Monday, February 17, 2014

Setting A Trap

What sorts of activities do you do with kids to help them avoid common mistakes?

There's obviously no easy fix for common mathematical mistakes. They're common for a reason. Helping kids avoid common math mistakes is synonymous with effective math teaching, and there's no recipe for teaching good.

OK, fine. But here's a move that I often make when I know that there's an especially nasty math mistake coming up.

Example: cos(a+b) = cos(a) + cos(b)

1. I ask kids what they think cos(a + b) is.
2. We get the wrong answer on the table, and we give it a name. In this case, I name it "The Distributive Property for Cosine."
3. I get kids to argue about the wrong answer until it's been disproved.
4. We give language to the truth: "The Distributive Property Isn't True for Cosine," and "Cosine Isn't Linear."
5. We practice the true thing a bunch in the next few days.

Trying to provoke interesting conversations amongst the childrens is a pretty big part of what I try to do. I also think it's important not to tip kids to the fact that you're setting a trap for them, so I try to hide my hand. (In this case, cos(a+b) = cos(a) + cos(b) was included in a long list of Always, Sometimes, Never problems.) I think that giving kids language to talk about this debate is absolutely crucial to them remembering it, so it needs to happen.

This is a go-to pedagogical format for me.

(New Blogging Rule? I only blog if it'll take less than 10 minutes or more than 2 hours to write the post?)


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  2. Michael

    I just wrote about this mistake over at my space yesterday. What I find with this mistake (and another distributive favorite (x + y)^n = x^n + y^n) is that most of my students acknowledge that this is not true when asked directly. However, in the face of not being sure of what to do, they will resort to these incorrect temptations. Helping them navigate their way past these treacherous mistakes is one of our primary challenges as teachers.

  3. I've been fascinated with a similar question, which is actually pretty deep and complex:

    when is the common misconception a/b + c/d = (a+b)/(c+d)?

    Try it!