Guillaume ParÃ©, in the very interesting comments of my last post where I urged us to reconsider mistakes:

I do agree with what is written, but I am still wondering what I’m supposed to do with that information and the student’s copy.

**T**: Oh this is so interesting! You’ve actually answered a *different* question correctly. Check this out:

**T**: How does that help you come up with an answer to the original question? Talk about it. I’ll be back.

That’s a *script* right there. It works for *any* incorrect answer. The script is all-purpose and all-weather but it has two challenging requirements:

- You have to actually believe that student ideas are interesting,Â especially ones that don’t correctly answer the question you were trying to ask.
- You have to identify the question the student answered correctly.

This is why I want to learn more math and more math and more math.

The more math I know, the more power I have not *just* to show off at parties but also to appreciate student ideas and to identify the different interesting questions they’re answering correctly.

Barbara Pearl, via email:

Can you write about it briefly again in a simpler way so I can try and understand it? When students make a mistake or answer something incorrectly, you want to …

I want to teach in a way that honors the specific student and also the general ways people learn.

So in any interaction with students, I need to a) understand the sense they’re making of mathematics, b) celebrate that sense, saying loudly “I *see you* making sense!”, and then c) help them *develop* that sense, connecting the question they answered correctly to a question they *haven’t yet* answered correctly.

Rachel, in the comments:

So, if we don’t call it a mistake, then what do we call it?

I don’t have any problem saying a student’s answer is incorrect, that they didn’t correctly answer the question I was trying to ask. But my favorite mathematical questions defy categories like “correct” and “incorrect” entirely:

- So how would you describe the pattern?
- What do you think will happen next?
- Would a table, equation, or graph be more useful to you here?
- How are you thinking about the question right now?
- What extra information do you think would be helpful?

How can you call *any* answer to those questions a mistake or incorrect? What would that even *mean*? Those descriptions feel inadequate next to the complexity of the mathematical ideas contained in those answers, which I interpret as a signal that I’m asking questions that *matter*.

**Featured Comment**

Denise Gaskins quoting WW Sawyer:

Most remarks made by children consist of correct ideas very badly expressed. A good teacher will be very wary of saying ‘No, that’s wrong.’ Rather, he will try to discover the correct idea behind the inadequate expression. This is one of the most important principles in the whole of the art of teaching.

“So, if we don’t call it a mistake, then what do we call it?”

THINKING

I find that I have to keep insisting that they restate the question in their own words. The culture of “right answer” is filled with shame and shaming, and students will try repeatedly to just give me the “correct” answer to the original question. But this is a missed opportunity for developing understanding, in my view.