3 – 1 = 2

What follows is a translation of a fragment from Igor Arnold’s (1900—1948) paper of 1946 Principles of selection and composition of arithmetic problems (Известия АПН РСФСР, 1946, вып. 6, 8-28). I believe it is relevant to the current discussions around “modelling” and “real life mathematics”. For research mathematicians, it may be interesting that I.V. Arnold was V.I. Arnold’s father.

Existing attempts to classify arithmetic problems by their themes or by their algebraic structures (we mention relatively successful schemes by Aleksandrov (1887), Voronov (1939) and Polak (1944)} are not sufficient [...] We need to embrace the full scope of the question,  without restricting ourselves to the mere algebraic structure of the problem: that is, to characterise those operations which need to be carried out for a solution. The same operations can also be used in completely different concrete situations, and a student may draw a false conclusion as to why these particular operations are used.

Let us use as an example several problems which can be solved by the operation

\[3 - 1 =2. \]

  1.  I was given 3 apples, and I have eaten one of them. How many apples are left?
  2. A three meters long barge-pole reached the bottom of the river, with one meter of it remaining above the level of water. What is the depth of the river?
  3. Tanya said: “I have three more brothers than sisters”. In Tanya’s family, how many more boys are there than girls?
  4. A train was expected to arrive to a station an hour ago. But it is 3 hours late. When will it arrive?
  5. How many cuts do you have to make to saw a log into 3 pieces?
  6. I walked from the first milestone to the third one. The distance between milestones is 1 mile. For how many miles did I walk?
  7. A brick and a spade weigh the same as 3 bricks. What is the weight of the spade?
  8. The arithmetic mean of two numbers is 3, and half their difference is 1. What is the smaller number?
  9. The distance from our house to the rail station is 3 km, and to Mihnukhin’s along the same road is 1 km. What is the distance from the station to Mihnukhin’s?
  10. In a hundred years we shall celebrate the third centenary of our university. How many centuries ago it was founded?
  11. In 3 hours I swim 3 km in still water, and a log can drift 1 km downstream. How many kilometers I will make upstream in the same time?
  12. 2 December was Sunday. How many working days preceded the first Tuesday of that month? [This question  is historically specific: in 1946 in Russia, when these problems were composed, Saturday was a working day --AB]
  13. I walk with speed of 3 km per hour; my friend ahead of me walks pushing his motobike with speed 1 km per hour. At what rate is the distance between us diminishing?
  14. Three crews of ditch-giggers, of equal numbers and skill, dug a 3 km long trench in a week. How many such crews are needed to dig in the same time a trench that is 1 km shorter?
  15. Moscow and Gorky are in adjacent time zones. What is the time in Moscow when it is 3 p.m in Gorky?
  16. To shoot at a plane from a stationary anti-aircraft gun, one has to aim at the point three plane’s lengths ahead of the plane. But the gun is moving in the same direction as the plane with one third the speed. At what point should the gunner aim his gun?
  17. My brother is three times as old as me. How many times my present age was he  in the year when I was born?
  18. If you add 1 to a number, the result is divisible by 3. What is the reminder upon division of the original number by 3?
  19. A train of 1 km length passes by a pole in minute, and passes right through through a tunnel at the same speed — in 3 minutes. What is the length of the tunnel?
  20. Three trams operate on a two track route, with each track reserved to driving in one direction. When trams are on the same track, they keep 3 km intervals. At a particular moment of time  one of them is at crow flight distance of 1 km from a tram on the opposite track. What is the distance from the third tram to the the  nearest one?

These examples clearly show that teaching arithmetic involves, as a key component, the development of  an ability to negotiate situations whose concrete natures represent very different relations between magnitudes and quantities. The difference between the “arithmetic” approach to solving problems and the algebraic one is, primarily the need to make a  concrete and sensible interpretation of all the values which are used and/or which appear in the discourse.


This to a certain degree defines the difference of problems where it is natural to request an arithmetic solution from problems which are essentially algebraic. For the latter, an arithmetic solution could be seen as a higher level exercise that goes beyond the mandatory minimal requirements of education. In many problems relations between the data and the unknowns are such that an unsophisticated normal approach naturally leads to corresponding algebraic equations. Meanwhile an arithmetic solution would require difficult, hard to retain in memory, algebraic by their nature operations over unknown quantities.

This happens, for example, in solution of the the following problem.

If 20 cows were sold, then hay stored for cow’s feed would last  for 10 days longer; if, on the contrary, 30 cows were bought than hay would be eaten 10 days earlier. What is the number of cows and for how many days hay will last?

Some basic understanding of relations between the quantities appearing in the problem suffices for its conversion in an algebraic form. But to demand from pupils that they independently came to the formula

\[(200+300) \div 10\]

means pursuing a level of sophistication in operation with unknown quantities that is unnecessary in practice and unachievable in large scale education.

[With thanks to Tony Gardiner]

Content and method

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.

Letter 1:

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.

 Letter 2:

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.

Letter 3:

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.

Children to be marked up for using long division in maths

By   in The Telegraph:

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.”

Former minister on "chunking" and long division

From a paper by Nick Gibb MP in The Telegraph:

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.

 Read the full article.

More on ban of calculators in tests for 11-year-olds

Press release from DfE and discussion of underlying statistics in fullfact.org. A table from fullfact.org

based on Trends in International Mathematics and Science Study (TIMSS):

From TIMSS 2007: England rank bottom of all sampled countries for restricting calculator use in the maths classroom


Calculators banned in primary school maths exams

By  in  The Telegraph:

Calculators are to be banned in primary school maths exams as part of a Government drive to boost standards of mental arithmetic, it was announced today.

Pupils will be required to complete sums using pen and paper amid fears under-11s in England are already more reliant on electronic devices than peers in most other countries.
The change – being introduced from 2014 – coincides with the publication of a draft primary school curriculum that recommends delaying the use of calculators as part of maths lessons.
Currently, children are expected to use them at the age of seven, but this is likely to be put back to nine or 10 under the Coalition’s reforms.
Elizabeth Truss, the Education Minister, said that an over-reliance on calculators meant pupils were failed to get the

rigorous grounding in mental and written arithmetic that they needed to progress onto secondary education.
Pupils should not use the devices until they know their times tables off by heart and understand the methods used to add, subtract, multiply and divide, she said.

Read full article

QAMA: The Calculator That Makes You Better At Math

From a post by Alex Knapp on Forbes blog:

QAMA [...] is a calculator designed to reverse the last several decades of education by actually improving students’ intuitive understanding and appreciation of math skills.  It does this in a deceptively simple way.

The name itself gives the method away – QAMA stands for “Quick Approximate Mental Arithmetic” (and in Hebrew, it means “How much?”). As with most calculators, to solve a problem with a QAMA, you first do what you’d do with a regular calculator: type in the problem. But rather than just give you the answer right away, QAMA asks you for one more step: you have to estimate the answer. If your estimation demonstrates that you understand the math, the calculator will give you the precise answer. If your estimation isn’t close, then you have to try again before you get the precise answer.

Quick – what’s the square root of 2? What do you mean you don’t have a calculator? Well, you can start guessing, right? So let’s work this through. You know you have an upper bound – it has to be less than 1.5, because 1.5 x 1.5 is 2.25.  And it has to be more than 1, because 1 x 1 is just 1. But 2.25 is pretty close, right? So what if you guess 1.4? Well, then you’d be pretty close. 1.4 x 1.4 is 1.96, and the square root of 2 is about 1.414.

But did you notice something? Without your calculator, you had to estimate. In order to estimate, you had to think about and engage with the math behind exponents and square roots. Which means, hopefully, that you came out of that first paragraph with a bit better understanding of math.

That’s the theory behind QAMA, which is a calculator designed to reverse the last several decades of education by actually improving students’ intuitive understanding and appreciation of math skills.  It does this in a deceptively simple way.

The name itself gives the method away – QAMA stands for “Quick Approximate Mental Arithmetic” (and in Hebrew, it means “How much?”). As with most calculators, to solve a problem with a QAMA, you first do what you’d do with a regular calculator: type in the problem. But rather than just give you the answer right away, QAMA asks you for one more step: you have to estimate the answer. If your estimation demonstrates that you understand the math, the calculator will give you the precise answer. If your estimation isn’t close, then you have to try again before you get the precise answer.

How close is close? Well, that depends on the calculation. If you put in 5×6, you have to estimate 30 – the calculator expects you to know your multiplication tables. For exponent problems, if you have an integer – say something like 23^2, the tolerance is such that you still have to be pretty close, but there’s a wider berth for say, 23^2.1, because non-integer exponents are a tougher problem.

Developing the calculator to have these different tolerances for different types of calculations was the key challenge for QAMA inventor Ilan Samson.

Read the original post in full.

A tale about long division

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

which means

12345 = (11 \times 1122) + 3

As simple as that.

School maths is failing children – a US and Australian perspective

A post by Jon Borwein and David H. Bailey in The Conversation. A quote:

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.

Nick Gibb defends the new curriculum

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.

Related posts in this Blog: an alternative curriculum and Robert Siegler’s paper.