Is it harmful to use the word "Cancel"?

Question

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)

Answer 1

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.

Answer 2

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 $\arcsin(\sin(x))=x$.

Answer 3

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?"

Answer 4

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.

Answer 5

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.

Answer 6

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 :-)

Answer 7

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.