Small children naturally think the answer is three plus five. Some never get over that misconception and move on to an understanding of the place concept. Nor can many young learners grasp the difference between addition and multiplication.
It was to work through such fallacies that academics in Western Australia devised First Steps in Mathematics, an approach to teaching that first identifies children's difficulties and then provides learning activities to address them.
Now primary schools in Torfaen are starting to use the approach, under the guidance of maths adviser Rod Cunningham, a New Zealander, who has been to New York to spread the gospel to schools there. "I think it'll make a step difference to children's learning," he says. "It has implications for behaviour too, because much bad behaviour is caused by frustration."
In contrast to England's national numeracy strategy, which Mr Cunningham describes as "outcome-based", the First Steps approach aims to tackle the child's understanding of basic mathematical ideas. "Children are often using up all their energy on coping strategies," he says. "Quite a lot are getting the right answer for the wrong reasons."
More than 60 primary teachers have been trained in the scheme so far, at least one from each of the authority's 40 primary schools, and 14 schools have signed up for three-day whole-staff training. The response has been overwhelmingly enthusiastic. One highly-experienced teacher admitted after training: "I don't know what I've been doing all these years."
At Hillside primary, a fast-improving school with 130 pupils in Blaenavon, reception teacher Jocelyn Nada was fascinated by the results of diagnostic tests. She found, for instance, that one child could happily count bears from one to nine but could not then answer the question: "How many bears are there?"
Mr Cunningham and his adviser colleague Jo Smith impress on teachers that they should start small - not revamping their schemes of work but just using some of the diagnostic activities. And he has words of comfort for those teachers who say they cannot address all the different wrong ideas they discover in a class of 30 children. "Misconceptions tend to come in groups," he says.
So the diagnosis can lead naturally to group work.