Is 2/3 bigger than 3/5?
How quickly and confidently you can answer this question may depend on your age, education – and possibly, on your native language.*
According to a growing body of research, the words that different languages use for numbers can affect how easily we learn to count and understand basic concepts such as fractions.
For children taking their first steps into the world of mathematics, this can mean that some are presented with additional challenges based on the language they speak, while others are offered a head start.
They may for example find it more or less difficult to answer seemingly simple questions like "Which number is bigger – 17 or 70?" or "How many quarters in a half?".
While the effect is subtle, exploring it can help us understand the deeper factors that shape our maths ability – and perhaps, allow the many children and adults who struggle with maths to see their problem in a new light.
A schoolgirl in Germany does her maths homework (Credit: Getty Images)
Is counting easier in Chinese?
Let's first consider the difficulties that a child may face with counting. In English, there is no systematic rule for the naming of numbers. After ten, we have "eleven" and "twelve" and then the teens: "thirteen", "fourteen", "fifteen" and so on. If you didn't know the word for "eleven", you would be unable to just guess it – you might come up with something like "one-teen".
Even more confusingly, some English words invert the numbers they refer to: the word "fourteen" puts the four first, even though it appears last in the number 14 (we'll look at the impact of such inversions later).
For multiples of 10, English speakers switch to a different pattern: "twenty", "thirty", "forty" and so on. It can take children a while to learn all these words, and understand that "thirteen" is different from "thirty", for example. In the meantime, they may unconsciously try to make the pattern more regular, by slotting words like "five-teen" or "twenty-ten" into their counting sequence.
Other languages have even more intricacies. In French, for example, numbers are named somewhat consistently up to 60, after which the system changes to a so-called vigesimal structure, meaning, it is based on multiples of 20. The French word for 71 is soixante-et-onze (sixty-and-eleven), for example, and 99 is quatre-vingt-dix-neuf (four-twenty-nineteen).
Even native French-speaking children seem to wrestle with this system: in one study, they performed worse in transcoding numbers over 60 compared with English-speaking children. Transcoding means correctly converting words into numbers and vice versa, for example, reading out 71 as seventy-one, or indeed, soixante-et-onze.
A primary school child in Cardiff, Wales, writes down maths excercises (Credit: Getty Images)
Chinese children were about a year ahead of American children in their capacity to count to higher numbers
In Chinese, the number words lack these irregularities. Once you know the words for one through 10, you can easily infer all the others.
For example, the word for one is yi, two is er, and ten is shi. Eleven is shi yi (ten-one), twelve is shi er (ten-two) etc. Twenty is er shi (two-ten), twenty-one is er shi yi (two-ten-one). This consistent characteristic is known as linguistic transparency by psychologists, and it was long thought to aid children's first steps to basic numeracy. (BBC Future has reported on a similar effect in writing.)
In the mid-1990s, Kevin Miller at the University of Illinois at Urbana-Champagne and colleagues put the idea to the test by comparing the numerical abilities of four- and five-year-olds in the US and mainland China. They found that the children from both countries were equally able to count up to 12, but the Chinese children were about a year ahead of the Americans in their capacity to count to higher numbers.
Further studies suggested that Chinese children find it easier to grasp the basic logic of our "base-10" counting system. Put simply, this is the fact that we use multiples of tens and units to represent numbers and that the order of the digits tells us which is which. In Chinese, this is more obvious: er shi, "two-ten" is easily understood as 2 x 10 = 20. The English word "twenty" doesn't spell this out so clearly.
To investigate whether this makes a difference to children's understanding, six-year-olds of various nationalities were given one set of blocks to represent tens and another set to represent units. Their task was to use the blocks to illustrate different quantities. Children in China and other East Asian countries with greater linguistic transparency were more likely to represent larger numbers using a combination of both sets of blocks, while those who spoke English, Swedish or French were more likely to count out the larger numbers in individual units.
As intriguing as these studies are, they couldn't rule out the potential influence of the different education systems in the different countries – it's possible that maths is just taught more effectively in some countries than others. However, a clever test of Welsh speakers in the UK managed to rule out this confounding factor.
Like their Chinese equivalents, Welsh number words have greater linguistic transparency. The Welsh words for one, two and 10 are "un", "dau" and "deg". Eleven in Welsh is "un deg un" (one ten one), twelve is "un deg dau" (one ten two), and twenty-two is "dau ddeg dau" (two tens two). Crucially, children in Welsh-speaking schools follow the same curriculum as those in the English-speaking schools.
When Ann Dowker, a lecturer in psychology at the University of Oxford, learnt about Welsh's linguistic transparency, she saw it provided the perfect opportunity to study the effects of the language's counting system on children's mathematical ability, without educational differences potentially muddying the results.
Dowker's findings were nuanced. She found, for instance, that six-year-olds who spoke Welsh at home and school made fewer errors when reading aloud pairs of two-digit numbers. They were also better able to point out which was the bigger of the two, compared to those who spoke English. "There was a significant advantage," she says.
However, these benefits didn't seem to translate to advantages in other measures of general mathematical ability. For this reason, Dowker concludes that the effects of language on numerical ability are subtle and specific rather than large and "pervasive". She certainly doesn't believe that linguistic transparence, alone, could explain why East Asian countries tend to be placed higher in educational league tables.
Cross-country comparisons within Europe support this position. Consider German, which shares many of the irregularities seen in English, including the inversion of certain numbers. Forty-five, for example is fünfundvierzig in German (five-and-forty). Some studies suggest that inversion confuses German children as they learn to write numbers as digits. (Hearing fünfundvierzig they might write 54, for example.) But that doesn't seem to hold them back for long. "Germany does rather well in international comparisons," says Dowker.
Visitors take pictures of an art installation named 100 colours by French architect Emmanuelle Moureaux (Credit: Getty Images)
Even if the influence of language does not extend to the whole of mathematics, emerging evidence suggests it might extend to a handful of skills beyond counting. So far, there is some evidence that language may affect how quickly children learn to use fractions. "When thinking about fractions, we have to look at the big part first and then see how much of that is in the numerator," explains Jimin Park at the University of Minnesota, whose PhD thesis concerns the linguistic representation of fractions.
In Korean, this relationship is explicitly spelled out. The term for 1/3 is sam bun ui il, which translates as "of three parts, one", and 3/7 is chil bun-ul sam, which translates as "of seven parts, three" – where the English terms "one third" or "three sevenths" do not make this so immediately obvious. And this seems to give young Korean children a slight advantage in matching named fractions to diagrams illustrating the quantity, before they have even been taught formal lessons in the idea. "When they have to verbally understand fractions, the Korean children definitely benefit," says Park. Intriguingly, when English children are taught to describe fractions with the Korean style of phrasing, it does seem to improve their intuitive understanding of the quantities.
Neither Park nor Dowker would suggest widespread revisions of how we name numbers, but the simple awareness of these linguistic quirks and obstacles may help teachers to give children the right support.
If nothing else, this research can help to remind us adults of the first steps of our intellectual journey, and take pride in having mastered something as unexpectedly complex as counting. Perhaps it will encourage those of us who feel we are simply bad at maths, to give it another try.
*Answer: 2/3 is bigger than 3/5
David Robson is a science writer and author of The Expectation Effect: How Your Mindset Can Transform Your Life, published by Canongate (UK) and Henry Holt (USA) in early 2022. He is @d_a_robson on Twitter.
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