[00:00:01.470] - Pam
Hey, fellow math-ers! Welcome to the podcast where Math is Figure-Out-Able! I'm Pam, a former thinker turned math-er.

[00:00:10.360] - Kim
And I'm Kim, a reasoner who now knows how to share her thinking with others. At Math is Figure-Out-Able, we are on a mission to improve math teaching.

[00:00:19.990] - Pam
And something else is happening.

[00:00:21.700] - Kim
Sorry.

[00:00:22.310] - Pam
We know that algorithms are amazing historic achievements, but they are terrible teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop.

[00:00:35.300] - Kim
In this podcast, we help you teach math-ing, building relationships with your students, and grappling with mathematical relationships.

[00:00:41.690] - Pam
And we love it that you're here to join us to make math more Figure-Out-Able. Kim, you okay over there?

[00:00:46.660] - Kim
Yeah, I'm good. I'm solid. I'm solid. Okay, so last week we started talking about explicit teaching and highlighted that there are two types of content in mathematics. Some of which is social knowledge, social-conventional. And some which is logical-mathematical. And that's the bigger category. And so, we talked about that we also are explicit in the classroom and said we'd share a little bit more about that today.

[00:01:16.170] - Pam
Yeah. And I probably should have last week given credit to Jean Piaget, who is the wonderful Renaissance man who did lots of work, some of which was in educational psychology. And he's the one that came up with logical-mathematical and social knowledge. Those two types of knowledge. So, we're borrowing his language to help us really get clear about what we think  excellent math teaching looks like.

[00:01:39.610] - Kim
Yeah.

[00:01:40.510] - Pam
So, last episode we defined excellent math teaching as, in part, having explicit goals, very intentional, explicit goals about both the math content and mathematical behavior. What it means to behave like a math-y person, a mathematician. Intentional teacher moves that are helping kids get high enough dosed on all the things and helping kids really learn to math. Clear vision. Teachers having a clear vision on the different types of knowledge, so they know when to tell kids the social-conventional stuff and they know when to involve kids in experiences and high-dose patterning of the logical- mathematical stuff. Kim.

[00:02:22.520] - Kim
Yep.

[00:02:23.100] - Pam
What does that look like in a classroom?

[00:02:25.270] - Kim
Okay, so we gave an example last week of a time where I said to a student, "Write that down." So, that has everything to do with me being direct about a particular behavior, a classroom management type thing. And then we also gave an example of me lobbing out, "Sure wish I knew something about quarters." And that was not as explicit. Meaning, I didn't direct them to tell them how they were supposed to do it. So, we brainstormed.

[00:02:55.590] - Pam
Ooh, ooh. But... Sorry, sorry.

[00:02:57.120] - Kim
Yeah.

[00:02:57.690] - Pam
But you were explicitly intentionally lobbing that particular thing out.

[00:03:02.580] - Kim
In that particular moment, yeah.

[00:03:03.940] - Pam
In that particular moment. It wasn't haphazard. It wasn't a guess. You knew the landscape. And so, you had an explicit goal. And part of your goal was, "Ooh, when can I get that in?" But you didn't directly say, "Go do this within these steps."

[00:03:17.600] - Kim
Yeah, I'm going to go back to your definition. There's different types of knowledge, and I knew that what they were working on was logical- mathematical. So I... And I was clear on my goal. I wanted to see ways that students would solve it, and I was interested to know who might use quarters. So, I had that intentional move to see if that could come out in our classroom.

[00:03:41.140] - Pam
Yeah. Yeah. Very intentional.

[00:03:42.980] - Kim
Okay. So, we think that there is at least a few categories of things that we are explicit about. You want to start with the first one?

[00:03:50.690] - Pam
Yeah, because if we can get clear what we're explicit about, then maybe we can help this unfortunate argument just die down because people can go, "Huh." If there are these two types of things happening, two types of knowledge, then if we can get clear about when to be explicit, then the rest of the time we are looking for really good tasks to help give kids a high enough dose of all the patterns. And blah, blah, blah. Alright, so one of the categories that we are super explicit about. So, different categories. One of the categories is mathematical content goals. So, kind of like I just said. You were super explicit that you wanted to see if quarters could be a strategy, a relationship that kids were using. So, another way of talking about mathematical content goals, the standards. So, add and subtract within 1,000, round up to millions, graph quadratic functions. Like, whatever the standards are, those mathematical goals, those are explicit goals. They are things that we are supposed to intentionally help our students develop understanding around. We add to a lot of the sets of... We don't have all the sets of standards memorized that are all around the states or all around the world.

But we add some other things that are mathematical content goals. We've had some episodes on specific fluencies. Like, we think it's super important that teachers explicitly help kids... Explicitly help. Explicitly have the goal that kids will get good at any number plus or minus 10. Like, that's important. They need to throw out a number and they can really think about the place value by adding or subtracting 10 from that number, multiplying or dividing that number by 10. And when I say that, I actually kind of have in mind plus and minus 10, plus and minus 100, plus and minus 1,000, times 10, multiply and divide by 10, multiply and divide by 100, multiply and divide by 1,000. Like all of those things are important fluencies that we want to intentionally develop in kids. Another example would be the 5 interpretations of rational numbers. We want kids to know that we can think about rational numbers as part-whole, but we can also think about them as a measurement context. Three-fourths is three 1/4s. We can think about it as the ratio of 3 to 4. We can think about it as an operator, three-fourths of something. We can also think about it as division, 3 divided by 4. Those five sort of interpretations. Thank you, Susan Lamon. Another example would be parent function. Short and long run behavior. Like, we need kids to... When I say something about exponential behavior, we need to have a feel for, you know, what does that look like and feel like. And I'll tell you what, Kim. That's one of the fluencies I see when I work with secondary teachers that we have not helped, we have not been good at as a discipline to make sure that we all own those. And here's where it comes out. I'll be doing a specific Problem String that I do with quadratics, and we start talking about long-run behavior of a quadratic, and so as we're going x to the right, what's happening with this very specific parabola? And teachers will often say, "Well, it's growing exponentially." And I'll say, "I think it's growing quadratically, polynomically, with a power of 2. It's not really a word. But it's not growing exponentially. Exponential behavior is very specific to exponential functions, not polynomial functions of degree 2. Anyway, so that's an area where, you know, maybe that's social that they're using a word that they don't quite understand. But the logical part of it. Anyway, short and long-run behavior of parent functions. That's another example of a mathematical content goal where the goal with all of these mathematical content things that I just mentioned, the goal is not in the air. It's not foggy. It's not ethereal. It's not nebulous. The goal's detailed, and specific, and, dare I say, explicit. Those goals are going to be very intentional about making sure that kids own these content goals.

[00:07:47.810] - Kim
And for many teachers, those goals are named for them by their state or by their country. Like, they're told. But for us, it's that they're aware of these goals, but not necessarily explicit about how you tell them how to do the things. So, you mentioned add and subtract within 1,000. We are clear. We are explicit that that is a goal we have in mind. But I'm not necessarily telling kids, "This is the way that you solve within 1,000. This is the one way."

[00:08:18.080] - Pam
These are the steps that you use to... Yeah. So...

[00:08:20.280] - Kim
Yeah.

[00:08:20.980] - Pam
We can have an explicit goal, but that doesn't mean that now to get kids adding and subtracting 1,000, you tell them, you treat that like social knowledge, and you tell them the steps to do it.

[00:08:30.780] - Kim
Yeah.

[00:08:31.380] - Pam
Not that.

[00:08:32.000] - Kim
So, as a teacher, I am explicit about what the goals are for that year. If I'm teaching 3rd grade, I know the goals, mathematical content goals for the year. But then I can scale that back and say for this 9 weeks, for this week, for this day, I have a goal for a particular lesson that's going to help me reach the goal for the year. And it may not be, you know, nuggeted down to a particular day, but you're heading towards marching those major goals out.

[00:08:59.670] - Pam
And those goals are explicit. So, there is a place where we're very intentional. We're going to meet those goals. What's another one, Kim?

[00:09:07.690] - Kim
Yeah, another category is that we have explicit goals about mathematical behaviors. So, we would be explicit about what it can look like to represent our thinking, so that others can understand our ideas. So, when we're focusing on attending to position, we would talk about particular models that we would all use, so that everybody can understand our thinking. So, you know, you and I have particular models that we think are hugely important at different grade levels. And so, I might say to students, "I'm going to represent your thinking on this particular model." And I am going to be very clear about if I'm using an area model, I'm going to name out loud, "And this side's 14, so it's going to be a little bit longer than this side that's a 10." Now, I'm not necessarily like forcing all my students to go back and redraw if they're not, but I'm naming out loud that I'm going to be proportional. I'm saying it out loud. I'm explicitly describing what I'm doing because I do think that that proportionality is important. So, I'm naming. We're using an area model. I'm naming that I'm drawing it proportionally. I'm naming that the area's going to go on the inside, and that the two dimensions are on the outside. Or that the product's on the inside and the two factors are on the outside. These are things that I need to tell students, so that we can all understand each other's thinking when we are modeling our ideas.

[00:10:42.680] - Pam
Nice, nice. So, in that case, somebody might have listened to last week's podcast and said, "Well, so, Kim, is this example of you saying write that down?" You know like, telling kids what to do? You're telling them some things, right? You're sort of speaking out loud. You're thinking out loud of what you're doing intentionally. And especially the social parts of it. Like, "This is the length of that." Is that the right way to say that? When we have a multiplication problem that's a 10 by 12. Or you said 12 by 10. Then we usually say the rows, the number of rows first and the number of columns second. That's social. So, we usually say it in that order. Yeah, maybe I don't even need to (unclear)

[00:11:23.950] - Kim
Well, I can tell you as a third grade teacher, if I had students who were new to representing their thinking on number lines, and every single jump looked exactly the same length, I was explicit to say, "Hey y'all, when we have jumps on a number line, if I have a jump of 10, we're going to represent that a little bit longer than a jump of 5. And it's going to be a bit longer than a jump of 1." Now, again, I didn't like grade on jumps. But I was explicit to say 10's value is longer than a jump of 1.

[00:11:56.140] - Pam
Yeah, yeah. And twice as long as a jump of 5.

[00:11:59.430] - Kim
Yeah.

[00:11:59.690] - Pam
Yeah. nice.

[00:12:00.380] - Kim
Yeah.

[00:12:00.490] - Pam
Nice, nice, nice. What's another mathematical behavior goal that you might have as an explicit goal and be sort of explicit about with kids?

[00:12:08.180] - Kim
Yeah. I think another one is how we persevere when solving problems. So, it's not demanding. You're going to solve the hardest problems that you can find on your paper. But I'm encouraging students to stretch themselves. And I think we model that. We model the disposition of grappling, and that it is okay to grapple, and that it is okay to sit with a problem for quite some time.

[00:12:31.950] - Pam
It's desirable even.

[00:12:32.960] - Kim
Right, right. So, we describe that we are persevering. We name it. We offer opportunities for that to happen. But we are explicit that math-ers wrestle with problems at times, and that it's helping us grow our brains.

[00:12:50.310] - Pam
And I could see you pointing it out when you see it happening as a, "Ooh, nice perseverance there. That's a mathematical behavior. Nice job."

[00:12:59.160] - Kim
Yeah.

[00:12:59.270] - Pam
And that is being explicit about the behavior that was happening to promote that kind of behavior continuing. Nice. I like it. Got any more?

[00:13:08.120] - Kim
Another one is just the idea that it's expected during a Problem String or a Rich Task, that you're looking for patterns, that you are trying to make use of structure and repeated reasoning. So, when we're delivering Problem Strings, we're naming, "Hey, we're looking for patterns here." When I introduce a Problem String, you know, sometimes people will say, "Well, how, what's the first thing you say?" We'll say, "We're going to do this thing called a Problem String. It's a series of problems. We're going to look for patterns and relationships that we can use to solve problems." Like, I'm telling you, you're looking for patterns and relationships.

[00:13:45.570] - Pam
Yep. Yep. So, you're being explicit about that goal during a Problem String.

[00:13:49.580] - Kim
Yep. Yeah.

[00:13:50.690] - Pam
Nice. Nice. It also comes up when people will ask me, "How do you get kids to stop using algorithms?" And I'll say, "Well, I don't tell them to stop. I let it happen. But then I very intentionally highlight and bring forward the thinking, the using relationships and reasoning that's happening. And then I will very intentionally say, "Did everybody understand Martha's strategy?" And when a kid goes, "I told you I did the algorithm." Then I very intentionally say, "Oh, that's okay. I didn't... Fine. But did you understand Martha's strategy?" "Well, I wasn't paying attention because I already got the answer with the algorithm." "Well, alright. Well, then go ahead and ask Martha. Do you want to ask her again? You weren't listening. That's okay. But you can ask her. Martha, would you mind saying it?" Martha says it again. And then I'm back to that kid. "Did you understand what Martha's strategy was?" So, I'm very intentionally setting the stage that what we do here is we listen to each other, and we understand each other's strategies. And that is a mathematical behavior that we want to have happen. Alright. So, we've had two categories we've talked about so far. We're very explicit about the mathematical content goals that we have, the mathematical behavior goals that we have. We are also super explicit about community behavior. Though, you might have been better at this than me. I don't know. But an example would be ways to interact with partners, so that you're sharing the conversation, that both of you are participating. That if you're going to do partner talk, especially depending on the grade level. I remember when I worked with Kit Golan in his classroom in New York. He was very explicitly helping kids realize how to work in partners. 

[00:15:25.000] - Kim
Yeah.

[00:15:25.640] - Pam
Yeah, I saw it happening as I was there.

[00:15:29.020] - Kim
Yeah.

[00:15:29.310] - Pam
Another example of community behavior is ways that we can show that we're listening or that we're thinking.

[00:15:35.000] - Kim
Yeah.

[00:15:35.530] - Pam
We're very explicit to say, "Give me a private signal when you're ready." And then often during presentations and in classrooms, we'll say, "But, you know, if today you don't want to share, you can give me this other signal, so I know you're ready." But, you know like, everybody can have a bad day. Everybody gets a chance to pass. Those are explicit, intentional ways of inviting kids to help us know they're ready and we can move on. And additionally, maybe they're having, you know, "Don't call me today." Now, if a kid can't pass all the time. We're going to have a conversation about that. But let's see, what's another community behavior? The fact that we can respect differences. Just even modeling. What we just talked about. You know, if a kid's doing an algorithm and a kid's doing a different strategy, we can respect the fact that they use different ways of solving the problem. And we might have two different strategies that come up that, you know like, "Hey, when might we use this strategy? When might we use that one?" Just in that very vein, of having the conversation about the fact that there were two different ways to do it. We're going to now discuss the advantages and disadvantages of each, gives credence to, kids can create their own strategies. It gives credence to, you can think and reason, and we can talk about the way you were thinking and reasoning. And it gives credence to the fact that we are different and that that's okay, but that we can learn from each other, and that our ways might have advantages and disadvantages, and then now that we own both, we can choose. That's a huge way in helping kids explicitly learn that we can respect differences.

[00:17:10.980] - Kim
You know, I think that, you know, if we have kinder or first grade teachers listening right now, I think this is their wheelhouse, right? They have young kids coming in and they are very explicit. "This is when we do this. This is how we ask this. This is how we engage with this." And I think so much of our students as they get older, we forget that there is that explicit teaching that can happen to make our communities run better. I think we forget that... Like we assume knowledge on their part about how to interact with somebody different than you. We assume knowledge. Maybe not everybody. But I do think that this is an area where we are explicit to say, "This is what it looks like to show listening." And it can look a lot of different ways. But we bring up the conversation on purpose, very explicitly talking about it because then we are expecting listening to your peers.

[00:18:05.860] - Pam
Absolutely. And I think you do it with your own personality. And another example might be how to ask for clarification. Like, what are ways that you can make that happen? How can you gather support from others when you need it? A very specific thing that I did to kind of get at—I don't know. It was my way of sort of generating my classroom management—was I just literally said to my high school classes, "This is a positive class." And then I kind of let things arise and I would say, "Ah, you know like, that thing that just happened, and this is a positive class. Does that fit? How does that fit? Ooh, that. That thing right there. That was super. You just helped your.... I love how you just helped your partner there. I loved how you just went out of your way to go get that for somebody." Or "I love how you just went out of your way to explain what you were thinking because the person next to you was like, 'What?' Like, I love how you... " So, we could, you know, this is a positive class. Kids would curse, and it was fun because pretty quickly the rest of the class would go, "Hey, this is a positive class." And they would just, you know, kind of help me manage things. But I had to explicitly bring up instances. I mostly let them arise. I don't know if this is smart or not at a high school level. I would let them sort of arise and then kind of explicitly like, "Let's weigh that out. Does that fit here? Because this is a positive class."

[00:19:24.070] - Kim
Yeah, that's a nice example. So, sometimes words get in the way of meaning, and sometimes we think we know that we are talking about the same thing that the person across the table from us are talking about. And so, we'd like to invite everyone to give examples about what you mean when you use words like "explicit teaching". We just named some ways that we are explicit in our teaching, and we invite you to do the same, so that when you can communicate with each other, you're not getting lost in the words that maybe you have different definitions for.

[00:19:54.050] - Pam
Yeah. And maybe some of this argument that's out there with at least some educators can even go away when we actually understand what we mean instead of just kind of using these high-tension words. Y'all, in the next few episodes, we're going to be explicitly calling out when we tell and when we help students develop, and what's the difference between the two, and the percentage of each that's happening. And I think it's going to be super awesome. So, whatever you do, don't miss the next few episodes. Y'all, thanks for tuning in and teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutable.com. Let's keep spreading the word that Math is Figure-Out-Able.