But after just over six months of regular practice using Rainey's grids for times tables and other basic arithmetic, the Year 5 students more than doubled their multiplication scores and trebled their division scores to 15 a minute. The Year 6 classes made similar spectacular improvements - from nine facts a minute to 20 in multiplication and from six to 19 in division. Teachers at the decile 2 school raved about their progress and said the children loved using the system. They were still behind but catching up fast and looking as if they might be able to cope with maths at secondary school.
Then the story turns sour. Both sides dispute what happened next but according to Rainey, the school's principal, Frances Nelson, returned from leave of absence on union duties, discovered the trial had been running in her absence and stopped co-operation immediately.
The tests were allowed to finish but Rainey says he was no longer allowed into the school to gather information and give feedback.
Nelson says Rainey merely told teachers how to use the programme but, as a non-teacher, he was never supposed to have any involvement in the classroom.
Asked for her reaction to the students' poor results, she replied: "I don't have a view to share with you one way or another ... It is absolutely right that in using his particular approach to learning basic facts, the children were successful over time. But we don't use the same programme all the time."
The school's results may simply be a severe example of a problem affecting many other New Zealand students who struggle with addition, subtraction, multiplication and division - known in education circles as "basic facts". New Zealand children at Year 5 - the same level as the Fairburn students - scored well below average in the the 2010/2011 Trends in International Mathematics and Science Study (TIMSS), ranking 34th out of 53 countries. Asian nations took out the top five places, while New Zealand effectively ranked bottom of the developed countries alongside Spain, Romania and Poland.
Education Minister Hekia Parata described the results as "seriously worrying" when they were released in December.
"We have many children who lack basic skills and knowledge, particularly in mathematics and science, compared to children of a similar age in other countries," she said. "We must pay urgent attention to what these studies tell us and tackle some system-wide challenges."
Our students scored especially badly on arithmetic. When the 9-year-olds were asked how many people were aboard a ship which had 218 passengers and 191 crew, only 52 per cent of New Zealand children got the correct answer (409). The international average was 73 per cent. Yet New Zealand students scored relatively well on questions such as interpreting graphs, which require more abstract thought.
New Zealand students scored even worse on a division problem in the previous test, which asked if 762 cars in a carpark were parked in six equal rows, how many cars were in each row. Only 39 per cent of all students got this right but New Zealand's success rate was only 8 per cent, compared with 36 per cent in the United States, 23 per cent in England and 12 per cent in Australia.
Education consultant Bob Garden, a retired Ministry of Education research director who now co-ordinates the TIMSS survey in New Zealand, said he believed the results were a concern, even though modern educationalists would say children could just use their calculators.
He strongly agrees with Rainey's emphasis on students knowing their basic facts from memory before they try to solve problems and with his contention that many teachers regard memorisation or "rote learning" as a dirty word.
He suggests we learn from the United States, which has repackaged the idea as "rehearsal" - the same way as a ballet student or a rugby player learns by practising the same key movements over and over. Their TIMSS scores, he notes, have since gone up.
Des Rainey knows he's an outsider challenging the educational establishment. He trained as a teacher but has never taught in a classroom. Instead, he became a psychologist for the old Education Department and performed folk music on cruise ships around the world with his wife in the 1960s. He was head of psychology at Auckland Hospital for 10 years and developed Kiwi Maths with his primary teacher daughter Anne in his retirement in the 1990s.
Under his system, children practise addition, subtraction, multiplication and division on grids in workbooks.
In multiplication, they start with the easy tables (2 to 5) before moving up to the hard ones (6 to 9). They can also write in any answers they don't know at first to ensure they get everything right. They only start to practise building speed once they have 100 per cent accuracy.
Teachers then arrange tests to see how many questions the children can answer accurately within a minute. This method, which Rainey calls Accurate Facts Per Minute (AFPM) allows teachers, students and parents to see at a glance how well they are progressing.
At first Rainey had plenty of official co-operation, thanks to an endorsement from then-Education Minister Lockwood Smith, who arranged for ministry experts to help fine-tune the system. He found many teachers willing to change even if their test results were embarrassing - like the horrified principal of one highly regarded school who overhauled teaching methods when he learned almost a third of his Year 6 students were at remedial levels similar to Fairburn's.
Initially, says Rainey, he got a similar response from teachers at Fairburn School. He still feels the research was a missed opportunity to help the children who struggled most and he worries about their future at secondary school.
Asked about Rainey's methods and his criticism of modern maths teaching, Auckland University maths education lecturer Peter Hughes agrees that our poor performance in arithmetic is worrying and says students must know their basic facts instantly. Otherwise they will not be able to handle more complex problems, because their brains are tied up on the simple calculations.
Hughes describes Rainey's method as a powerful way to learn basic facts and says repetition is the best way to learn. He adds that probably a third of primary school students are not numerate when they reach secondary school and are very unlikely to catch up as secondary teachers are far too busy trying to teach advanced algebra and geometry.
But Hughes - a former secondary maths teacher who for eight years ran the Ministry of Education's Numeracy Project, which pioneered the current problem-solving approach - urges parents not to focus on times tables as the real measure of maths ability. His definition of numeracy is being able to apply those skills to real life.
For instance, he asks, can you instantly work out what is 4.9 per cent of $200,000? That's the sort of question people need to know when they're dealing with money and it requires a combination of instant recall and knowing the right strategy. (The quickest way is to estimate the answer by dividing $200,000 by 10, then halving that to get 5 per cent.)
Fretting over our failure at long multiplication and long division in international tests is pointless, says Hughes, because calculators mean no one has to do these calculations any more. Schools now teach a more logical way of calculating these problems but unfortunately many parents, who were brought up the old way, don't understand it.
He also questions our obsession with arithmetic as the gold standard of maths knowledge, saying if we really want to address a shortage of engineers, geometry and measurement are even more important than numbers.
Hughes maintains the TIMSS survey is more basic and less searching than the Programme for International Student Assessment (PISA) run by the OECD. In the last survey in 2009, New Zealand ranked 11th out of 65 countries for maths.
The real elephant in the room, he believes, is that many primary teachers are not confident at teaching maths.
"That's why so many kids are falling through the system. You get one bad year in maths, you know you're going to have it very difficult to catch up."
Meanwhile Rainey's success at Fairburn School has caught Parata's eye and she has asked Ministry of Education officials for a report on his methods to see if they can be applied in other schools.
Asked whether she agrees with his view that maths teaching has swung too far away from memorising the basics, Parata says she does not want the Government to dictate any changes in teaching style, as this is up to individual schools.
However, she notes that an Education Review Office report this month on maths teaching in primary schools found most schools needed to change their methods to help slow learners. For instance, it criticised the way teacher aides were often assigned to the struggling pupils who most needed an experienced teacher's help.
Parata says other important factors are school leadership, teacher training quality and changing social attititudes towards maths, which affect many teachers.
"If you've got a cultural attitude that maths is hard and reading is easy, then you go into it with that mindset."
Rainey says he'll be delighted to talk to Parata's officials, adding that he doesn't particularly care whether schools buy his teaching materials or use another similar system.
"I'm only around for a couple more years. I'm happy to give them all sorts of stuff."
The questions that stumped New Zealand kids
1) A ship has 218 passengers and 191 crew. How many people are on board?
Success rate
International average - 73%
New Zealand - 52%
2) There are 762 cars in a carpark in 6 even rows. How many cars are in each row?
Success rate
International average - 39%
New Zealand - 8%
(Answers: 409; 127)
