We're often in the position of talking to teachers who have heard about the materials and are evaluating them, or whose district has adopted them and they are just learning about them. I'm putting together a FAQ for people in our organization so they are prepared for questions we know they will get. I am thinking to hash some of the Q's out in blog form, first. So theoretically this one in the first in a series. If you want to fight with me on anything I have to say, please speak up!

Imagine this scenario: you demonstrate a problem-based activity with a group of teachers. You let them know that this is a grade 6 task where students have already learned to use double number lines and tables to represent a set of equivalent ratios. By this point, students are also familiar with recipe contexts; they know that an equivalent ratio of a recipe tastes the same. Here is the task:

Lin and Noah each have their own recipe for making sparkling orange juice.

- Lin mixes 3 liters of orange juice with 4 liters of soda water.
- Noah mixes 4 liters of orange juice with 5 liters of soda water.

How do the two mixtures compare in taste? Explain your reasoning.

The task is launched with a notice and wonder, they start happily working away, and you monitor what they are doing. You invite a few of them to make their reasoning visible to everyone, deliberately selecting them to share in a way that highlights a particular nuance you want to make sure everyone will understand, making mathematical connections between their approaches. (If you're savvy, you'll recognize this structure as Smith and Stein's 5 Practices, though my short description here isn't really doing it justice.) After conducting this discussion, many voices have contributed. Earlier in the day, you did another activity that loosely followed this same structure. You think, hey, I've done a pretty good job demonstrating the basics of how a problem-based classroom is meant to operate.

Then you get the question, maybe timid but very curious, "So, when do I

*teach*?"

So here is a response that I'm turning over.

Can you say a little more about what it looks like when you teach, as it looks in your mind, here? Okay, it sounds like synonyms for what you are describing might betellingorexplaining. Is that fair? Okay. It's expected that you'll do some telling and explaining when using our stuff as it's meant to be used. The difference is in the timing. Let's think about what we did in the sparkling orange juice activity. You had a chance to work on a task, a few people shared their approaches, and then we made some observations about their approaches. What do you think the mathematical learning goal of that activity was?

"Well, I remember seeing two sets of equivalent ratios represented with a double number line and with a table, and then so-and-so explained how she computed how much orange juice for 1 liter of soda water for both mixtures. It seemed like the point was that when you want to know which mixture tastes stronger, you need to create equivalent ratios so that one of the quantities is the same for each mixture. For example if orange juice to soda water is expressed as $15:20$ and $16:20$, you know that the second recipe tastes stronger."

Okay cool. Do you think you got out of that activity what was intended? Does that mean you learned something? Does that mean teaching happened?

There's still telling and explaining. Mathematical playtime is awesome, but a problem-based classroom is not just about mathematical playtime. We have clear learning goals for the course, each instructional unit, each lesson, and each activity.

The way it's different than you might be used to iswhenthe explaining happens. Perhaps you are used to first explaining something, and then kids do some work on the thing you just explained. In problem-based instruction, this is reversed. Kids have a chance to try and figure some stuff out first, you see what they come up with, and then after they've had a chance to get good and familiar with the context, the question being asked, the constraints, and they at least make some progress. . . then you take steps to make sure the relevant learning goals are made visible. Sometimes this part looks like explaining or telling.

I'd suggest that teaching is a really broad and complex set of skills and behaviors, and telling or explaining is just one of them, and that telling or explaining isn't the only way to help kids understand something. In fact, does that approach work well for every student? How much do your students remember of what you explained the next day, or the next week? If you're completely satisfied with how things are going, awesome, but I bet you're here because either you or someone in your school endeavored to look for ways of conducting a math class that might work better for more kids, so that things made sense to them and the learning stuck around.

Did I miss anything to address this particular question? (Please note that this is one vignette from two days of learning, and we spend time on a whole

*bunch*of other things as well.) Does any of that come across badly? I want to acknowledge the person's completely understandable discomfort but also not shy away from asserting that teaching and learning happen in a problem-based classroom, and that we did it this way because we think

*better*teaching and learning happen.