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