R'ithmetic

The LA Times has discovered why Americans can't do math. In Why Johnny can't calculate David Klein and Jennifer Marple blame those who teach the teachers.
The root cause of the LAUSD's shortcomings in math is its failure to place its best math teachers in charge of math policies. Cronyism substitutes for knowledge of subject matter. The district should systematically require those in authority over math policies to pass rigorous math tests and interviews at the chalkboard before a panel of university mathematicians and veteran math teachers.
They also complain that the curriculum committe ignored the advice of, among others, Caltech mathematicians. I personally would treat that recommndation a bit lightly, since I have it on good authority that most of those guys can't even teach their own 800 Math SAT students.

The item stirring some controversy in the region of the blogosphere that I read concerns fractional division as repeated subtraction. Here's Kevin Drum's reaction:
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.
On the other hand, Brad Delong thinks it makes perfect sense:
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?").

...

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?"
WTF?

Hey, I can understand division as repeated subtraction. It even looks kind of sensible when you express fractions in terms of a common denominator (1/4)/(2/3) = (3/12)/(8/12), but figuring out that 3/8 of 2/3 goes into 1/4 once requires knowing the answer in advance. I think it's much more useful to learn the idea of an inverse operation, and that multiplying by the inverse is the same as dividing.

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