Should long-division be the pinnacle of primary maths education?
Many people asked me to comment on the UK government's draft primary curriculum in maths, and the Department of Education's response letter. Rather than compare computerbasedmath.org with the new curriculum, I'll instead make a few initial observations for those who've followed its ideas.
When governments talk maths, they seem intent on convolving hand-calculating with rigour, rigour with understanding, calculating with numeracy, maths with calculating, rote-procedure learning with the vital conceptual and intellectual requirements of today's real-world maths.
I read the response letter first. It fitted this mould rather too well.
Then I scanned the curriculum itself. It seemed much better. I agree with many problem-solving aspirations and indeed many of the skills cited. I like its not-too-prescriptive approach, as I understand it, giving leeway for lots of different ways to achieve the teaching outcomes including (though this is not specifically cited) basing it on technology. Yes, I'd like this to be much more radical and programming to be included as a core skill, but I understand the difficulty of hard-coding it at this stage. I also understand why there's little reference to technology on the basis that its use isn't an outcome but a highly appropriate (I'd argue essential) tool to reach the outcome--outcomes which I think could have been bolder if computers were the default assumption for calculating.
Where my support starts to diverge is with procedures for multiplying fractions (when did you last use this formally eg. 3/16 x 7/8?) and there's a gaping chasm by the time we get to long-division (ever need to use that?).
Not only are these examples mechanics-led outcomes, not problem-centric (in the end it's problems that maths is there to solve not its own mechanics), but the mechanics in question is in practice obsolete ie. it's not in use in the real world nor do I believe it empowers understanding that is.
This saps student's time, energy and motivation. But I'm concerned about a far more serious problem: the lowly government portrayal of maths.
Should placing long-division or learning your times tables really be portrayed as the pinnacle of achievement in maths at primary school? Worse still, why imply that those tedious procedures are what maths is primarily about?
This is about the worst maths marketing you can do to prospective students--and in the long term to parents. Perhaps it's a good short term vote-winner for some, like brands that consistently do special offers improving sales short-term, but it's not a good long-term strategy for building a quality image of maths in our society or one that's aspirational. It's using long-division as a badge of honour of what the government call rigour when in fact it's a prime example of mindless manual processing.
And more than ever, it presents a broadening chasm between government's view of maths and the real-world subject.
The nub of real maths isn't rote-learning procedures nor does it depend upon them. It's not calculating, but the highly challenging mathematising of ever more complex situations for a computer to calculate, de-mathematising the results and validating their worth. It's creative, applied, powers some of the most successful ideas and developments of recent centuries and can even be fun and engaging!
A useful analogy is with survival skills. In the past your life would depend on rubbing sticks together to make a fire. Now those aren't likely to be life or death. Instead basic survival is how to cross the road or handle money. What are today's maths survival skills? What's at the pinnacle of today's maths?
Instead of rote learning long-division procedures, let's get students applying the power of calculus, picking holes in government statistics, designing a traffic system or cracking secret codes (so topical this month with Alan Turing's anniversary and his computer-based code breaking). All are possible, all train both creativity, conceptual understanding and have practical results. But they need computers to do most of the calculating--just like we do in the real world.
Examples from Wolfram Demonstrations site.
I hope these sorts of examples will all be encouraged under this new curriculum, and crucially that the assessments will highly value the skills they require, utilising computers so problems can be harder, more realistic and far more engaging.
One country will take this computer-based approach first and leapfrog others' technical education. This change will happen. The question is when not if. I worry that the UK won't be in the leading edge of this, but in so many ways it still could be.
Meanwhile, I won't be much help with my daughter's long-division homework. I've actually never learnt how to do long-division; I don't think it's disadvantaged me one iota.
P.S. Paradoxically, I would include times tables in a curriculum: they're still useful. Surrounded as I am by computers, I still find mentally hand-estimating helps me make quick judgements on information and I often do approximate multiplications to achieve that. Of course I could get my computer to do this---but for the moment, it's that bit slower. I do not think times tables give one valuable inherent understanding; but they are useful. Long-division doesn't and it isn't.
Tuesday, June 12, 2012 at 10:46AM
Reader Comments (8)
Long division is a specific algorithm. Good to teach algorithmic methods but why choose an obsolete example when there are so many more useful ones?
I have a primary-age daughter and have worked as a programmer. Surely primary education in mathematics is all about understanding number - yes, the mechanics of it, actually, but more than that - as Kevin Hewitson mentions in relation to times tables - the recognition of patterns.
Long division reveals something unique about numbers - not maybe of direct practical use, but learning it is still a process of discovery for the young mind. If taught well it will help to inspire a joy in pattern-recognition - problem-solving.
In the end it doesn't much matter whether young children are taught long division with a pen and paper or harder problems using a computer, if the teaching is adequate the result is the same - developing the ability to recognise patterns and solve problems.
Many people in education today see technology as a panacea, when in fact it can often just be a distraction.
School is not (or at least not primarily) about teaching the direct skills we need in adult life - most of those are more easily picked up as an adult as they are more relevant to an adult. Children (especially primary school children) do not need to be burdened with the world of work.
The demonstration examples are all very interesting and all, but they have not been made by 8-year old schoolgirls! Exactly what use are they to a primary school educator? Sure they can interact with them in the sense of pressing buttons to see what happens but they can't understand the words used to describe what the models represent, so they have no more idea what they mean than they are able to program in java as a result of playing an online shoot-em-up.
In any case it isn't either/or. Long division can be learned by a 10 year old in an hour if they have the basic building blocks. It has nothing to do with rote-learning and everything to do with understanding how numbers actually work.
I would suggest to Conrad Wolfram: learn long division, it'll only take you a minute and you might see something beautiful in it you never realised before!
Its a very good point, and I think I'm going to use this post as a reading for my adult ed maths class this autumn. Having said that I find that some of my students enjoy working the algorithm, and I remember my eldest daughter as a small child being impressed that she could cover a board with real mathematics just by trundling through the process. One final thought, the long division algorithm does have some slight further applicability when we get on to algebra and dividing polynomials, and this is the hook I hang it on when I am presenting it to my grown up students.