Place-value notation

“Place-value notation” is the third chapter in “Maths in 100 Key Breakthroughs”, by Richard Elwes. It is preceded only by “The evolution of counting” and “Tallies”. Note, the author said columns
…the point being, to use place-value notation but to not allow young children to add and subtract in columns is utterly perverse. This is precisely what the system was designed for.

And before we hear any more dogma, let’s go back to the facts. In 2011, 48% of 5572 New Zealand students averaging 9.9 years of age could not answer correctly a question that involved adding two three-digit numbers.

Come on, people, let’s give these kids a fighting chance.

Alberta, Canada brings back column addition!

Meet the inimitable Dr Nhung Tran-Davies, a concerned parent who started the “Back to Basics Math Petition” in Alberta, Canada, and has gathered a whopping 16,000+ signatures. I’ve always thought of Canadians as pretty laid back, but now I see they are passionate about things that really matter, and their passion brings about change.

The petition (and a protest outside the Alberta Legislature) forced their Minister of Education to open his doors for discussion. He has agreed to the following curriculum changes:

  1. To remove the need for students to use multiple strategies to solve problems.
  2. To remove all disparaging languages regarding memorization, practice, and standard algorithms.
  3. Re-integration of the multiplication table in the curriculum.
  4. Explicit expectations for children to recall basic math facts.
  5. The re-inclusion of the “traditional algorithms” as a listed option for teaching.

Nhung’s work hasn’t finished yet – she won’t stop until column addition and the other column-based methods are mandated. Her dedication to this important cause sets an example to us all.

The Canadian media have played a huge part in igniting public debate and instigating change. What will it take for the New Zealand media to show a similar interest?

Dr Audrey Tan, Mathmo Consulting
June 2014

Alberta government makes progress on fixing badly broken math curriculum, but… | Edmonton Journal
The Great Canadian Math Debate, Pt. 44: Tran-Davies has met with Education Minister Jeff Johnson and made real progress on fixing our math curriculum, but standard arithmetic still needs to be a curriculum imperative …

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

The perversity of Reform Math

“The essential perversity of all Reform Math programs is to introduce the complex ahead of the simple. It’s a bulletproof way to confuse little children, and our Education Establishment keeps exploiting it.

Clearly, the all-pervasive problem is that we can’t trust our so-called experts. They are ideologues first, educators second, and therein lies the tragedy. These pretenders design curricula to achieve ideological goals. What we need from now on are curricula that achieve educational goals.” – Bruce Deitrick Price


Progress in Parts

Adding in Parts - An Example

Sophie (not her real name) is a Year 6 student who does not find maths easy. Each week, she comes to see me and I always ask her what maths she is learning at school. One week, she described what I believed to be subtraction as addition in reverse, i.e. starting with the smaller number and working out how much more is needed to make the bigger number. What she wrote down wasn’t correct, but I was able to decipher it and got her computing differences correctly by writing them out as jumps along the number line. It helped her a lot and she was quite happy with that.

Thinking of subtraction as addition in reverse, and viewing these sorts of calculations on the number line, is a very worthwhile strategy. However, the following week, Sophie wrote down something which I can only guess was meant to be “Adding in Parts”. It was difficult to know. She drew a fork underneath one of the numbers but simply didn’t know what numbers to write.

In case you don’t know, Adding in Parts is one of the main additive strategies that over-complicates things for children. One number is split up so that the other number can be made up to (nominally) the next “tidy number”, then you add on whatever is left. For example, 27 + 5 = 27 + (3 + 2) = (27 + 3) + 2 = 30 + 2 = 32. Implicitly, this requires computation of the next tidy number (30), two subtractions (30 – 27, 5 – 3) and a new but hopefully easier addition (30 + 2). This type of (“part-whole”) thinking is highly regarded in the Numeracy Project, and is notably the glorious hoop that students must jump through to gain permission to learn/use column addition at school. Sadly some students never quite make it. It’s not a hoop, it’s a barrier. If you really want to understand how stupidly inefficient and cumbersome Adding in Parts is, try programming a computer to perform addition in this way.

I explained to Sophie what I thought needed to be done, but it didn’t ring any bells, nor did the task get any easier for her after numerous attempts. I admired her determination to master this strategy, but it upset me to see her so close to tears. All the while, I tried to reassure her that she really didn’t need this strategy! Sophie is perfectly capable of adding numbers using column addition and she executes the method well. There is nothing to be gained by her taking twice as long, struggling (or even failing) to add two numbers using this highly overrated strategy. In the end, I managed to persuade her to stick with column addition and she agreed that she felt better using this method.

With Sophie’s permission, I wrote to her mother and recommended that Sophie should not continue with Adding in Parts at school at this point in time. If it was deemed absolutely necessary for the purpose of assessment, then perhaps she could come back to it later, but it was harming her mathematical development and confidence to persist with it now.

Sophie’s mother arranged a meeting with Sophie’s teacher, and this is what happened:

“Our meeting went well. Sophie’s teacher listened and took on board everything we said. She sympathised with Sophie as we talked about her difficulties learning maths. I think the teacher is now going to use some of your strategies with the class. She is just so thrilled that we are working together and backing each other up.”

I am equally thrilled. Maybe the whole class will benefit. Hats off to Sophie’s teacher. I hope other teachers will follow suit.

Dr Audrey Tan, Mathmo Consulting
June 2014

Dreamworks Heroes Action Cards

Are your kids collecting the Dreamworks Heroes Action Cards from Countdown? Here’s a little game that you could play with them to practise multiplication and division by 6.

Part 1: Get them to fish out all the cards with numbers that are multiples of 6, i.e. the numbers in the 6 “times table”. Next, turn these cards over and look at the die/dice patterns on the back. Do they notice anything interesting?

Part 2: With the cards that are left, take a card and show them the number on the front of the card. Ask them to divide that number by 6 and calculate the remainder. Then turn the card over and see if their answer matches the die pattern on the back of the card!

In playing this game, it would be great if children progressed from skip counting in 6s to jumping straight to the required multiple of 6. So if the card number was 20, a beginner might say “Six, twelve, eighteen…I need TWO more to make 20”, while a seasoned player might say “Six goes into 20 three times with TWO left over”.

Parents can try this at home, or teachers (who haven’t banned the cards!) can try this in the classroom. Have fun!

Dr Audrey Tan, Mathmo Consulting

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

“[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!