*Learning To Think Mathematically*:

As teachers, we want to value every question that comes out of a kid's mouth. But not all questions are equally valued in math -- asking mathematically productive questions is a skill that successful math students learn.

Take this enormous badminton puck or whatever you call it:

You ask most students: "What do you wonder when looking at this picture?" Tell me if I'm wrong here, but I think that they'd respond like this:

- What is that?
- What's that thing in the back?
- Why did someone put it there?
- Is that a sculpture?
- What's it made out of?
- Where is this?

And so on. Compare that with the list of questions that a bunch of teachers came up with:

- How big would the racket need to be?
- How big would the person who could hit this thing be?
- What time is it?

With time and practice, the students will more closely resemble their teachers. They'll notice what questions get picked up in class for lengthy discussions. They'll compare their questions to the questions their teacher asks. Slowly, they'll get a nose for the sorts of questions that animate mathematicians.

Asking a good question in math isn't natural. It's something that people learn how to do.

Which leaves me with a big question: how important is it to play to students' natural curiosity? Should we follow Annie Fetter's lead and ask students what they wonder? Should we follow Dan and ask kids what the first question that pops into their head is?

On one hand, the case for natural curiosity seems strong. If we care about what engages kids, it's important to know what kids are interested in. If we care about assessment, we shouldn't want kids to hold back. And if we care about including everyone, there should be a low barrier to participation.

On the other hand, the ability to ask an interesting mathematical question is something that is learned, and it is important. And if something is important and capable of being learned, shouldn't we teach it?

**Questions:**

- Should we also be worried about teaching productive question-asking? Can we teach it effectively if we always play for our students' natural curiosity?
- From Max Ray's
*Powerful Problem Solving*: "Noticing and wondering is something students get better at over time." How do kids get better at being naturally curious? Are we changing what they're naturally curious about? - Are there times when we want kids to be naturally curious and times when we want them to be unnaturally curious? When would those times be?