As far as “moving beyond explanations,” I can totally agree that we should move beyond seeing explanations as the key component of math instruction. But it still seems to me that explanations are a pretty significant piece of what we do to help students learn math, and I have no reason to believe that it’s a particularly well-understood aspect of teaching.

(A trick that some teachers pull off is to avoid offering teacher explanations and instead creating situations where students provide explanations to each other. It seems to me that it’d be a mistake to say that this is moving beyond explanations.)

Teaching is a deeply interconnected activity. If you listen differently, you explain differently. So I think we can’t move beyond explanations, any more than we can move beyond classroom management. My experience has been that once I started listening and learning more about how kids think about a topic I’ve been able to more accurately understand what my students limitations of learning are. This allows me to point out and explain in new and (yes) exciting ways. And all this requires slightly differently classroom structures and management techniques, etc.

Teaching is one big knot, and I’m all for focusing on just an aspect of the work at a time. But if we’re moving “beyond” anything, I think we have to move “beyond” placing one aspect of teaching above any other. It’s all connected.

(related: Nothing Works)

Do we really have to move “beyond explaining”?

I think so. Not in the sense of “leaving behind,” but rather “progressing beyond.” In my early years, I relied pretty heavily on explaining. I figured that honing my skills here would be a (the?) major ingredient in boosting classroom success (as measured by student learning).

That being said, I appreciate your encouragement to think more carefully about what it takes to give a good explanation in math, and how explanation fits into a teacher’s overall skill set.

Side note: I think it’s time for me to learn more about CGI. What I actually know about it is pretty limited. Is Children’s Mathematics (Heinemann) the best place to begin?

I love Max’s distinction between listening for and listening to. So good.

I’m excited to see where an orientation towards listening takes you in your work as a curriculum designer.

Me too.

Are there principles for giving good mathematical explanations? Granted, there’s more to teaching than explanations (a lot more). Still, it doesn’t seem to me like the MTBoS is brimming with pieces thinking about what it takes to give a good explanation in math. In fact, I’ll bounce your question back — if you know of any good pieces on how to give good mathematical explanations, I would love to read them!

As far as listening goes, though: absolutely! It’s so important.

One thing that I’ve learned that listening isn’t just about “listening.” Part of what helps me listen more carefully to my students is understanding more about their thinking. The model for me, here, is Cognitively Guided Instruction. I think of CGI as a theory of how students think about arithmetic word problems.

Max Ray-Riek distinguishes between listening for an idea and listening to our students. In that sense, CGI helps me listen for different sorts of thinking, but that opens up space for me to listen to my students in more patient ways. I think it’s because having a mental model of student strategies helps me listen to those strategies, instead of just listening for correctness/incorrectness.

I’m excited to see where an orientation towards listening takes you in your work as a curriculum designer.

To make a connection to a previous post, I think these questions we ask students (which lead to them discussing and we as teacher listening) are easier to create with a high level of content area knowledge. I great lesson to me is where I design activities and discussions, and students work with one another and guide one another to the correct techniques and solutions.