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?

goldenojThe danger to me is that cancel somehow infers that because two things are the same, we can make them both disappear. It hides, in the multiplication case, the factor structure that’s crucial to understanding those expressions. (x+2)/(x+3)=2/3 is two slash marks away.

That said, I trust you and believe you know your students and what their understanding is. You’re intentionally making a choice based on that, which is the height of teaching.

dkane47Post authorI definitely hear you on this. But if I use the language “divide to 1” isn’t there a chance students interpret that as meaning (x+2)/(x+3)=(1+2)/(1+3) ?

Thought-provoking though. Reminds me that no matter how precise the language I use, I need to take time to unpack exactly what the language means and why it makes sense. Those moments diving deep into the why seem really important, probably more important than everyday language when we’re using one tool in service of a broader problem.

CraigFor a moment, I thought this was going to be a post about “cancel” culture. 🙂

GWI’m all about the “Nix the Tricks” philosophy myself. Whenever possible, I try to make sure my students understand why we do what we do in math. But can we go overboard with it? Possibly.

I agree with your take on the fact that there is nothing wrong with saying “cancel” as long as the students understand the concept. When we cancel, like you said, we are applying an inverse identify of multiplication or addition. This is also the same concept of “undoing” something. If I multiply by a number, then divide by that same number, the net effect on the value of the expression has become zero (i.e. unchanged). If a business person lost their company $1,000 on one deal, but earned their company $1,000 on a different deal, one might say that their two deals “cancelled each other out” in term of net profit so that the aggregate resulting profit was zero. Apply the same principle to one’s carbon footprint. How does one become carbon neutral? Most likely, they will have actions that contribute carbon dioxide to the environment. But they will also take actions that remove carbon dioxide which in effect cancels out the contributions, making that individual or company carbon neutral. This concept that two actions that are equal in magnitude but opposing in direction (adding/subtracting, multiplying/dividing, etc.) are fairly common and mostly intuitive to students, particularly at the high school level. I don’t see the use of the word “cancel” as short-circuiting their understanding of mathematical process.

A follow-up question for you. What do you say instead of FOIL when students are multiplying two binomials together?

Thanks!

dkane47Post authorOne way of framing tricks that I like is “rules that expire.” Taught well, cancel doesn’t expire and generalizes to lots of future situations. I don’t mean that we should exclusively use the word cancel, just that avoiding it in favor of clumsy language like “divide to make one” is often unnecessary if I’ve done a good job unpacking that rule elsewhere. I say distribute for binomials – it reflects multiplying a binomial by a monomial, and generalizes to multiplying polynomials with more terms. Happily, it isn’t much longer to say either. I try to reinforce this by using the box method and connecting to broader ideas of multiplication sot hat multiplying binomials doesn’t seem like an isolated skill, but connects to other areas of distribution, past and future.