A few things are popular in online-math-education-land. One is trying to be precise with the mathematical language we use. I often advocate for these ideas. When my students say “FOIL” or “cross-multiply” or “add a zero when you multiply by ten” or “keep-change-change” I’m likely to go off on a micro-tirade. That language is ambiguous in cases, doesn’t support understanding, and doesn’t generalize well.
But recently I’ve noticed how often I try to use something instead of saying “cancel.” “Divide to make one” or “subtract to make zero” are fine, but I find them a little obtuse and distracting in the moment. I mean, they’re correct, but they’re also not language students are familiar with. It can be distracting not to use a simple word students recognize when I’m trying to make a broader point about integration or sums of series. I often try to use more precise language, but at times it seems simpler just to say “cancel.”
Thinking more about “cancel,” it actually has a fairly precise meaning. I use it most often in two cases: when two terms divide to make one, or subtract to make zero. I don’t think it’s crazy for one word to mean two things. That happens all the time in math. And those are actually two instances of one mathematical idea: using inverse operations to make the identity. In this case, either the multiplicative identity or the additive identity. From that perspective, “cancel” generalizes well to the case of squaring a square root or exponentiating a logarithm. And I don’t think it’s a bad thing to have a general word to refer to that collective idea.
Tina Cardone’s book Nix the Tricks has a great epigraph:
“I would say, then, that it is not reasonable to even mention this technique.
If it is so limited in its usefulness, why grant it the privilege of a name and
some memory space? Cluttering heads with specialized techniques that mask
the important general principle at hand does the students no good, in fact it
may harm them. Remember the Hippocratic oath – First, do no harm.”
– Jim Doherty
The thing is, “cancel” is a technique with broad application that is popular among students and teachers. If I taught Algebra I, I might think differently. But in Algebra II, Precalculus and Calculus, I don’t think I’m doing harm by using the word cancel, and helping students to use it consistently. Students typically come in with both accurate and inaccurate ideas of what it means to cancel. Rather than expunge the idea and try to plant several new ones in its place, why not embrace the useful and build on it a robust understanding of what mathematicians mean when they say cancel?