One word is being scapegoated for student misunderstanding.
A recent trend in mathematics education is the idea that we should never, ever say the word “cancel”. The argument for the prosecution is reasonable, and I will lay it out first.
But I think the problem isn’t the word itself, but rather how we approach language in mathematics class.
I’ll begin with the argument against using the word “cancel”. Historically, there are two places that the word is typically used in mathematics classes.
We “cancel” when we add the opposite value. When solving 4x + 3 = 7, the first step in the standard Algebra Class algorithm is to subtract 3 from both sides:
Teachers describe this various ways. The sloppiest is to say, “First, we have to move the 3 to the other side.” I regret to admit that these words have slipped from my mouth, and I absolutely agree that they are the wrong thing to say.
Teachers can also say, “We subtract 3 from both sides because we want to isolate the variable on the left, and whatever we do to the left, we must also do to the right.” This is the root of “balancing”, which is ultimately what the word “jabr” (whence “algebra”) refers to.
But what do we say about “+3–3”? We could say that “negative three cancels positive three”, or we could say we “cancel the threes”. I’ll return to this distinction later.
More commonly, we “cancel” when we multiply the reciprocal value. When simplifying (2x²)/(4x), we factor the numerator and the denominator, and then we remove the factors that are common to both:
It is common to say we “cancel” those common terms.
The main argument for the prosecution that I’ve heard is that “cancel” is vague. Students then think they can “cancel” willy-nilly, as in:
Also, students get confused about why addition “cancels” to 0 while multiplication “cancels” to 1. Finally, statements like “we cancel the twos” don’t make any sense because, as one person recently put it, we cancel appointments, not numbers.
This makes up an excellent argument for why “cancel” is problematic, and why we need to be careful with how we use the word. It is not, however, a complete argument for getting rid of the word entirely.
What is the meaning of “cancel”?
Merriam-Webster offers this four-part definition for the main sense of the word:
1a: to decide not to conduct or perform (something planned or expected) usually without expectation of conducting or performing it at a later time
b: to destroy the force, effectiveness, or validity of : ANNUL
c: to match in force or effect : OFFSET — often used with out
d: to bring to nothingness : DESTROY
I’m deliberately ignoring the mathematical definition (the third sense provided). I want to stress that there is a sense of the mathematical act of cancelling that is consistent with the main (natural language) sense of the word.
In general, if g(x) is the inverse of f(x), then x=g(f(x))=f(g(x)). Which is to say, inverse functions (including inverse operators) have the effect of annulling or offsetting the original function.
But we have to be careful with our language. When we subtract two in order to offset an added two, we’re not “cancelling the twos”, we’re cancelling the effect of the addition.
Adding negative two offsets the addition of positive two, which we could say as “Negative two cancels positive two” or “Two is cancelled by its opposite, yielding zero”. When we do this, we’re cancelling by subtraction. What we’re not doing, though, is “cancelling the twos”.
Likewise, in the case of multiplication, the operator-value pair /2 offsets the operator-value pair *2, leading to “cancelling by division”:
This leads me to a rebuttal of what I see consistently as the main reason given for why we should get rid of “cancel”: It confuses students.
On the one hand, I don’t think we should needlessly confuse students. On the other hand, though, I don’t think “It confuses students” should ever be a main reason to get rid of something that otherwise makes sense.
At the very least, we should ask ourselves: Why does it confuse students?
If students truly understand the mathematics, there’s no reason to avoid accurate language. And if students don’t truly understand the mathematics, avoiding a term isn’t going to fix the underlying problem.
Here’s an example of a situation where mathematical language and natural language conflict:
When we discuss simplifying radicals, what we mean is writing the radical so that anything that can properly be removed from the radicand is. But certainly it’s simpler to write the left hand side above (the square root of negative fifty) than the right hand side (five times i times the square root of two).
Likewise, we say that the opposite of a number is its additive inverse. Why is the reciprocal likewise not an opposite? After all, isn’t division the opposite of multiplication? This is another case where natural language and mathematical language conflict.
As I’ve already established, though, the mathematical use of “cancel” doesn’t conflict with several of the prevailing natural language senses.
Furthermore, consider the language for rewriting 3/6 to its equivalent fraction, 1/2. This is called “simplification” or “reducing fractions”, by which we “cancel the threes” or “divide out/away/through the three”.
Of all of these terms, “cancel” is the least in conflict with the natural language, although we’re really cancelling the multiplication, not the threes. 1/2 is as many symbols as 3/6, so it’s not immediately obvious how it’s simpler. 1/2 is not less than 3/6, so “reducing” and “divide” are both misleading.
If, instead of getting rid of “cancel”, we focused on replacing it with the greater detail of “cancel by division” and “cancel by subtraction”, it would reinforce what’s wrong with this:
We can’t cancel by division in this case because addition is in the way.
Indeed, shifting our language here supports a greater shift, away from mnemonics like PEMDAS towards an understanding that there are three basic levels of operators (addition, multiplication, and exponentiation), and we have to be much more careful when we cross those levels than when we work entirely within one of them.
We can only cancel by division when we’re talking about a factor that is shared by an entire dividend (numerator) and an entire divisor (denominator). We can’t simply pull any factor out of some term and treat it like a factor of the entire expression.
Which means we need to be careful not to do this:
and do this instead:
We cancel multiplication by dividing by the same value. We cancel addition by subtracting the same value.
I’ll wrap up by referring you to Concepcion Molina’s excellent book, “The Problem with Math is English”. I say this knowing he disagrees with me about “cancel”, of which he writes: “it implies that something is deleted, eliminated, or otherwise brought to nothing” (24).
I assert that we are eliminating something: We’re eliminating an operator by introducing its inverse. Rather than avoiding confusing language, we should dig into why we use it and fix the language (if need be).