Marilyn Burns ( @mburnsmath ) posted an interesting example of how one child divided 56 by 4.
It just so happens that I’ve a whole collection of different ways pupils tackled this very question and got the correct answer. (There’s one incorrect answer, but it’s another interesting method.) ((PS Everyone has been far too polite to point out this is complete nonsense – Marilyn’s example is actually 56÷14 rather than 56÷4, but I should have spotted that some time ago. My apologies.))
Mr Nick Gibb, Minister of State at the Department for Education, objects to certain methods of tackling questions because he claims they’re inefficient. He’s making it plain that he wants to see children using traditional calculation algorithms, and he’s particularly had the grid multiplication method and the chunking process for division in his sights. Now I see how more children tackle these processes than most, and all the evidence that I have points to grid multiplication and chunking having brought about something close to a revolution in children’s understanding and fluency. Throwing away these valuable gains in favour of a purely notional increase in efficiency seems to me little short of crazy.
A dozen years ago I’d mark literally hundreds of children every summer who found it was the traditional long multiplication algorithm that was totally inefficient – because they couldn’t remember where or why they were supposed to be parachuting a zero or two into their operation. By contrast, the grid method allows each separate sub-product to be given the correct magnitude.
Exactly this situation occurred in yesterday’s lesson; the pupil is attempting to multiply 543 by 12. He uses the traditional algorithm and makes the classic error of multiplying first by 2 and then by 1 rather than by 10. He checked using the grid method and all the place value problems disappeared and the correct answer was reached. Which of the two methods was the inefficient one?
Today, every one of my average-attainment pupils will use the grid method with total efficiency and understanding. What’s more, unlike the standard algorithm, they use it as a basic and natural part of their toolkit to call upon when faced with new challenges. They can use the same method for every situation – three-digit by three-digit? No problem. Decimals? Lots of zeroes? And even fractions – for which the standard method is little help – they’ll use the same method for all of these. It’s not just my few pupils, either – today when I mark hundreds of pupils in their SATs papers I see the grid method used naturally and effectively as a general and reliable multiplication method.
A few weeks after Mr Gibb spoke, Freda and Joe, who were both on the level 4/5 borderline (i.e. a bit above average but by no means outstanding) got deep into triangular numbers. Effectively they were summing arithmetical progressions. They set themselves bigger and bigger challenges and surely even Mr Gibb would have been impressed how effectively they used the grid method. At one point we were walking down the corridor and Joe set himself the challenge of working out the 79th triangular number, which involved multiplying by 39½, but the grid method was so well established that he did the whole thing in his head. Go on, you have a go.
Here’s the board they filled with their calculations as the ideas and the challenges just flooded out. See how understanding the grid method is so complete that it’s become a tool that allows them to do their workings speedily without a second’s delay. All the work you see – along with more which got rubbed out – took place in a hectic burst of about 10 to 15 minutes. (One of yesterday’s pupils said “I haven’t got the time to rule a grid, I’m too busy working”.) Not much sign of inefficiency here, Mr Gibb.
With long division it’s much the same story. In this year’s Key Stage 2 Tests the long division question was 936÷36. In the conventional algorithm you first decide that you’re going to remove 720 (20 lots of 36); this leaves you with 216 which is 6 further lots of 36. This, I guess, is Mr Gibb’s preferred method – a similar calculation is done this way in a document National Curriculum Mathematics_Appendix_1 published in 2013.
Since this method involves removing first a batch of 20 lots of 36 and followed by subtracting a batch of 6 further lots of 36, someone will need to explain to me how it differs from an “inefficient” chunking method. I’m guessing – and it is only a guess – that it’s deemed as efficient because just two chunkings are used and the first of these is a multiple of 10.
I asked the statisticians for the information about the 600 000 responses to the 936÷36 question and they told me that 96% of pupils attempted the question, and 53% got the correct answer (another 9% used a valid method but made an error). These are figures that are far, far higher than anything we’ve ever seen before, and it was the chunking technique that gave children the confidence and the ability to handle the question. I can claim something like expert knowledge here, because this summer I marked no fewer than 3000 of those responses to this very question – which is a little more than 1 in every 200 children in the age-group.
Rather than a weakness, it seems a great strength of the chunking technique that you can do your chunkings however feels right to you – for example, I marked several cases where there were three chunkings, of 10, 10, and 6 lots of 36. Perhaps Mr Gibb believes using three chunkings is inefficient, though it doesn’t seem to me to devalue the method. By the way, many pupils preferred to chunk up to 936 and not down to 0. Again, that option seems a strength rather than a weakness.
Whoever is going to have to write the rules that define an approved method deserves some sympathy. When you’ve got 600 000 pupils tackling a question not everyone wants to use anything resembling the standard algorithm in the first place. For example, it’s easy to see that 936 is divisible by both 9 and by 4, so what about the children who chose to tackle division by 36 as a two-stage short-division process, dividing by 9 and then by 4? Or dividing by 6 twice? They’re not using Mr Gibb’s method, so are they being inefficient – or are they being insightful and creative?
And the response I found most memorable was the child who first subtracted 36 from 936 to leave 900, and then called upon their recognition that 900 is 25×36, so immediately getting the answer of 1+25, or 26. It’s hard to see anyone could be more insightful or efficient than that. But wait a minute, hasn’t the child has actually done a chunking, which Mr Gibb says is inefficient? Even worse, they’ve taken the smallest chunk off first, which seems to be against Mr Gibb’s procedure.
And of the 600 000 children in the cohort, some will be seriously clever and will work the question out mentally. When you can work out 936÷36 in your head, how can anyone possibly have the nerve to criticise you for being inefficient because you didn’t use a standard method?
None of this is intended as a criticism of traditional algorithms. If teachers and children see them as their preferred choice and can use them effectively and with the fluency and understanding that the National Curriculum requires, then that’s fine with me. But my observations about the grid method and chunking don’t point that way. We subject well over half a million 11-year-olds to the Key Stage 2 Tests each year, and we use the results to grade children and their schools. I’d be far more comfortable if we used the huge mass of data to tell us about the mathematics children can do and the methods they use.
I’ve heard it said that Nick Gibb does some long division practice every day before breakfast. That may or may not be true, but it may be more relevant than you think, because Mr Gibb is Minister of State at the Department for Education and it’s certainly in keeping with many of his views. Politicians do seem to attach almost mystical importance to long division, though perhaps it’s rather like doing the laundry. Everyone knows the traditional way is to go down to the riverbank and bash your washing with a rock, but I bet it’s a long time since Mr Gibb attacked his shirts with a large stone.
It might be rather nice if Mr Gibb and those in his circle – and the Education Secretary Ms Nicky Morgan certainly seems to be one of them – could be persuaded that there might be other ways of tackling mathematics in the 21st century than automatically applying a mechanical procedure. After all, I’m at the end of a very long line of people who’ve pointed out that if you want a mechanical procedure it makes a lot more sense to get a machine to do it.
I’d guess one reason Mr Gibb enjoys long division is because division feels different from the other operations. In addition, subtraction, and multiplication you have your two numbers, you do things with them (conventionally beginning with the digits of least value) and out pops the answer.
With all but the simplest divisions it works differently. You do something (starting this time with the digits of greatest value rather than the least) to find part of the answer; then you do another something and find the next part of the answer, and so on. And since finding each of these numbers may involve trying one value, and then another and perhaps another, it’s all akin to a trial and improvement process and yes, it does feel a different kettle of fish entirely.
There’s a parallel to another process. In the sixth form I learned an algorithm that was reminiscent of a long division algorithm on steroids. You thought of a number, did several things with it, did it all several times more, and eventually you found the square root (or more likely, an approximation of the square root) of the number you started with. This process was even more arcane than long division – I recall very little apart from the fact that at one point you needed to double the number you first thought of. In fact, and I may be unique here, I actually used the even more complicated procedure to find cube roots (though I’d certainly appreciate it if you didn’t ask me for any details).
Even in 1958 this was the most pointless activity you can imagine. At no time in my life did I ever need to employ these algorithms and they’ve long since passed into history. Fortunately politicians have never heard of them, so I’ve never heard even the most diehard traditionalist demand that extracting square roots ought to be a fundamental part of the mathematics curriculum. Otherwise they might still be asking students to find square roots by hand, and indeed do so in the examination room.
Don’t get me wrong; I do actually have some sneaking regard for Mr Gibb, Ms Morgan, and others in their profession. They have qualities which I totally lack, they’ve got to the top of a hugely competitive profession, they’re prepared to tackle big problems, and I don’t doubt they’re motivated by a desire to make things better. But I do wonder how Mr Gibb’s advisers (or indeed the audience) could allow him to speak dismissively of “… inefficient methods such as the grid method for multiplication and chunking for long division …” (London Thames Maths Hub Primary Conference, 27th March).
Now I’m prepared to assume we’re going to devote some curriculum time to the processes of long multiplication and long division and I want to write about these “inefficient methods” in part (ii).
I recently wrote about a pupil who uses a subtraction technique he invented for himself; this is a rather similar situation.
There are two reasons I particularly remember the inspection I did at Hamphill Middle School. The first happened when I sat down at the back of the classroom. Now, like most of us, I’m pretty experienced at sitting down, so I wasn’t concentrating on the process very hard, but I became aware that something strange was happening. I couldn’t work out what it was, and it continued to happen very much in slow motion until I found myself on the floor – one of the back legs of the chair had given way. I was hugely impressed by the reaction of the children, who were most solicitous and greatly concerned for my welfare even though no inspector has ever made a more undignified and hilarious spectacle.
After this dramatic start the lesson continued with the children doing some division examples. Partway through, I noticed a girl with her back to me, behaving in a manner so furtive it would have resulted in instant arrest in the outside world. She looked around to check the teacher wasn’t watching, hunched herself up to conceal what she was doing, and scribbled something on a piece of paper. She wrote something in her exercise book and scrunched up the paper and stuffed it into her pocket.
The calculation she was doing was 95÷5 and her method is totally transparent. She’s seeing it as a sharing, with 95 to be distributed into five packages. She makes an initial distribution of 10 into each package and follows this with a further 5. She’s keeping track of the amount distributed, and decides she can make another distribution of 5 into each package. You can almost hear her exclamation of frustration as she realises that will be too many and the 5 needs to be downgraded to 4.
At that point she knows she’s distributed all 95; she adds the 10, 5, and 4 in any package and has the answer of 19. She transfers this to her exercise book in the approved format, and the piece of paper has now served its purpose and goes into her pocket.
I talked in detail with her, and what she said went along these lines:
(1) My teacher’s taught us how to do these, but I don’t understand her method, so I’m not a very good pupil.
(2) However, I can do them using my own method, which of course is really cheating, because it can’t be as good as the correct method.
(3) I love my teacher, and if she thinks I can’t understand her method she’ll be disappointed because she’ll think she’s let me down.
(4) So I’ll work them out my way and write them up as if I’ve used her method.
Poor Justine! Her emotions involved her perception she was a failure who could only use a method she saw as inferior, coupled with a very real concern that she didn’t want her teacher to be disappointed either in herself or in Justine.
My reaction was of course exactly the opposite. As far as I was concerned, Justine was actually doing better maths than many others who were simply following a rule without insight or understanding. The example shown is pretty low-level, but it’s extensible. As understanding grows, you can divide – should you need to do so – both larger and smaller numbers using the same starting point.
Tim and his friend were both happy to spend their first few minutes with me chatting about themselves. The friend had a pretty good feeling about himself and said he felt he was good at mathematics. Tim, on the other hand, said “I’m pretty bad at maths” – so even though school feels they’re of pretty similar abilities they’ve clearly got very different self-images.
A few minutes later the calculation 74 – 46 came up. Tim gave it some thought and gave the correct answer, 28. He’d done it mentally, so I asked him to explain what he’d done. “6 minus 4 is 2” he said.
“Erm, aren’t you doing 4 minus 6?”, I asked.
“6 minus 4 is 2”, he repeated. “That tells me how many I’ve got to take away from 70. So I get 68, and then I take off the 40, and 28 is the answer.”
He did several more the same way and some three-digit subtractions as well, using the same method and getting each of them correct.
Now, for something like 70 years whenever I see something like 74 – 46 a little voice in my head says “4 minus 6 you can’t do, so you borrow a 10 ….” But the voice in Tim’s head is saying “4 minus 6 of course you can do, it’s negative 2”.
So Tim’s method is actually not a convenient trick, it’s actually better than the method I was taught. It’s based on clarity and understanding rather than confusing and misleading terms like “borrow” and “pay back”.
I’ve heard anecdotal accounts of children who’ve used this method before, but Tim is the first of my pupils who’s discovered it and articulated it to me. As you’d expect, I took some pleasure in telling Tim that that far from being bad at maths, his method represents much more insight and achievement than is needed by those pupils (like me in 1950 or thereabouts) who simply follow a rule given by the teacher.
(I recall with considerable embarrassment the first talk I ever gave at a parents’ evening. It was a new school and the hall was full with parents who wanted to know about about teaching. I demonstrated a subtraction example – and with 150 people watching I got in a total mess. Never in my life had I needed to think about what I was doing, I knew the rule and applied it automatically – until now. As I floundered around, I knew what every person in the audience was thinking, and I knew it wasn’t complimentary.)
I asked Tim when he’d devised his method – probably about 7, he thought. What did his teacher say about it, I asked. As I rather suspected, Tim had never previously disclosed it to any teacher, believing they wanted him to use the traditional written method – so Tim has always worked his subtractions mentally, and then writes out the sum with borrowing and carrying figures so the teacher won’t suspect he’s done anything unusual.
PS: I’ve just discovered this in my files. I can’t remember anything about Shelley – she may have been someone I met or someone who a colleague alerted me to – but clearly she’d used the same reasoning as Tim but recorded her working in a different way: