When it comes to teaching children to read, there is plenty of evidence about the sequential steps that need to be taken, how each step works and the science behind why it works. But when it comes to maths, according to Professor Daniel Ansari, we are a long way from that kind of certainty.
“Many people have been trying to find what underlies arithmetic, what needs to be in place for kids to learn this skill,” he says. “But it quickly became apparent that it was not going to be as straightforward as it was in reading. Maths is a much broader domain in terms of the processes involved.”
Ansari, who is a professor in the department of psychology at the University of Western Ontario in London, Ontario, where he heads the Numerical Cognition Laboratory, has concentrated his efforts on trying to unpick one specific area of maths: the origins of number and how we teach children to count.
Whether we are "natural" counters is, he says on this week's Tes Podagogy (see player directly below), a topic of much debate.
“Some people argue we are born with an approximate sense of number, so just like we can see colours we can see numbers,” he explains. “But other research says it is not numerical quantity we are born to process, but all sorts of quantity – length, height, density, area – and that out of this more generalised magnitude system we build a precise numerical system.
“I think we are still trying to work out the origins of our numerical ability and there remains a lot of controversy in the field around that.”
How we learn to count
Where Ansari and others are starting to make some progress is unpicking one of the very earliest stages of maths: counting.
“We have been grappling with this question about whether counting comes naturally to us.
"There is, as mentioned, a rudimentary sense of quantity that we are born with and which we share with other species – and we can study the mechanisms for this – but it is only humans who have a precise and infinite count sequence to represent the exact number of items in the set.
“So when we learn numerical symbols we have to, in a way, construct new brain circuits, or recycle existing circuitry to create a new system. So in the same way children learn to read by using existing brain structures, to build a reading brain, some would argue the same happens when children acquire symbolic representations of number, starting with count words and going on to Hindu or Arabic numerals. These are new systems we are not born with.”
Steps for maths learning
The most effective sequential process of learning to count is, he argues, starting to become clear.
“We are moving towards trying to understand how children learn numerical symbols and how that provides an important scaffold. Learning numerical symbols is a very complex thing, as not only do you have to learn the symbols – what 4 or 5 means in quantity – but you also have to recognise and decode that symbol into verbal representation then understand its ordinal position.
“When it comes to the development of counting and understanding the meaning of counting – the cardinality principle – we do have good evidence that their ability to recount the count sequence comes before their understanding of the meaning of the count sequence.
“This tends to be a very gradual process – so they first become "1" knowers, so they know that 1 refers to one item. Then they become 2 knowers, and three knowers, four knowers and then they become what we refer to as cardinality principle knowers.
“That transition from procedural ability to count – but not understudying what that means – to having an understanding that counting refers to quantity is one of the most fundamental steps in early numeracy.”
Role of the teacher
What should the role of the teacher be in this process? Ansari explains that children will make the shift from procedural counters to understanding the meaning of counting at different ages – anywhere between the ages of 2 and 5.
He argues that trying to introduce written number representations before the verbal understanding has been reached tends to go against the evidence.
“The most important thing early on when it comes to developing number is first the verbal number sequence and then written number symbols after that. Because that means the process is only transcoding, it is just mapping the understanding you have already on to the written symbols,” he says.
He adds that teachers need to “acknowledge the non-linear nature of development” but that this “is very challenging for early childhood educators” – not least because when children tend to hit Year 1, they are usually expected to be able to use numerical symbols for basic arithmetic when, for some children, they might not be at that point yet.
Ansari believes such expectations may need to be looked at more closely.
“I certainly think we should not think too narrowly about ways of getting an idea of what children know about maths, number, numerical relations early in primary school,” he says. “As with reading, children come into the early years with huge variability in their knowledge and skills, and we need to give children who may have not had the opportunity to learn about Arabic numerals in the home to show what they know about quantity. That way, teachers can scaffold them adequately. These children should be acknowledged for what they know and given opportunities to grow their knowledge.”
What might be the best way for teachers to scaffold the early maths learning process? Ansari says we don’t actually know with any certainty, but he does feel that play-based learning is too often misrepresented.
“One of the common misunderstandings is that play-based teaching applies a purely constructivist approach to education, in which children discover concepts through play. But, actually, the most effective play-based work is where teachers use play as an opportunity to carry out direct instruction, to direct a child’s attention to numerical relations,” he says.
“A combination of play and intentional instruction can be very effective. Play-based can be more engaging for children, and more age-appropriate.”
Ansari goes on to discuss how spatial skills are key in early maths learning and what impact screening for difficulties in maths may have on long-term maths attainment.
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