Basic arithmetic is more complex than simply putting two and two together. Growing links between educational research, psychology and neuroscience have found that doing even simple addition, subtraction or multiplication uses several different parts of the brain, some not previously thought to be associated with mathematics.
But are teachers making the most of the links between different parts of the brain when they teach? The complex way the brain processes numbers may mean that teachers could use different approaches to teach numeracy successfully to pupils who struggle to grasp number concepts, experts believe.
"In recent years, child psychology and neuroscience have finally begun to shed light on the nature of one of the most elementary components of mathematical knowledge, the concept of number and how it develops with school experience," explains Stanislas Dehaene, Professor of Experimental Cognitive Psychology at the College de France in Paris.
Professor Dehaene trained as a mathematician and computer scientist before turning to psychology and neuroscience, and is now one of the world's leading experts on numerical cognition. He believes such work can help teachers to understand which bits of maths are intrinsically hard for children to grasp and which are easier.
It's not always obvious, but because of the linkages in the brain, basic maths may also help children with other kinds of learning and vice versa - learning grammar or even taking a nature walk can help children develop mathematical skills. Learning a musical instrument or spending time on computer games have also been shown to improve mathematical recall.
Other research shows that learning words in a foreign language has a similar effect.
Moreover, just as reading aloud involves every part of the brain, doing mental calculations out loud also has powerful effects. "You move your mouth, you hear what you say, which involves the auditory part of the brain, and you see the board. The end result is that children get better at mental calculation, but also improve other functions of the brain, including memory," says Bruno della Chiesa, head of the Organisation for Economic Co-operation and Development's Centre for Education Research and Innovation (CERI).
"If you look at the (brain) map of someone performing mental calculations, everything in the brain is active," he adds. Some neuroscientists have described it as watching the brain light up like a Christmas tree.
Teachers in Japan, Taiwan and elsewhere in Asia verbally detail the thinking process as they solve a mathematical problem on the board, helping children to understand the strategy of mathematics. This may even be a key reason for these countries performing so well in international comparisons of mathematics.
In 2007, CERI published its report Understanding the Brain, the Birth of a Learning Science, pulling together research on neuroscience, psychology and learning. The section on maths suggested that mathematical abilities were distributed in several different parts of the brain, which children learn to co-ordinate through their maths lessons in school. This went against the thinking prevalent in the 1980s and early 1990s, that maths was processed in a separate part of the brain from language. It also meant that teachers could not assume that, because some children had difficulties with one topic in arithmetic, they would also have problems with other topics.
For example, neuroscientists working with educational psychologists have found that subtraction is dependent on one brain circuit, while addition depends on another, and multiplication a third. Other research has shown that neural circuits holding algebraic knowledge are independent of those related to mental calculation. "There are multiple circuits for learning mathematics. If you learn multiplication, you use more linguistic circuits than if you do subtraction," says Professor Dehaene.
Similarly, learning multiplication affects different parts of the brain. Drilling times tables uses one area, while learning strategies for multiplying uses others. So the now unfashionable practice of learning times tables by rote may not be as bad or ineffective as we think. Anything that is memorised is consigned to the back (memory) part of the brain, freeing the front of the brain (the learning area) for more learning, according to Brian Butterworth, Professor of Cognitive Neuropsychology at University College London and author of The Mathematical Brain. "Processing is done at the rear as you become more experienced," he explains.
Jan de Lange, director of the Freudenthal Institute for Science and Mathematics Education in the Netherlands, runs the institute's Mathematics Education and Neurosciences project (MENS) and believes that many teachers underestimate what children can do in maths and science. "What I am interested in is to be able to see in the brain what processes - not just multiplication, but real processes of deductive reasoning - are going on and whether we are missing something that we need to address in education," he says.
Some years ago, Professor Dehaene devised the "triple-code model", which demonstrates what happens when children see that the written numeral 3, the spoken word three, and three objects are one and the same thing. He describes this as a kind of "Aha" moment in education - the Arabic numeral is recognised by one part of the brain, speech activates another, and the notion of magnitude activates a different one still.
"There is a triangle linking numerical sense, spatial sense and grammar in the brain," says Professor de Lange. "It is an area really worth exploring further. I don't know any curriculum in which this relationship is used. Arithmetic and geometry are completely different areas of teaching and learning. If the connection is true, then teaching should change a little."
He adds: "We know already that algorithmic processes have strong links with the same areas of the brain as grammar. This leads to the question of why we teach maths the way we do. Should we be teaching logical thinking, which is very close to grammar?"
Professor de Lange also believes that pupils have an innate sense of navigation and space, which could help them to develop their maths skills. "Kids are born with a GPS in their minds. It is the two- or three-year-old in the back of the car who suddenly notices if you do not take the usual route. But we lose these innate navigational skills we have as young children, because we do not use them. I believe we can develop these skills very simply, say, through nature walks.
"The same is true of grammar. There is now some evidence from both empirical studies and neural studies that spatial skills are related to grammatical development."
De Lange and others believe that, when children have difficulties with basic mathematics, it may be due to poor bridging between their innate numerical and spatial sense (what he calls "informal knowledge") and school mathematics.
"Children often make the same types of errors in school mathematics," observes Paul Ernest, Professor of Mathematics and Psychology at the University of Exeter. "Their errors are systematic."
Professor Dehaene's work on children's innate numerical sense helps to explain this. He has found that even infants can compare quantities and detect simple errors of addition before they have learnt arithmetic. They also have an innate spatial sense, which helps them to recognise shapes and understand positioning. Space and orientation are the foundations of geometry.
Professor Dehaene believes that games such as snakes and ladders can be used to enhance these cross-links. He and his colleagues have also designed a computer game, the Number Race, at his Paris laboratory, to help develop the neural links important for mathematical skills.
Children trained with such games "show generalised and lasting gains in school-based arithmetic. These games seem to be particularly helpful when they are used at a very early age and in children from poor neighbourhoods, who seem to be particularly at risk of deficits in arithmetic," he wrote in a recent paper, The Calculating Brain.
There are various explanations of why this might work. "Anything you do changes your brain. Anything you do systematically will train it. That's true for batsmen and musicians as it is for mathematicians," explains Professor Butterworth, "The question is whether training in other things helps your maths even by a small amount."
"Research suggests that playing computer games helps your ability to make rapid mathematical estimates of objects on display, and that is a predictor of ability to learn arithmetic. Dyscalculic people, for example, are bad at this."
It perhaps comes as no surprise, then, that being good at maths requires more than activation of the supposed "mathematical side of the brain". It also means that the idea of left-brained mathematical and logical individuals and right-brained creative, emotional types no longer holds sway. Research in Japan has shown that language can move to the left hemisphere of the brain when the right hemisphere is damaged. Mathematical-logical thinking may even be able to "change sides".
So, in truth, when someone says they can't do maths, they're really making a statement about how they learnt to co-ordinate their thinking. As Professor Butterworth concludes, "If you are good at maths, it probably means you have had a decent education."
Number Race game: www.unicog.orgnumberrace
LAST CHANCE TO ENTER
Don't forget to get your entries in this weekend for the TES ConnectAssociation of Teachers of Mathematics resources competition for a chance to win an iPhone or a digital camera.
If you haven't yet submitted your maths resources the deadline is Monday, September 14. Go to www.tes.co.ukmaths-competition to find out more and upload your resource.