Show your working – it really helps

“Show your working” is a common request in high school maths exams.  Yet, in the modern learning environment, where worksheets and workbooks are common place, students don’t get a lot of practice at this important communication skill.  With so much content to deliver, there is little time left for teachers to teach students how to craft a good solution.  Furthermore, textbooks with answers at the back generally do not show any working.  It’s no wonder so many students find writing out mathematics both difficult and puzzling.

So, why should you bother to show your working?  Here are some points worth considering.

  1. In a written exam, there is no kudos in working everything out in your head or getting the wrong answer quickly.  It gets you no extra marks.  In fact, if working is required, writing down only the final answer will cost you marks.
  2. Getting the right answer is not as important as understanding the steps you took to get there.  That’s not to say that getting the right answer isn’t important.  Quite the opposite.  It’s so important that you want to be confident you’ll be able to get the right answer every time you encounter the same type of question.
  3. Showing your working is not supposed to be a nuisance.  It is meant to explain, with justification, how you arrived at your final answer.  Rather than being a nuisance, it should actually help both you and your teacher/marker to understand your work better.  Whoever is marking your work is not a mind reader.  Even you might have trouble understanding what you did when you look back at your work, sometime later, if you haven’t provided enough detail!  But if you have made a mistake and the steps are written out clearly, it will be much easier to identify where the misunderstanding occurred and correct the mistake.
  4. Use your brain power wisely.  Most students are faster and more accurate when they write things down because they can focus on the accuracy of each step rather than holding everything in their heads.  If you think that not showing your working will save you time, think again!  What if you make a mistake?  You won’t save any time if you get a wrong answer quickly and then have to re-do the calculation.  And that’s only if you notice.  If you’ve written down your steps, you are more likely to spot any mistakes and only have to re-do part of the calculation.
  5. If, in an exam, you’ve written down nothing but an incorrect answer, you’ll get no marks.  However, if you’ve written down an incorrect answer but have shown your working, there is a chance that your marker will be able to find something in there to justify giving you a partial credit.  In fact, if you made only a tiny numerical slip, it might be overlooked and you might still be awarded full marks.

Hopefully, you are now convinced that showing your working is a good idea.  Here are some tips on how to get started:

  1. Read quality worked solutions, and then practise writing out solutions in a similar manner.
  2. Articulating your ideas in writing helps you to understand better what you are doing.  You need to start practising this skill with the easy questions, when you know what all the steps are.  If you can’t do it for the easy questions, you won’t be able to do it for the harder questions either!  In actual fact, the harder questions might not be as hard as you think, if you are well practised at breaking down a problem into smaller steps.
  3. Knowing how much working to show requires a bit of judgement.  Again, refer to quality worked solutions for guidance.  Here is a good rule of thumb: when you are writing out a solution, imagine that the person reading your solution is someone in your class who doesn’t know how to solve the problem.  Would that person be able to follow your solution and then understand how to solve the problem?  Try it yourself: swap solutions with a friend and see if you can understand each other’s solutions.  It might give you a better appreciation of what it takes to write a clear and coherent solution if you try reading someone else’s work.
  4. Solving word problems is generally a test of whether you can extract information from the question and apply it correctly. So, identify when you are making explicit use of information or numbers supplied in the question and make this clear in your solution. For example, if you are calculating the area of a circle with diameter 6cm, a good written solution might be:

    Radius r = × 6cm = 3cm
    Area of the circle = πr2π × 32 = 28.27cm2 

  5. Understand the precise meaning of the equals sign (=) and use it correctly. The equals sign does not mean “Here is the answer:” or “Here is my next line of working:”. The equals sign means “is equal to” or “has the same value as”. Naturally, as the words suggest, it only makes sense to use the equals sign to indicate that two expressions (not equations) are equivalent or have the same value. For example, if I wanted to multiply 3 by 4 and then add 17, I would write:

    3 x 4 + 17 = 12 + 17 = 29

    and not: 3 x 4 = 12 + 17 = 29. The latter is what I call a running calculation.  It might be how we think through the calculation in our heads, but, as written, it is nonsense because it says that 3 x 4 is the same as 12 + 17.

The benefits of writing out your mathematics solutions in full cannot be understated. With continued practice, you will understand things better and have more certainty of your answers, particularly when attempting harder questions. Ultimately, you will feel better prepared and more confident when you go into your exams.

Dr Audrey Tan, Mathmo Consulting

What are your chances of collecting a full set of DreamWorks Heroes Action Cards?


There’s a highly infectious illness sweeping through the country at the moment.  It’s called DreamWorks Heroes Action Card Fever.  Symptoms begin innocently enough with excitement at opening a little packet and pulling out a new card, to disappointment at receiving a duplicate, followed by card swapping at school or organised meets, or even ridiculous bidding wars on Trade Me.  If you shop at Countdown, your child has probably been infected and may now be desperate to complete the set of 42 cards.

As far as we know, completing the set is the only cure, but here, we offer some pain relief for parents. You can find plenty of maths in these pretty little cards if you’re sufficiently interested.  They can actually help your children to practise their multiplication and division by 6.  Also, at any point in the collection, you could ask your children to lay out their cards in a neat rectangular grid.  How many ways can they do this?  This will get them thinking about factors.  For example, a 12 card collection could be laid out in a 3 x 4 grid, a 4 x 3 grid (aha, multiplication is commutative!), a 2 x 6 grid, and so on.

So now your children are doing some maths whilst playing with the cards, but what about us parents?  We’ve calculated some probabilities so that, at any point in time, we know (theoretically) how many packets need to be opened to find a new card, and ultimately, how many packets it will take to collect the whole set.

Intuitively, it should make sense to most of us that, the more cards you have, the harder it will be to find a new one.  For example, if you have only one card, then the probability that the next packet contains a new card is = 97.6%.  But if you have 41 cards, then the probability that the next packet contains a new card is  = 2.4%.

But the big question is, as the collection grows and you encounter more duplicates, just how many packets will you need to open to get a new card?

Let’s consider a concrete case.  If you have 14 cards in your collection, then there are 28 cards that you don’t have.  To calculate the expected number of packets you will have to open to get a new card, we take a weighted sum of probabilities based on the number of packets it might really take.

The probability that the next packet contains a new card is = 66.7%.

The probability that the next packet contains a duplicate but the following packet contains a new card is = 22.2%.

The probability that the next two packets contain duplicates but the following packet contains a new card is = 7.4%.

As you can see, the probabilities are decreasing quite quickly, so if at this stage in the collection it took, say, five packets to get a new card, you’d be very unlucky since the theoretical probability of this happening is = 0.8%.

To calculate the expected number of packets that need to be opened to get a new card, we calculate the infinite sum:

 

 

 

What, adding numbers forever?!  Well yes, but the good news is that while some sums will grow to infinity, this one doesn’t because the numbers we’re adding are diminishing quickly enough.  We actually get a sum that will barely change if we go out far enough.  This is what we call convergence.

Fortunately, this type of infinite sum is well-known.  It’s an arithmetico-geometric series with sum

 

 

 

So, in theory, with a collection of 14 cards, you would have to open 1.5 packets to get a new card.  Of course, this is a nonsense number in the real world, but it does tell us that, up ‘til now, we would have expected every packet you’ve opened to produce a new card, but now, you are on the cusp of encountering your first duplicate card.  From here on, it is quite likely that you will have to open at least two packets to get a new card.

This calculation can be generalised so that we know at any point in the collection how many more packets it will take to get a new card.  If you have n cards in your collection, the expected number of packets is a delightfully simple expression:

 

 

Rounding these expected numbers to the nearest whole number, we can now see how hard it will be to complete the collection, especially near the end:

 

 

 

 

 

 

 

 

 

In summary, assuming Countdown has distributed the cards fairly and you’re not doing swaps all the while, you’re going to need about 180 packets to collect the whole set, amounting to a $3,600 spend (not allowing for bonus promotions).  For a six week promotion, that’s $600 per week.  Some families might manage that, but we certainly won’t.  Instead, we’re simply delighting in how our “simulation” is panning out remarkably closely to the probabilities we’ve calculated, and accepting that the only realistic way to collect a full set is to swap, swap, swap, or go mad on TradeMe.  As we write, the highest Buy Now price for a full set is $250.  Quite a bargain, when you think about it.

There is another strain of the virus called DreamWorks Heroes Action Card Album Fever, whereby people are willing to pay more than ten times the retail price for a $6 album.  We have no expected numbers for that one.

Dr Audrey Tan, Mathmo Consulting
June 2014

Automaticity and why it’s important to learn your ‘times tables’

Recent press articles, both here and in Canada, suggest that the debate over teaching mathematics in primary schools today boils down to one tiny matter: whether or not we get students to memorise their ‘times tables’. It’s kind of sad, since there is so much more to learning maths than just knowing your multiplication facts, but nevertheless, it is an important and necessary step to becoming numerically fluent. I’ll explain why later.

In February this year, The New Zealand Listener had a few weeks of amusing banter. In the cover story “The superstar learner”, Professor John Hattie, the highly acclaimed New Zealand educationist and co-author of “Visible Learning and the Science of How We Learn” (which discusses the importance of instant retrieval of number combinations), said

“I’m a great fan of teaching kids the times table by rote. Why would you want to understand why six times nine is 54? It is, just accept it. Then you can use it.”

Gasp, did he say rote? I’ll come back to that. He went on to call the Numeracy Project “a debacle” that “pooh-poohed” automaticity.

Predictably, one of the lead researchers of the Numeracy Development Projects rose to the bait. Associate Professor Jenny Young-Loveridge disputed the NP’s “pooh-poohing” of automaticity, but did say

“Current mathematics education reform emphasises thinking and reasoning mathematically, rather than mindless recall.”

Er, who said anything about mindless recall?? In all fairness, I find Jenny Young-Loveridge to be one of the more honest maths education researchers in this country. See her final longitudinal study paper “A Decade of Reform in Mathematics Education: Results for 2009 and Earlier Years“, in which she concedes that the results are disappointing and “we have underestimated just how difficult [changing the way mathematics is taught] is.” (So, why did the Ministry of Education at that time not ask some tough questions about the return on their $70 million investment?)

The icing on the cake was the prize-winning letter in the following week’s issue. David Prior-Williams, a (retired?) high school maths teacher, wrote about Little Doreen who “was never going to set the mathematical world on fire”, but she knew her ‘times tables’. He doubted that she would have understood an explanation of the sort proposed by Jenny Young-Loveridge, who of course provided not one, but two strategies to explain why six times nine is 54! (If you’re curious, you’ll find them at http://nzmaths.co.nz/resource/easy-nines.)

“[Little Doreen] would have lost interest in 10 seconds and been none the wiser. She didn’t need to know why; but behind the counter in the dress shop a few years later, when a customer wanted six buttons at 9c each, she would have been on top of the situation immediately.”

The rest of his letter were the words of a wise teacher who learned, through pure experience, that trying to explain why certain things were true was “not only too difficult, but unnecessary. …In fact, in many cases, the penny will drop as a consequence of repeated use.”

Out of the three points of view, I find Mr Prior-Williams’ to be the most compelling. He describes someone who understands multiplication well enough to apply it quickly and correctly, in context. In other words, Little Doreen is efficient and effective in the real world. I rather like the sound of that. Incidentally, I wonder how many shop assistants these days would be able to cope with a cash transaction in the absence of a calculator, built in to the till or otherwise?

Over to Canada now. Both Manitoba and, more recently, Alberta have inserted into their curricula a specific requirement for children to memorise their times tables. This has sparked a huge public debate, including one secondary math teacher in Ontario who asked “Knowing Your Times Tables Is Nice. But Is It Necessary for Success?

You bet it is! I am surprised that a secondary math teacher would think that “having the memorization of multiplication tables removed from the Ontario math curriculum is a step in the right direction…”. Where is the evidence that “through a ‘variety of’ strategies and tools,…the eventual memorization of multiplication tables will come”? [Update (Dec 2016): The author updated his article earlier in the year and has retracted these statements.]

In my experience, children who do not put some direct effort into memorising their ‘times tables’ never learn them, and it really holds them back. They are slow at multiplying, and even slower at dividing. They all but grind to a halt when simplifying fractions, finding least common multiples or highest common factors. Consequently, they struggle with elementary algebra, e.g. expanding/factorising algebraic expressions including quadratics, working with algebraic fractions. This is no surprise. Research by Robert Siegler et al. has shown that knowledge of fractions and division is a strong predictor of success in high school mathematics.

In case the previous paragraph isn’t clear, understanding multiplication in and of itself is not the ultimate goal. What is far more important is getting students to understand division and the inverse relationship between multiplication and division, the precursor to proportional reasoning. Not once did the secondary math teacher refer to division, except to quote an expectation from the Ontario curriculum. Instead, he focussed on multiplication strategies that exploit the distributive property of multiplication over addition, i.e. at the end of the day, students are still adding. Even our own educationists recognise that the transition from additive thinking to multiplicative thinking is important, and yet they condemn students to remain stuck in the world of addition by promoting these strategies! Don’t get me wrong, they are sometimes useful, (especially as a backstop when a basic multiplication fact has not yet been committed to memory,) but this is not where the emphasis should be. Furthermore, in my experience, students who rely on distributive strategies show no more aptitude for grasping the abstraction of these methods, e.g. expanding quadratics, than their memorisation peers when they reach high school. A common mistake is to think that (a + b)2 is the same as a2 + b2. So where is the deep understanding there?

Believe it or not, it is actually possible for children to memorise the multiplication tables and understand what multiplication means. Plenty of us have done it, despite what the educationists would have you believe. Sure, we all have anecdotes about children who rattle off their tables but don’t have a clue what it’s all about. Memorisation isn’t the problem here, failing to bring meaning to the concept of multiplication is the problem. That’s why good teaching is essential. (Actually, most children don’t have a problem with the early concept of multiplication as groupings. It’s the concept of division that needs more work; again, groupings is a very good place to start.)

As soon as we hear the word ‘rote’, we gasp in horror at the thought of mindless recall, lacking in engagement. But in the strictest sense, ‘rote’ simply means memorising by repetition. Does repetition necessarily mean mindless recall? Definitely not; good teaching should monitor for that and prevent it. In a broader context, especially if a student improves with repetition, one might call it practice.

Learning by rote may not necessarily promote understanding, but memorisation does promote understanding, as cognitive science tells us.

Daisy Christodoulou: “Facts are not opposed to understanding; they enable understanding. This is because of the way that our minds work. Our long-term memories are capable of storing a great deal of information whereas our working memories are limited. Therefore, it is very important that we do commit facts to long-term memory, as this allows us to ‘cheat’ the limitations of working memory. The facts we’ve committed to memory help us to understand the world and to solve problems.”

Marc Smith: “Memorising facts can build the foundations for higher thinking and problem solving. Constant recitation of times tables might not help children understand mathematical concepts but it may allow them to draw on what they have memorised in order to succeed in more complex mental arithmetic. Memorisation, therefore, produces a more efficient memory, taking it beyond its limitations of capacity and duration. …[There] exists a considerable body of evidence to suggest that a memory rife with facts learns better than one without.”

I am not a traditionalist who harks back to the days of old, when children used to chant their ‘times tables’ in the classroom every morning. I support the memorisation of multiplication tables because it truly helps. It doesn’t have to be done by mindless recall. There are plenty of creative resources out there to make the task more fun and engaging. Aside from the direct benefits, it helps to create ‘muscle memory’ in the brain. Quite simply, students who retain what they learn are able to make faster progress. Now who wouldn’t want that?

Dr Audrey Tan, Mathmo Consulting
May 2014

Further reading:

Thank you very much to the writers who choose to link to this post. Please let me know when you do. Let’s keep the discussion going!

What have we Achieved?


We have a spat between our tertiary engineering schools and our secondary schools/NZQA. It’s time to bang some heads together.

Unfortunately, it’s true. I respect NCEA, but its structure does not support student achievement in algebra, and hence calculus, and recent revisions to NCEA standards have reduced the examinable content in these core topics. It’s a real concern because New Zealand desperately needs more science, technology and engineering.

On the other hand, the engineering schools should raise the bar if students attaining “Achieved” grades are under-prepared. The bar should never have been lowered in the first place! Every NCEA student wanting to continue with their studies should be aiming for “Merit” or higher.

But wait a minute. Secondary students are under-prepared for their studies too! PISA 2012 results are out and New Zealand’s rankings have plummeted (and not just in mathematics). It all starts at primary school…

New Zealand mathematics education is in trouble.

We need to turn things on their heads if we want to prepare school students adequately for tertiary study. The impetus must come from the top. University lecturers should influence what is taught in secondary schools, secondary school teachers should influence what is taught in primary schools. There needs to be a division of responsibility in designing a school mathematics curriculum. The mathematicians should determine the content, the educationists should determine how to deliver that content and ensure that teachers deliver it effectively.

This is my idea for a brighter future for maths education in New Zealand.

Dr Audrey Tan, Mathmo Consulting
December 2013

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

Counter-responses

“[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