I got down pretty hard on inquiry in my last post, but I do think it presents some value. I’m still using my loose, initial definition of inquiry, as students figuring things out. Obviously this definition isn’t very rigorous, but I think it reflects the broad conceptions of inquiry that are floating around, and motivates an examination of what about inquiry, exactly, helps students learn.

**Generative Learning**

The research on generative learning is fascinating. There’s plenty unresolved or ambiguous, but there are a few key pieces that I really believe in, and that I think are embodied by inquiry. One popular one at the moment is the generation effect, cited in Make It Stick. The premise is that attempting to answer a problem, successfully or unsuccessfully, before being provided with a solution improves retention and understanding. My interpretation of the research is that casting about for a solution activates the reasoning centers of the brain (System 2, from Daniel Kahneman’s work), as well as activating the relevant schema that the new knowledge will be assimilated into. But feel free to check out Make It Stick’s references if you’d like to read further (first, second, third).

The second principle is that students must be actively making connections between ideas and creating their own interpretations of math in order to truly understand it. Lee, Lim & Grabowski write: “Only through learner’s generation of relationships and meaning themselves can knowledge be generated that is sustainable — this is the essential process of meaning making by the learner … A variety of studies reporting on results of generative strategies have shown that, in most cases, active learner involvement produced increased gains in recall, comprehension, and higher order thinking or improvement in self-regulated learning skill.” (111-112). This is not impossible in a teacher-directed classroom, but the values teachers tend to associate with inquiry — students doing the figuring out — facilitate exactly what is being described here. There are definitely still challenges, but this seems to me a concrete benefit of an inquiry approach to teaching.

**Beliefs About Math
**Jo Boaler’s norms for math class

These are the values that we want students to have, beliefs about math that are productive for their learning. Inquiry is neither necessary nor sufficient for facilitating these beliefs in students, but I think it’s pretty self-evident that a successful inquiry-based classroom sends these messages about what it means to learn math. I’ll spare the growth mindset spiel, but I’ve watched students figure things out –both students who have done well in past math classes, and students who have struggled. The joy and pride they take in that work underscores for me the power of an inquiry approach. Math class is not just about the content students learn, but about what they believe mathematics is all about, and these beliefs are worth investing time in.

**Learning Is Messy
**There’s a fascinating debate to be had about whether students in a class are more alike or more different. Let’s bypass the policy debate for now. My stance is that students are alike enough that, in the vast majority of classrooms, I can expect all students to grapple with the concepts we’re working on — whether they be fractions, functions, or derivatives. I understand there are exceptions, and that this requires great teaching and tasks that have a low floor and multiple access points. All of these differences mean that, while every student can access a new concept, they will all think about it in different ways, make different connections, and build their knowledge through a unique path. An inquiry approach, that allows for students to make their own sense of content, acknowledges this reality. Learning is messy, and starts and ends at a different place for each learner. This isn’t an argument for infinite differentiation, or tracking, or any structural change — just an acknowledgement that, if learning is owned by the student, and the student is doing the figuring, we do a better job of meeting that student where she or he is at.

**Some Challenges**

These aren’t answers to the question of how to teach. At best, they’re principles that I can use to gut-check a lesson or idea and see if it matches my values. And, in practice, they provide some pretty significant challenges. If I’m going to move away from the “guess what’s in my head” approach to inquiry, I’m introducing a huge amount of uncertainty into my classroom. Where will we go next? Not knowing is a scary idea. Going through with that approach is a big step, and requires a flexibility in teaching and planning that is hard, but acknowledges the enormous difficulty in getting a room of students to engage with mathematical ideas.

In the research I cited above, the writers acknowledged that not all studies showed positive effects of generative learning. I think this is critical to acknowledge. An inquiry approach, even with the best methods, likely increases the variance in whether students learn — it could be fantastically successful, or it could fall flat. Explicit instruction is surely a safer route. This tension, as well as the tension between what is best for strong students and what is best for weak students, is one I want to try and find some answers for.

**An Example**

I’m always struck by this video of Jo Boaler teaching a 6th grade class:

This is pure inquiry. Jo does no instruction, and the students do some fascinating thinking. But it’s important to acknowledge (and this is true of other teaching demonstrations of hers) that there’s no content objective students reach in this lesson. This is pretty scary. Maybe some teachers will throw this in as an addition to the curriculum, but imagine using a set of lessons like this to teach key concepts. That scares me. But I have no doubt that, while any content objective I might ascribe to this lesson is well below 6th grade, the students in the room did some significant mathematical thinking and mathematical learning. What would a classroom that embraced these ideas every day look like? I have trouble imagining it.

Martin Joyce (@martinsean)That is the one part of Number talks that seems intimidating, is that where in the content objective? I think part of it is the culture and expectations it builds. There are multiple ways to see it giving multiple students chances to share how they see the dots. I’m reading Making Number Talks Matter right now and it has grade level aligned talks. At the beginning they show an example of a number dot talk in a geometry class where the students bring up lines of symmetry, hexagon, and more academic vocabulary. So, I’m hopeful it will stretch my students, build up confidence, and call on people with their thumbs up ready to explain their solution. I also saw a graphic on twitter about hand signals during a number talk and will try to implement that.

dkane47Post authorI like that example of a pretty high-ceiling number talk in geometry. That’s gotta be a huge part of it. Similar to you, I’m pretty far from trusting these strategies to actually teach new content, but I’m thinking more and more about how that happens and what that classroom would look like.

Joe SchwartzDylan,

I love the work you are doing, Thanks for sharing your thoughts and insights, and for pointing me in a direction where I can find out more about inquiry.

I would agree with Martin that what Jo Boaler is up to in the clip is building culture and expectations. In my mind it’s not a lesson, not in any formal sense. I would call it an activity, and I do dot card number talks like this in the primary grades. There, in addition to building culture and expectations, it promotes fact fluency, an important content standard. You can do them as a warm-up, or at the close of a lesson, or even at the end of the day while the kids are waiting for their buses to be called.

I’m interested in your statement, “Explicit instruction is surely a safer route.” For the teacher, yes. I’ll admit that I’m coming from personal experience, but all I got as a student was explicit, direct, safe math instruction. I learned very little except how to hate math, and I don’t believe I’m alone.

Keep experimenting and writing about what happens as you work out these ideas in your classroom. Inquiring minds want to know.

dkane47Post authorThanks, Joe. You used the word activity — that’s one I’ve had trouble nailing down recently. I used to think that activities were fluffy and meaningless, and call things “tasks” instead. I’ve swung in the opposite direction recently, and think of activities as something kids get absorbed in that has mathematical meaning beyond a single problem. I’d love to think more about how those distinctions can be useful in planning.

Thanks for sharing your experience — I particularly enjoyed your most recent post which tread on similar ground. I will keep experimenting, and I’m looking forward to putting some of these ideas to work with my students. I’m looking forward to it, and hopefully hearing about how you do the same.

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