|Source: Illustrative Mathematics|
Last Year Pershan: "What does the next step look like? What would the 10th step look like?"
This Year Pershan: "What are some interesting questions we could ask about this?"
And here's what the kids came up with:
"What comes before [the first picture]?"
"How many cube blocks would you need to build this [third picture]?"
"What the next one going to look like?"
"I think it's going to be the power of 3."
"What would the _____ step look like?"
"How many squares could go in the 4th step?" (To clarify, he was asking how many squares could fit on the grid of the 4st step in the pattern. So that includes all squares, 1x1, or 2x2, etc.)
At the beginning of the school year, I said that I was going to help my students learn to pose their own mathematical problems. As you might predict, this failed repeatedly for me until it didn't, and here's what I've learned since then.
There's an enormous difference between "Any questions?" or "What do you wonder?" and "What's an interesting question we could ask?" There's value in all of these prompts, but when I started this year I wasn't attuned to their differences. "What do you wonder?" makes a play at natural curiosity, and comes off as not so different from "Are you wondering anything?" It's easy for a student to answer this sort of question in the negative, but it would be sort of awesome if they were in the habit of wondering about things all the time. "Any questions?" comes off as concern for the kids, not different from "Can I help?"
"What's an interesting question we could ask?" is different. It's less about natural curiosity and more about imagination and the way a student sees the mathematical world. Coming up with an interesting mathematical question is often difficult for kids, it can involve a great deal of creativity, and it gives me a window into their mathematical world-views.
I see this as a distinction between "question asking" and "problem posing."
As X gets more familiar, asking questions about X gets easier. This is a bit of a "duh," but at the beginning of the year I was trying to prompt kids to ask spin-off questions at the beginning of studying some object, scenario, or problem. This failed, and I think part of the reason why is that they weren't familiar enough with the sorts of questions we could ask (and answer) about these objects. As the year has gone on, and kids have seen what sorts of questions we ask in class, asking questions has gotten easier for them.
For a couple of kids, this can be a great way to get them to create their own extension problems. For a few -- maybe two or three? -- of my students, I always have to worry about them finishing some task quickly. Worry is the wrong word, maybe, but as I'm planning I'm always thinking about whether I have enough interesting stuff for them to think about. Once we got in the habit of posing our own problems, though, it's been nice to share the responsibility of coming up with an interesting question with the kid. We look for an interesting follow-up problem together, and it feels like a truer collaboration with a student than most anything else that I do.
This has worked better with 4th Grade than with high school, though it's also worked in Precalculus a bunch. I just feel like I should be upfront about this.