Do kids need help learning how to ask interesting questions? (Continued)

"Let's call out the elephant in the room: we're in math class. Chances are extremely high that mathematical questions are preferred. I think this should be discussed with students AND modeled by the teacher." (-Andrew)

"I think kids think it is reasonable to focus on math in math class, especially if they believe us when we tell them it's useful and lets us find out cool things." (-Julie)

"So when we ask for questions, we honor all the questions. We're grateful for all the questions. Then we model how mathematicians think, how mathematicians ask questions, and we ask our OWN questions that we know lead towards productive mathematical goals." (-Dan)

I was grateful for the comments on this earlier post. It seems to me that while many of you agree that mathematical questioning ought to be taught, we're caught between two different teaching strategies.

Strategy 1: Implicit Instruction

Honor every question or curiosity. Get excited by what the kids are excited about. But then always pick up on the questions that are mathematically productive. Say things like "We'll try to answer as many of these as we can." and "I love all these questions, but today we're going to focus on..." Say these on many days.

Show that you're excited and curious about your own question. Show that you have questions that animate and excite you too, and they just happen to take us to interesting mathematical places.

The kids will pick up on this, if not at first then with time. They'll want their questions to lead to the interesting mathematical discussions. If only mathematically productive questions lead to prime-time, they'll learn to ask mathematically productive questions. They'll be standing at the edge of a culture, peering in and trying to fit-in. It'll take time, but it'll stick.

Strategy 2: Implicit Instruction + Explicit Instruction

Help kids see things that they'd struggle to notice. Speed along their learning process by pointing out aspects of questions that make them mathematically purposeful. Say things like "What interesting mathematical questions could we ask?" or "I love questions that aren't just a matter of opinion."

Honor questions, yes, but also use praise, questioning and advice to help guide students towards productive lines of questioning: "I love 'How many?' questions."

Asking mathematically productive questions is hard. If we rely on implicit instruction then maybe some of our students will learn to ask great questions, but it's unlikely that all will. And if we want equitable outcomes, shouldn't we care about helping all students get there?

Explicit instruction can be done well and need not involve being dismissive of a student's natural curiosity. We use explicit instruction to teach all sorts of concepts, skills, strategies and practices. Why not use it to help students ask productive questions as well?

Questions:

1. When is implicit instruction preferred over a mix of implicit and explicit strategies? Why?
2. Is "modeling" a form of "implicit instruction"? Are they synonyms? Are there are other forms of implicit instruction?
3. The concern with explicit instruction seems to be that any explicit instruction on questioning would necessarily show that natural curiosity isn't valued by the teacher. But implicit teaching techniques aim to alter student questioning, anyway. Is the difference whether students notice that they're being guided? If not, is there some other difference?
4. Is there necessarily a trade-off between equity and discovery in teaching?