Pam  0:00  
Hey, fellow mathematicians. Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris.

Kim  0:07  
And I'm Kim Montague.

Pam  0:08  
And you found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But it's about making sense of problems, noticing patterns, and reasoning using mathematical relationships. We can mentor mathematicians, as we co-create meaning together. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keep students from being the mathematicians they can be. Kim, do you ever have students who say, "I don't know?" Like, you ask them a question, and they just look at you? And they're like, "I don't know."

Kim  0:45  
Yes, yes. Yes. Yes. I think the answer for every (unclear) has been or is, "Yes." And you know what? Another group or a ton more of students that also say, "I just know. I just know that." Right? Like you ask them to explain your thinking a little bit. And they go, "I just know."

Pam  1:04  
"I just know." Isn't there, like, an initial? There's like letters that you just put J.K. "Just know."

Kim  1:10  
Yeah, so both are typical responses. And the trick of it is sometimes when students say, "I don't know," they do know. I mean, they know something. Right? And then, sometimes when kids say, "I just know that," or put J.K. on their papers, they maybe have some sort of inkling, or maybe they do know a lot about what you're asking. But they sometimes use J.K., or "I just know too much". And I don't... You know, I don't know if any other teacher ever had students who wrote J.K., but when I was in the classroom, kind of early in the beginning of me having students attempt to represent their thinking, sometimes they didn't know what to do. And so, I would have a conversation with them. And sometimes... You know, if it's a problem, like 2 plus 2 is 4, and you're teaching fifth grade, then there's not a lot to talk about, really. And so, I would say things like, "Hey, if it's that kind of problem, you can write J.K., "just know", but at least put some sort of effort into letting me know the ones that you're really thinking about." And that absolutely lead to a lot of, "I don't know," or J.K. all over the papers. And so, what I had to do as a teacher was take the time to sit down with those students and listen, and ask questions, and model for them what was really happening. Because what I found was a lot of kids were writing J.K. because they figured it out, and they solved the problem, but they didn't know how to represent. And so, they wrote J.K. meaning, "I know it now, but I don't know what you want me to put on my paper."

Pam  2:54  
Sure. And it wasn't even... Maybe some of them were being lazy. You know. 

Kim  2:58  
Oh, sure.

Pam  2:58  
"Oh, I can get away with just putting J.K."

Kim  3:00  
Yeah. 

Pam  3:00  
But more than that, there were students who literally didn't know how to put down what was in their head, or on paper, and that's real. In fact, I remember this day so well. You and I were at a thing. I don't actually remember where we were, but we were at a thing together. And you turned to me. You're kind of intense. I don't know if you (unclear). 

Kim  3:19  
I can be, yeah.

Pam  3:19  
You turned to me. You kind of like, bore your eyes, like bore into me. And you're like, "Pam, we can all do more. We all own more than we can say. And we can say more than we can represent." 

Kim  3:32  
Yeah.

Pam  3:33  
I'll never forget you. And I was like, "Say that again." 

Kim  3:35  
Yeah. 

Pam  3:36  
And you looked at me, like, "You know, everybody knows that." And I was like, "No, no. I think that's actually like really, like important. And, like, let's... "Like, we can say more, we could do more, than. Sorry. We could do more than we can say. And we can say more than we can represent, we can put down on paper. And I think that's important and noteworthy. And, you know, what's funny? So, I was just in Canada the other day, and they were joking around with me a little bit and they all were like kind of snickering and I was like, "What?" And they're like, "That's noteworthy." And I was like, "What's so funny about that?" They're like, "You say that." And I'm like, "Okay." And they're like, "No one says that." 

Kim  4:15  
Pam does.

Pam  4:15  
"Oh, alright." And they're like, "So, is it noteworthy that you're the only one who says it?" I was like, "Alright, alright. Can we be done with it?" Anyway, they just kept representing. So, let's dig in today to this idea that we can all do more, we own more, we understand, we can have things happen in our head, that it's difficult to talk about. Like, to say that stuff is hard and. But yet, we can all say more, talk about more than we could put on paper, that we can represent in some way, that we could write about. Like, if you've ever tried to write anything you are clear, it is difficult to write what you mean. That's why we edit and revise, right? Revise and resubmit. That is a huge part of what writing is because it is so hard to get what we actually mean out, to represent it in some way. So, Do, Say, Represent. Let's talk about those a little bit today. Would you agree, Kim, that there are times when you solve a problem that I say, "What did you do?" that you say, "Uh..." 

Kim  5:18  
Yeah.

Pam  5:18  
Like, there's this pause. And that talking about how you solve that problem is actually quite tricky. One of the things that we want to talk about today is, we think that there's a miss in some early elementary curricula out there that said, "Show me your thinking in words, numbers, and pictures." 

Kim  5:39  
Yes. 

Pam  5:40  
Too soon. 

Kim  5:41  
Yes. 

Pam  5:42  
Because students would be solving problems, but then the directions would say, "Show your thinking." Now, was it brilliant to say, "These are think-about-able. You don't have to rote memorize something to get these." Absolutely. So, I'm not I'm not dinging these early curricula and saying, "You're bad, awful, horrible." I'm not saying that. Trying to and said those two words together. We're not saying that. In fact, they had some really, some great ideas and definitely propelled things forward. But I think early on, we didn't know what we didn't know, which was that when kids have stuff going on up in their heads, it can actually be quite difficult to represent what's actually happening in their heads. 

Kim  6:21  
Yeah.

Pam  6:21  
And we saw an awful lot too much, where kids would either put J.K., or they would put, "I don't know", or they would represent a less sophisticated strategy than they were actually using. 

Kim  6:33  
Absolutely. 

Pam  6:34  
Because they knew how to draw it. Like, we saw kids that would draw one by one counting, when they could do something much more sophisticated, but they didn't know how to write it. And when they would read the words, "Show your thinking in words, numbers, or pictures." "Pictures! Pictures! Bam, I can do pictures." And so, they would draw these like way too detailed pictures that represented counting strategies, which was actually not what their brain was doing or was capable of doing, but it's what they could draw. 

Kim  7:02  
Yeah. 

Pam  7:03  
And so, let's keep digging into how do we, what do we do about Do, Say, Represent? How does that influence how we teach? What are some tricky things that can kind of come up with that? Let's keep talking about it.

Kim  7:18  
It's funny that you mentioned that because I have a piece of paper from when one of my son's was in first grade. And he solved a problem but as a first grader didn't really know a lot about what to put on his paper and his brilliant, brilliant teacher. There's notes on his paper about the conversation that she had with him and helped him to represent because he had like, I don't know, squirrels running up a tree or something like that. 

Pam  7:46  
Yeah, I totally remember that problem.

Kim  7:47  
From that piece of curriculum. Yeah. And when I had kids who would write J.K. What I was trying to do was (unclear).

Pam  7:55  
Actually, can we say a little bit more? You're about to move on. Can I say a bit more about that paper? 

Kim  8:00  
Yeah. 

Pam  8:00  
So, the problem was, "There's 14 squirrels playing in the park. 6 of them ran up the tree. How many squirrels are not up the tree?" Or something like that. 

Kim  8:08  
Yeah. 

Pam  8:09  
And I'll never forget that he drew this gorgeous tree, and six squirrels running up the tree with every leg on every squirrel. Like, the detail on the squirrels. And 8 squirrels still on the ground. Now, a less sophisticated teacher... So, Sarah Hempel, we give you a lot of credit for this. 

Kim  8:30  
Yeah.

Pam  8:31  
A less sophisticated teacher might have said, "Oh, you're using a counting strategy. Clearly, you counted all 14 squirrels, and then..." But see, I'm pausing because I don't even know how to. Like, how did he know to draw 6 squirrels? Or if the question was how many squirrels are still on the ground, how did he know to draw 8 squirrels still on the ground? Like, he almost would have had to have solved the problem first. 

Kim  8:51  
Yes. He absolutely did. 

Pam  8:52  
In order to draw it that way. Yeah. 

Kim  8:54  
Yes. 

Pam  8:54  
Brilliant teacher, know your content, know your kids, knows her content, and knows your kid well enough...

Kim  9:00  
Yeah. 

Pam  9:01  
...that she said, "Wait, wait, wait. How did you actually solve it?" And I'll never forget this because what's written on that paper was. She said something like, "I asked Luke what he actually did, and he..." Was it Cooper or Luke? I think it was Luke.

Kim  9:11  
No, it was Luke. Yeah. 

Pam  9:12  
Yeah. "I asked Luke what he actually did, and he said, 'Well, yesterday we did the problem 8 plus six is 14. So, today, I knew that 14 minus 6 is 8.'" 

Kim  9:21  
Yeah. 

Pam  9:21  
And she wrote that down on the paper, and then she wrote, "We're still working on representing our thinking." It was brilliant on paper. Well done, Ms. Hempel.

Anyway, sorry. I just had to kind of. Like, what was the... Why am I so excited about what she did? Know your content, know your kids. She knew him well enough to go, "Oh, yeah. You're not counting by ones. I know you. I know what you're capable of." 

Kim  9:44  
Yeah.

Pam  9:45  
"At this moment. Not what you're capable of like. Like, I'm being not equitable here. Not like I'm disparaging you. I know what the relationships you've been using. I know you're not counting by ones. Let's see if we get that actually out of you. And then, I'll help you represent it. I'll show you a way that you can use equations to represent what you were thinking." 

Kim  10:04  
Yeah, yeah. You know, it's really easy to go, if you have a paper. I'm like making a worksheet with my hands. If you have a worksheet, paper, whatever, where it has some math problem, and we hand it out to kids, and we say, like, "Here you go," the only thing that we're asking them to do is go straight to the representing thinking part. And what she did was interrupt that cycle of you go straight to representation, and she had the part in the middle, where really a lot gets worked out. Kids are doing things in their heads all the time, like all the time. You know, we've talked about how, you know, you and I are exercising, and we're thinking about stuff, how we're in the grocery store, and we're doing stuff. Kids are doing stuff in their heads a lot of the time too, and I don't know that we give enough credit to what's happening there because we don't... First of all, we don't have as much time as we should to be able to talk to them. But I don't know that necessarily we make a big deal about how that conversation piece really has to happen in order to get a clearer picture of what's going on in their heads. We skip straight to representation and...

Pam  11:13  
Can I add to that? 

Kim  11:14  
Yeah.

Pam  11:15  
I'm sorry. Finish your sentence.

Kim  11:16  
No, go ahead.

Pam  11:16  
Well you said, it's so important that we... How did you say that? That we have the conversation, so we get clear about what's happening in their head. But I'm going to add to that, and I know you agree with me here, that it's so important to have the conversation, so that the kid gets more clear on what's happening in their head. 

Kim  11:32  
Yep. 

Pam  11:32  
So, there's an awful lot of talk right now about discourse in math class, and we should... There's talk moves, and there's ways to get kids to talk in mathematics. There's books about math talks. And I would suggest A, if you're teaching algorithms, and it's all about rote memory, there ain't a lot to talk about. And if that's your perspective about what math is, then you might very well, very logically, very sensibly going, "What is there to talk about?" And you also might be saying, "If I'm going to ask this kid to explain it to everybody, then why don't I just explain it? Like, surely I'm going to explain it more clearly. So, if it's about getting the most clear explanation, I should just do the explaining. And then, the kids will understand it better." If math is about transmitting, if it's not about helping gain clarity. So, one of the reasons we want to talk today in this podcast about this sort of hierarchy that we can do more than we can say, we can say more than we can represent is part of what's built into that, that's underneath it, that's like seeping through that is, that it's our job to help pull out those words. That kids are doing things. It's our job to say, "Oh, let me help you think about, let me help you gain clarity on that by talking about it. Say more about that?" And as we pull out their words, as we help them verbalize what they're thinking, they get more clarity, other students listening can ask questions, they try to understand what the kid's saying, they try to put it in their own words that helps them get more clarity, all before we make a big deal about them representing. 

Kim  13:14  
Sure, sure. And, you know, it's funny. As I'm sitting here listening to you, and I'm thinking it's. You know, we're talking about mathematics, but it's so true in so many different areas. I mean, I'm picturing myself as you and I are talking about different things that we want to do in different topics. I'm pacing my house sometimes just like rambling on, and I'm just talking and you're listening. And you're like, "Hey circle back to that. Say more about that." And that's just the way conversation happens. And if mathematics is a second language to everybody, which it is, then they need an opportunity to...

Pam  13:49  
Beat it out. 

Kim  13:50  
Beat it out, and to have somebody probe a little bit further. And so, listen, we were just in Oklahoma. And I saw.

Pam  13:59  
Had a great time, by the way. Thank you, Oklahoma teachers for having us come. Yep.

Kim  14:02  
You masterfully...Is that a word?...worked with teachers and students. And some of them would say things like, "I just know." And sometimes we saw teachers say like, "Oh, okay." You didn't let that happen when you were working with students. You followed up with, "Tell me more about that. Did you..."

Pam  14:24  
(unclear). Oh, go ahead. 

Kim  14:25  
No, go ahead. 

Pam  14:26  
Well, which is not trivial. Right? 

Kim  14:29  
Yeah. 

Pam  14:29  
We had the opportunity to watch some teachers teach. And then, I also did some model teaching. And Kim took great notes and gave everybody some feedback. And one of the things that we noticed with a couple of the teachers were that they would say, "Well, how do you know?" And the kids would go, "I just know," or "How did you do it?" "I just know." And then, often it kind of appeared the teacher didn't really know how to follow up with that. They weren't sure, and so they were just like, "Oh, great, great. Glad, you just know it." Well, I was there going, "No, I actually want to know what the kid did. I'm actually curious what the kid did, and I'm really pretty clear that there are students in this room, that that's not sufficient for." When the kid says, "I just know it," well even though that might be true, that's not going to help anybody else in the room now be able to follow why that's a reasonable solution to that problem, why that's a reasonable place to go in the problem. We need to get more out. We need to have more floating for more kids to be able to grab on to, have access to what's happening. So, let's talk about how do you pull out students' thinking because you got to know your content, know your kids. You have to know the content well enough that when you're saying, "What do you know?" you have to, like, have some sense of the content to not just stop there to then say, "Well, like, what do you know? Like, how are you doing it?" And they go, "I just know it." "Well, how do you know that? What do you know in there. Let's pull that out. Let's parse that out."

Kim  15:53  
Yeah. And I saw you say things like, "Do you know, some eights?" (unclear) if it was multiplication problem, you would say things like, "Do you know some eights?" And the kid would, like, kind of nod or kind of shake their head, and then you would follow up with, "Some sevens?" And you got them started with some common suggestions or strategies that people would use. It may not have landed the very first time, but you knew some common ways that people tackle that problem. And you would say things like, "What about this? Did this ping?" and wait for a kind of a glimmer in their eye. And then, they would go, "Yeah, yeah, yeah. That, that, I did that." And then. And it was a. You were facilitating a conversation between the two of you and letting everybody in on what was happening in their head. And so, when people would say, "No, not that," you knew enough to lob out another thing that they might have done.

Pam  16:47  
Yeah. And just to put a fine point on that, I didn't just say, "Oh, so..." Let's go back to your 7 times 8. "...do you know some sevens? Do some eights?" And then, when they say, "No," or they say, "Well, maybe," I didn't just say, "Oh, so you did 5 times 8, plus 2 times 8. Is that what you did?" I don't give the whole thing. 

Kim  17:03  
Yeah.

Pam  17:03  
I go, "Oh, so you do some eights." Like, "Did you know, like, five 8s?" And then I pause. We kind of call it the "trail off method" where I just say just like, "Do you know some eights?" If they run with that, "Oh, yeah. Like, I knew five 8s and two 8s," then I let them go. Or, "I knew eight 8s." And if they. If all I say is, "Do you know some eights?" and they go, then I let them go. If I say, "Do you know some eights?" and they kind of look at me like, "Maybe?" then I go like, "Do you know five 8s?" If they have that glimmer, then I'm like, "Well, how did five 8s help you with eight 8s?" Or I might say. If they look like, "No, not five." Then, I might go, "Well, do you.. 8? Were you think about eight 8s?" And then, if that glimmer shot, "Oh, yeah, yeah." And again, if nothing, then I might, "Oh, well, maybe we're thinking about sevens." "Yeah, yeah. I think I was thinking about sevens." Like, "Well, how many sevens were you thinking about?" They pause. Again, I'm just lobbing just enough out. 

Kim  17:53  
Yeah. 

Pam  17:54  
And then, I pause and see if they can run with it. It's not about, like, handing them, "Is this the strategy you used?" Just enough. Because then again, if all I do is lob that little bit out, "Well, were you thinking about sevens? Were you thinking about 10?" (unclear) pause, "Sevens?" Pause. And they're like, "Oh, yeah, because I was thinking about 70, and then I just got rid of some extra 7. Because I was looking for eight 7s." I'm like, "Well, how did ten 7s help you with eight 7s?" "Well, I just had two extra 7s." "Oh, so you were thinking about 70?" Pause. They're like, "Yeah, minus 14. Those two 7s." "Oh, so 70 minus 14, that got you to 56. So, that's how you were..." And then, I might record it. Then, I might model it on the board. So, it's the sort of lob, like you said. Lob something out, see if it kind of pings in their zone of proximal development, does it hit with kind of what they were thinking about. Is that what they were doing, and then the kid can kind of run with it from there. And as they run with it, they get more clarity on. As they have to put words to it, they get more clarity. Everybody else gets a chance to hear this kind of muddy description. Wait, is that a good thing? Yes. Because as they hear the muddy description of what the kid was doing, they get a chance to try to clarify it and make it make sense in their head. And all of that grappling and clarifying is learning occurring. Woah! Yeah?

Kim  19:14  
Yeah. 

Pam  19:15  
Also, while I was there, I did a bit of work with some Algebra teachers. In fact, I think I'm actually thinking of a different presentation that I just did, where I was working with some Algebra students. We were thinking about writing the equation of the line. And so, I might say, if I was in a rote memory, math is rote memorizable perspective, I might say, "Okay, what's the equation? Alright, you don't remember the equation? Here it is." I'll give it to them. "So, what do you... You know, what are you going to plug in where?" If that's a rote memorizable. But if I'm in a figure-out-able, where Math is Figure-Out-Able, I might say, "Alright. So, like, what do you need? What do you know? Do you know how fast he was walking? Can you find that? Like, if he was walking 3 feet. We know that. If he covered 3 feet in 2 seconds, what do you know? Like, is that fast? Is that slow? How do you know? How do you know that's fast? How do you know that's slow? Like, if I told you to walk that, what would it look like? Does that help you think about that rate? Like, how would you know if you were walking that rate? What if I only gave you 1 second to walk that rate? How would you?" I mean, I'm really kind of going down this rabbit hole but to help kids, like, think about what's happening, what does rate really mean, then we can sort of put that information with what they already know about the equation of a line, So, there's a couple of examples. In fact, Kim, I'm also thinking this could be a time when you could refer to an anchor chart. 

Kim  20:38  
Yeah.

Pam  20:38  
You could say, "Okay, so based on this problem... Hey, have we done this before? Like, we anchored some thinking over there? Oh, yeah. Like, as you're thinking about fast this kids walking, there's an anchor chart we created over there for the equation of a line based on somebody walking. Oh, we need to know how fast they were going? Well, we've just been talking about that, but what else do you need? We need to know their beginning point. We need to know that y intercept at time equals 0? Well, can you find that?" So, whatever you have, if you have anchor charts, can you reference those? And as you do that, like when you're asking students, "What have you been building that could ping for you right now?" As you're looking around the room, and you're like, "Is there an anchor chart?" You're helping demonstrate for kids. "Oh, like, when I hit a problem I don't know how to solve, I can use what I know. Even if it's anchor charts that are hanging on the wall. If we've developed those together, I can. Oh, yeah, that might help something ping for me. I can look at that anchor chart. I can sort of draw on that."

Kim  21:37  
Yeah, absolutely. And I think that this idea that you can attach things that you're kind of... It's kind of swirling around in your head, and looking around, and attaching it to something, a conversation that's been happening in the classroom before. You know, we give way too much credit, I think sometimes, to having the right, perfect, few word answer that summarize it's all and like nail on the head. But again, you've said this, verbalization is so much about helping kids grapple and make sense and resolve disequilibrium, and helping other mathematicians in the class to do that too. We talk about sharing our thinking as if the goal was to just help others. It's about helping ourselves and about solidifying what's going on because at some point, we don't want it to just be this like swirling around thought. We want kids to feel solid and to be able to say, "I know that. I'm confident in that."

Pam  22:39  
But as learning occurs, it might not feel like you're always on solid ground. 

Kim  22:43  
Yeah. 

Pam  22:44  
But we do want to get to solid ground. 

Kim  22:46  
Sure. Yep.

Pam  22:46  
Yeah, I think that's super important. Cool. And as we pull that thinking out of students, we want to represent that thinking and make it visible. And I'm looking at the time. This has already been a great podcast about Do, Say, Represent. Let's spend some more time talking about representing modeling soon. We'll do a podcast on modeling and making thinking visible. That representing part is so vitally important. So, that is coming. I'm still doing a little thinking about that. So, let me. I'm still grappling with a few things to bring some clarity to modeling. Don't worry that's coming. So, stay tuned to the podcast because it's coming. So, important.

Kim  23:25  
But for now, can I just lob out that we do think it's a little bit tragic that the most common way that we assess our students is by the way that they represent. So, we're just going to lob that out for just a second about something that we're thinking about and... And, yeah.

Pam  23:43  
And we're grappling with.

Kim  23:44  
 Yeah. 

Pam  23:44  
Yeah, yeah. That the only thing they ever get kind of credit for is what they can represent, even though we already have just admitted they can say a whole lot more than that, and they can do even more than they can say. 

Kim  23:55  
Yeah.

Pam  23:57  
Hey, ya'll, thanks for joining in. And thanks for 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!

Transcribed by https://otter.ai