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## Tuesday, July 2, 2013

### Another long division befuddlement

In "The case Against Long Division" (see it here), Tony Ralston gives this example of an alternate method for division:
Then, at the first step, multiply by 10, record the integer portion of 10A/B as a, subtract a from both sides to get .bcdefg… on the right and then repeat this idea to get b, c, … . When B has one, or even two, digits, the subtraction can be done mentally. Thus, for example, to compute the decimal equivalent of 3/7 we would set 3/7 = .abcdefg… and proceed as follows:

30/7 = 4.bcdefg… so 2/7 = 30/7 – 4 = .bcdefg…
20/7 = 2.cdefg… 6/7 = 20/7 – 2 = .cdefg…
60/7 = 8.defg… 4/7 = 60/7 – 8 = .defg…
40/7 = 5.efg… 5/7 = 40/7 – 5 = .efg…
50/7 = 7.fg… 1/7 = 50/7 – 7 = .fg…
10/7 = 1.g… 3/7 = 10/7 – 1 = .g…
30/7 = 4. …

and now it is clear that the sequence 428571 just repeats since once one digit of the quotient is repeated, subsequent digits must also repeat. Note that in practice students should be expected to write down just the successive quotient digits with all other calculation done mentally.

I, for one, don't picture this as easier than long division, but it doesn't really matter, as the steps taken are EXACTLY the same as in long division, just written out differently. In this case I think the presentation creates a greater chance for careless error, but that is just my opinion.

Mr. Ralston, further down in his writings, claims that the long division algorithm is no better than the almost defunct pencil-and-paper square root algorithm. From my perspective, the reason the square root algorithm was marginal in school mathematics is because 1) it is time consuming, with a number of false starts for weak arithmetic students; 2) rarely tested, probably because of its time consuming nature; and 3) rarely needed since perfect squares and radical notation "ruled".

What I find scary about this long division issue is that the reformers are claiming that long division itself is irrelevant. By this logic, no one will learn it, helping to firm up the calculator dependency that is taking over. Skills will mean less and stock in batteries will mean more.

PS: I should add that Tony Ralston is (was) a computer science professor at SUNY Buffalo. His algorithm as exampled above is easily programmable for a single digit divisor, provided it is recognized that the integral part of the quotient is identified first. Otherwise, his algorithm needs many more examples in order to be understood.