The great maths division

By Andrew Laxon

Weekend Herald stories about how children learn maths at school have prompted a flood of responses from readers and experts. Andrew Laxon explains what the argument is about

Our ability to mentally calculate seems to have fallen away in the past few decades as calculators and computers do the job for us. Photo / Thinkstock
Our ability to mentally calculate seems to have fallen away in the past few decades as calculators and computers do the job for us. Photo / Thinkstock

Why are we talking about this now?

New Zealand 9-year-olds ranked 34th out of 53 countries - and effectively bottom equal among developed nations - in an international maths and science test last December. Education Minister Hekia Parata called the results "seriously worrying". At the same time Auckland education consultant Des Rainey approached the Weekend Herald about his research on the basic maths abilities of 9 and 10-year-old children (years 5 and 6) at Fairburn School in Otahuhu, Auckland. Rainey found most students could not do simple times tables when first tested but their correct answer rates doubled or even trebled after six months of practice using his multiplication and division grids. He argued that the children were failing because of our back-to-front maths teaching philosophy, which asked them to solve problems before they knew their basic facts.

In February Parata said she had asked ministry officials to examine whether Rainey's methods could be used to improve performance in other schools.

Last week, New Zealand's foremost mathematician, Sir Vaughan Jones, agreed with Rainey's criticism of the back-to-front approach, saying children had to learn to multiply and add before they tried to grasp bigger mathematical concepts.


Are New Zealand kids really that bad at maths?

They are at arithmetic. Only half the New Zealand children in the 2010/2011 Trends in International Mathematics and Science Study (TIMSS) could add 218 and 191, compared to 73 per cent internationally. In 2007 only 8 per cent could divide 762 by six. The overall pass rate was only 38 per cent but China, Hong Kong and Singapore had pass rates in the 80s and we scored worse than the United States (36 per cent), England (23 per cent) and Australia (12 per cent).

Rainey's initial testing at Fairburn School found about 90 per cent of the 9 and 10-year-olds were in the "at-risk zone" - unable to answer simple multiplication and division questions within four seconds and therefore unlikely to make it through intermediate and secondary school maths. All the experts the Herald spoke to - including those who support current teaching methods - agreed this was too slow. About half the year 5 students could not even answer five questions correctly inside a minute, suggesting that five years of maths teaching had made very little impact.


Has it always been like this?

Many adults would admit they're weak at basic maths so, as University of Auckland mathematics lecturer and curriculum reformer Peter Hughes puts it, there was never a "golden age" when everyone knew their times tables. But our ability to do calculations in our heads or quickly with pen and paper seems to have fallen away in the past few decades as calculators and computers do the job for us. Maths liberals tend to see this as inevitable, conservatives as a dangerous trend that must be stopped.

Ironically, the new maths teaching methods now under attack stemmed from a horrified political reaction to New Zealand children's poor performance in the 1995 TIMSS test, in which we finished 18th out of 24 countries. Several reviews, including a Government-appointed taskforce, said the main problem was that maths teachers did not understand the subject well enough themselves, especially at primary level. The Ministry of Education responded in 2001 with a Numeracy Development Project, designed to lift student performance by improving teachers' understanding.


What went wrong?

Critics say the new system became overloaded with theory at the expense of practical knowledge, confusing many teachers and their classes. Veteran Carterton principal Kevin Jephson complained in NZ Principal magazine that the project assumed all children were budding mathematicians, eager to explore new concepts, when most just needed simple rules they could understand. Rainey said the Fairburn children - and thousands like them at other New Zealand schools - couldn't possibly progress because they had "nothing in their heads" to work with. He even considered abandoning the research at the start because the children lacked any basic understanding of how numbers worked, an essential foundation for times tables.

Despite a number of positive academic reviews on the Numeracy Development Project, results from the ministry's own National Education Monitoring Project (NEMP) show the ability of year 8 students (12-year-olds) to answer a series of multiplication questions correctly within four seconds dropped from 47 per cent in 2001 to 37 per cent in 2009.


What do the critics want to change?

New maths methods emphasise having a range of strategies to answer equations. For instance, many people would solve 19 x 6 by mentally converting the problem to 20 x 6 and subtracting 6, which is now encouraged. The problem, say the conservatives, is that this can become the default position, even when it is slower and more complicated than doing the traditional calculation. Many New Zealand students would probably have tackled the 218 + 191 equation by adding 200, then subtracting nine, as they had been taught. Yet for many, it would be just as easy to add the three columns of numbers vertically in the old-fashioned way. Another example is converting 15 + 6 into (15 + 5) + 1, which critics say sidetracks children who haven't yet grasped 5 + 6 = 11.

Several add that the multi-strategy approach also confuses many primary teachers, who lack confidence in their own maths ability. There are many anecdotal reports from both professionals and parents of baffled children having to learn up to four different strategies to solve one problem, when all they need is one method that works for them.


But surely you don't need strategies to learn your times tables or 2 +2 = 4?

Actually you do but opinions differ on the best way to get this across. Primary maths teaching is now based on counting, so children are supposed to learn 5 + 3 by counting on "6 -7 - 8" and then converting this to 5 + 3 = 8. In the same way they learn multiplication through "skip counting", so 2 - 4 - 6 - 8 - 10 is the basis for learning 5 x 2 = 10. In theory they may also learn to say "I know five lots of two is 10" but Margi Leech, director of private maths teaching agency Numicon NZ, believes this final step is often neglected, so many children fail to learn the true value of each number and relationships between them. She uses different-coloured geometric shapes to teach children numbers from one to 10 - a visual rather than conceptual learning style that she argues works best for most children and teachers.


Why don't schools teach long multiplication or division any more?

Some do but most give it far less emphasis - a stance encouraged by Hughes, who says history has bypassed the need for humans to do complex number crunching, as we now have calculators and computers. When schools do teach long multiplication they tend to use the horizontal method, e.g. 25 x 26 = 400 + 120 + 100 + 30 = 650. This is supposed to be easier for students to understand than the old vertical method, where numbers are often "carried" to the next column, and useful training for algebra in future.

However, the 2009 NEMP maths report shows a big drop in long multiplication performance from 2005 to 2009, especially when students using the new maths strategies had to carry numbers or handle two multi-digit numbers. It says students were also weak on estimation and fractions, especially once they got beyond halves and quarters.


So why don't we go back to the old ways of teaching maths?

Because they put most people off maths for life, says Professor Bill Barton, the former head of Auckland University's mathematics department and outgoing president of the International Commission on Mathematical Instruction, the global body for maths educators. "We'd abandoned them because they'd been useless. If you talk to 100 people of any age, more than 50 per cent are going to say they hated maths and were useless at it and dropped it at school."

Education consultant Len Cooper agrees. He says traditional maths teaching methods never explained how the skills applied to the real world - like asking an apprentice builder to measure and cut pieces of wood and nail them together for a few years, without ever telling him he is building a house.

Or as British maths education expert Professor Afzal Ahmed put it, "If teachers of mathematics had to teach football, they would start off with a lesson on kicking the ball, follow it with lessons on trapping the ball and end with a lesson on heading the ball. At no time would they play a game of football." It's a good analogy, but as most football coaches would agree, it doesn't hurt to get the skills right either.

- NZ Herald

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