Teaching to the negative space

I’ve been troubled ever since David Cox asked why students needed to learn the names of the algebraic properties (associative, identity, and so forth). Certainly I see misunderstandings: students confronted with

(3 + x) + 2

not aware they can combine the 3 and 2 with associative and commutative properties, or the classic


leading the student to cancel the x terms rather than consider the distributive property. (*)

Clearly these things are being taught, but what’s going awry when they are used in practice? And why do students learn the names of these things?

It struck me that students only get taught the definition in a positive sense, memorizing (for example) that

a + b = b + a

and identifying 2 + 4 = 4 + 2 or (2 + x) + 3 = 3 + (2 + x) as the commutative property, and leaving things at that.

From my experience students can eventually identify a particular sequence of symbols with particular name but with no transference to actual practice. I’ve seen books address this by forcing students to laboriously label the property with each step of an algebra problem, but even after such an activity the knowledge is compartmentalized and then forgotten.

What I believe is missing is for students to know when not to use a property, that is, the “negative space” of the definitions. The use of the distributive property, for example, is often done with pattern matching other than understanding, giving results like:

2 \cdot \sqrt{x + 3}
\sqrt{2x + 6}

Rather than labeling each algebraic manipulation in a problem, I’d suggest students correct erroneous algebra while stating which rules are being used wrongly.

I feel the negative space gives sufficient justification for learning the properties of names, so it’s possible to say that “here you can’t use the associative law like you’re trying to” rather than trying to redefine the positive space just to talk about the negative space, or worse, establish as a specific rule “you can’t do that” individually for each type of mistake. It’s easier to use the positive definition as a tool to anchor the negative examples as a single category.

(*) I don’t understand why it’s rarely made explicit that the distributive law includes division. I have yet to find a textbook that does this. While a can be a fraction in a(b+c), most students do not recognize this on their own.

14 Responses

  1. The properties are a 7th grade standard in CA and we rarely go beyond identifying the property used. Occasionally you’ll see something where the commutative property is disguised as the associative property (eg. (2+3) + 4 = (3 + 2) + 4) because everything with parentheses is associative, right?

    I’ve been using the properties this year to talk about mental math and students are using it to give vocabulary to why we can do what we do.

    But I love the negative space idea. You nailed it, thanks. That’s exactly what I needed to make these things sing. I guess it’s kinda like the #needaredstamp in practice.

    • Occasionally you’ll see something where the commutative property is disguised as the associative property (eg. (2+3) + 4 = (3 + 2) + 4) because everything with parentheses is associative, right?

      I sometimes feel like this was solely designed to mess students up on standardized tests. It’s a valid question, but I feel like the energy I have to spend for students to get this one example right could be better spent elsewhere.

      • I think that the question is an OK one, as it tests understanding plus naming rather than rote memorization. I would much rather that the naming component was dropped entirely though, and only the understanding was tested.

  2. I learned somewhere in grad school that young kids learn definitions by seeing examples AND non-examples. I always try to do that when I introduce new terms. For example, if you show a kid a plant and lettuce and say “green”, the kid thinks they know what green means but they might be misconstruing it. Showing them carrots and green vs. red apples, and green vs. blue shirt, would help clarify the definition for them.

    • Any memory where I could find the reference?

      Pairing examples with non-examples is a popular trick in geometry classes (I believe I first saw it in the Discovering Geometry textbook) but not as common elsewhere.

      What I’m talking about here is slightly different than the geometry use. Students won’t be confusing a triangle with a quadrilateral; the non-example in geometry is to make sure the students have every part of the definition correct. Here I’m suggesting that students look at the particular places where they will apply the definition incorrectly.

  3. In regards to the “distributive law includ[ing] division,” I’ve wondered if it goes back to students mindlessly learning PEMDAS and the silly mnemonic devices that go with it. I think students focus too much on P=parentheses, instead of realizing the importance and structure of various groupings, including numerators and denominators consisting of expressions of multiple terms.

    • PEMDAS is first taught before expressions with fraction bars, and it seems when they get introduced (and square roots, and absolute values, and etc.) teachers never go back and modify the P.

  4. In Australia, or at least my state, we don’t teach the names of these properties and don’t focus on identifying their use, other than examples in the context of mental shortcuts. Of course, the misconceptions you mention are very common here, and the idea of focusing on negative examples is excellent.

  5. This is just my two cents, so please take it for what it’s worth.

    I find that my students need to do more than just memorize the names of properties — just as they need to do more to master a skill than just memorize an algorithm.

    They actually need practice in using terms in different contexts. Once I started insisting (gently) that students use the correct mathematical terms for things — and (unfortunately) once they started getting credit for it — their understanding of the properties themselves started improving. Dramatically.

    Names have power. Mathematical language has power. Most of my 9th graders have managed to get by using baby talk in their math classes. Now some of them are beginning to see that they’ve been cheated out of understanding by not being held to a higher standard.

    It’s true that a toddler doesn’t need to know that we call the downward property “gravity” in order to make use of it. But knowing that we call it “gravity” sure makes it easier to talk about once you’re trying to work with it abstractly.

    • Agree. I tell my kids to learn the terminology of ANY subject they are learning, and to create and annotate glossaries as they proceed through the subject matter. Mind maps work great for this.

      But this does not give them everything they need. Definitions are required, but then you need to show them examples, positive and negative, AND provide interpretation along the way. Jason’s idea of teaching to the negative space is a great idea.

      Let’s even go a step further and take a workshop approach to teaching math, the way we do with literature studies. Students DO need guided practice working through analysis of subject matter. They need the opportunity to challenge the ideas they are presented with, because deep understanding of new concepts is more like forging than acquiring.

      Great discussion.

  6. […] Teaching to the negative space […]

  7. […] Seeing this pattern as linear at first could be productive for the first goal, as the contrast between the linear and exponential model could make their differences seem even more stark. (See “Teaching to the Negative Space”) […]

  8. […] understanding of exponents. I’m stealing from Jason Dyer’s ideas here “teaching to the negative space” — that we need to teach what a concept isn’t, as well as what it is. The […]

  9. Reblogged this on Capture Their Interest and commented:
    Nice piece, Jason. Often our students will make enough mistakes for us to “capitalize” on the ‘negative space’. Really it’s a good reminder of the value of mistakes.

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