Now I think I’m talking myself into “counterbalance” instead of “neutralize”. Interesting side note: “Algebra” is probably best translated as “restoring”, although there’s disagreement on that… which would support “undo”. ;)

As Corey suggests, I think the important thing is that we’re being mindful about our word choice and making sure we understand why we’re saying what we’re saying.

This is the most cogent argument against “cancel” that I’ve read. However, I still don’t like “undo” because it implies that we’ve actually done something (when we may not have).

My rebuttal is that “cancel” is ambiguous and CAN mean “to neutralize the effect”, but it’s a fair point that some if not most students don’t see it that way.

How about: “When we apply the inverse operation, we neutralize the values”? I want a word that describes what we’re doing — cancelling, neutralizing, counterbalancing, offsetting. If “cancel” has problems, let’s find a word that doesn’t.

The problem with “cancel” is, in its usage, it appears to describe some changes on the paper where we keeping tract of the arithmetic we do. It refers to “marks-on-paper” objects not mathematical ones, rather like cancelling keeps track of postage in a mail process. “Cancel”refers to the account we are keeping of the arithmetic process, not to the mathematics in that process. We will fight about the accounting of it, but the mathematics itself need not be a war of words.

When students say “cancel” it means “cross things out.” In my experience they are not thinking of anything remotely mathematical. Things just disappear. When you say “apply the inverse” they have to think about what that means. It begs the question “Inverse of what?” So they must think about the operation they are inverting. That’s thinking mathematically.

I suppose you could teach them to think “cancel” that way, making them think about why things “cancel” but (again, in my experience) that’s not generally what’s done and it’s not generally how students think of it.

Yes.

We just started getting into inverse Trig Functions. I think its interesting how some students readily accept it as an extension of “cancellation” (ex: asin(sin(A))=A ) while others insist on an “inverse” interpretation (ex: sin(Angle)=Ratio -> asin(Ratio)=Angle ). And there are those who are uncomfortable with either interpretation, even though they seem perfectly capable of solving things like 3x+2=8 or x^3=8->cuberoot(8)=x.

And then there’s e^(ln(x))=x …that was one “cancellation” I always felt uncomfortable about as a student.