The Association of Teachers of Mathematics is dismayed at the programmes of study for mathematics just published [...] The curriculum as presented will result in more attention spent on developing technical competence in outdated written methods for arithmetic at the expense of developing secure foundations for progression through mathematical concepts and skills. [...] Appendix 1, entitled formal written methods for multiplication and division, but including addition and subtraction as well as multiplication and division, is a complete travesty and needs to be removed.
Gove’s decision on the EBCs took attention away – initially at least – from another big announcement made on the same day, as England’s helter-skelter reform programme continues: the detail of the entire draft the new national curriculum for first teaching from 2014.
But it is a fair bet that controversy on that front is not going to let up. One person seriously unhappy with one aspect of the proposed primary maths curriculum is Anne Watson, professor of maths education at the University of Oxford.
Watson was involved in the drafting of the document, but says that concerns about the inclusion of long division in the new programmes of study, registered by her and most of the maths teaching community including the overarching Advisory Committee on Mathematics Education, have been ignored by ministers.
Writing to the Guardian, Watson argues that long division is a “ping pong between the government and maths educators”, most of the latter believing that specifying it in the curriculum is not the best way of preparing children for secondary education.
“Why on earth is a government interfering at this level with the teaching of a subject?” she asks, adding that there appears to have been a “blatant disregard” for what is known about how children learn maths by either ministers, their advisers, or both. The government has defended long division as the “most efficient” calculation method.
This approach [...] was called progressive because it moved away from a set hierarchy of knowledge – literary canons, mathematical proofs, scientific laws, musical exercises and artistic traditions towards a new emphasis on “learning to learn”.
The EBacc squeezed out creativity, some claimed. So does that mean scientists from Rutherford to Dawkins are arid and uncreative mechanics? Mathematicians from Pythagoras to Turing are enemies of creativity?
[...] unless you have knowledge – historical, cultural, scientific, mathematic[al] – all you will find on Google is babble.
So our new curriculum affirms – at every point – the critical importance of knowledge acquisition. [...] There is new and detailed content on the mathematical processes every child should master – including early memorisation of tables, written methods of long division and calculations with fractions – which was either absent or obscure before.
Three letters published in a recent issue of TES (1 Feb 2013) under the heading
The junking of chunking is bad news for maths pupils highlight what, in my opinion, remains, a serious flaw in the current debate on mathematics education: confusion between the content and methods of teaching.
The recent speech by education minister Elizabeth Truss and subsequent articles about mathematics (“Time to knock chunks out of KS2 maths, minister says“, 25 January) fill me with fear for the next generation of primary children.
Her straw man argument mischievously rubbishes well-tested methods currently being taught. So-called “gridding” and “chunking” are logical learning developments which help children later to understand formal written long multiplication and long division respectively. Teaching these new methods has relieved the problem of the failed maths teaching of the past century: many children who were taught traditional methods of calculation, without understanding how they worked, had little confidence in their arithmetic and became fearful of maths.
I would instead draw ministers’ attention to the most significant problem facing maths education now – the lack of high-quality maths teachers who are willing to enter and stay in a profession which is endlessly dictated to according to the career aspirations of rising ministers, eager to impress their political masters.
Ralph Manning, Lecturer in primary mathematics education, University of East Anglia, and primary teacher.
It would be very optimistic, or educationally naive, to imagine that we could find one definitive method for multiplication and division and that all children could successfully learn it that way.
Finding the most “efficient” method may be an easier task, but there is a difference between efficient and effective when one considers the individuality of pupils. The chunking method often requires more steps but that may be a trade-off for other disadvantages that some children experience, most notably the tendency not to try the task at all if it is considered “too hard”.
That was the less worrying part of the article. The bit that is truly fascinating is the way in which children and teachers will be encouraged to take a narrow view of learning maths. Children’s efforts will be judged on a basis that can be summed up as “no marks for thinking differently from me”. I feel that we are entering an almost Orwellian world where “Orthodoxy means not thinking – not needing to think”.
Steve Chinn, Bath.
Your article on primary maths raises the issue once again of whether or not politicians should be able to prescribe teaching methods. The legal situation is unclear. The Education Reform Act 1988 does proscribe the education secretary from prescribing teaching methods. But there is an ambiguity. Is doing long multiplication by traditional methods part of the content of the proposed new curriculum or is it one of the methodologies by which that curriculum is taught? If the former, then it can be prescribed by the government. If the latter, it cannot.
If challenged, Michael Gove would probably say that he won’t be prescribing how traditional long multiplication is taught but that it will be taught. I’m afraid the system lost the chance to challenge this issue when it capitulated on synthetic phonics.
Colin Richards, Spark Bridge, Cumbria.
Long division and multiplication will make a return to maths exams as part of a Government drive to boost standards in primary schools, it will be announced today.
Pupils aged 11 will be given extra marks for employing traditional methods of calculation in end-of-year Sats tests, it emerged.
Children who get the wrong answer but attempt sums using long and short multiplication or adding and subtracting in columns will be rewarded with additional points.
Ministers insisted the changes – being introduced from 2016 – were intended to stop pupils using “clumsy, confusing and time-consuming” methods of working out. [...]
Elizabeth Truss, the Education Minister, will outline the plans in a speech to the North of England Education Conference in Sheffield on Thursday.
Speaking before the address, she said: “Chunking and gridding are tortured techniques but they have become the norm in recent years. Children just end up repeatedly adding or subtracting numbers, and batches of numbers.
“They may give the right answer but they are not quick, efficient methods, nor are they methods children can build on, and apply to more complicated problems.
“Column methods of addition and subtraction, short and long multiplication and division are far simpler, far quicker, far more effective and allow children to understand properly the calculation and therefore move on to more advanced problems.”
Eighteen months ago, I visited a London primary school hailed as exemplary in its maths teaching. A year 6 pupil (aged 10) showed me how she calculated 432 ÷ 12. She hadn’t been taught long division and used a method known as “chunking”. This involves deducting multiples of the divisor (12 in this example) from the 432 until all that is left is a zero or a remainder, like so:
432 – (20 * 12) = 432 – 240 = 192
192 – (10 * 12) = 192 – 120 = 72
72 – (6 * 12) = 72 – 72 = 0
The answer is derived from adding the 20, 10 and 6 to make 36. It is a form of repeated subtraction and its advocates believe it aids the understanding of “place value”. Ofsted, however, views this method as “cumbersome” and “confusing” and it did seem to take the pupil a long time to complete. A similar approach is taken to multiplication. Instead of learning and practising the long multiplication algorithm, numbers are deconstructed into their component parts and formed into a grid:
Imagine the size and complexity of the grid if you wanted to multiply a seven-figure number by 12.
These methods are now universal in our primary schools, with strong resistance to the teaching and practice of traditional algorithms amongst many in the educational establishment.
The Advisory Committee on Mathematics Education (ACME), for example, believes that “the details of longer algorithms are easily forgotten” and
dismisses their importance because they require time and continuous practice.
Anne Watson has continued discussion of the role of long division by posting a comment to one of the earlier posts. It is awkward for me to talk about long division: I teach at university, it is difficult for me to have an opinion on at what age and at what Key Stage schoolchildren have to learn long division. But I believe in the educational value of written algorithms for addition, subtraction, and especially long multiplication and long division — because the latter is a tremendous example of all important recursive algorithms.
My approach to school level mathematics education is very practical: I teach a course in mathematics for Foundation Studies, to students who wish to study hardcore STEM disciplines, but have not taken, or dropped out, or failed mathematics A levels. I work at the receiving end of the GCSE. And I have to make sure that my students master long division (with remainder!) of polynomials. Why? Because relatives and descendants of the long division, various versions of the Euclidean Algorithm (including the ones for polynomials) saturate information processing around us; for example, they are used every time when we pay in supermarket by a credit card. Of course, the user of a credit card does not need to know Euclidean Algorithms, but the society needs some number of people who know how credit cards are working, and therefore understand long division.
I believe that we should give a chance to learn long division to every child. I do not know what is the best way to achieve this. But I make my modest contribution: I give my students a second chance to learn long division, this time long division of polynomials. And I start this topic with a brief review of long division of integers, largely with the aim to alleviate fears and psychological blocks accumulated by many of my students in their KS 1-4 studies. I intentionally do that in a lighthearted and semi-improvised fashion, engaging students in a direct dialogue.
What follows is an example which I improvised for my students in my lecture in December 2011; it was published in my blog on 9 December 2011. Most likely, my example it is not suitable for use in school level teaching, but, judging by response from my students, it appears to serve its purpose to help those students who learned long division at school, but forgot it, to refresh their memories and move to the next level of learning, to long division of polynomials. Also I think that the fact that we have to remind long division to university students suggests that we cannot avoid some discussion of its place in the school curriculum.
A fable about long division. An innumerate executor of a will has to divide an estate of 12,345 pounds between 11 heirs. He calls a meeting and tells the heirs: “The estate is about 12 grands, so I wrote to each of you a cheque for 1,000 pounds.”
The heirs answer: “Wait a second. There is more money left” — and write on the flip chart in the meeting room:
“Ok” — sais the executor – “there are about 13 hundred left. So I can write to each of you a check of 100 pounds”:
“But there is still money left in the pot” — shout the heirs and write:
“Well,”– says the executor, — “it looks as if I can give extra 20 pounds to each of you”:
“More! More!” — the heirs shout. “I see” — said the executor — “here are 2 pounds more for each of you”:
“I deserve to get this remainder of 3 pounds and buy myself a pint. And each of you gets 1122 pounds”:
After finishing my tale on this optimistic note, I commented that the whole calculation, which looks like that:
is usually written down in an abbreviated form:
And we say that
12345 gives upon division by 11 the quotient 1122 and the remainder 3
As simple as that.
Pedagogy and mathematics
It is undeniably important that mathematics teachers have mastered the topics they need to teach. The new Australian national curriculum is misguidedly increasing the amount of “statistics” of the school mathematics curriculum from less that 10% to as much as 40%. Many teachers are far from ready for the change.
But more often than not, the problem is not the mathematical expertise of the teachers. Pedagogical narrowness is a greater problem. Telling that there is a correct idea in a wrong solution to a problem on fractions requires unpacking of elementary concepts in a way that even an expert mathematician is not usually trained to do.
One of us – Jon – learned this only too well when he first taught future elementary school teachers their final university mathematics course.
Australian teachers at an elite private school could not understand one of Jon’s daughter’s Canadian long-division method nor her solution techniques for many advanced school topics. She got mediocre marks during the year because of this.
From the speech by Nick Gibb, State Minister for Schools, at the annual meeting of ACME, 10 July 2012:
[T]he draft programme aims to ensure pupils are fluent in the fundamentals. Asking children to select and use appropriate written algorithms and to become fluent in mental arithmetic, underpinned by sound mathematical concepts: whilst also aiming to develop their competency in reasoning and problem solving.
More specifically, it responds to the concerns of teachers and employers by setting higher expectations of children to perform more challenging calculations with fractions, decimals, percentages and larger numbers. [...]
As it stands, the draft programme is very demanding but no more demanding than the curriculum in some high-performing countries. There is a focus on issues such as multiplication tables, long multiplication, long division and fractions.
Last month, the Carnegie Mellon University in the US published research by Robert Siegler that correlated fifth grade pupils’ proficiency in long division, and understanding of fractions, with improved high school attainment in algebra and overall achievement in maths, even after controlling for pupil IQ, parents’ education and income.