The example appears to be an attempt to make sure that the child recognizes that the carrying is occurring from one register to the next higher one, and to focus the child's attention on the proper place for the carried digit when writing it. The extra horizontal line seems to be an attempt to make sure the child sees the carried digit(s) apart from the original ones.
I agree with this, it's how I emphasised place value at first. And having pictures of manipulatives is helpful, too. That's how we started with division. And I'd kill to have my daughter putting the new value for a number above it where it's supposed to be. My husband used a crazy technique where you don't write anythign down, so she won't be able to show her work, instead of the proper way where you borrow 1 from the tens column, cross out the 4 and write 3 there instead, then change the number where you've brought the ten, too. he rhas her writing a 1 in the place where you'd write something you're carrying.
Teaching the same concept in six or seven different ways seems unnecessarily involved when all the kids really need is a bucket of Cuisenaire rods.
But you need all six mothods to make sure all the kids in the class get it. What if the one method they chose made no sense to you at all?
eta: I just reread your OP, and you have the opposite problem, they're writing the carrying number where a borrowed number should be!
