In talking to math teachers about an Inquiry- , Project- , or Problem-Based approach, these are the following questions that come up most often.

1) *“*How am I supposed to cover all the standards using this approach?*“*

and,

2) “So, when do I *actually* teach?”

An attempt at the first question is reflected in the Great Inquiry-Based Curriculum Mapping Project, from a couple weeks ago.

The second question can be a bit loaded, especially when you move the emphasis from word to word (as I did, by emphasizing “*actually*“).

We can discuss the second question a bit more in-depth going forward, but I’d like to attempt to simplify the nebulousness of “inquiry-based instruction.” When teachers ask “when do I actually teach?” I think they’re asking when do they stand up in front of the class and demonstrate examples and processes? And is such a time ever appropriate for an inquiry-based classroom environment?

But before we get into the weeds of such a rich topic of discussion, let me posit this to you: I would suggest that the change from a “traditional” approach to an “inquiry-based” approach may be as simple as moving from this

to this:

(apologies for the computer science jargon)

Now, obviously there’s a lot more to it than just a couple diagrams, but the point is this: **instruction still happens**, but it simply happens **after** students have attempted a problem and within the context of a problem. Instead of saying “Today class, we’re learning about slope, here’s a lecture,” followed by a lecture, followed by a problem set, the practice is in some sense, **simply reversed**: “Today class, here’s a problem,” followed by instruction about,say, slope.

So yes: instruction is still useful and necessary for an inquiry-based environment. And I would also say yes: lecture or direct instruction is often a appropriate tool to transmit mathematical knowledge in an inquiry-based environment. (Although, I would warn against it’s overuse, lest it become the default mode of instruction.)

The deeper questions of **when** and **how** do I instruct is a bit more of a dance that I hope to at least partially address in the coming posts. But in the meantime, let me hazard a broad-brush answer at these.

**When**: after students have had a goodly amount of time to discuss the problem with each other, and at least begin to attempt a solution. Maybe at least 30 minutes?

**How**: it depends in part on the number of students struggling with the content. If every group is having difficulty even starting the problem, then a whole-class lecture may be appropriate. If half the class is struggling, maybe some share-out, gallery-walk, and/or group-student-exchanges may be appropriate (or better yet: Kate’s “Speed Dating” activity). If only a few students are unable to jump into the problem, a small workshop may be necessary, while groups discuss and assess their solutions.

But these are broad-brush, haphazard solutions to potentially a much bigger question. I’d love to begin aggregating and categorizing math scaffolding activities and to have a discussion about when they may be appropriate.

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I cannot overstate how impressed I am with this whole summary and thinking-out project. You are taking on a transformative project and are making steady, meaningful progress in overhauling your entire model of teaching — and I, for one, am extremely grateful. People don’t appreciate how hard it is to “chip away” at an enormous and overwhelming project like this, and I want to celebrate your work by yelling, “YAYAYAYAYAYAYAY!” here in the comments section.

I’ve been kinda lurking on your blog for the past several months, reading, digesting, thinking, learning, practicing, struggling. I am over the moon about your curriculum mapping process for integrating all the different CCSS pathways and standards of mathematical practice. It is exciting to see how you’ve woven in wonderful lessons from the Math Twitterblogosphere.

Thank you for doing this work and keep going!!!

– Elizabeth (@cheesemonkeysf)

Thanks for the kind words Elizabeth. I’m a huge fan of cheesemonkey wonders. 🙂

Yeah, I need to revamp or re-organize things, including this post. Now that we’ve got a library of complex, CCSS aligned tasks, I want to learn more about facilitation and facilitation techniques. Maybe it’s using in-class video? Maybe it’s sharing more tools? Maybe it’s doing something with the CCSS Math Practice standards? Maybe it’s making a timeline – sort of like I tried to do here? I’m not sure but it’s given me something to think about!

Thanks again Elizabeth – so THAT’S your name! 🙂

— Geoff