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.
Sasha continues the discussion about long division, which is valuable. I myself like to point to its relation to continued fractions, and also therefore to the Euclidean algorithm, and of course we need to understand how to express rational expressions in equivalent forms using ‘the algorithm’ to do so. I also like asking people if they think 222222 is divisible by 13, and once they have sorted that out using an algorithm to ask if 2222223 is divisible by 13 – for which they should not need an algorithm. But none of this leads to an argument that ALL 11 year olds should have to learn a particular long division algorithm rather than more usefully spending time learning more deeply about the meaning of the multiplicative relationship – too often in the past the ‘doing’ has been given priority over the meaning of division, and what remainders tell us (rather important for dividing polynomials) and so on. Sasha, I am fairly sure that most of the students to whom you teach the traditional algorithm (some version of it) for new purposes have met it before, but that does not mean it is in the bank as a mathematical investment for the future. It is more likely to have been ‘got out of the way’ for some test or other and then never used again. The point of my earlier posting was not to say that long division has no place, but to say that the tests on which the ‘finding’ that long division was necessary for future success did not, in fact, include any need for long division algorithms or binary fraction arithmetic. The assumption that these studies somehow proved a need for the algorithm and for fraction arithmetic at a particular age is based on an inability among the psychologist authors and politicians to grasp that being able to perform a method is not the same as having a conceptual understanding, and might even get in the way of choosing and using appropriate reasoning.
It certainly gets in the way of discussing how to help young students get a sound and flexible understanding of division that allows them to grasp ratio and proportionality, which are immediately necessary for secondary school mathematics.
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