Conrad Wolfram


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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 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.

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Reader Comments (8)

I feel just the same about mechanical rote procedures but teaching children to set their maths out well and systematically is a useful skill. Marking exam scripts I can see a lot of muddled thinking with bits of sometimes irrelevant calcuations all over the place. We want children to understand as much as possible but if we fail in this, our children often end up with nothing.

June 15, 2012 | Unregistered CommenterBarbara Edwards

Let me start by saying I find maths fascinating, I love the way it approaches and tries to solve problems. As a student I recall buying a book with the title "Amusements in Mathematics" by H E Dundeney first printed in 1917(yes I still have the book). My peers thought me "different", how can you enjoy maths! I did not become a maths teacher, instead I became a Technology teacher although still seeking to solve problems. After retiring I turned my attention to developing ways of helping others enjoy maths, although at the prime reason was to get their SAT or GCSE grades. The result of research into learning, the maths curriculum and working with young learners has been a great journey. The result is a new (if there is such a thing!) approach to helping students not only in maths but literacy and understanding their learning needs. One way I approach coaching students in maths is to treat it as a language. The result of this approach with adults has resulted in tears (of joy and frustration) when they realise they have, as one person put it, "wasted so much time in my life and turned down so many opportunities because I though I could not do maths". This is the cost of playing with a curriculum and getting it wrong. I wish people (politicians) would demonstrate their understand of this instead of use education for their own personal political ends.

I have some examples of the coaching packs I have used in our centre over the last 12 months if anyone is interested. You can contact me through my website at

I too believe in the times table not only for its convenience but because it helps develop number pattern and recognise relationships between numbers. I find this an essential tool when problem solving using maths.


June 15, 2012 | Unregistered CommenterKevin hewitson

Thank you for wrestling with critical areas to move student thinking ever higher. In elementary school, being engaged to some level in the foundational understanding of how calculation works, to take the mystery out of the functions that need to become automatic, is important. But as you say, " the end it's problems that mathis is there to solve, not its own mechanics."

50 years after learning about solving quadratic equations I still have not found a use for them! I seem to remember I liked the idea there was a solution, a bit like solving a crossword, but as a life skill, which I presume is what learning is about....useless. Conversly I think hardly a day passes when I don't use my times tables to estimate something, basic numeric relationships should definitely be learned.

June 15, 2012 | Unregistered CommenterShena
This is an interesting discussion as well and gives a different perspective from which to have a rich conversation about mathmatical learning in elementary grades.

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