Kevin Drum is confused by math "gurus" David Klein and Jennifer Marple:
The Washington Monthly: FRACTION DIVISION.... In the LA Times today, math gurus David Klein and Jennifer Marple tell us that one of the reasons high school kids in Los Angeles aren't learning math is because the teachers themselves get rotten training. This particular anecdote struck me as especially bizarre:
Too often, the math that teachers are taught at district training sessions is just plain wrong. For instance, middle school teachers are erroneously taught that fraction division is repeated subtraction. This makes sense only for special examples such as 3/4 divided by 1/4 . In this case, 3/4 may be decreased by 1/4 a total of three times, until nothing is left, and the quotient is indeed 3. Understanding division as repeated subtraction, however, is nonsensical for a problem like 1/4 divided by 2/3 because 2/3 cannot be subtracted from 1/4 even once. No wonder students have trouble with fractions in high school.
"Fraction division is repeated subtraction"? I don't even get that. Even in the example that "works," how does getting nothing somehow translate into 3?
And what's the point, anyway? It's one thing to try some weird new technique when the old one is difficult to understand, but fraction division is simple. Why would anyone spend any time trying to come up with some new way of teaching it?
Well, division can be thought of as--in fact, is--repeated subtraction. That's one way of defining what division is, just you can define multiplication to be repeated addition and exponentiation to be repeated multiplication (and taking roots to be repeated division; the cube root is the answer to: "what number can I divide this by three times to get one?").
You can see that division can be thought of repeated subtraction most clearly in long division: 50,008/14, say. First we subtract 3,000 fourteens from 50,008, and get a remainder of 8,008. Then we subtract 500 fourteens from 8,008, and get a remainder of 1,008. Then we subtract 70 fourteens from 1000, and get a remainder of 28. And then we subtract two fourteens from 28, and get zero. Voila: we have subtracted 3,000 + 500 + 70 + 2 = 3,572 fourteens from 50,008--we have divided 50,008 by 14 and gotten 3,572.
The idea that "division is repeated subtraction" is much better when a student is first confronted by division by a fraction--3/4 divided by 1/4, say--than is the alternative of "division is dividing into piles." You divide 50,008 into piles of 14 and you have 3,572 piles. But you divide 3/4 into 1/4 of a pile and... a student who thinks "division is dividing into piles" is immediately lost. By contrast, if the student starts out thinking that "division is repeated subtraction," it is easy for him or her to see what 3/4 divided by 1/4 is: how many times can you subtract 1/4 from 3/4 before you get zero? And the answer is three.
It even works with 1/4 divided by 2/3: you can't subtract a whole 2/3 from 1/4 and get zero; but you can subtract 3/8 of a 2/3 from 1/4 and get zero. I at least, think it is more intuitive to think of 1/4 divided by 2/3 as "what fraction of 2/3 can you subtract from 1/4 to get zero?" rather than "suppose you divide 1/4 into 2/3 piles, how much is in each pile?"
The fact that David Klein and Jennifer Marple claim that it is erroneous to say that division is repeated subtraction makes me extremely skeptical about their qualifications. Math is hard. People learn it in lots of different ways.