What’s in a name?

This year, I didn’t teach students to factor the difference of squares.  Don’t get me wrong, I did teach how to factor the difference of squares, but I held off on giving this particular case of a quadratic a label.

For the past several years I have been reflecting a great deal on the pedagogy of teaching mathematics and how precise language factors into best teaching practices.  So I want to talk about this decision around difference of squares.

Let me start by saying I am a huge believer in using precise and robust mathematical vocabulary.  Being able to name something allows us to organize our thoughts better and to communicate effectively with others.

Case in point: this year, I used the language of input-output extensively in reference to functions of all sorts and in all representations.  When we got to the midterm exam and students were identifying a function by looking at graphs, I expected an explanation of “it passed the vertical line test.” But to my surprise and delight, out of 30+ students, fewer than 5 left their reason as simply “it passed the vertical line test.”  The rest went beyond that to give some variant of “it passed the vertical line test, which showed that for each input there was exactly one output.”  Bravo, students! Bravo!  I never emphasized memorizing the exact words of the definition of a function, but through using language carefully and consistently, students were able to recall this definition with ease.

Let me mention a second example. Removing the word “canceling” (a valid, but confusing word for students) and replacing it with “these factors divide to one” or “these terms add to zero” allows us to clear up a very befuddling topic.

And one last example: the controversial acronym FOIL.  It’s a useful in a way, but it is so incredibly limiting.  How do you apply FOIL to a binomial times a trinomial? And what if we have something even more daunting, like a trinomial times a trinomial!?  I have completely eliminated FOIL from my teaching vocabulary, and just call it what it is: multiplying polynomials (or distributing).

Normalizing the use of a robust mathematical vocabulary in the classroom is empowering for students.

So why did I teach “difference of squares” without giving it a name?

Because sometimes, language is a barrier and a stumbling block rather than a help. Sometimes, when things have separate names, they have separate mental compartments. And sometimes, giving something its own special name is a signal to students to memorize rather than to connect to prior knowledge.

Difference of squares is one such entity.  It deserves to have its own name because it is just so darn cute and special.  But in my experience, students who learn “difference of squares” as A THING tend to memorize a pattern instead of understanding why.  They start to get confused about x^2+4 vs. x^2-4, especially when they learn about sum and difference of cubes.

But factoring difference of squares without a name forces students to focus on what it really is.  A trinomial in disguise.  Where the middle coefficient is zero.  Students who learn to factor difference of squares by visualizing the zero term are at a distinct advantage, able to connect this type of factoring with all their prior knowledge.

I will admit it.  It took me many years of teaching factoring to realize what harm it was causing students to give difference of squares a label too early.  I would look at student quizzes and tests and wonder how they missed certain problems.  We had, after all, been over many times why the difference of squares worked the way it did.  I was confused why so many students would factor x^2+16 and on the next page miss x^2-16.  It began to dawn on me that the very thing I thought would help (the name, the label, the category of difference of squares) was actually the problem.  I started to see how students were just memorizing a form which led to applying it in all the wrong places, rather than knowing the form inside and out because they had connected it to what we had just done: trinomial quadratics.

Without giving the label early on, students who try to factor these special cases must rely solely on their comprehension of the logic of the way difference of squares is factored.  No longer do they fall into the trap of thinking that x^2+16 can be factored.  And it really is beautiful to see.

Do the students deserve to know this pretty little binomial’s real name?  Sure.  But I would argue it’s worth waiting until students have the depth of understanding to appreciate why this special case gets its own name.

I would love to hear your reflections on mathematical vocabulary.  In what ways have you tightened up your own vocabulary to help students understand better?  Are there any cases where you’ve noticed the label/vocabulary has caused confusion rather than the intended clarity?  Comment below!

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3 thoughts on “What’s in a name?

  1. Paul Jorgens says:

    Wow! I just returned from a walk reflecting on a polynomials unit we completed yesterday. A common mathematical vocabulary is critical. Terms like polynomial, binomial, trinomial, greatest common factor, factor and product are foundational to the unit. Students should bring that vocabulary from previous experiences entering a study of polynomials.

    I think much opportunity is lost though when a polynomials unit is presented as several techniques independent of each other. Terms like differences of squares, perfect square trinomial and factor by grouping as well as a=1 (or not) can enter the classroom too soon. They can inhibit deeper understanding when taught as techniques independent of each other. It can prevent the opportunity to begin with shared informal experiences that promote access for all students. Starting with informal experience through algebra tiles or area models will not only result in greater access but also more opportunities to push the depth of knowledge.

    We did this desmos activity after students had some of those informal experiences multiplying polynomials and before any formal introduction to factoring.

    https://teacher.desmos.com/activitybuilder/custom/587647001bcf814405d19f6c

    On screen 3 they are asked to give a product of two binomials that results in a binomial. On screen 4 they are asked how they did it. A few responses follow.

    “I made sure there were only two terms in the final product because if you make two of the amounts add up to zero they will cancel out and you will have only two terms”

    “I made sure that there were only two terms in the final product because the two middle terms of my unsimplified four terms added up to zero.”

    “I used an area model to find that the middle terms add together to get 0.”

    “I created two terms that would cancel each other out. The full equation is 9x^2-6x+6x-4. The “-6x” and the “+6x” together equal zero, so those can be removed from the equation.”

    “The second number in the parenthesis must be opposites”

    By the end of the unit we had named the pattern that they had observed as a difference of two squares as well as naming other techniques. The early shared informal experiences with area combined with pushing the depth of knowledge led to virtually all students finding success on the summative assessment.

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  2. Ashley Henderson says:

    This is my exact teaching method of this unit!! Early on I taught the separate vocabulary terms as you stated and then struggled to get them to see they were still doing the exact same thing they had been doing when dealing with trinomials! It’s like giving it a special name made them think the solution method was completely different. I don’t call it anything special until we are well past the unit. I also don’t separate coefficients of a=1 from those with coefficients with a>1. I used to teach a=1 first then move to a>1. But again they acted like we were doing something entirely new. Now we do them all as a mix on the first day.

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