## You Can’t Break Math

We were solving linear equations in Ms. Warburton’s eighth grade class last week and I learned (or re-learned, or learned at greater depth) a couple of truths about mathematics.

B suggested they multiply by two as a “fraction buster.”

(One small pleasure of guest teaching is trying to identify and decode the vernacular of each new class. I heard “fraction buster” more than once.)

R asked, “But do we multiply this by two or the whole thing by two?”

If you’ve taught math for a single day, you know the choice here.

You can tell them, “You multiply the whole thing by two.” That’d be helpful by the definition of “helpful” that includes “completing as many math problems as correctly and quickly as possible.” But in terms of classroom management, I’ll be doing myself no favors, having trained B and R to call me over whenever they have any similar questions. More importantly, I’ll have done their relationship with mathematics no favors either, having trained them to think of math as something that can’t be made sense of without an adult around.

Variables like x and y behave just like numbers like -2 and 3.” I said. I wrote this down and said, “Try out both of your ideas on this version and see what happens.”

After some quick arithmetic, they experienced a moment of clarity.

In the next class, students were helping me solve 2x – 14 = 4 – 2x at the board. M told me to add 2x to both sides. One advantage of my recent sabbatical from classroom teaching is that I am more empathetic towards students who don’t understand what we’re doing here and who think adding 2x to both sides is some kind of magical incantation that only weird or privileged kids understand.

So at the board I was asking myself, “Why are we adding 2x to both sides? What if we added a different thing?”

Then I asked the students, “What would happen if we added 5x to both sides? What would break?”

Nothing. We decided that nothing would break if we added 5x to both sides. It wouldn’t be as helpful as adding 2x, but math isn’t fragile. You can’t break math.

BTW

• I haven’t found a way to generate these kinds of insights about math without surrounding myself with people learning math for the first time.
• One of my most enduring shortcomings as a teacher is how much I plan and revise those plans, even if the lesson I have on file will suffice. I’ll chase a scintilla of an improvement for hours, which was never sustainable. I spent most of the previous day prepping this Desmos activity. We used 10% of it.
• Language from the day that I’m still pondering: “We cancel the 2x’s because we want to get x by itself.” I’m trying to decide if those italicized expressions contribute to a student’s understanding of large ideas about mathematics or of small ideas about solving a particular kind of equation.

Here is an awesome sequence of comments in which people savage the term “cancel,” then temper themselves a bit, and then realize that their replacement terms are similarly limited:

I have a huge problem with ‘cancel.’ It’s mathematical slang, and I’m OK with its use among people who really understand the mathematics. But among learners it obscures the mathematics and leads to things like “cancelling terms” in rational expressions.

I think the word “cancel” is misused in math when teaching students. We are not canceling anything we are making ones and zeros.

I never say cancel. I’ve worked hard to eliminate it from any teaching I do. Same with cross multiply, never say it. Instead I say “add to make zero” or “divide by or multiply by the reciprocal of to make one”.

We use “cancel” to mean too many things and so they use the term anytime they want to get rid of something or slash something out. The basis for my concern: when I ask kids “why can you do that?” they often can’t explain.

However, when it comes to squaring a square root, what is most accurate to say? I don’t correct the kids in that case, and I tell them that cancel means the same thing as “undo”.

If the point is to be rigorous, “apply the inverse” is more rigorous and technical than “cancel”.

More miscellaneous wisdom on language in mathematics:

In the topic of “get x by itself”, I’ve started saying “We want to say what x is. What would that look like?” They usually say “It would look like x = something”. Then they’ve chosen what the final equation ought to look like for themselves.

I wonder what would happen if we had an equation and then asked them to find out what, say, 2x+1 was.

Even in my Advanced Algebra 2 classes I have started using the phrases “legal” and “useful”. In the original post adding 5x to both sides was definitely “legal” just not as “useful”.

One of my refrains is that an algebraic step is “correct, but not useful.” Inspired by my dance teacher, I also talk about how a particular procedure is lovely, just not in today’s choreography (which is geared towards solving for an unknown/simplifying a trig expression/ finding intercepts, etc).

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

1. #### Greg

February 8, 2017 - 5:41 pm -

I’ve definitely never been impressed by “cancel” or “getting a variable by itself.” But I think this ends up just returning to whether you’re doing “math” or solving an actual problem, where x just isn’t a variable but represents some unknown *that you are interested in knowing.* So it’s not either “canceling” or “isolating a variable” but finding out how many ducks fit on the pond or how much you should charge for lemonade…

• #### Chester Draws

February 8, 2017 - 8:37 pm -

where x just isn’t a variable but represents some unknown *that you are interested in knowing.*

I don’t see the difference. If we are doing pure maths, then the variable we are solving for is the what we are interested in knowing. Why is that less real than some completely fake question about ducks in a pond?

I tend to say that solving is the process of finding the value of the unknown (and I prefer “unknown” not “variable” because it works better for keeping no distinction between “pure” from “applied” questions.)

To do that, I explain, we need to have the variable in only one place, because we can’t figure out its value if it is in two places. If they read that, and some will, as “getting the variable by itself” then so be it. We might tiptoe around it being very careful to use what we consider the exactly correct language, but since the process is basically getting the variable by itself it’s hard to see how that does any harm. (Are there any algebraic ways of solving that don’t work by isolating the variable?)

2. #### Greg Ashman

February 8, 2017 - 5:54 pm -

This discussion centres around the principle of equivalence. One thing that seems to be missing from it is the model of a set of weighing scales. I think I have read somewhere that projecting a faint image of a set of scales in the background of the equation can cue the equivalence principle. I don’t have a reference to hand – It may be from a Willingham paper.

It’s then quite straightforward to explain by analogy why you have to add or multiple by the whole of a side rather than apart of it.

• #### Greg Ashman

February 8, 2017 - 6:27 pm -

I found the reference. It’s not to a specific study:

“As concepts become more complex, familiar examples from the students’ lives become harder to generate, and teachers may use analogies more often; a familiar situation is offered as analogous to the concept, not as an example of the concept. Thus, a teacher might tell students that algebraic equations may be thought of like a balance scale: the two sides are equivalent, and you maintain their equivalence so long as you perform the same operation on both sides. Laboratory studies have revealed several principles that make analogies especially effective: familiarity (e.g., students know what a balance scale is), vividness (actually having the balance scale for students to see), making the alignment plain (e.g., writing the two sides of the equation over the two sides of a drawing of a balance scale), and continuing to reinforce the analogy (e.g., by referring to the scale at appropriate times as the equation is solved). Some data indicate that math teachers in Hong Kong and Japan (where mathematics achievement is consistently high) are especially effective in using analogies according to these principles.”

• #### Dan Meyer

February 8, 2017 - 7:31 pm -

I buy it. You’re describing an analogy to a balance scale. I’m describing an analogy to numbers.

• #### Greg Ashman

February 8, 2017 - 8:14 pm -

I wonder whether the problem with proving things by substituting numbers is that kids might get the impression that it only works for those particular numbers (because that’s what happens in most equations they meet – there’s only one number that x can be). I think the balance scale might be superior because it leads to a deeper, more generalised understanding of the principle of equivalence. I might be wrong there – would be good to see some of the research.

• #### Chester Draws

February 8, 2017 - 8:27 pm -

How does a balance scale help when they get to quadratics? Exponentials?

I’m happy to use the balance scale to explain why a single term doesn’t alter the “balance” of the equation, but if they thing solving in terms of balance scales they are stuck at linear equations.

• #### Paul Hartzer

February 9, 2017 - 9:02 am -

I use the balance scale as well, but it’s problematic with negative terms. How does one meaningfully represent “there there three units less than two objects”?

3. #### Paul Hartzer

February 8, 2017 - 5:57 pm -

“R asked, “But do we multiply this by two or the whole thing by two?””

A thought: Neither. Multiply every term in the statement by two.

I’ve been exploring this recently. Ironically perhaps, this came out of my watching students completely give up on systems of equations but not having the same problems with “add two to both sides” (for instance). But if we shortcut solving “2x – 14 = 4 – 2x” by adding “2x + 14” to both sides, we’re basically doing the same thing as adding two different equations. That is:
2x – 14 = 4 – 2x
2x + 14 = 2x + 14
ADD like terms on each side to get:
4x = 18

2x + 1 = 3y – 7
2x – 5 = y + 8
SUBTRACT like terms on each side to get:
6 = 2y – 15

That got me thinking about how equality is a certain type of relationship, how an equation is a certain type of thing, how terms are a certain type of thing, and so on. Adding (and subtracting) to balance involves one object in each “section” of an (in)equality (they usually have two sections, but could have more), while multiplying (and dividing) to balance involves applying that process to each term.

This thinking works for me, and seems to have “clicked” with some of the students who don’t get the “multiply both sides by the denominator, and then distribute” method.

• #### Kyle Pearce

February 8, 2017 - 7:00 pm -

Paul – I think your idea to be explicit about what “multiplying both sides” really means is so important.

Mathematics is abstractions built on abstractions and we often just back up one layer when students often need us to head right back to the beginning.

• #### Belinda Thompson

February 9, 2017 - 9:02 am -

Paul, I like where you’re going with this, and I think students get stuck on the idea that “adding the same thing to both sides” means that those things have to look alike, rather than the idea that those things have to have the same value. I’ve tried to impress on students that we’re invoking if a=b, and c=d, then a+c=b+d has to be true. It’s really hard for them to believe!

4. #### Corey Andreasen

February 8, 2017 - 6:21 pm -

I don’t have a problem with ‘get x by itself.’ I don’t want students to think that’s exactly the same thing as solving the equation, but it is a strategy that is useful in solving many kinds of equations. But it is one strategy.

featured

I have a huge problem with ‘cancel.’ It’s mathematical slang, and I’m OK with its use among people who really understand the mathematics. But among learners it obscures the mathematics and leads to things like “cancelling terms” in rational expressions.
• #### Melissa Lechleiter

February 8, 2017 - 6:51 pm -

featured

I think the word “cancel” is missed used in math when teaching students. We are not canceling anything we are making ones and zeros.

If we focus on the meaning of inverse then our students will have a better math vocabulary and a better understand that we can build on. We will need to watch what we as teachers say so that students use the correct words and not our math slang words. Maybe have the student say do we multiply both terms instead of the whole thing. Then they might answer their own question.

• #### Aaron

February 9, 2017 - 7:35 am -

Hey Corey. My problem with ‘get x by itself’ is that it doesn’t really push on the bigger more generalizable idea of equivalence. If the goal is get x by itself then it’s kind of a mystery how we might accomplish that (at least to the novice). If the goal is equivalence then there is all sorts of room for what Dan describes by adding 5x to both sides and the implications of that. I think equivalence explains a whole lot of mathematics that a lot of times is boiled down to procedures that have to be “remembered”. Curious what you think.

• #### Bowen Kerins

February 10, 2017 - 12:57 pm -

“Get x by itself” is not descriptive enough to solve an equation.

2x + 3 = 5x – 12
(subtract 3 from each side)
2x = 5x – 15
(subtract x from each side)
x = 4x – 15

This is more commonly an error in rational expressions than in linear equations.

5. #### Susan

February 8, 2017 - 6:38 pm -

featured

I never say cancel. I’ve worked hard to eliminate it from any teaching I do. Same with cross multiply, never say it.

Instead I say “add to make zero” or “divide by or multiply by the reciprocal of to make one”.

• #### Sam Shah

February 8, 2017 - 6:56 pm -

Similar thing here! My kids know the “C word is bad” (I explain why the first time it naturally comes up in class), and they now catch themselves when they say it. My rationale I share with kids to explain my pet peeve:

featured

we use “cancel” to mean too many things… and so they use the term anytime they want to get rid of something or slash something out…
The basis for my concern: when I ask kids “why can you do that?” they often can’t explain.

So we do a lot of talking about what we’re doing. And so kids often say “divide to make 1” or “add to zero.” And sometimes they or I go further and say “divide to make 1, and anything multiplied by 1 remains unchanged.”

My favorite moments are when kids catch themselves or each other using “the C word” and refine what they’re doing to be more specific. It makes me so happy when that happens.

February 8, 2017 - 9:36 pm -

I second this motion (to say “making one” or “making zero”). Lani Horn shared this with our class and I’ve taught it to my students for the last 11 years.

featured

However, when it comes to squaring a square root, what is most accurate to say? I don’t correct the kids in that case, and I tell them that cancel means the same thing as “undo”.
• #### Aaron

February 9, 2017 - 7:30 am -

@Jeremy Is it the idea of inverse that’s missing? So “making one” is employing multiplicative inverse, and “making zero” employs additive inverse? So for a square root we employ the inverse again which is squaring? I agree that making zero and making one are better than canceling, but maybe we should be pushing further on the idea of inverse. It seems like we might get more eventual bang for our buck.

• #### Belinda Thompson

February 9, 2017 - 9:03 am -

Oh goodness, how about the hybrid of “cross-cancel”?? I’m not sure where that came from, but I tried to train it out of my students sometimes to no avail :(

• #### Paul Hartzer

February 9, 2017 - 9:43 am -

One of my problems with “making 1” are the students who insist on the needless step of 4x/4 = 1x and 1x = 1.

featured

If the point is to be rigorous, “apply the inverse” is more rigorous and technical than “cancel”. I’m fine with that.
6. #### Kyle Pearce

February 8, 2017 - 7:09 pm -

Having spent much of the past 5 years working in grade 9 with both strong and struggling math students, my big “ah-ha” came when I realized that most of my students only understood rules to apply, but didn’t know what these big groups of symbols actually represented.

While I think there is a lot of benefit to having students grapple with abstract problems like what we see above, I have started to try unpacking what these algebraic expressions/equations really look like. I’ve found that if students can clearly articulate visually and verbally what the symbols on the page represent, that is half the battle. The problem is finding the time to go all the way back to simplifying simple numerical expressions using order of operations and working our way up to incorporating variables.

Here’s a playlist where I’ll try to articulate a bit of what I mean:

7. #### Paul Hartzer

February 8, 2017 - 8:03 pm -

I don’t understand the problem with “cancel”. In common parlance, “cancel” has two main meanings: To end (“I cancelled my subscription”; “they cancelled my favorite show”) and to negate the effect of by using an opposite force (“noise-cancelling headphone”). The latter is what is generally called “cancelling” in mathematics as well: 3 – 3 = 0; 3 / 3 = 1. Can someone clearly articulate the problem with this?

• #### Paul Hartzer

February 8, 2017 - 8:05 pm -

My second example should have been “noise-cancelling speakers”. Obviously “noise-cancelling headphones” would be the first sort of cancellation.

• #### Sarah

February 8, 2017 - 11:23 pm -

I don’t allow the word cancel in my classroom because students don’t understand what is happening. They think things just “poof” disappear. Have you ever posed a question where everything “cancels”? Most of the time you’ll get an answer of “0”, not “1”. I explain that the “bar” actually means “divide”. And division undoes multiplication, so you need to have multiplication in order to undo it. (You can’t untie your shoes if they aren’t already tied) This seems to help with the desire to “cancel” the 7s here. (3+7)/7 Usually presented without parentheses. However *dividing* the 7s here is totally okay. (7*3)/7

• #### Paul Hartzer

February 9, 2017 - 3:55 am -

“Cancel” means “undo”. So if division “undoes” multiplication, then “cancel” is an appropriate term.

Personally, I find “undo” to be less clear than “cancel”.

• #### Karen Campe

February 10, 2017 - 12:09 pm -

I find UNDO to be incredibly powerful in the context of Inverse Operations: operations that undo each other. All of our algebraic solving strategies from Algebra 1 that we “do to both sides” are accomplished via inverse operations. Later on, we include squaring/square rooting, inverse trig functions, and exponents/logs to the list. This is one of the “Big Math Ideas” that connects concepts throughout secondary math.

8. #### David Butler

February 9, 2017 - 12:46 am -

I really like the idea of letting the students know they could do any number of things to make other equations, but only some of them will actually progress towards a particular goal. A similar idea came up while teaching my university students to rotate conic sections to turn them into standard form.

featured

In the topic of “get x by itself”, I’ve started saying “We want to say what x is. What would that look like?” They usually say “It would look like x = something”. Then they’ve chosen what the final equation ought to look like for themselves.

I wonder what would happen if we had an equation and then asked them to find out what, say, 2x+1 was.

• #### Belinda Thompson

February 9, 2017 - 10:01 am -

Ooh. As part of a research study we asked community college students in a developmental math class called Algebra something like this: If 2(x+3)=14, what is the value of x+3? As you might guess most students (about 75%) applied the distributive property, found x (only 25% of those found the correct value for x), and then used that value in x+4 (or just reported the value they found for x).
Very few recognized that this equation basically stated “2 times some number is 14, so that number must be 7.”
“Doing something to find x” was very strong with this group.

• #### Aaron

February 9, 2017 - 10:07 am -

I feel like this is another place where a focus on equivalence would make a difference. What are some equations that are equivalent to 2(x+3) = 14, and do any of them help us figure out the value of x+3?

• #### William Carey

February 10, 2017 - 9:48 am -

> I wonder what would happen if we had an equation and then asked them to find out what, say, 2x+1 was.

For what it’s worth, I do this all the time in Algebra II. The kids generally are perplexed by it at first. But after one or two examples, decide that it’s *more fun* solving for something that’s not even in the original equation than just for plain old x.

9. #### Dick Fuller

February 9, 2017 - 4:58 am -

I don’t think I understand the notation. If I have (12+8x)/2 -1 = 2(2x+5) then I have 6 + 4x -1 = 4x + 10 or 5 + 4x = 4x + 10 which is satisfied by no x unless 5 = 10. Or maybe x is infinite.

• #### Dan Meyer

February 10, 2017 - 8:39 am -

Hi Dick, it turns out this equation has no solution, a fact which didn’t seem germane to this post but which was definitely germane to the classroom discussion.

• #### Dick

February 10, 2017 - 11:33 am -

Dan;
This has been a good discussion. I know your point addresses reduction or rewrite”, but there is some nice, simple math here that might be useful to students. And it is simple. Your expression is an equality. As such it it has one of the values True or False, and that value is the answer to “did I understand the, or at least a, problem when I asserted x was the answer?”. Reducing it as you did is a proof that it is False, a perfectly good result. If you formulate your own problems, this is the most important thing to know. I have learned to value it. Thanks

10. #### Joel Penne

February 9, 2017 - 7:10 am -

featured

Even in my Advanced Algebra 2 classes I have started using the phrases “legal” and “useful”. In the original post adding 5x to both sides was definitely “legal” just not as “useful”.

Many times in algebra 2 there misconceptions with reducing rational expressions such as (2x+5)/x. Large amounts of students, even the best students, will try to reduce the x’s here when it isn’t “legal”. Then you can tell them it would be “useful” but unfortunately not “legal”. I always try to show super simple counterexamples whenever a “law is broken”. Such as 9 + 4 = 13. Everyone agrees. Now take the square root of all 3 items which students really love to try with radical equations. Obviously the square root of 9 is 3 and 4 is 2. But adding these up will yield 5, not the square root of 13.

• #### Scott Leverentz

February 11, 2017 - 11:44 am -

Are there means by which to get students to do this type of thinking independently? I’ve greatly enhanced my own understanding of lots of mathematics by learning to substitute values in place of variables to help make something murky temporarily more concrete. I’ve utterly failed to get my students to value that kind of reasoning enough to independently do the same though.

11. #### Mark Schommer

February 9, 2017 - 7:31 am -

Language in math is so inconsistent and short-changed it is a wonder students carry knowledge from one year to another. We cause more misconceptions than I feel most are aware of.

what I find most intriguing is a similarity that I am going through with Dan. I also have taken some time away from the classroom to observe and pursue other interests in curriculum. It has made me a significantly better teacher…much more aware of what I say and how I say it. More importantly, how information is discovered not presented (never was one to “present” but now it bothers me to see it all the time).

It is truth that as teachers we want to save our students. We don’t like to see them struggle. However, that productive struggle is where learning occurs. I liked your use of alternative situations…something I do need to work more on. Nice post!

• #### Dan Meyer

February 10, 2017 - 8:40 am -

I also have taken some time away from the classroom to observe and pursue other interests in curriculum. It has made me a significantly better teacher…much more aware of what I say and how I say it.

“Much more aware of what I say and how I say it” feels apt.

12. #### Laura Hawkins

February 9, 2017 - 8:50 am -

featured

One of my refrains is that an algebraic step is “correct, but not useful.” Inspired by my dance teacher, I also talk about how a particular procedure is lovely, just not in today’s choreography (which is geared towards solving for an unknown/simplifying a trig expression/ finding intercepts, etc).
13. #### Anders

February 9, 2017 - 9:02 am -

“You can’t break math” is a pretty strong claim. Take the equation x = 1 and square both sides to obtain x^2 = 1. Broken.

Yes, probably best to save this for later in the year, but when you trot it out I think it would be important to circle back and reflect. Something like, “We thought until now that equations can’t be broken, but now we see that we can break them. How do we amend our earlier claim? ‘Certain operations don’t break equations?’ How specific can we be?”

• #### Paul Hartzer

February 9, 2017 - 9:07 am -

@Anders Likewise, the defined domain of y = x is infinitesimally larger than that of y = x^2 / x

• #### Dan Meyer

February 10, 2017 - 8:42 am -

Take the equation x = 1 and square both sides to obtain x^2 = 1. Broken.

By what definition has math been broken? Math did exactly what you told it to do. It isn’t math’s fault the operation you told it to do isn’t invertible.

• #### Anders

February 10, 2017 - 11:17 am -

By the definition that I take from your discussion of adding 2x vs adding 5x to both sides of a linear equation in x, something along the lines of “no longer equivalent.”

You write, “Nothing would break if we added 5x to both sides.” I read this to mean that adding 5x to both sides results in an equation equivalent to the original, and inferred that an operation that did not have this result would count as “breaking” math. No?

14. #### Dick Fuller

February 9, 2017 - 12:46 pm -

Dan,
Let me try again. I am missing something. Your expression is false. You assert: for some x
( 12 + 8x) /2 – 1 = 2(2x +5), which can be rewritten as
6 + 4x – 1 = 4x + 10. To see simplier expression let
z = 4x then
5 + z = z + 10, or
z = z + 5 is false for any finite z.

I like your original suggestion: try something, but students should not expect every problem they encounter to be well founded. I would like to hear thoughts on how instructors would handle a student discovery of one.

• #### Scott Leverentz

February 11, 2017 - 11:46 am -

I’m guessing that the class Dan was visiting was working on solving equations which included some exercises where there were no solutions (this case).

• #### Dan Meyer

February 11, 2017 - 4:22 pm -

Thanks for picking up my slack there, Scott.

Dick, Scott’s right. They were solving equations that had one solution, no solutions, and infinite solutions.

15. #### Lori Mains

February 9, 2017 - 12:53 pm -

I like the terms that Joel uses: legal and useful. I tell my advanced 6th graders that it is like they are a wise wizard- they can figure out what to do to both sides of the equation (useful) but they must follow the rules of the kingdom (legal).

I can remember clear as day sitting in 8th grade algebra myself and earnestly asking Mr. Garrett “Is it wrong if you don’t do the right thing to both sides of the equation?”. The entire class laughed and I am not sure if he even answered me. It is a great question. I agree with the other comments- try it and see what happens!

This is a tangent, but one thing I also point out to my students is that in grades K-5 they have always been use to working left to right. They see 2 + 5 = and they write in the answer to the right of the equal sign (2 + 5 = 7). Now, with the algebra, they will work *down* the page.
I have them draw a vertical line through their equal sign and continue to write the = sign on that vertical line for each line. Now if you had 2 + 5 = x, you would write underneath it “7 = x’. I know these all sound like mechanical, procedural things but I think that learning comes from multiple different angles- after doing it for a while you start to understand the why.

16. #### Joey Kelly

February 9, 2017 - 3:34 pm -

I agree with the squeamishness about “cancel” at the Algebra 1 level, but I think “cancel” can be useful in Algebra 2 or Precalculus courses to capture the connections between solving linear equations and solving equations using other inverse operations, such as roots/exponents, exponents/logs, trig/inverse trig, derivatives/integrals. “Cancel” can be a way to informally capture the similarity before diving into the nuances and technicalities.

17. #### Jane Taylor

February 9, 2017 - 9:06 pm -

I find it helpful to talk about what operations were applied to “x” and then think about “undoing” those operations one at a time, starting with the last one and working back to the first. That is a helpful way to think about inverse operations and inverse functions.
PS. I spent far too much time agonizing over minor changes, too. After 17 years of teaching I am finally learning that what I did last year worked well and I can definitely use it over again and I don’t always have to reinvent the wheel.

18. #### sean

February 10, 2017 - 1:00 am -

man. i think i’d tell the kid “multiply the whole thing by 2” and then just walk away! check back in maybe 2 mins later to see if she handled the distributive property correctly.

the “variables behave like numbers” explanation strikes me as too deep in the context of a narrowly focused lesson. maybe hit that after school on some longer-form tutoring thing. develop that for like 20 or so minutes.

this is a good post, dan. thank you.

19. #### Dan Meyer

February 10, 2017 - 9:59 am -

Thanks, everybody, for the insightful comments on language that’s helpful and limiting. I think it’s interesting to watch a lot of us come down hard on “cancel,” urging “making 1’s” or “making 0’s” as a replacement, only to later realize that this new language also limits us when we arrive at later math like exponentiation.

I’ve selected a bunch of comments for the main post that illustrated this trajectory pretty well.

• #### thaslam

February 15, 2017 - 8:17 am -

We just started getting into inverse Trig Functions. I think its interesting how some students readily accept it as an extension of “cancellation” (ex: asin(sin(A))=A ) while others insist on an “inverse” interpretation (ex: sin(Angle)=Ratio -> asin(Ratio)=Angle ). And there are those who are uncomfortable with either interpretation, even though they seem perfectly capable of solving things like 3x+2=8 or x^3=8->cuberoot(8)=x.

And then there’s e^(ln(x))=x …that was one “cancellation” I always felt uncomfortable about as a student.

20. #### Dana

February 10, 2017 - 10:54 am -

If you ensure understanding of equivalence, mathematical properties, and inverse operations, does it matter whether you say isolate the variable, get x by itself, find a solution, or solve for x?

We want consistency among the classrooms learners will predictably (grade promotion) or likely (common moves within a city/county/region) migrate from/to but the “right” term is one that is understood within a learning community and isn’t misleading or wrong.

On that note, I would argue that “cancel” is less misleading than “making ones”.

When simplifying the left side expression (12+8x)/2 -1 to 6 + 4x -1 , I think it’s more accurate to say cancelling the (common factor) 2.

I concede that I’ve taken advantage of (2/2)*6 + (2/2)*4x -1, but I didn’t “make” those ones, they were part of the original expression.

21. #### Alexa

February 10, 2017 - 11:13 am -

I stray away from using cancel. I use the word “undo” when manipulating an equation. For example, I will say, how can we undo a positive 2x? Students will say, by subtracting 2x, or by adding a negative 2x. We undo certain things to isolate x. I do not have a problem with “getting x by itself.” However, “canceling” seems like an unnessary word that does not contribute to student understanding.

22. #### Paul Hartzer

February 10, 2017 - 2:13 pm -

“Variables like x and y behave just like numbers”

This implies that unknown values AREN’T numbers. I’m wondering if this is a productive tack. What IS a number? Is “3” a number, or is it the representation of a number? Is there a difference between a number and a numeric value?

I was thinking about this recently while teaching how to calculate the length of a trapezoid’s midsegment. The diagram had bases of x and 6, and a midsegment of 4. The students wanted to “take the numbers and find the average” (hence, 5). It occurred to me that I saw three numbers in the diagram, where the students saw two. That got me thinking about what a “number” is in the first place.

23. #### Judy Hante

February 10, 2017 - 11:11 pm -

I was amused by your comment about spending so much time putting together new things when you the material you have is perfectly good. I do exactly that, spending excessive hours and sometimes leaving myself only a few hours of the night to sleep. Occasionally the work gives me a great 1.5 hr hands-on exercise, but too often I really overdo it and don’t get nearly enough value out the elegant thing I created. I think it is a love of teaching that takes me down so many rabbit holes, but I have to recognize sooner when I’ve bitten off more than I can chew :)

• #### Scott Leverentz

February 11, 2017 - 11:50 am -

This seems to be to be one of the big problems of improving math education at large… how do we get teachers strong instructional materials that they can trust in and worry more about individual students, providing feedback, and building strong cultures of learning than designing instructional materials?

In my case at least, a large part of my fault is my conviction that a more perfect plan will result in a more perfect lesson. While that might be true, it’s surely a case of diminishing returns.

24. #### Thaslam

February 13, 2017 - 6:22 pm -

Funny thing, mathematics already has the terminology for this (or at least a name for the notion):
-apply the “inverse”
-to get the “identity”
-so you can “solve for x”
…I guess for whatever reason though the terminology is a bit intense, or something.

25. #### Dick Fuller

February 14, 2017 - 5:28 pm -

It has taken me a long time to appreciate the great opportunity algebra provides for us to communicate the mathematics we think about. Where would we be without “Let x be….”, or “cancellation is”. Let me try to communicate a difference I’ve been trying to understand. I admit I was trying to undo Sarah’s undoing of “cancel”. Instead, as usual in this sort of thing, I have come appreciate her insight.
There are two ways to look at the result of an action: “what happens if I did not do what I intended to do?” and “what would be the result if I undo what I did do?”. I think “cancellation” is the former, what happens if I cancel the visit, or the subscription, or the subtraction?

“Undoing” is the latter.
cancel: blob + M = blob + (M – N) + N -> blob + (M – N) /+ M\ ->
blob + (M – N). “/x\” denotes “strike x”.
undo: blob + M = blob + M + 0 = blob + M + (-N) + N ->
blob + M (-N) /+N\ = blob + M + (-N).
We used the additive inverse: 0 = (-M) + M.

For multiplication we use the multiplicative inverse 1 = (1/M) x M to get
cancel: blob x M -> blob x M / N.
undo: blob x M -> blob x M x (N^-1).

I are looking for meaning. It is the difference between “subtraction by” and “addition of the negative of”, or “division by” and “multiplication by the inverse of”. One is striking out the forward action, and the other is removal of the result of acting forward. I first felt the difference when I was trying get my grandson’s tardy mark expunged. I could not undo the fact that he was tardy, I just wanted wanted the record of it cancelled. Didn’t work.

• #### Paul Hartzer

February 14, 2017 - 6:37 pm -

We need to be cautious about using an argument against mathematical jargon that is based on how a word is used outside of mathematics. There are plenty of words in mathematics that have a related but different meaning to what those words mean outside of mathematics.

That said, “cancel” has multiple meanings, including “to offset”. Indeed, the earliest modern meaning is “to cross out”; the Latin word means “to make to resemble a lattice”. So when we draw a line through a factor of 5 on each side of an equation, for instance, that’s exactly what we’re doing: Cancelling. Crossing out.

I fail to see anything in the meaning of “cancel” that prevents its use as appropriate mathematical vocabulary as “to offset through inverse operations”. As I say elsewhere, “apply the inverse operation” is a more rigorous way, but I think that’s a response to the goal.

Q. What is one way to solve for an equation?
A. Cancel (i.e., strike out) everything that’s not the variable on one side of the equation.
Q. How do we do that?
A. We could apply the inverse operation of each operation, in reverse order of precedence.

Now that I’ve written this, I’m even more bothered by “undo”: That implies that we’ve already done something with, say, 3x + 5 = 26. The point of solving an equation, ostensibly, is that we want to know what the original value is. For instance, perhaps we want to buy some number of items at \$3 each, and a second item at \$5. We have \$26 in our budget… how many items can we buy? In this case, we’re not undoing anything because we haven’t done anything yet.

Perhaps I shall remain the last hold out on this issue, but I have yet to be convinced that “cancel” is truly problematic.

• #### Dick Fuller

February 15, 2017 - 6:54 am -

I agree. We do students no favor suggesting “cancel” is somehow misused in mathematics discourse. It is so widely used that our efforts should go toward learning to use it as students will find it used.

My point was there are two senses in which we alter an expression. I would conjecture this is the reason we are uncomfortable using one word. I felt more comfortable when I found that the two forms for mathematical operations, at least the ones that apply in this thread, for altering an expression line up with these two meanings. With algebra, this is the sort of discussion we can have. Or not have. I just thought it was interesting that mathematics itself could have something to say. You guys are very nice to tolerate me. Thanks.

• #### Corey Andreasen

February 14, 2017 - 6:49 pm -

Here’s my problem with ‘cancel.’ And it has nothing to do with the fact that above the box I’m typing in, I see “Cancel reply.” :)

You cancel when the same thing is on the top and bottom of a fraction. So (x + 1) / (x + 4) = 1 / 4. And other similar misapplications. I think this is because we aren’t describing the mathematics of what is going on. We aren’t emphasizing that we divide the numerator and denominator by the same factor.

I have no problem with using ‘cancel’ among people who understand the mathematics. But algebra students are not those people.

As far as ‘undo,’ one approach to solving an equation is to think about starting with an equation like
x = 4. Now add 3 to both sides.
x + 3 = 7. Now multiply both sides by 8.
8(x + 3) = 56.

The left side of the equation keeps track of the things that were ‘done to’ x. So thinking in terms of ‘undoing’ makes sense.

I don’t think of ‘cancel’ as mathematical jargon. It’s mathematical slang. And just as slang can be confusing for anyone learning a language, it can be confusing for students learning the language of mathematics.

• #### Paul Hartzer

February 14, 2017 - 7:38 pm -

Jargon is also confusing for people learning a new language. Think of the “normal” of the “curve” y = x (which is its own “tangent”).

I already expressed my opinion of “undo”, which you have not swayed. :D

“You cancel when the same thing is on the top and bottom of a fraction. So (x + 1) / (x + 4) = 1 / 4. And other similar misapplications. I think this is because we aren’t describing the mathematics of what is going on. We aren’t emphasizing that we divide the numerator and denominator by the same factor.”

So explain the mathematics. Emphasize that words in mathematical contexts don’t always mean the same thing as words elsewhere. As I’ve said, if the goal is to ratchet UP the vocabulary, so we insist on “apply the inverse” instead of “cancel”, okay, great, wonderful goal.

If someone personally wants to avoid “cancel”, that’s up to them. My problem is with utter bans of the word “cancel” by anyone in the classroom, only to be replaced by equally casual-sounding terms like “undo” or “make into 0/1”. That sort of linguistic nitpickery strikes me as counterproductive.

• #### Corey Andreasen

February 14, 2017 - 8:02 pm -

Except “undo” has a similar meaning to “apply the inverse operation.” That’s what applying an inverse operation does. But ‘cancel’ is so general. 1 + -1 so they cancel. 4x/4, the 4s cancel. Using the same word for so many different contexts obscures meaning.

• #### Paul Hartzer

February 14, 2017 - 8:40 pm -

You just gave two examples of inverses in order to exemplify how “cancel” is too vague, but “apply the inverse” is okay.

• #### Corey Andreasen

February 15, 2017 - 6:22 pm -

When students say “cancel” it means “cross things out.” In my experience they are not thinking of anything remotely mathematical. Things just disappear. When you say “apply the inverse” they have to think about what that means. It begs the question “Inverse of what?” So they must think about the operation they are inverting. That’s thinking mathematically.

I suppose you could teach them to think “cancel” that way, making them think about why things “cancel” but (again, in my experience) that’s not generally what’s done and it’s not generally how students think of it.

• #### thaslam

February 15, 2017 - 8:05 am -

“It is the difference between “subtraction by” and “addition of the negative of”, or “division by” and “multiplication by the inverse of”.”

Interesting thought. You can subtract/divide by anything as long as you do it to both sides of the equation, but only one thing–the inverse–will “cancel”

26. #### Paul Hartzer

February 15, 2017 - 6:50 pm -

I have nothing more to add at this point, except I wish people would stop misusing “begs the question.”

27. #### Susan Jones

February 19, 2017 - 11:45 am -

In my experience, cancel has more meaning to students than “apply the inverse” … but not a good meaning. I usually start with their language, though, and encourage them to build meaning. So, “Yes, that ‘cancels’ because it adds up to zero,” or “cancels because we’re multiplying by one now”….and working to build mathematical meaning.
I also encourage the “would that work with real numbers?” strategy for figuring out if things will work with variables…

28. #### Dick Fuller

February 20, 2017 - 7:35 am -

I assert: mathematics is powerful because it can work without leading to word wars. I can define words to mean what I need them to mean. Definitions must ultimately refer to mathematical objects whose meaning I can know is shared by those I am addressing. That started with the whole number system which we understand in terms of its arithmetic based on counting.

The problem with “cancel” is, in its usage, it appears to describe some changes on the paper where we keeping tract of the arithmetic we do. It refers to “marks-on-paper” objects not mathematical ones, rather like cancelling keeps track of postage in a mail process. “Cancel”refers to the account we are keeping of the arithmetic process, not to the mathematics in that process. We will fight about the accounting of it, but the mathematics itself need not be a war of words.

• #### Paul Hartzer

February 20, 2017 - 8:01 am -

This is the most cogent argument against “cancel” that I’ve read. However, I still don’t like “undo” because it implies that we’ve actually done something (when we may not have).

My rebuttal is that “cancel” is ambiguous and CAN mean “to neutralize the effect”, but it’s a fair point that some if not most students don’t see it that way.

How about: “When we apply the inverse operation, we neutralize the values”? I want a word that describes what we’re doing — cancelling, neutralizing, counterbalancing, offsetting. If “cancel” has problems, let’s find a word that doesn’t.

• #### Paul Hartzer

February 20, 2017 - 10:24 am -

Now I think I’m talking myself into “counterbalance” instead of “neutralize”. Interesting side note: “Algebra” is probably best translated as “restoring”, although there’s disagreement on that… which would support “undo”. ;)

As Corey suggests, I think the important thing is that we’re being mindful about our word choice and making sure we understand why we’re saying what we’re saying.

29. #### Corey Andreasen

February 20, 2017 - 9:39 am -