but has an upper limit of just three.

Yet, as with adults, six-month-old infants can discriminate between larger numbers of dots when there is a large ratio between them (that is, eight versus 16 dots, say, but not eight versus 12 dots). This ability improves with age so, for example, when infants are nine months old, they can discriminate between eight and 12 dots but not eight and ten dots (see research by Fei Xu and colleagues at the University of California, Berkeley). So, just like adults, infants have an ANS that is imprecise, depends upon ratios and improves during their development.

IMPORTANCE

Studies that have investigated which abilities predict our number skills later in life have found strong correlations between large number discrimination and maths skills, and that ANS abilities are a better predictor than counting ability or subitising skills.

For example, children who were better at deciding which side of the screen showed more dots in an ANS game before they were old enough to enter formal education scored higher two years later during formal mathematical tasks.

Similar correlations have been found in adults, pre-schoolers, and children with mathematical difficulties. Thus, the better your ANS ability, the better you are at mathematics, regardless of overall intelligence (see research by Lisa Feigenson and her colleagues at Johns Hopkins University, Baltimore).

It is unclear why ANS is important for mathematical ability, but one theory is that our ANS abilities work as a checking system that provides us with feedback on whether or not a sum is correct; therefore greater ANS acuity results in a faster and better error-checking system.

MORE INFORMATION

- Feigenson, L, Dehaene, S, Spelke, ES (2004) Core systems of number. Trends in Cognitive Sciences, 8(7), 307-314

- Mazzocco, MMM, Feigenson, L, Halberda, J (2011) Impaired acuity of the approximate number system underlies mathematical learning disability (Dyscalculia). Child Development, 82(4), 1,224-1,237

- Mazzocco, MMM, Feigenson, L, Halberda, J (2011) Preschoolers' precision of the approximate number system predicts later school mathematics performance. PLoS One, 6(9): e23749

- Xu, F (2003) Numerosity discrimination in infants: Evidence for two systems of representations. Cognition, 89(1), B15-B25

SUMMARY

- Both infants and adults are good at estimating numbers and we use two different cognitive mechanisms to do so.

- For small numbers (up to four), or displays we have seen many times before (for example, a die), we can use our subitising abilities, which provide us with an exact number.

- For large numbers or displays we have never seen before, we use our approximate number sense (ANS).

- However, the ANS is imprecise and relies on ratios so it cannot give us an exact number.

- ANS ability has been found to be a strong predictor of maths ability later on in life.

Part 2 will discuss how we can improve ANS abilities in young children, in Nursery World 10-23 March

Dr Jo Van Herwegen is a senior lecturer in psychology at the School of Psychology, Criminology and Sociology at Kingston University London