I’m exhausted, so I’m going to make this short and sweet! I just want to archive it.

I think this was a good teacher move in my precalculus class today. It wasn’t so much about the question, as much as the question and what I did afterwards. Background: this is advanced precalculus, and our third day of class.

I was working on binomials expansions with my kids… And we got to a question like .

When walking around, I saw a number of students write initially for their first term. Most of them, when talking with their groups, recognized their error, and changed it to .

My question: “How many of you initially wrote for the first term? Even if you changed it?”

Two hands hesitantly went up.

I threw them each a piece of candy.

Then four or five more hands. I threw them each a piece of candy.

I told them that I love that they made that mistake because it is an opportunity to learn, and perhaps making that mistake and recognizing it will help you remember! And I love that you weren’t ashamed to share something you did wrong. Because you shouldn’t be ashamed. You’re learning. *We’re* learning.

(Okay, so I might have written that a tad more eloquently and dramatically than what I actually said. C’est la vie.)

I’m trying to build a classroom culture that promotes risk taking and sharing of not-fully-formed ideas. And I loved that this on-the-spot teacher move actually helped do that.

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A short blip in similar spirit (though sadly candy-free):

Today I gave a warm-up problem of,

Solve for A and B in 2AB_8 = 136.(On the left-hand side: Those are digits 2, A, and B, and the number is in base 8.)

I managed to get multiple answers; most students expanded 2AB_8 as:

2×8^2 + Ax8^1 + Bx8^0 = 128 + 8A + B.

Setting this equal to 136, they arrived at:

8A + B = 8.

But then there was a bit of talk about whether there were multiple solutions. (Could there be?!)

The consensus was A = 1, B = 0 gave a solution: 210_8 = 136.

Another student suggested we could

alsohave A = 0, B = 8; so: 208_8 = 136.Her classmates talked her out of it (“what digits are allowed in base 8?” they asked) and she was quick to grasp the error. But when she said

no, never mindabout her suggestion, I was prepared to remark that it was a very reasonable mistake — and in fact I thought it worth discussing even if it didn’t arise organically, and had written it down just in case!No candy, but at that point I handed her a piece of paper with “208_8” already written down on it. (Probably there is a thin line here between “gotcha!” and “thank you for helping to reinforce some of the key concepts with your openness to voice something possibly incorrect”; I sure hope it tilted towards the latter…)

If this interaction seems reasonable to you, then surely you could do the same the next go-round with “3x^2” etc.

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