Review of March 2013

I thought I would try to collate some of the interesting and thought-provoking comments on the Bring Back Column Addition Facebook page, to help bring new readers up to date with the discussion, but also to respond to some of the points that have been raised.

Parents’ comments

The message from parents is loud and clear. Parents are being “told off” for teaching their young children column addition, and frustrated that their children aren’t allowed to use it. Other examples of friction between parents and teachers are cited.

“I have an 8 year old who is struggling with Maths and all these strategies. I absolutely identify with this article – I taught her the column adding in the school holidays and was told I shouldn’t have.” – Erika Nunns

“I taught my kids column maths who felt more comfortable and achieved more, but also struggled to learn the way their school taught to guide them, so they could ‘appease’ what the school expected them to.” – Denver Leung

“I feel as if my kids will come out of school without even basic maths skills. I don’t think they will ever be mathematicians, but they gotta know simple things like addition. My daughter gets told off for using this method. We taught her, she understands it clearly, why can’t she use it? These rubbish methods are confusing and I resent my daughter not being allowed to use a column method.” – Leigh Payne

“My son gets taught the column method at Kip McGrath (and I’ve been told off by 2 teachers about this over the last 2 years – that they don’t learn this way until later – he’s year 6 this year). I understand the new method is supposed to teach them the concept of numbers, how to take numbers apart and put them back together again. My reasoning is that if one method doesn’t work, isn’t it best to look at a method they can understand instead of failing completely?!” – Karen Burgess

“I really think this is a great idea, how many parents struggle with trying to understand the ‘new’ method of teaching maths? I was told I was not allowed to show my son our ‘old’ way of adding, and yet I find he does not even know the times table!…How do they expect kids to get more advanced maths problems when they don’t even have the basic facts in their heads?!” – Sue Hoare

“Yay!, this would be great. At least parents might have more idea of how to help their children at home.” – Kerry Scott

“My oldest child learned to believe she was no good at maths because of all the crazy strategies they tried to make her learn. Actually she is very mathematically strong – they just couldn’t support her with these silly imaginings!” – Michelle Cavanagh

“Some children just want to get on with it instead of sitting in circles endlessly discussing how they worked out the problem and which other strategies they could have used.” – Maryanne Newton

“This ‘new’ maths that my children are learning is HIDEOUS! They’ve got to go through so many steps to get a simple answer. AND, what’s worse is when your child gets marks downgraded because they worked it out ‘old school’ and therefore didn’t show the ‘74’ steps needed to do a basic multiplication!…Give them the option for goodness sake….Of note is that none of my children have a problem with maths, but when I have helped them out, doing it the old way, they tend to grasp it a lot quicker and to them it makes more sense. How many other kids out there, if given the option, would choose old over new?” – Teri Findlater

“I taught my son how to add using columns, when he went to school, he was told he could not do it that way. On asking his teacher what the new method was, I was in disbelief. I am for change, but in this instance it does not work as well as the old, straightforward method. Why is getting an answer the easiest way possible not good enough for some?” – Penny Whitelaw

“While having a discussion with our child’s teacher about mathematics as pertaining to our daughter she asked us how my husband and I were at maths. Shocked, we replied that we were both successful at maths at Uni, but we understood the implication was that we may lack competence at math, and hence so would our daughter.” – name withheld

“Broached subject at both year 8 parent teacher interviews, explained concerns particularly re multiplication and subtraction. Teacher was dismissive, explained that subtraction was done by equal addition and checked by rounding and compensation or vice versa. Explained that subtraction was checked via addition i.e. a-b=c so c+b=a. She did not take this well.” – Matthew Conibear

Educators’ comments

Educators on this page tend not to acknowledge parents’ frustrations. Their views are much along the same lines: children should not learn column addition (and other column-based methods) until they are ready to understand how and why it works.

“If kids do not comprehend place value then column addition or subtraction is like a foreign language to them … it is about creating an understanding of place value before introducing column addition/subtraction” – Dave Harrison

“It is important for children to have place value knowledge first and be ready for bigger numbers. When adding 469 and 29 they need to be able to estimate the correct answer first, then use an algorithm to check it.” – Megan Gooding

“Addition, subtraction, multiplication and division algorithms are taught [in the later primary years]. The difference being that children have a better understanding of why we carry numbers and rename etc.” – Stephen McLean

“[Column subtraction] needs to be understood, not just applied.” – Tarquin Smith

“What we have to be careful with is that they understand how it works. So many children solve equations in the written from by adding on their fingers, and ‘carrying’ to the next column, with absolutely no understanding of what they are doing (often because it has been introduced too early).” – Gail Ledger

“Children, who in earlier times were given the traditional algorithms (with all the strange carry marks to jog the memory), had little or no understanding.” – Murray Britt

“Column addition does not actually require the student to know much about place value. By following the procedure, an answer will be reached but it does not always mean that the student really understands what is happening with the numbers involved.” – Marg Farrelly


“[Murray Britt’s] comments about children of my generation not understanding the arithmetical methods taught to them are both incorrect and patronizing.” – Matthew Conibear

“Column addition does not use a different algorithm for addition from the masses of handwriting across the page advocated by Murray Britt. They use the same algorithm, of adding units and tens and hundreds and so on. Column addition differs in that it sets out the same algorithm in a neat and comprehensible manner. Competent teachers using column addition were able to teach students to understand, and better than those who advocate the mass of confused handwriting.” – Peter Oakley

“There is value in getting the right answer, and value in a method that gives the right answer. Perhaps getting the right answer is enough in some cases. If you want to study mathematics, though, it would pay to get an understanding, too. But the understanding need not precede the acquisition of a method to get the right answer.” – Peter Oakley

“Understanding, as such, usually follows success with concrete skills, and we don’t place enough value on early mastery of mental skills in whatever form they are taught.” – Margaret Nicholas

“While you maintain they don’t understand place value, and I think you might be wrong here as I always show them the place and value of the numbers when I am teaching, do you always have to know everything before you use it? Do you know how an electric light works or a computer? But you still use it.” – Mark Newman

“[In the late 1960s], written column addition was learnt in conjunction with abacus work; I don’t recall it being conceptually difficult. During that same year – standard one, in old currency – the subtraction column algorithm was taught. This was conceptually more difficult, but with repetition and abacus work – together with checking our answers by addition – we got it. My understanding was developed, over a matter of months, from actual use. I can’t see how this is wrong in principle.” – Matthew Conibear

I think these responses sum up my own position pretty well, and I’m glad that people can hear these things from people other than me. Column addition can be taught with meaning to children from an early age. Educationists who say otherwise are scaremongering. It might have been true (in the 1980s perhaps) that arithmetical methods weren’t always taught with meaning, but I have no intention of letting that happen again. If the teachers on this page are anything to go by, they are very eager to promote understanding, which is great. Coupled with our modern culture of encouraging children to question how our world works, I remain optimistic that mistakes of the past won’t be repeated. But let me address some of these points myself.

Place value: For anyone who uses place value as an excuse for delaying the teaching of column addition until Stage Six, I have to ask: which of the mental “strategies” introduced before Stage Six do not require a firm grasp of place value?? Teaching place value in and of itself needs to be done, but it’s only when we actually try to work with numbers that place value takes on more meaning. It is entirely natural to align numbers in columns, precisely to emphasise the place value of each digit.

Approximation: Personally, I would compute the answer first and check its reasonableness using approximation. Yes, this is ultimately where we would like our children to be, but approximation is a completely independent skill. (We should be encouraging approximation when using a calculator too.) Approximation is not a prerequisite for learning column addition.

Ready for bigger numbers: Admittedly, the confectionary Hundreds and Thousands tends to be called Sprinkles these days (- I hope this wasn’t a Ministry of Education mandate!), but no matter how hard we try, children’s natural curiosity will propel them to discover larger numbers before we think they are “ready”. It is our responsibility to ensure they have the skills to keep up with that natural curiosity. If a young child wonders how many 125ml cups of water will fill a 500ml jug, what sort of teacher would I be if I said “Forget it, kid, you’re not ready for large numbers”? I, for one, would find any possible way of helping this child to work out the answer. If column addition simplifies the task and achieves the goal, well frankly I think that’s fantastic.

Understanding: Now this is the big philosophical question! How much should anyone understand of a method before they are permitted to use it? Mark Newman’s real-world analogies are perfect. My own favourite: do we insist that everyone learns about the mechanics of a car before they learn to drive? Should we insist that teachers learn some Abelian group theory and commutative ring theory before they are permitted to teach the “strategies”? Don’t worry, I don’t expect everyone to understand that last question, and the good news is you don’t have to! In practical terms, one only needs to understand enough to do the current job well. If the next job is a little bit harder, then we might need to understand a little bit more. This is a very natural way of learning and it suits children very well.

We can teach children to perform a task well, but we flatter ourselves if we believe we can make them understand it conceptually. Understanding is intangible and organic, and those “lightbulb” moments we sometimes see are mostly due to their natural maturation. All we can do in the meantime is support them with age-appropriate explanations and good role modelling to promote better understanding in the future. For something like column addition, that might mean teaching in conjunction with materials, promoting understanding through the language we use, wise placement of the “carry digits”. These things should help children to understand place value.

Perhaps teachers feel column addition is a single page in the instruction manual, to be taught once and only once. We need to foster a culture of collective responsibility, i.e. continued support of a pupil’s understanding across many years; where it is okay for teachers of younger students to teach a method with an expectation of proficiency, but not necessarily an expectation of “full” understanding. However, the method should be revisited many times, not least because its merits will need to be discussed relative to other methods of computation. Teachers of older students should have a higher expectation of understanding from their pupils, especially those who have exhibited proficiency for some time. On the other hand, assuming their pupils are proficient, teachers may feel time is better spent understanding applications or higher-level concepts. There needs to be some flexibility here.

Further direction

I would welcome more comments from (intermediate and) secondary school teachers, since the ultimate goal is surely to prepare primary school students for higher-level learning.

“I am a secondary school teacher and I have had students in tears at secondary school because they could not remember the numeracy thinking tricks and still made mistakes in adding and subtracting in years nine and ten. Now some ideas about numeracy are very good but I felt this was just wrecking these particular students’ self esteem and made them feel like failures for too long. I taught them old fashioned column methods and they were away laughing and have never looked back. Too bad, I thought, I will do what is right for the individuals in front of me.” – Sheena Charleson

“I have watched [intermediate] pupils trying to add [using a Numeracy Project strategy]. By the time they do this half the lesson is over and their Asian counterparts are on question 5 using basic algorithms and getting them all right…I have yr 7 pupils who start the term with no idea how to add, multiply and subtract…I am spending so much time teaching yr 7 and 8 [pupils] stuff they should have done in primary school…If only our feeder primary schools would get back to basics my job would be so much easier.” – Mark Newman

Notes of optimism

As my campaign aims to redress the balance between written methods and mental methods, it is encouraging to see the recent removal of the sentence “Teachers should debate whether they will introduce the written form at all.” from I am also heartened by this teacher’s comment:

“Children should be writing down their working when they are doing Maths. This is critical, and some teachers seem to think children shouldn’t be doing this. I’m not sure where they get this idea from, but think it might be an over-reaction to our old dependence on written Maths (or algorithms).” – Tarquin Smith

I also like these constructive suggestions. Did either of these happen in the late 1990s, prior to the roll-out of the Numeracy Project?:

“Maybe the solution is to go into schools like ours and find out what is working in numeracy…” – Leslee Allen

“Perhaps we need to look at countries where students do well in maths.” – Margaret Nicholas

Some points for us to ponder:

  1. Polarised views between teachers and parents can hardly be conducive to a successful home & school partnership. I firmly believe that parental involvement is critical to a young child’s success with maths, or any other subject for that matter. Parents and teachers need to be on the same page, and parents need to be involved as much as possible. I’d really like to see better alliances being forged.
  2. Teachers, if you don’t like parents teaching their children column addition, then get in first! Make sure it is taught well; make sure your pupils are thinking about things in the right way. I’ve suggested a few ways to promote understanding. What other ideas do you have?
  3. “Algorithms” and “strategies” are divisive labels. We should abandon them. They are all methods, each with their own advantages and disadvantages. We should help students to use the methods that work best for them.

Finally, my favourite quote so far:

“When the Tans speak, people should listen.” – Matt O’Connell

Well, what more can I say?  :o)

Dr Audrey Tan, Mathmo Consulting
April 2013

A bit of progress

Great news! Earlier in the week, I pointed to the FAQ:

Q: “When should I start teaching the written form?”
A: “Teachers should debate whether they will introduce the written form at all.”

Within days, this sentence was removed. In fact, the revisions are becoming something of a daily event! Today’s answer did not include the sentence “Early teaching of the written form often locks students into low-level thinking from which they never emerge.”

You never know, perhaps the answer will keep changing every day until we’re happy. :o)

Update May 2015: Sadly, the old answer still remains in certain places. In anticipation of it being removed, we have taken a snapshot for posterity:

Update June 2015: On 4 June 2015, this controversial answer was exposed again in Rose Patterson’s report “Un(ac)countable: Why millions on maths returned little”. The Ministry of Education has tidied up their website, so there are no longer three versions of the FAQ and the amended page from 2013 no longer exists.

Dr Audrey Tan, Mathmo Consulting

Less Numeracy, More Maths

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.

If you do not have a Facebook account, but would like to support the here campaign, please feel free to contact us. We would love to hear from you.

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.