This strikes to me as massively confused.

Keeping track of cancelled values is not required as long as you're working with a group, that is, a set(like reals), and an operation(like addition) that follows the kinda rules addition with integers and multiplication with non-zero real values do. If you are working with a group, there's no sense in which those canceled out values are left dangling. Once you cancel them out, they are gone.

http://en.wikipedia.org/wiki/Group_%28mathematics%29 <- you can check group axioms here, I won't list them here.

Then again, canceling out, as it is procedurally done in math classes, requires each and every group axiom. That basically means it's nonsense to speak of canceling out with structures that aren't groups. If you tried to cancel out stuff with non-group, that'd be basically assuming stuff you know ain't true.

Which begs a question: What are these structures in advanced maths that you speak of?