Ep 213: Making Progress - Representing Your Students' Thinking

Pam  00:01

Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris, a former mimicker turned mather.

Kim  00:09

And I'm Kim Montague, 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.

Pam  00:18

We know that algorithms are amazing historic achievements, but they're not good teaching tools because mimicking step-by-step procedures can actually trap students into using less sophisticated reasoning than the problems are intended to develop, and it can trap students into thinking that math is a bunch of procedures to mimic and memorize. (unclear).

Kim  00:39

You just made your sentence even longer. 

Pam  00:41

I did. 

Kim  00:43

In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships.

Pam  00:50

We invite you to join us to make math more figure-out-able. Bam! Alright, Kim.

Kim  00:55

Hello.

Pam  00:56

Let's dive into today. Hey, how's it going?

Kim  00:58

It's good. How are you?

Pam  00:59

I'm a little stuffy today. Hopefully I don't sound too... No, it's fine. I don't know. Allergies. Allergies in Texas, you never know. You never know what's blooming, what's going, what's... Yeah. (unclear). (unclear).

Kim  01:11

I'm super, super jazzed about this series, where we have laid out. For the last two weeks, we have laid out the first two stages?

Pam  01:21

Places?

Kim  01:21

Parts of a journey, places. Yeah, we should come up with some better words for that.

Pam  01:25

Levels of experience. 

Kim  01:26

Yeah. So,

Pam  01:27

Yeah. 

Kim  01:27

in stage one when you join the Math is Figure-Out-Able movement, you are building your numeracy, you are making sense of mathematics, and you are thinking about ways to make it more reasonable. In stage two, you are thinking about the major models and strategies for each operation. So, what's next? When you can build your own numeracy and you know the major models and strategies, where too? 

Pam  01:55

Bam! 

Kim  01:55

And so stage three is all about listening to students and representing student thinking. 

Pam  02:04

Yeah.

Kim  02:05

So, this is so important to us because it's the connection between teacher and student to move math forward for each individual student. So, in this stage, you are listening to students thinking, and it's crucial, and it requires time and practice. So, just like we said, last week about not rushing through stage two, we would highly encourage you not to rush through stage three as well because this is so important. If you're a leader, you're working with teachers to build their confidence by embedding opportunities to play the role of both the teacher and the student in the interactions, where the sharing of strategies and modeling occurs. If you are a teacher working with your students, then you are diving on every opportunity to ask questions, so that you can hear what they're thinking, so that you can represent their thinking as well. 

Pam  02:56

Mmhm. And if I can just... I'm going to step outside again, and just kind of compare.

Kim  03:02

Yeah.

Pam  03:03

Stage one is kind of about mathematics.

Kim  03:05

Yep.

Pam  03:06

You really own your own. Stage two is mathematics for teaching. And then in stage three, it's mathematics for teaching with students. 

Kim  03:14

Yep.

Pam  03:14

Sort of bring in the students now, and like listening, eliciting strategies for students, and modeling representing their thinking. 

Kim  03:21

Yes.

Pam  03:22

That's where we are here.

Kim  03:23

So, if anybody hasn't like made the connection we're layering on, right? As we move through the stages, we're layering on a little bit more accountability, a little bit more depth, a little bit more to consider as you move through this stage. So, you're more equipped all along, and we're just giving you a little bit more to consider.

Pam  03:41

Yeah. If you didn't listen to stage one and stage two, we'd highly encourage you to go back and listen to those two. We just like zipped right through them. Today, we're really going to dive into this stage. Were eliciting and representing student thinking. And, Kim, I feel like I just cut you off (unclear)

Kim  03:55

No, no, no. Well, I was just going to say that, you know, we've been talking about stage one, stage two. And I don't think what we've said yet is that people join us. You mentioned in stage... Or in the... Sorry, stage one. In the first episode. That we meet people at all of these different stages every day in groups, and in workshops, and all the things that we do. It's not as if we meet everybody, and they're like, "Hi, I'm building my numeracy." People are all over the place, and so we're just kind of framing the progress that we think people make, so they can identify where they think they are. And we help them figure out where they are, so that they can move on to the next stage.

Pam  04:36

And in a particular area of mathematics. So, when you say we meet people that are all over the place, they're all over the place in different areas of mathematics. So, I might meet a person who, as I'm diving into proportional reasoning might sort of be at stage one where they're really building their own numeracy. Whereas, in multiplication and division, they're totally at stage five. Which we haven't even talked about yet. So, people are not only all over the place as in you know like, you are here and you are here. But a particular person probably is in different stages in different areas of mathematics.  Yeah. We're suggesting it can be helpful to recognize kind of where you are in a particular area of mathematics, so that you can know how to work in that area and progress to the next one. And it gives you choice. It gives you sort of agency to say... It's not just this sort of vast, you know like, crazy mathematics out there, and where do I start? Nah, like, we're giving you some action things that you can do to keep progressing to make math more figure-out-able for you in specific areas of mathematics.

Kim  05:38

Yeah.

Pam  05:38

Yeah.

Kim  05:38

Absolutely.

Pam  05:39

Cool.

Kim  05:39

Okay, so at stage three, we've identified some major milestones, right?  So, in this stage, you are asking and understanding how students use relationships to solve problems. So, you've built your own numeracy well enough or you're working on it still. But you are seeking to understand how students use relationships. So, they might not be as clear to be able to communicate it, right? Just like we aren't in the beginning as well. But you're seeking to understand and helping them figure out how to say what they're saying or how to say what they're thinking about.  Yep. 

Pam  06:18

And part of what makes that possible in this stage is that in the stage before...

Kim  06:24

Yeah.

Pam  06:24

...when you learned the major models and strategies you have those top of mind now. So, even as a student sort of haltingly kind of says these like things. Because you own the relationships, you own those major models and strategies, you're able to go, "Oh, I see the relationships you're using," and you're able to choose a model that fits that well. And so that's part of why this builds on that, right?

Kim  06:49

Right, right. 

Pam  06:50

Yeah. Super important to be asking not only your students, but everybody you meet. I mean, as much as you can. Like, having conversations about like, "How do you think about..." Throughout a problem. You know like, "99 plus 47. How do you think about that problem?" And then listen. Like, this is a stage where teachers really can practice the art of listening. Yeah, to hear.

Kim  07:14

Yeah, because it's definitely different to represent your own thinking than it is to hear what someone else says. Whether it's a student or another adult. Or teacher, if you're a leader. To hear what someone else is saying. And in that moment of hearing it going, "Okay, I have to think about what they said, and I haven't figured out how to put it on paper. But like you said, since you're building off of stage two, you are asking, understanding, and you're representing appropriately.

Pam  07:40

Yeah, so this idea of what does it mean to represent student thinking appropriately is a place where we kind of differ from some other really good people out there. There's some really good people out there, who would... How do I say this? Who I've seen videos of them, and I heard him talk, and I've read their books where they say, "So, a student says that I did an area model," and then the student will walk through like an Over strategy. And the teacher will then feel kind of obligated to use the area model or an open array to represent that Over strategy, even if the other the two strategies shared before were students, say, in a ratio table, or students, say, using equations or something. And the teacher kind of feels obligated to use the model that the student is saying they used when we are going to suggest that actually this is part of the mathematics for teaching with students is that the teacher in that moment gets to go, "Let me see if I can figure out the relationships you're using. And now let me choose a model that will represent that, that will make it visible in such a way that now it's comparable to other students thinking, that we can now have a conversation about how these two strategies are related, how they're the same, how they're different, how students were using relationships. We've seen... And we're not recommending this, ya'll. But we've seen teachers, where they'll do some sort of, "You know, how did you do it?" And the student says, "Well, I did it this way." And they put the relationships on the board the way the students said to. And then the second student says the exact same relationships, but using a different model, and so then the teacher puts that on the board as if it's a different strategy and just sort of plays, "Oh, okay. So, we've got Kim's strategy over here and Pam's strategy over here." When in reality, it was the same strategy. So, that's part of the work that needed to have happened in stage two is what's the difference between a model and a strategy, and being clear on the relationships enough to be able to hear students, and go, "What in this case would be a great model to use to make this thinking visible in such a way that we can progress towards our goal? So, Kim, I can actually foresee a case where a teacher might say, "Oh, this kid said that they did this thing in a ratio table. And this kid said that they did the exact same kind of relationships but on an open array. As a teacher, I might in that moment put them both up and then go, "What do you guys think? Same strategy? Different strategy?" Like, that could be a conversation. But we need to... That's coming from a place where the teacher actually understands the difference, understands that since they were using the same relationships, it was the same strategy. And so, now I'm going to try to move the math forward in some way. One way could be putting them both up, discussing if they're the same. But another way could be, "Oh, you used the same relationships but just a different model? Okay. Did anybody actually use a different strategy?" Like, that's where the teacher is kind of clear on the difference between models and strategies and how you use them as you're working with students. I kind of went on for a long time there. Kim, you still there? 

Kim  07:40

Yeah, I am. 

Pam  08:31

Sorry.

Kim  09:25

It's okay. So, what you just described are some challenges that we see people have when they're trying to facilitate Problem Talks. And just in the excitement of, "Oh, my gosh! These kids know things! And they're saying things!"

Pam  10:55

"Whoo!"

Kim  11:00

And in this stage, we would assume, and hope, and encourage people to facilitate Problem Talks. But to do them well, in such a way that the teacher still is highly engaged in making choices, so that the math is appropriately modeled. And facilitating conversation that's actually useful and not just filling a board with a bunch of stuff that is maybe connected, but they don't see it. And, you know, just anything goes. And so, there's still a purposeful choice that teachers are doing when they're facilitating Problem Talks. And all of that comes from the understanding the work they've done in stage one and stage two. Another thing that we want... 

Pam  11:51

Hey, Kim, before you go on there. I just actually had a conversation with Cathy Fosnot the other day, and one of the things that she said is that often we'll find kind of what you just were describing is that people will do Problem Talks for the purpose of creating talk. And she said, "That's not enough."

Kim  12:07

Yeah.

Pam  12:07

We don't want to just do things for talk. Of course, we want to talk. But toward the end of building the mathematics. And in a huge way, that can be... So, we want to do Problem Talks purposefully. But it's also super important to realize the difference between a Problem Talk and a Problem String, where it's a series of problems, where I... Now, I'm going to quote Brendan Scribner, our friend, who said, "We're going to high dose pattern kids, so that kids really get hit with his high dosage of patterning in a Problem String. And that's where they really construct strategies." So recognizing the difference between a Problem Talk and Problem String happens in this stage. And how do I use a Problem Talk to its advantage? And how do I use a Problem String to really... Maybe I'll say that differently? How do I use a Problem String to really construct relationships in the learners heads, so they own strategies. And then use a Problem Talk to compare already constructed strategies? So, it's super important in this stage to kind of understand the difference between those two routines and how to use them both to their advantage? 

Both Pam and Kim  13:12

Yeah.

Pam  13:12

Cool.

Kim  13:12

Okay, so if somebody is in this stage, what are some things that they can do? We mentioned talks and strings. What are some other action items they can have? 

Pam  13:12

Yeah, so literally any opportunity that you can take a deep breath and just ask people on the street, in the grocery store line, on the subway. Like pick. Church. Wherever you are. Where you can just say, "Hey, how do you think about 99 plus anything? And 99 times anything?" And you might give them a number. You know like, "99 plus 27. Or 99 times 67? Like, how do you think about that?" And then just wait. Just listen. Don't in that moment feel like you have to teach them anything. Like, you can actually just really get good at asking questions and listening. And then you might, maybe try to make their thinking visible. Like, choose a model. Don't feel the pressure to choose the best model, the most appropriate model. You might not even do it with them. You know, you're at church. You don't have to pull out your iPad and like, or your church bulletin and be sketching (unclear). Later, you can be, "Huh. Let's see. How did that person use those relationships? How would I represent that thinking?" Throw those experiences into the Math is Figure-Out-Able teacher Facebook group. Say, "Hey, I just did this, and that person said this. How would you represent that? Is that one of the major model? I don't even. Or strategies? I don't even recognize this thinking. Like, this would be a place to really join in with other like-minded teachers who are in this stage, who are really trying to figure out how do we elicit thinking? And then once we've heard it, how do we make sense of it and how do we represent it? What would be a good model in this situation? Is there a different? Could I hear that same strategy and choose a different model sometimes? So, the Math is Figure-Out-Able teacher Facebook group would be a fantastic place. If you're not in there yet, join that group, and then dive in, and share with us the kinds of experiences that you're having, and get feedback from other people about the way, the things that people are saying as you're asking them questions, and how you would represent them. You can also download our "How do you reason?" paper. We've got a guide for you where we give you certain questions to ask that can really help you identify how students are reasoning. Are they reasoning additively? Are they reasoning multiplicatively, or proportionally, or functionally? So, our "How do you reason?" free download would be a fantastic. Gives you fantastic questions for you to ask. You're like, "Pam, what other kinds of questions that I can ask?" Well, bam, here's a great resource that you can download. We'll put that link in the show notes. We also have Problem Strings on the website. We have Problem String books. Ya'll, if you want resources for Problem Strings, we have got them! Check out mathisfigureoutable.com. Go to the store, and click on "Books". We have got Problem String books coming out for grade levels, grade bands. You can check out those Problem Strings, use those Problem Strings, listen to student thinking, represent that thinking. You are right in this stage if you're doing all of those things.

Kim  16:08

Alright, so stage one, building numeracy. Stage two, learning about the major models and strategies. If you haven't heard those, go back and listen to the last couple of weeks. And stage three, all about listening to and representing student thinking. 

Pam  16:21

Bam. Ya'll, 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 And keep spreading the word that Math is Figure-Out-Able!