In an earlier post, I shared Michael Fenton’s scenario and categorized the responses he got on twitter.

There were at least seven distinct responses that teachers offered to Fenton’s prompt. Wow! This makes me think two things:

1. Fenton’s scenario was so thought-provoking that it yielded an amazing variety of responses.
2. How come there was so much disagreement about how to act in this scenario?

Part of the disagreement, I think, comes from what went unspoken in Fenton’s mistake. We didn’t know if this mistake was shouted on in a discussion or found on a piece of paper. We don’t know if this is one of those times when we can afford to have a one-on-one conversation with a kid in response to her mistake, or if our response will be scrawled on her paper and returned. Was this a common error, or an isolated mistake? Could our response be an activity for the class instead of a chat?

While one-on-one conversations are crucial in teaching, they are hard to talk about. By their nature, they’re improvisational and somewhat unstructured. I’d also argue that opportunities for one-on-one conversations can be rare, and they get rarer as the number of students in your class grows larger.

Revising the Scenario

So let’s add some details to Fenton’s scenario. This was a mistake in an Algebra 1 class. Smart kids, thoughtful teacher, but when she collects papers after an ungraded check-in she finds that about half her class made Fenton’s mistake. Oh no! She decides that she’s going to launch class the next day with a brief activity to help advance her kids’ thinking.

Her first idea is to try a string of equations. She has three different drafts. Which one would you choose, and why?

Equation String 1

Equation String 2

Equation String 3

Other Activities

Then, she has some other ideas. Maybe equation strings aren’t the right move? She comes up with three other activities: Working With Examples, Which One Doesn’t Belong and Connecting Representations.

Working With Examples

Connecting Representations

Which One Doesn’t Belong?

Commentary

The meta-question here is about the conversation. Can we have a conversation with so many options? I don’t know. I worry that maybe I should have just limited discussion to the equation strings.

What would do?

My first reaction is that I like the equation strings, because it most directly gets at the issue of overextending the zero-product property to other equations.

But what I really want to do is lay out a sequence of 3-4 activities that I could do in sequence to develop this idea for a class.

And do I know enough to answer that question? Wouldn’t that depend on the math that we’d already studied and the math that’s coming up next?

Do we learn anything from thinking about these questions?

• Howard Phillips

This is a case of multiplication in the sense of “times”, and the multiplication is an “AND”.
Consequently the product is “ANDed” before the rhs is considered.
To give the details, (x-2)(x-3) is (x-2) and (x-3), result 3 * 2, not 2.

• Michael Fenton

Michael, thanks for capturing/interpreting/summarizing all of these responses and adding your commentary. I figured the original prompt would spark *some* conversation, but I had no idea it would generate this much discussion. But therein lies the power of a good mistake. More generally, thanks for maintaining this website. I think my “mistake radar” has been tuned and strengthened more by what you’ve posted here than by anything else.

• So much grist for the mill here.

What is the teacher’s goal? I’ll add my own typical idea of one: that her students can use factoring to solve quadratics. Other possible include SMP3, symbolic fluency, representation, meaning of a root or solution, etc.

If the goal is to solve quadratics, I want my students to check their answers. Which they refuse to do like it’s against their religion. The closest to this goal is the second situation. The teacher move I like here is giving the answer. “This is wrong.” So what else is there to attend to but how do you know? If I have time for two lessons, I do the representation one first, providing more things to notice or think about when doing the symbolic.

What I wanted to know was how to make more misleading factoring like this. I think that would go to symbolic fluency. I think this would be hard for most students, and would need to think about how to equip students to try it. My long term goal would be a class that could tackle something like that without supporting.