Algebra Middle School

3 Phrases from Math Class we Need to Expunge.

A brief twitter exchange last night between myself and the great NY math educator Mike Pershan caused me to get off my rear to assemble a post which I had kicking around my head for some time now, a list of terms and shortcuts we use in math class which, while well-intentioned and used everyday by many math teachers, aren’t necessaily helpful in causing kids to understand their underlying math concepts.


In a recent in-service with middle-school math teachers, I used a video by Phil Daro (one of the authors of the Common Core math standards) to have colleagues reflect upon the practice of “answer getting”, short-term strategies employed by teachers to get students through their immediate math assessment, but with little long-term value in math understanding.  Click on the “Against Answer-Getting” tab for the video.

So, here is my first list of nominees for elimination, and some strategies for helping students develop underlying algebraic ideas.  It probably won’t be my only list, and I welcome your candidates and thoughts.


This is a device I often see in pre-algebra classrooms, often times as a poster for easy reference, other times as a mantra for the students to help complete worksheet problems.  From the site

TIP: For subtracting integers only, remember the phrase

“Keep – change – change
So, we have a short and snappy device which helps us with just one specific type of integer problem.  It’s not wrong, just too specific, and do students understand why it works?
What to do instead:
Let students develop their own summaries of integer problems, and create their own posters which describe their findings.  Use integer zero-pair chips or online applets, like from the National Library of Virtual Manipulatives (search for “chips”).  Number line applets can also help students visualize addition and sibtraction problems.  Have students write stories about given integer and subtraction problems, and have students peer-assess work for proper use of math terms.  Eventually, have students debate the possible equivalence of integer pairs:
  • 5 – (-2) and 5 + 2
  • a + (-b) and a – b
  • a – b and b – a


The ad-laden math site Coolmath gives its own snazzy description of foil:

We’ve got a cool little trick called “FOIL” for multiplying binomials….it’s really just an easy way to do the distributive property twice, which would be really messy and confusing to do.

YEY!  You mean I can multiply stuff without that nasty and scary distributive property, without actually talking about the distributive property!  Yey shortcuts!  I’m in! {insert sad face}

Folks, ditch FOIL, and use the opportunity to talk about the double-distributive property.  Re-write the binomials as an equivalent expression and multiply.  Set the stage for factoring and note how much more understanding factoring by parts takes on.  And, now we can tackle those “messy” trinomials too.


Try this exercise tomorrow: take a class tht has been through Algebra 1, and as an opener tomorrow ask them to explain what the phrase “Cancel Like Terms” means when dealing with a rational expression.  Or, if that is a bit too scary, simply ask your students what it means to recude a fraction.  This is a nice activity to do as a Google form, and have students assess the explanations.  Many students will give an example as a definition, which is not what we are looking for here.  How many students discuss factors, GCF’s, numerators or denominators?

Reducing a rational expression means to divide both the numerator and the denominator by the greatest common factor of both numerator and denominator.  (Incidentally, also try having your students provide steps for finding a GCF.  This one also reveals what your students understand.)  The great part about this procedure for reducing is that it works equally well for each of the following expressions:

To many of our students, cancel is digested as “cross-out stuff”.  We have better vocabulary for it, so let’s encourage its use.


By Bob Lochel

HS Math Teacher. Hatboro-Horsham School District, Horsham, PA.

15 replies on “3 Phrases from Math Class we Need to Expunge.”

I agree with all of these! My BIGGEST issue with the “cancel like terms” is that it’s not even accurate. We don’t cancel terms, we cancel factors! And we’re not just magically “canceling” them – we’re dividing them and getting 1!

I hate all of these terms also… foil only works in one case and not transferable, cancel? what? Let’s use math tems for an ability to accurately communicate math and let students know what is going on from math class to math class!!! What a concept! We have math vocabulary for a reason!

Awesome. Thanks! Feel free to change the format so people are more inclined to contribute. A bunch of people viewed today since I tweeted it along with the link to this post, but no one added anything yet.

OMG, this is so timely for me. In my Math Support class, I’ve been actively expunging the “keep-change-change” that a colleague insists on teaching — even though it IMMEDIATELY degrades in students’ minds into a thousand perverse variations.

Over the course of one day, I can hear students in the same class say, “Keep keep change” and “Keep change change” and “Keep keep change change.”

They know what I will tell them: “I don’t know what that is. It is NOT mathematics.”

They know that I will pull out my number line boards and my +/– dice and invite them to think through the concepts of adding and subtracting on the number line, then adding and subtracting positive quantities, then adding and subtracting negative quantities.

They know that I will insist that they use their God-given REASONING SKILLS rather than some silly trick that always betrays them in a pinch.

But they keep pressing that button, hoping it will yield a food pellet, even though it only gives them an electric shock.

The ones who want to succeed always thank me later for disabusing them of their confusion. But what I want to know is, WHY IS ANYBODY PLANTING THIS CONFUSION IN THEIR MINDS IN THE FIRST PLACE???

– Elizabeth (aka @cheesemonkeysf)

Great post – proper math verbage is a never-ending battle we must fight daily! 🙂 Ideas like FOIL, “cancel”, and process mantras are tempting to everyone because so much of what we do wants to feel like a success. Battling through DISTRIBUTION instead of FOIL might not bear fruit until the next semester, or even the next year’s class, which can be frustrating for me and the student in the present.

If all of the math teachers in your school are on board, you can fight – if one or two are NOT, parents will find out who’s class to try and switch their kids into so its “easier” and as we know, kids in those “easy” classes will experience future frustration in other classes; it delays the problem for later, and then you turn around and there are kids who’ve gone their whole math career getting through with tricks.


[…] Last fall, I participated in an Ingite session with the awesome folks from the Math Forum at the Association of Math Teachers of New Jersey conference.  Thanks to Suzanne for her kind invitation to share with a group of math leaders I have admired from afar.  The videos from the Ignite are finally available. Enjoy my flailing arms in a talk related to a blog post from long ago on math phrases I’d like to expunge. […]

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