Today my twitter feed has been filled with talk about this article by Professor Marina Ratner of U.C. Berkley:
http://online.wsj.com/articles/marina-ratner-making-math-education-even-worse-1407283282
Dividing fractions is a subject with a bit of history for me. On the funny side, as a kid I missed the week of school when this particular subject was taught and I never seemed to be able to catch up from that missed week. My pals on my high school math team loved giving me a hard time about always having to go back and figure out how dividing fractions worked. Even now they’ll needle me about it when it comes up in one of the videos I do with the boys.
When I was going through this subject with my older son I’m sure my approach was pretty much all over the place. The primary reason is that I’d never had to explain dividing fractions in detail to anyone – much less a kid. That alone assures a lot of stumbling around. Another reason is that although we were following Art of Problem Solving’s Prealgebra book when the subject came up formally, much of the teaching I do with my kids doesn’t follow a textbook and many important subjects come up almost out of the blue as we discuss various math problems. I certainly wasn’t aware of the different (and sometimes strongly held) beliefs about teaching fraction division when I was talking about it with my son.
Teaching the same subject to my younger son was a little different. Hopefully I’d learned a little bit from going through this subject once already (ha!), but also I’d begun to follow lots of teachers and math ed folks online so I’d seen some approaches to teaching fractions that were different that what I’d done on my own. I still used the approach in Art of Problem Solving’s Prealgebra book as the starting point, but I supplemented it with a couple of other ideas. Here are those three approaches from back in January. Having watched all three of this videos again just now, I’m perfectly happy with what we did and I believe that all three approaches have merit:
(1) Art of Problem Solving – define division as the reciprocal of multiplication and understanding fraction division just boils down to understanding what the reciprocal of a fraction is. Note also that he notices that you need multiplication to be commutative for this approach to work (!):
(2) Talking about patterns. Here we look at this sequences of divisions: 8 / 8, 8 / 4, 8 / 2, and 8 / 1 to help form a guess about what the value of 8 / (1/2) might be. We also use dimes and nickels to illustrate the division:
(3) Drawing rectangles / using snap cubes to talk about division. This is the approach that appears to have motivated today’s WSJ article -“Who would draw a picture to divide 2/3 by 3/4?”
The best response that I saw to the WSJ piece was this simple tweet from David Radcliffe:
Finally, if you don’t find fraction division to be an interesting topic to think about, perhaps you’ll be more interested in this delightful problem that Radcliffe posted earlier in the week:
Mike's Math Page 
Interesting post…yup that is pretty much where many folk start to lose hold of basic math….dividing fractions. Somehow it was my daughters 3rd grade teacher that introduced me to the fact that between “inverse multiplication” and “sets of” (one or the other or both) our brains can make this not so mysterious. Both descriptions even tend to cover the unsettled feeling one gets when asked 42 divided by zero. Great Post. Love your stuff.
Does anybody take the trouble to ask “What is one half divided by one half?” ? If this is not seen as “How many halves in one half? “, with the stunningly obvious answer “One” then something has already gone very wrong. Otherwise the rest follows fairly painlessly.