Claire Kenyon asks the following question on Lance Fortnow's blog:

Yesterday my sixth-grade son, doing his math homework, asked Is there a faster way to divide a number by another other than with long division? I had no good answer to that. Any suggestions for an alternative to tedious long divisions?Here's a good alternative: not doing them.

A few years ago, I taught my then-six-year-old to do division, essentially by binary search. If you want to distribute 4832 cookies among 23 children, look at the sequence 23, 46, 92, 184, 368, 736, 1472, 2944, 5888, to figure out that each one will get at least 128, and you'll be left with 4832-2944 = 1888 cookies; of course, you can now give each one 64 more, using up 1472 and leaving you with 416, etc. (Of course, you will do this with powers of 10 if you are less of a sadist than I am). A year later, he probably forgot all about the mechanics of doing division, but still carried with him the ideas of division as distribution, perhaps the underlying "recursion", etc. Which is just as well.

Tony Ralston (an ex-president of the ACM!) has a rather interesting take on the theme of paper-and-pencil arithmetic, where he advocates mental arithmetic and basic number sense.

Two excerpts (PPA = paper-and-pencil arithmetic):

So is a facility with PPA necessary - or even desirable - for later study of mathematics? If a student arrives in secondary school, say in a first algebra course, unable to do PPA, is that student ipso facto disadvantaged compared to students with PPA skills? I take it that it is not PPA skill itself which critics of calculator use consider important. After all, there is little secondary school or other mathematics which requires much computationand on long division:per se. Rather it must be the ancillary benefits of developing PPA skill which are considered important. These are usually subsumed under the rubric ofnumeracyornumber senseand include, in addition to the obvious ones of knowing the addition and multiplication tables, such things as knowing which arithmetic operation to use when, having a good sense of number size and knowing strategies to check the answers to arithmetic operations. These are all important. Is there anything about them which a calculator-based curriculum could not instill? I think the answer is, no...

I should say a word about division. Although it is over 15 years now since the Cockcroft Report [1982] recommended that long division no longer be taught in British schools, this recommendation been, at most, spottily implemented. The California Board of Education standards [California, 1998, p43] would require students to master long division. This is mind-boggling. The only excuse can be that those who promulgated the California standards believe that long division is good for the soul. Not only does being able to do long division have no practical value whatever but, in addition, the time required to teach this algorithm to students is far, far in excess of any benefit which might accrue from learning it.Of course, students must learn what division is, when to apply it, what remainders are and how to do simple division problems mentally. But teaching long division is pertinent to none of these aims; it is as nonsensical as teaching the square root algorithm which was staple fare until recent times. I cannot help but believe that those who favor teaching long division in elementary school (and these include some research mathematicians [Klein, 1998]) are in the grip of some fantasy about what is important and useful in school mathematics^{6}

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## 11 comments:

"A few years ago, I taught my then-six-year-old to do division, essentially by binary search."

You realize that what you were teaching him was in fact long division, just in binary?

And doing it with powers of 10 is just long division in base 10, with an inefficient method for finding the next digit in the quotient.

The more important point of it was to realize that division was distributing something as evenly as possible into some number of portions, and that it could be done by successive approximations -- a way to reason about it so that the correctness of the algorithm is "obvious" to a 6-yr old. As opposed to following a mechanical recipe which is what happens in much of early education.

Well, it would be a little better if we weren't going on people's gut feelings here, and there were actually some research as to which elementary math education is the most effective in preparing students for mathematical reasoning later on.

Perhaps long division - which isn't that difficult, and is the first kind of inverse problem students usually run into - is bad training for learning number sense, and perhaps it's good. Perhaps number sense can be instilled with calculator use, and perhaps a calculator acts as a crutch so that it's even more difficult to teach the proper lessons. Perhaps the best lesson from long division is not

number sense, but how to neatly organize a problem. Perhaps it's like practicing scales in music, where one is learning agility - but here it's mental agility as opposed to finger agility. Perhaps it's a waste of time. I'd rather see some hard data.

Well, my son already understands of course how to do divisions by successive approximations. When he is free to do it in his own way, he does it a little differently from the usual algorithm, though, approximating either above and below, whichever appears more convenient: for example, to divide 179 by 6, he would first approximate by 30, then divide 1 by 6 and substract the result from 30.

One motivation for long division is that it leads one to notice phenomena such as periodicity of rational numbers and understand why that occurs.

Claire wrote:

One motivation for long division is that it leads one to notice phenomena such as periodicity of rational numbers and understand why that occurs.Excellent point. The question is, which subset of the student population (and at what level) need to learn this? One could argue that the notion of successive approximation is much more basic than the long division algorithm, whose correctness is likely less obvious to a pre-highschooler. In other words, should we treat long division a little bit like faster (than n^2 time) algorithms for multiplication, that is, as "advanced material"? Perhaps it is an excellent algorithm to teach in introductory discrete mathematics (like

Karatsuba's multiplication algorithm). Perhaps it would be a good idea to teach such "simple" algorithms for multiplication, division, sorting, string matching, etc., in high school, emphasizing their proof of correctness.

This method looks like the same as yours, but much more simpler: http://www.doubledivision.org/

By the same token, decimal long division gets a 1 digit under-approximation to the right answer very quickly. These days, estimation is a much bigger component of things than it used to be.

Maybe the real reason we teach PPA is to prepare students for the notion of an algorithm/working on an assembly line/etc. Seeing a variety of algorithms: addition, multiplication, long division does give a better gut-level understanding of the notion of executing an algorithm. Maybe we can replace long division with Sachertorte recipes and get the same effect...

On a related note, when I was in school we also learned a pencil and paper algorithm for computing square roots. (I remember it only vaguely but it was similar to long division.) If long division is so important why don't we teach square root algorithms, too?

Well, it would be a little better if we weren't going on people's gut feelings here, and there were actually some research as to which elementary math education is the most effective in preparing students for mathematical reasoning later on.

Perhaps long division - which isn't that difficult, and is the first kind of inverse problem students usually run into - is bad training for learning number sense, and perhaps it's good. Perhaps number sense can be instilled with calculator use, and perhaps a calculator acts as a crutch so that it's even more difficult to teach the proper lessons. Perhaps the best lesson from long division is not

number sense, but how to neatly organize a problem. Perhaps it's like practicing scales in music, where one is learning agility - but here it's mental agility as opposed to finger agility. Perhaps it's a waste of time. I'd rather see some hard data right here www.forumvadisi.com

Well, it would be a little better if we weren't going on people's gut feelings here, and there were actually some research as to which elementary math education is the most effective in preparing students for mathematical reasoning later on.

Perhaps long division - which isn't that difficult, and is the first kind of inverse problem students usually run into - is bad training for learning number sense, and perhaps it's good. Perhaps number sense can be instilled with calculator use, and perhaps a calculator acts as a crutch so that it's even more difficult to teach the proper lessons. Perhaps the best lesson from long division is not

number sense, but how to neatly organize a problem. Perhaps it's like practicing scales in music, where one is learning agility - but here it's mental agility as opposed to finger agility. Perhaps it's a waste of time. I'd rather see some hard data right here www.forumvadisi.com

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