This article is intended to provide some explanation for the motivation behind the campaign to Bring back column addition (and other vertical column-based methods) to New Zealand’s early primary maths curriculum.
According to the New Zealand Herald, New Zealand 9-year-olds finished last-equal in maths among peers in developed countries in an international survey published in December 2012. Almost half could not add 218 and 191 in a test. In 2007, only 8% could divide 762 by 6. Whether or not you are a supporter of the Numeracy Project, you must agree that these results are very worrying.
The Numeracy Project has been around for more than 10 years now. Admittedly, New Zealand children have never done particularly well at maths, but the results are showing a clear downward trend, i.e. the Numeracy Project has failed to provide the intended improvement. I’d really like to see more than 8% of 9-year-olds being able to divide 762 by 6.
I am hardly surprised by the findings; the statistics reflect what we are seeing on the shop floor. In a letter published by North & South magazine in 2007, I wrote “We have a generation of children who quickly run out of steam when faced with large numbers or complex calculations because they have neither the mental capacity nor the written skills to cope.”
For all the emphasis being placed on numeracy, children are less numerate than ever before. Even if they come up with a valid mental “strategy” (to use current terminology), they can be very slow at applying it. In many cases, it would be much faster to use written, column-based methods (currently referred to as “algorithms” in schools), which are perfectly good methods refined over thousands of years specifically to make a complex calculation quick and easy. When I show my students how to use them, their faces light up and they say “but that’s so easy!!”, and away they go, able to work with numbers far larger than anything they were ever able to work with in their heads.
In mathematics, we try very hard to reduce the complexity of a problem by breaking it into smaller, more manageable tasks. Column addition is all about reducing complexity and therefore ideal for “emergent” pupils, to use a Numeracy Project term. It’s quick and easy for young children to learn, and it gives them the confidence to work with numbers of any size. As far as strategies go, it’s a great one for reinforcing the very important concept of place value.
Take, for example, the sum 89+15=104. Column addition reduces the complexity of the calculation by effectively replacing it with two single-digit sums: 9+5=14 (ones) and 8+1+1=10 (tens). I think most children, with practice, would be able to do this.
A straightforward mental strategy would be along similar lines: 80+10=90, 9+5=14, and so 90+14=104.
A more sophisticated strategy would be to “transfer” some of the weight from 15 to 89, to make 89 up to 100, and then add on what’s left after the transfer. This requires two intermediate calculations: 100-89=11 and 15-11=4, and then we have a new sum to compute: 100+4. Have we reduced the complexity of the calculation? For some children, 100-89 will be just as difficult as 89+15, so all we’ve done is replace one calculation with another calculation of similar complexity…and there are still more steps to complete, if we can remember what they are…
It is precisely this sort of multi-step strategy that I believe is currently hindering New Zealand children’s learning and enjoyment of maths – not necessarily because of how it’s taught, but because of when it’s taught. Most young children lack the capacity to hold so many steps in their heads. On the other hand, many of them will pick up these strategies more easily in a few years’ time, when their brains have matured a little, so not much time would be lost by delaying the teaching of these strategies and teaching them the column-based methods first. Some pupils will always struggle with the mental strategies, but if we can equip those pupils with the “algorithms” and at least give them one way to work with numbers, before they lose confidence in their abilities altogether, then we will be better off.
My ideal curriculum would start with concentrating on the speed and accuracy of single digit addition and quickly leverage that skill to adding larger numbers using column addition. The next big focus would be speed and accuracy of single digit multiplications. We see many children who don’t know their times tables well enough, and it really slows them down. Learning times tables has far wider implications than just knowing a few basic facts. The discipline of memorising some basic facts for instant recall plays a crucial part in training the brain to absorb and retain information. In essence, this is what learning is all about. It is excellent “brain exercise” and the earlier our children get onto it, the better. And to those who frown upon rote learning, it does not need to be considered rote learning if it is part of a well-designed curriculum that promotes the understanding of the basic operations at the same time.
Laying down these foundations gives most children a good chance of using numbers effectively and being able to apply them to mathematical concepts and problem solving, which is where the emphasis really needs to be. Numeracy is only one aspect of mathematics. The study of mathematics is really about understanding patterns and processes. Some of the mental strategies can, and should, be introduced later – some pupils will take to them, others may prefer to stick to pen and paper. Personally, I’m fine with that. At the end of the day, I simply want all New Zealand children to be able to work effectively with numbers, one way or another. We need a curriculum that gives every young child the chance to succeed with maths, and teaching mental strategies at the expense of teaching the “algorithms” (or any other written methods) is really unfair on the ones who are struggling.
I also question how much time is being spent exploring all these different whole number strategies – they carry on all the way through to Year 8, when decimals, fractions and percentages should have already moved to the forefront. The Ministry of Education justifies its curriculum by saying that “Employers are increasingly looking for staff that have problem solving skills and an understanding of concepts, rather than just the ability to follow rules for calculating. The increasing use of technology has also meant that a calculator or computer is almost always available in the workplace for larger calculations.”
If that’s the case, why are we spending so much time working on numeracy?? Why aren’t we spending more time on problem solving? Our children’s time would be better spent learning when and how to apply the four basic operations so that they really can solve some maths problems. (It wouldn’t surprise me if a significant proportion of the 92% who failed to answer the division question failed not because they couldn’t divide, but because they didn’t know it was a division problem.)
So far, you’ve read my mathematical justification for the campaign, but I have a social justification for it too. Another aspect of the Numeracy Project that worries me is the message being delivered to children through their assessments. Whether or not it was the true intention, many pupils feel they must do it all in their heads, they are not allowed to write anything down, and therefore writing things down must be bad. We should commend and encourage every child who finds any valid way to solve a problem, and if they need to use pen and paper, is that such a bad thing? It must be pretty demoralising to get the correct answer, only to be told that it was done the “wrong” way. Some parents have reported to me that they were asked by their child’s teacher to refrain from teaching the “algorithms” at home, to avoid a conflict of approach. Many parents don’t understand the mental strategies, and so they feel unable to help their children at all. This disconnection between parents and their children’s learning is perhaps the most worrying aspect of the whole thing.
So that’s the motivation for the campaign. It’s not a case of going back to the old system, which wasn’t working very well either. I see things more as a pendulum that has swung from rote learning with little understanding, to strategic learning with better understanding (for some) but inefficient, if not ineffective, in practice. I am aiming for something inbetween! A let’s-get-on-with-it approach that allows our children to quickly get up and running with numbers, before they lose confidence and interest, so that they can focus on understanding mathematical concepts, e.g. measurement and geometry, and applying their skills to real-world problems.
The Ministry of Education will only take this campaign seriously if they can see there are a lot of people who support it. So please, show your support. Let’s really try to make a difference and help our young New Zealanders to be successful with maths.
Dr Audrey Tan, Mathmo Consulting
March 2013
For a list of further campaign material, click here.
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We are honoured to receive the public support of our campaign from Professor Sir Vaughan Jones. Please click here to read his statement.
We are honoured to receive the public support of our campaign from Professor Victor Flynn. Please click here to read his statement.
We are honoured to receive the public support of our campaign from Distinguished Professor Gaven Martin. Please click here to read his statement.
Mathmo Consulting 