Is it harmful to use the word "Cancel"?

Elsewhere, among a group of high school math teachers, I encountered a discussion of the term 'cancel'. Most (>20) people in the discussion had very strong feelings about why the term should be eliminated. I searched among posts on this stack to find the word common used and its meaning seems well understood.

The objection the the word's use included the concern about errors that might result from dividing to cancel and thinking that zero results.

My response was that if a student's work reflects accurate use of subtracting from both sides, dividing properly when attempting to simplify, etc, that what the student calls it is far less important. I even suggested that it would be preferable to teach the proper name for what's occurring, e.g. 'The additive inverse property' but as long as the student is doing the math correctly, and typically, silently, that there's no harm in using a word that's less than a textbook definition.

(Given the use of the word here, over 100 posts, and no pushback that I saw when skimming over a dozen, I suspect I know the answer. I'm here for confirmation)

• Perhaps a better term would be "to factor out 1" Commented May 16, 2023 at 21:10
• ... just don't try something like this ... $$\frac{64}{16} = \frac{4}{1}$$by canceling the $6$s. Commented May 17, 2023 at 5:30
• I have no strong opinion on whether cancelling the word "cancelling" is good pedagogy, but I wonder if your colleagues might be coming up against a version of the "law of universal linearity". (See also: Whence the "everything is linear" phenomenon, and what can we do about it?) Commented May 17, 2023 at 10:53
• I noticed that I actually use phrases like "and this $x$ kills the $x$ over there" or "and the term from the $du$ will take out that numerator", and now I wonder if my language might be seen as inappropriately violent. Commented May 18, 2023 at 5:20
• @Mazura - the first sentence of my question should make my intent clear. Commented May 21, 2023 at 0:32

Coincidentally, a colleague at my community college brought up this same issue one week ago (and I'd never heard such a complaint before then). My colleague directly tells their students that "canceling" is simply not a proper thing in mathematics. Therefore I was doing a brief literature review in the last few days.

In short, "canceling" is a customary and well-defined term in standard textbooks of any level I could find, from the most elementary algebra to graduate abstract algebra texts. Students will be set up to have a hard time interfacing with those standard works if they're warned away from use of the word "canceling". A few examples where I found it:

• Bennett & Briggs, Using & Understanding Mathematics, A Quantitative Reasoning Approach (Sec. 2A, Working with Units, Brief Review of Common Fractions, and problems throughout the text).
• Sullivan, Algebra & Trigonometry (presents two Cancellation Properties as basic axioms of real numbers in Sec. R.1)
• Thomas' Calculus (explanations in work on basic limits and derivatives)
• Rosen, Discrete Mathematics and its Applications (explanations in problems on induction and combinations).
• Hungerford, Abstract Algebra (as an analog to Z, Theorem 3.7 says a ring is an integral domain if and only if it has a "cancellation property"; likewise Theorem 7.2 asserts that "cancellation holds" in any group).
• Rudin, Principles of Mathematical Analysis (Proposition 1.14 describes the basic property of fields as "a cancellation law").
• Lang, Algebra (describes a "cancellation law" in the context of Grothendieck groups, Sec. 1.7, as well as work in basic fractions and factoring), and also his Complex Analysis.
• Serre, A Course in Arithmetic (Theorem 4 on quadratic forms described as a "cancellation theorem").
• Wikipedia has articles on cancelling out in elementary algebra, and a cancellation property in abstract algebra.

... and so forth.

My colleague's chief complaint was along the lines of "there's not 'a' cancellation law, there's two, one for adding, and one for multiplying, so we shouldn't use that term", but I find that unpersuasive -- that's just a reflection of the fact that in elementary algebra on the field of real numbers, we have two commutative properties, two associative properties, two inverse properties, two identities, and so forth.

I suppose language changes over time, and sometimes ideologues wind up winning campaigns like this. But I broadly don't see the benefit of such a change, and as a description of current standard mathematical language it seems to be broadly incorrect (and would leave students in the lurch about interfacing with this common usage in a variety of texts throughout the discipline). In regards to my colleague's issue, I would see this not as something to shy away from, but as an opportunity to embrace a careful awareness of our two basic operations in real numbers.

• I suppose language changes over time -- Possibly the concern raised by your colleague was motivated to provide some distance from unintended connotations suggested by the phrase "cancel culture". Nonetheless, I've occasionally heard such concerns about "cancel" for several decades, and in particular it came up from time to time in some Math Forum discussion groups I used to participate in (e.g. ap-calculus and math-teach, roughly 2000 to 2013). What you've concluded is pretty much my view, with more careful wording used if deemed necessary. Commented May 17, 2023 at 12:54
• @DaveLRenfro It shows up at least as early as 1932. From The Mathematics Teacher: "When pupils employ without understanding such phrases as canceling, transposing, clearing of fractions, and removing parentheses, their work amounts to nothing more than meaningless juggling of symbols and guessing answers... To most pupils ['cancel'] seems to mean nothing more than the crossing out of something. 'I can cancel in $\frac{3a}{b}\times\frac{5}{6a}$, ... why not in $\frac{3a}{b}+\frac{5}{6a}$, or in $\frac{3a}{b}=\frac{5}{6a}$?'" Commented May 17, 2023 at 13:18
• I hope there's been a Q&A on "imaginary", the greatest misnomer in all of math. Commented May 17, 2023 at 16:10
• @JustinHancock: Re different levels of books: IMHO it does a disservice to students to just focus on teaching the curriculum. That's how you get students who think they're "good at math" but flame out in college or grad school (when they discover that "math" does not mean "following rote instructions to manipulate algebraic expressions"). Commented May 17, 2023 at 23:22
• @JustinHancock if you can't expect students to remember how canceling works in those fractions, why would you expect them to remember how addition works there? Commented May 18, 2023 at 1:09

In the book Nix the Tricks, written by Tina Cardone in collaboration with numerous other math teachers, "cancel" appears in section 6.4. The book gives this reason for "nixing" it:

Cancel is a vague term that hides the actual mathematical operations being used, so students do not know when or why to use it. To many students, cancel is digested as “cross-out stuff” by magic, so they see no problem with crossing out parts of an expression or across addition.

The following "fixes" are suggested: $$\require{cancel}$$

• Say two factors "divide to one" for multiplicative cancellation: $$\frac{4x+2}{2}=\frac{2(2x+1)}{2}=\cancelto{1}{\frac{2}{2}}\hspace{-0.75em}\cdot \frac{2x+1}{1}=2x+1.$$
• Say two terms "add to zero" for additive cancellation: $$4x-4(x+2)=4x-4x-8=\cancelto{0}{4x-4x}\hspace{-0.75em}-8=-8.$$

I don’t think the word “cancel” is harmful, and I wouldn’t penalize students for using it. As others have suggested, it may even be helpful to encourage students to use “cancel” after they’ve attained some fluency. But in my experience, students who have learned to make erroneous cancellations are less likely to do so when given clearer language and notation to use.

Edit: I've also commonly heard "cancel" used by students in the context of inverse functions, and they often make erroneous statements like $$\sqrt{x^2}=x$$ and $$\arcsin(\sin(x))=x$$.

• I've seen people cancel the sin in sin(a)/sin(b). :-/ Commented May 17, 2023 at 5:49
• @EricDuminil Oh my, what a great sin! Commented May 17, 2023 at 7:56
• "divide to one" and "add to zero" are really good descriptions, BTW. Thanks. Commented May 17, 2023 at 8:07
• @EricDuminil There was a rumour going around at my school that somebody had simplified $\frac{\sin(x)}{\cos(x)}$ to $\frac{in}{co}$. Commented May 18, 2023 at 5:16
• @MartinKealey This is a real argument that comes up in Calculus when dealing with $\varepsilon$ values, where $\frac{\textrm{sin}\left(a+\varepsilon\right)}{\textrm{sin}\left(a\right)} \rightarrow \frac{a+\varepsilon}{a}$. Commented May 18, 2023 at 14:44

Use of the word "cancel" is not a bug. It's a feature.

A year or two after high school, while manipulating equations in later training, I found myself "canceling" in various ways and doing it automatically. I wasn't even using the word "cancel" in my head. I just knew that I could simplify expressions according to different rules.

At one point I stopped and wondered "What allows me to do that?" And after a short reflection, I remembered what I was actually doing: Adding or subtracting identical terms from both sides of an equation, or multiplying a fraction by another fraction that equaled 1.

But that's how it's supposed to be. Alfred North Whitehead once said (paraphrase) "Civilization advances by the number of operations we can perform without thinking." That's why we drill students. We want the rules of algebra, trigonometric identities, boolean algebra, DeMorgan's laws, matrix operations, integration formulas, etc. to become automatic. And if some of those rules fall under the generic word, "Cancel," well that's fine. That's a sign the rules have been internalized. It's a sign of fluency.

To be sure, we don't want students to just memorize the rules by rote. We want them to understand why they work. But they shouldn't have to derive a often-used formula every time they apply it.

As others have pointed out, trying to eliminate the word "Cancel" from the student's vocabulary will put them at a disadvantage when communicating with others mathematically. For those that feel strongly that "Cancel" is too ambiguous for mathematical use, I'd suggesting teaching students to learn to live with the ambiguity instead of taking the burning books approach. Maybe analyze the word "Cancel" with them. Itemize the different operations that fall under that category. Or when students use the word in class, ask them, "That's right. What kind of canceling is that?" or "That's right. What allows you to 'cancel' in that situation?"

• Automaticity is a characteristic of expertise, but emulating expert practice from the start isn’t always beneficial. For example, children first learn to skip count, then they memorize multiplication facts. In the same way, I think it’s reasonable to at first expect students to use language like “add to zero” and “divide to one” with the understanding that these processes will eventually be reified through the metaphor of “cancellation.” Commented May 17, 2023 at 16:50
• +1 I actually wrote the Whitehead quote as a comment here myself earlier today (then deleted it for brevity). Commented May 17, 2023 at 21:37
• For those that feel strongly that "Cancel" is too ambiguous for mathematical use, I'd suggesting teaching students to learn to live with the ambiguity instead of taking the burning books approach. Amen to this; mathematics, like any other intellectual pursuit, is an art of figuring out and communicating meaning. Commented Jun 26, 2023 at 9:49

Language that points to the actual reasoning helps math make sense. I might say cancel sometimes (and I would never ding a student for saying it), but I do think it's better to say: We factor, and then the common factor makes a 1, which is multiplied by the rest of it, so we don't need to keep that one.

That's more than I'm going to say every time, but I definitely say it if a weak student asked the question.

I think the word "cancel" is a useful one, particularly the phrase "cancel out" as applied to two terms, factors or operations which are inverses of each other ─ that is, I would always rather say that the terms cancel out, or the factors cancel out, or a term cancels with another term ─ not that we cancel them.

This distinction is perhaps important because terms (additive expressions) cancel additively whereas factors (multiplicative expressions) cancel multiplicatively. But it's also perhaps important because it frames the algebraic simplification as something inherent to the formula being simplified, rather than a way that we choose to change that formula. And also it can make sense that terms cancel in a different way to how factors cancel.

If students are having trouble with the word "cancel" e.g. because they're simplifying multiplicative inverses to zero instead of one, I would suppose that the problem is not the word used to discuss the concept, but the way that concept is being taught and presented. If a student incorrectly simplifies e.g. $$4 + \frac{x}{x}$$ to $$4$$, this student's misapprehension is that they know division "cancels" but they haven't identified that it needs a multiplication to "cancel with". And indeed, this formula doesn't contain any multiplication. So we might introduce one: $$4 + 1 \times \frac{x}{x}$$. The student can agree that multiplying the fraction by 1 doesn't change its value; but now there is a multiplication for the division to cancel with, and the formula can correctly be simplified to $$4 + 1$$. (Of course, it must be given that $$x \ne 0$$.)

The argument has been made in another answer that "cancelling" is not a formal concept so it shouldn't be taught. I would argue instead that mathematics education is not just about teaching formal concepts, it's also about developing intuition, and the ways the subject is practiced and talked about by real people.

The concept of two inverse components of a formula being simplified is an important one which reflects the way people intuitively think about algebra, and it's useful to have one word which acknowledges the connection between the different ways that things can be inverses of each other. The word "cancel" also happens to be a word that is commonly used for this by people who do mathematics or talk about it.

• I’m curious what the advantage is of students viewing “the algebraic simplification as something inherent to the formula being simplified, rather than a way that we choose to change that formula”? I’m thinking of the Common Core 7th grade math standard 7.EE.A: “Use properties of operations to generate equivalent expressions” and algebra standard A-SSE.B.3: “Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.” These seem to emphasize that “simplifications” are just ways that we choose to rewrite expressions. Commented May 17, 2023 at 11:16
• @JustinHancock The way I would see it is that they are two different formulae and they are related in that they are equivalent due to inverse operations cancelling out, like two nodes connected by an edge in a graph. The difference is in perceiving algebraic simplifications as things rather than actions ─ a common theme in abstract thinking. The two formulae must exist "at the same time" in order to analyse things like whether they are equivalent; if perceived as an action then there is only one formula and it changes when acted upon. I can't comment on the US curriculum as I'm not from there. Commented May 17, 2023 at 11:28
• I like your analogy of nodes in a graph. For me and I think for many students I've taught, the word "cancel" by itself doesn't evoke this imagery, but instead the physical act of crossing things out. The original meaning of "cancel" is "to cross out with lines." Phrases like "write an equivalent expression" or "rewrite in an equivalent form" have been more effective for me to get students to see a node and its adjacent nodes. I can then say that there are different ways of producing equivalent expressions, cancellation being one of them, but only very specific cancellation rules are allowed. Commented May 17, 2023 at 14:16

Like any other word, it comes handy when the students are already proficient with the standard cancellation operations: reduction of common factors in a fraction, removing equal summands/factors from both sides of an equation, removing the summands with opposite signs from a formula, and so on.

In the beginning, one is much better off with solving something like

$$5(3x+6)+18=15(7x-5x+8)+18$$

by saying

Subtracting 18 from both sides, grouping the terms with the common factor $$x$$ on the right, and dividing both sides by $$5$$ afterwards, we can simplify the equation to $$3x+6=3(2x+8)=6x+24$$. Subtracting $$6x+6$$ from both sides, we get $$-3x=18$$. Dividing both sides by $$-3$$, we get $$x=-6$$.

However, once one gets accustomed to such tricks, one can certainly say just:

Cancel $$18$$, then cancel $$5$$, then cancel $$5x$$ in parentheses, then open the parentheses and cancel $$3x$$ and $$6$$ to get $$0=3x+18$$, whence $$x=-6$$.

That's how we really think of the whole procedure ourselves and there is no reason to abstain from that language once one is ready for it. It is just a shortcut language with one word for many different operations having the common underlying idea. When you encounter some longer and more complicated formula, like $$xy+3(x^3+y^3)-zt+abc-3x^3-4y^3+tz+y^3\,$$ it is much easier to say "canceling similar terms with opposite signs", than to describe what exactly is being done step by step, but that phrase requires that $$-zt$$ and $$tz$$ immediately strike your eyes as essentially the same thing, that you can automatically open the parentheses in $$3(x^3+y^3)$$ and combine $$-4y^3$$ and $$+y^3$$ into $$-3y^3$$, etc. Before that stage is achieved, it is, probably, better to be more detailed on how you transform the formula into the "obvious" $$xy+abc$$.

P.S. Looks like I mostly just rephrased what Sintax Junkie and Daniel R. Collins said already, but once I typed it, let it be :-)

• You say "whence" in class?? Commented May 18, 2023 at 11:00
• @ryang Yeah, what's wrong with that? Commented May 18, 2023 at 16:43
• Whence did you get the impression that my comment was disapproving? ;) Commented May 18, 2023 at 16:45
• @ryang from two question marks after the sentence :lol: Commented May 18, 2023 at 16:47
• @RustyCore Indeed, your objection is completely valid because two equality signs have different meanings there, which might confuse the student. However, most of us routinely write it in that style in our research papers, so it might be a somewhat lesser crime than you suggest. If it had appeared in a student work and everything were correct, I wouldn't bother to attract the student's attention to it :-) But yeah, for struggling students it can make a difference. I agree with you here. Commented May 23, 2023 at 13:01

Just because a term can be used without really understanding it doesn't, in my opinion, mean that the term should be avoided. One good practice in such cases would be to sometimes ask students what they mean by the term.

Whenever we teach the concept of canceling in fractions, we definitely need to communicate what is really going on (that we simply have a multiply by 1 that we no longer have to write). When it's for addition, there simply is an addition of 0 that we also don't have to write.

One advantage of using the word "cancel" is that a similar thing really is going on in both addition and multiplication. Using the same word for analogous things is a good thing.

1. In addition to the other pro using "cancel" answers, it's important to realize that there's a vast amount of dimensional analysis ("railroad track problems") in science and engineering classes. Especially and extremely prominent in college or even high school general chemistry classes (stoichiometry, gas laws, etc.) "Cancel" is a completely normal term in those contexts.
1. I think there's a tendency of the "humans as computers" thinking to propose that errors (sign errors, linearity, transcribing, etc.) can be prevented by alternate terminology. This is generally NOT the case. It's not how the human brain works. Drill and repetition and rapid feedback are required to learn things, whether it is math procedures or rebuilding an engine. Those errors will come out, especially with new or weak (or new AND weak) students, regardless of instruction. In fact, making the language too formal and precise can overcomplicate things for new learners. It's a balance, sure, and I'm not "Every Which Way but Loose" on instructions. But in general, the errors students come up with are not so much instruction errors, but natural (human, flesh brain) errors, which need to be drilled out. So thinking you can preempt them is naive.

2. One term that does bug me is "flip to other side of equation and change sign", when used for new learners of solving linear equations of x. This is a profound beginning stage of algebra. And at that time, it is better to be very clear to the students, that the thing about an equality*, is that you can do same thing to both sides. So, don't "flip to other side of equation and change sign", but subtract the quantity from both sides and then cancel (addition cancel). This is not a problem say for 11th grade physics students, who have extensive experience in algebraic manipulation. Fine to "flip to other side of equation and change sign" for them. But for 7th or 8th graders, first "doing math letters", don't be so cavalier.

*Later on, in a precalculus class, they will learn about how to manipulate inequalities.

A dictionary like Cambridge or Merriam-Webster defines the word "cancel". Why not use the word "vaporize" instead of "cancel"? At least vaporize would be interesting and closer to what truly happens in the operation of removing common factors in a fraction.

• Although you may have good intentions, this is a bad idea: canceling, like $\frac{6}{4}=\frac{3 \times 2}{2 \times 2}=\frac{3}{2}$ is something, teached at the age of 8 years old (third year in primary school). "Vaporising" is only teached at a later age (around 10 years old, fifth year in primary school), so you cannot replace "cancel" by a word, not known by the students yet. Commented Jan 8 at 7:32
• I do not use the word "cancel" because nothing is canceled. reducing common factors to an implicit "1" is not cancelling - though it is widely used term. Use "cancel" all you wish. If I am in the room, I will call you on it. Yes, I have been through all the college math courses -- if you are wondering. I should have applied for a minor in math while I was there but I did not think to do so.
– Rojo
Commented Jan 8 at 19:56