What Do You Do When a Math Trick Gets to Class Before You Do?

It’s time for another Teacher Tune Up.

I was late to class the other day and I didn’t realize it.

I don’t mean that I was late to start the class. I mean that students had figured out a mathematical trick before I got there which meant they didn’t have the same need to understand the math which, unfortunately, was my goal for the class.

Floating in Groups is a lesson designed to help students understand one of the most counter-intuitive truths of mathematics: that multiplying two negative numbers together results in a positive number. It does that with a helpful metaphor and helpful feedback.

The metaphor is that multiplying two negative numbers is like removing several groups of anchors from a submarine, which makes the submarine go up. (“Removed” is the first negative number. “Anchors” is the second.) That’s a nice metaphor. Then we pair that metaphor with feedback designs that let students play with the metaphor, subtracting and adding groups of floats and anchors dynamically.

We spent the lesson developing that context and its connection to an important mathematical idea. I was feeling great. Then the lesson asked students a question:

Use the floats and anchors scenario to explain why it makes sense that (-2)(-4) is positive.

And I felt less great. Here’s video of the moment.

Students resorted to a mnemonic that was far removed from the context we’d been developing the whole class and also really, really fragile.

because 2 negatives make a positive

because a negative times a negative = positive

a negative plus a negative is a positive

because negatives make a positive

When a student says “because a negative times a negative = positive,” it’s possible their understanding of operations on numbers is robust but it’s very likely they’ll see a question like, “What is -2 - 6?” and answer +8 because they think that two negative numbers in any operation produce a positive number no matter what. (One student actually says that more or less explicitly: “a negative plus a negative is a positive.”)

In case it bears mentioning: none of the adults in the room had offered up any of those mnemonics. 

My response (which you can see in the video above) felt clumsy enough that I wanted to bring it back to y’all here. What do you do when a trick or mnemonic has already taken hold of a class? How do you help students strengthen a fragile understanding they believe is very strong?

2023 Feb 17

The commenters come through every time! One suggestion was to put a little extra pressure on the mnemonic by either asking “is it always, sometimes, or never true?” about statements like “Two negatives make a positive” and “A calculation involving adding always gives you a positive answer.” (Thanks, anne!)

Also other commenters suggesting keeping two numbers fixed, changing operations and signs around them, and asking, not for the result, but for the sign of the result. Like:

-2 -4 (-2)(-4) -2 +4

Thanks, Leandra, Marty, and DK!

2023 Feb 21

Dylan Kane has written up a useful response here, including this paragraph:

So instead of telling students “tricks are bad,” I propose we show students the limits of that trick, and help them understand why the trick stops working where it does. This might seem obvious. But there’s a huge difference between telling a student “don’t use FOIL, tricks are bad” and showing them its limits with a thoughtful set of problems.

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