ALEKS problem: subtracting with common denominator with renaming. Student: no idea. (Also non-native speaker so might have ideas w/0 language to express that I’d understand.) ALEKS’ explanation involved renaming 4 2/9 as 3 + 1 + 2/9 = 3 + 9/9 + 2/9 and … since it’s all just stuck on the page and done in a space-efficient way, it’s pretty instantly confounding. (However! I think they’ve gotten better, in general, at letting things take up a little more space.)
I write 41 – 28 or the like and ask what he’d do. He puts the 3 underneath and then changes 4 and puts the right answer down.
I tell him that he just “borrowed” because we didn’t have enough in the ones … and that fractions work the same way except that with our decimal numbers, it’s *always* ten that we’re borrowing… but with fractions, we’re borrowing any number of pieces in the world… and they tell us with the denominator. So we go over and grab a whole ( I did the “borrowing” first) and we add those 9/9 to the 2/9.
The thing I did differently from any other time was: I put “+9” with the two on the fraction bar, and showed the “9/9” as being the “1” we’d just borrowed, now adding to the 2/9. I’d like to dream that this even introduces to the brain that both terms were bound to the denominator, eh????
He totally got it and went on to get ’em all right.
Hmmm. Would he have gotten there with a few more examples/mistakes? Probably. However, I think that guiding from what he knew and yes, showing and telling the connection was better than that. There are so many weird mental models and visual dances students invent w/ fractions that get one little set of problems done but don’t really make sense.