Daniel Ansari pauses to think about our knowledge of the mental processes that underpin children’s learning to read, and he despairs. Not because he believes it to be lacking, but because it is so good. In fact, the head of the Numerical Cognition Lab at Western University in Ontario is perturbed because he knows that we are currently nowhere near having an equivalent understanding of maths.
“We always borrow from the reading field because they are so much further ahead than we are,” he laments. “Sometimes I say we are trying to look for the ‘phonological awareness’ equivalent or equivalents [in literacy] in the domain of maths.”
Research into reading has developed insights into the individual steps that children need to take to understand how words are constructed from individual sounds, and how they are represented on the page. Those who can understand the sound structure of words are more likely to go on to become fluent readers. This insight has underpinned the rise of phonics as the pre-eminent method of teaching to read.
Ansari’s career has been dedicated to discovering how the process of learning about numbers works, in the hope of making similar advances. “I wanted to find ways of identifying the key building blocks of maths, numeracy, and to be able to measure those early in development to see the extent to which individual differences early on go on to predict later individual differences,” he says.
Ansari only arrived at this question after entering the field by chance: he responded to a job advert for a PhD student to research numerical development among children with Williams syndrome – a genetic disorder in which individuals typically have relatively high verbal ability, but mild-to-moderate learning disability. He then became gripped by his research into numeracy, partly being hooked by its practical importance, but also the intellectual puzzle it presents.
“From a more basic science perspective, numbers are really interesting,” he says. “Just like letters, they are cultural symbols, they change over the course of cultural history, so [the research] brings up really interesting questions of how children undergo this learning process.”
The meaning of counting
Using numbers isn’t inevitable. Many researchers studying animals believe that they only distinguish magnitude rather than number; while they may have an understanding of whether something is greater than or lesser than something else, it is not from counting. Similarly, some isolated human communities do not have words for numbers – they just use terms such as many, more and most.
“If you watch children develop, you see that the biggest milestone early on is when children understand the meaning of counting,” says Ansari.
At about two years old, they might be able to count, but having counted objects – one, two, three – they may not be able to say how many they have in total. Understanding that counting tells you the total number of objects – the cardinality principle – is the first step in developing a symbolic representation of numbers, says Ansari. “It takes children a long time to get there,” he adds. “Some researchers have argued that children go from being a one-knower, to a two-knower, to a three-knower, to a four-knower, and then they eventually have this insight.”
Later, they begin to relate the counting words that they have already learned to visual, symbolic representations, in the form of Arabic numerals.
“Early symbolic number learning, which is what I’ve spent most of my time looking at, is very complex. It may seem to an adult to be very trivial, but it takes children a long time,” says Ansari.
They need to learn that the symbol “6” refers to six of anything, that it forms part of a sequence and lies between 5 and 7, that it means the same thing whatever size of typeface it is printed in, and that it forms part of multidigit number sequences where it means something different – 6 or 60 or 600 – depending on its position.
Since many of these skills are acquired before school, Ansari says, he believes that teachers should carry out diagnostic tests when children start school to see how much they understand about numbers, with follow-up tests to check their progress.
He compares this to phonics screening, such as the Year 1 check introduced in England in 2012 – although that test is also an accountability process to ensure schools are teaching phonics effectively, and not solely a means of identifying who is in need of extra support. “I do believe there should be more screening of maths at an early age, but definitely not in a high-stakes way and it shouldn’t take too much time,” he says.
Ansari has developed a two- to four-minute test, called the Numeracy Screener, which has been used by some Ontario schools; he is currently creating a cross-cultural version for eight countries, including Belgium, Singapore and Chile. The Numeracy Screener tests abilities, such as translating patterns of dots into the number symbol representing the correct quantity. His research has shown that this level of ability correlates with children’s general performance in maths years later.
“Children come into the classroom with such different levels of preparation for what they are expected to learn, and teachers don’t always have the tools to understand which groups of children already have a really important concept, and which children do not and need more attention,” says Ansari.
Those who are most in need of attention – children with dyscalculia who have severe difficulty in making arithmetical calculations – have traditionally struggled to get the support they need, at least compared with the research and support systems that exist for those with dyslexia. Both conditions have a similar level of prevalence in the population, but in one 10-year period, dyscalculia attracted less than 2 per cent of the amount of funding allocated for reading difficulties.
The good news is that dyscalculia doesn’t necessarily require a different approach to other maths teaching, says Ansari, although it may demand more intensive support.
“I don’t think that we have any evidence to suggest that children with dyscalculia are qualitatively different from other children,” he says. “So probably the same approaches that work for children with normal maths skills will also work for children who have dyscalculia.”
He says that he is reluctant to tell teachers how they should teach maths – “I’m just a researcher” – but he believes the evidence shows that it is important to help children to understand numerical symbols early on, and not to just rely on countable objects or graphical representations of numbers.
“Helping them to make the link between manipulatives [physical learning aids such as interlocking blocks] and symbols in a playful way – I think that’s really critical,” he says. “We’re seeing how much children’s understanding of what numerical symbols mean, how they relate to quantity and even things such as being able to name a number symbol early on, is a predictor of their later performance.”
One debate that Ansari doesn’t think is worth having is one that often crops up in the discussion over the forthcoming times-tables test: the question of whether it is more important to develop a deep conceptual understanding of maths, or to learn “maths facts” fluently, so they can be rapidly recalled.
“I think those two things go hand in hand,” he says. “It’s sort of a false dichotomy, which crops us time and time again, and [the debate] gets quite emotional. People argue very strongly against the notion that children should be able to fluently and quickly retrieve their maths. There’s lots of research that has shown that procedural fluency, like being able to retrieve your times tables, facilitates your acquisition of a conceptual understanding – that indeed it helps you later on when you’re solving complex problems. You can quickly retrieve intermediary steps and so forth.”
Ansari is speaking over the phone from Singapore, where he is currently working as a visiting professor at the National Institute of Education on a five-year project aimed at optimising the numerical processing abilities of preschoolers. The experience has given him an insight into the methods that propelled the country to the top of the international rankings. “The one thing that always surprised me about the south-east Asian way of teaching maths is that you have real expert teachers teaching maths from first grade onwards,” he says. “You have dedicated maths teachers that undergo constant professional development.”
Singapore’s curriculum systematically focuses on connecting symbolic representations of numbers to children’s understanding of quantity, which they call the symbol and the model method. “I think that’s very powerful,” says Ansari.
Maths also has a different cultural position in Singapore. “In Western countries, I think that we still don’t care about maths as much as we care about reading. We’re more likely to admit that we’re not very good at maths, whereas no one would admit at a cocktail party that they can’t read,” he says. “I don’t get that reaction here. Parents care deeply about maths and do a lot of tutoring at home, and I’m sure that has an effect.”
But those effects can also be troubling. Ansari recalls walking past the country’s learning centres, in which many children receive extra tuition. “Sometimes you walk around at eight or nine o’clock, you see the little shoes outside the door and you know children inside are still doing homework,” he says. “I know that many Singaporeans and many people in the Ministry of Education find that problematic, too.”
Joseph Lee is a freelance writer