Mr Gibb and Long Division etc (part ii)

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.

.

.