A brighter future for mathematics education in New Zealand

In the NZ Herald, Peter Hughes asks how TIMSS 2011 and PISA 2009 can produce “such wildly contradictory results” reflecting New Zealand children’s performance in mathematics.

We can argue endlessly trying to compare two assessments of children of different ages, taken at different times, assessing different aspects of understanding mathematics. It is pointless, not least because PISA 2009 points to a population no longer representative of 15-year-olds in New Zealand today. Whatever PISA may tell us, it should not detract our attention from the “very depressing” results produced by New Zealand primary school children who have known no other approach to learning maths than that of the Numeracy Project, the very curriculum Peter Hughes helped to write.

Hughes says we urgently need to work on algebra and geometry rather than “number”. I couldn’t agree more. So much for the pioneering curriculum that was supposed to develop “algebraic thinking” in children! Why are our primary school children spending so much time taking numbers apart and putting them back together again? This introspective over-analysis of numbers is not a good use of a young child’s time.

Curriculum co-writer Vince Wright says the Numeracy Project’s failure to deliver improved pupil performance is due to the insufficient maths knowledge of our primary school teachers. We can continue to beg for more funding to up-skill our teachers, but wouldn’t it be more practical to simplify the curriculum to meet the skill set of the teachers we have, present and future?

A bright future for maths education in New Zealand depends on bright beginnings, hence my campaign to Bring back column addition to New Zealand’s early primary maths curriculum. The aim is to redress the balance between written and mental methods of computation, and to make the curriculum more accessible to a wider range of students. I do not advocate teaching column-based methods exclusively, but it’s a good place to start. As Sir Vaughan Jones says, “We have this wonderful decimal system which took tens of thousands of years to bring to perfection and to not take advantage of it for basic operations is nothing short of folly!”

When recent untested ideologies about learning take precedence over the principles of mathematics, which by their very nature are the most logical of all, one has to wonder where things are heading. There should be a division of responsibility in writing a mathematics curriculum: content and delivery. Those with true, long-term mathematical knowledge and experience should determine the content. The educationists should determine how to deliver that content, and ensure our teachers deliver it effectively.

Our university mathematics lecturers should influence what is taught in secondary schools. They, together with well-qualified and experienced secondary school teachers, should influence what is taught in primary schools. This is my idea for maths education reform, and a brighter future for maths education, in New Zealand.

Dr Audrey Tan, Mathmo Consulting
April 2013

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 nzmaths.co.nz. 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