This discussion at Kevin Drum's and Brad DeLong's over the utility of regarding division as repeated subtraction reminds me of something funny. I sort of thought I had blogged about this before, but google says I haven't. When I was in 5th and 6th grade I spent a lot of time doing long division. Pages of long division. This eventually got boring, and so I developed--short division! Except now I can't remember what it was. I wrote subscripts on the number being divided, I think. It worked, but my teacher forbade it as being too hard to check (which isn't crazy, since if I were to make some trivial error it would be hard to see at what point I had gone wrong.) Still, I felt rather aggreived that she wasn't more excited about it; I felt it was an exciting and mysterious process. Ah, I see I can re-learn the magic here. I don't suppose I'll bother until Zoë gets to division in school...

Moments Later: On reflection, she was just being dumb. It would have been just as easy to check. It's almost as if Winyah Academy in Georgetown S.C. sucked.

I think you're talking about the same way I did short division, which is just like long division except you're not writing it all out -- i.e. instead of writing the product of the current digit and the divisor on the line below the dividend and subtracting, you do the product and subtraction in your head and scribble the answer next to the next digit of the dividend. More difficult for a teacher to grade because you are not showing your work. Handy though.

My complaint with the whole way division of fractions is taught, is I think it obscures the nature of fractions, which are quotients of two numbers -- the best way to think about multiplying or dividing with fractional operands, I think, is to think of it as an iterative process -- you are multiplying integers and taking the quotient -- "dividing" is the same as "multiplying by the reciprocal" and "reciprocal of n" is "the quotient of 1 and n", once you can get that in mind I think the process makes much more sense than just following a rule to "flip over" the divisor.

Posted by: Jeremy Osner | September 27, 2005 at 09:20 PM

I was really chuffed when I figured out why "numerator" and "denominator" were so named. Like, the numerator numbers the fraction!

Onefifth!Fourfifths! And the denominator gives it its general name—fifths, because it's five!It blew my 20-year-old mind.

Posted by: ben wolfson | September 28, 2005 at 01:46 AM

Ooh that

iscool. Though this naming convention obscures the fact that they are simply a dividend and a divisor. Thanks for the observation, I will take that one with me.Posted by: Jeremy Osner | September 28, 2005 at 02:31 AM

I had something of the opposite experience in second grade. After teaching us "long multiplication", the teacher told us there was a "short cut" but wasn't going to tell us what it was. I banged my head against it for a while but didn't get anything. Later she mentioned that some people had figured out that you didn't have to write down the digits that you didn't have to write down every single digit you carried. Yeah, you save a little bit of time, but not sure it merits being a shortcut.

I'll have to remember the fraction naming thing, though, Ben.

Posted by: Paul | September 28, 2005 at 05:22 AM

the only way to teach division correctly is to say rather quickly "it's the opposite of multiplication" and greet all further enquiries with an aggressive glare. I think that the ability to generate a sufficiently lethal glare was part of the teacher's exams in Wales in the 1970s.

Posted by: dsquared | October 01, 2005 at 10:35 AM

A similar tactic also works well when you're teaching how to take roots.

Posted by: Jeremy Osner | October 01, 2005 at 09:32 PM

In Spain we *only* learn short division @ school. Never knew there was a "long division".

we write it this way, though:

Dividend | Divisor

---------

Quotient

Posted by: eurocent | October 09, 2005 at 05:29 PM

Oops, comment got formatted the wrong way.

the quotient goes below the Divisor, not the dividend.

Posted by: eurocent | October 09, 2005 at 05:30 PM

Belle, what years were you at Winyah Academy? Your name sounds familiar. I would have been class of '87 had the school not closed.

Posted by: David Leland | February 09, 2006 at 04:26 AM