Pam  0:01  
Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able! I'm Pam Harris, a former mimicker turned mather.

Kim  0:10  
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  0:19  
And we're glad that... Oh, my goodness. And we're glad you're with us on that mission because we know that algorithms are amazing achievements. But they are not good teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems could be developing.

Kim  0:36  
In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships.

Pam  0:43  
Thanks for joining us to make math more figure-out-able. 

Kim  0:46  
Hey, there. 

Pam  0:47  
Hi. How's it going? 

Kim  0:49  
It's good. 

Pam  0:51  
Is it cold? Where you are?

Kim  0:53  
I mean, I'm like 20 minutes away from you.

Pam  0:55  
15 miles down the road from me. I haven't been outside. I think it's cold.

Kim  0:59  
That's hilarious.

Pam  1:00  
Well, cool, maybe.

Kim  1:01  
You know, I do have teenage boys. And so, you know, it's January, and mine rolled out of the house with shorts and a t-shirt on. And I said, "Hey, just, just so you know, it's probably a little cold outside. But you do you, buddy.

Pam  1:15  
I mean, here in Texas, you never know, right? Like, it could be...

Kim  1:18  
Yeah, I mean, come on...

Pam  1:19  
...up and down.

Kim  1:20  
Put a jacket on. I used to... You know, when I taught elementary school, I would see these kids. And I'm like, come on, parents. Like, put your kid in a jacket. You know? Oh, man. I'm like eating my words.

Pam  1:33  
Did you have to eat some words? Yeah.

Kim  1:34  
Oh, 100%. Like, whatever. You want to be cold? Be cold.

Pam  1:37  
Yeah, one of the first things I ever tell new parents is like never in your life ever say the words, "My kids will never". Like, just strike that from your vocabulary because as soon as you say the words out loud, karma will strike.

Kim  1:50  
Yeah, yeah.

Pam  1:50  
And that will be the exact thing that your kids will, you know, yeah. Right? Right?

Kim  1:54  
Well, and you know, they need experience to learn, right? I could shove a jacket on him, and that might be a fun fight. But learning, learning...

Pam  2:04  
Learning is a good thing. Yeah. Yeah. Young parents, don't ever say the words, "My kids will never". Just don't do it. Alright.

Kim  2:04  
We're going to talk math today.

Pam  2:14  
What are we doing in this podcast episode? Oh, we get to do math. I like math. Math is good. Okay, math, yeah.

Kim  2:18  
Yeah, okay. So, you know, we are big fans of Problem Strings as a routine.

Pam  2:23  
Big fans. 

Kim  2:24  
And we do Problem Strings quite a bit. But not too terribly long ago, somebody asked if we had a video of a good Number Talk, which we call Problem Talks because they're about problems. And we thought that we would, today, describe the way that we think a good Problem Talk could go in a classroom.

Pam  2:47  
Yeah, nice. Number Talks can be good, right? They can be good. We just think that if you're going to do them, you're going to do Problem Strings a whole lot more than you will a Number Talk or what we call a Problem Talk.

Kim  3:00  
Say why. Say why.

Pam  3:01  
Problem Strings give kids a higher dose of the underlying relationships and the strategies, a high enough dose that kids actually can build them. Number Talks, most of us, when we first saw a Number Talk, we were like, "Oh, this is amazing. There's more than one way to solve a problem! Look at all the ways people are thinking! And I can learn from that," and didn't get a high enough dose. And it felt like, in that... So, see if everybody agrees with me. But it could have felt like in that moment that there's this vast, unknowable, innumerable, indescribable set of strategies that's like, "Whatever anybody does is cool. And I didn't even follow that one. Do I have to learn that? Do I have to teach my kids that?" Instead, we're suggesting there's just a small set of major relationships that lead to a small set of major strategies. We need to know those as teachers, so that we can high dose kids with those relationships to lead to those strategies, so that then everybody is mathing the way mathy people math. Number Talks don't do that well. Problem Talks. What do they do well? Well, they compare strategies well. So, if we do a good Number Talk, Problem Talk, once we've planted the seeds of these important relationships and kids have developed a few of the strategies, that's a great time to do a Number Talk or what we call a Problem Talk because now we can compare strategies for efficiency, and we can get kids better at the strategies in that comparison. We can help kids develop intuition about when to use which strategy.
Is anything? Did I leave anything out? 

Kim  4:34  
Yeah, I'll add on that there are some nice times to do a Problem Talk. Although I agree with you that a Problem String would be happening much more often in my classroom. A Problem Talk is a nice chance to put up a problem every once a while to see what strategies are being used by students. The beginning of the year might be a good time to put up a problem and say like, "What do my kids think about? What are they owning before I start the work that I'm doing with them." So, there are a couple of nice opportunities to use Problem Talks, and today we're going to chat about what we think they could look like.

Pam  5:09  
I have a question for you. 

Kim  5:10  
Yeah.

Pam  5:11  
When you say that it could be a good opportunity, say like at the beginning, to sort of see what your kids are doing. How much time would you spend in the discussion of that Number Talk or Problem Talk, comparing those ways that kids were?

Kim  5:28  
Are we talking like beginning of the year?

Pam  5:30  
Yeah. Like, you don't know the kids yet? 

Kim  5:31  
Yeah. I don't know that I'm going to spend too much time. That's more formative for me. That's like,"Hey, what..." (unclear).

Pam  5:37  
You're almost kind of pick. You're almost kind of gathering that. Like, (unclear).

Kim  5:40  
Yeah. Yeah (unclear).

Pam  5:41  
Like, you're not spending...Yeah.

Kim  5:42  
Do my kids talk about their thinking or is it pretty quiet? Do they break apart numbers? Do they... Like, what kinds of things are they saying? And then I ... kind of gives me some information about where to start. Like, if they have fantastic strategies, and like tons of kids are nodding, and they're into it, and they're chiming in, and adding on. Then, you know, it's just one way to assess what they come into the grade level with.

Pam  6:09  
Nice. What I'm not hearing you say is that this is like a teaching time. 

Kim  6:12  
No, unh-unh. No, not. 

Pam  6:14  
Like, where you're really, "Like, okay, everybody. Got this one? Like, really good, really down?"

Kim  6:19  
That's not how I use Problem Talks.

Pam  6:22  
It's more formative. 

Kim  6:24  
Yeah.

Pam  6:24  
Yeah, okay, cool. So, let's say that we are in a place where we think a Number Talk is a good one to do. So, Kim, let's do a Number Talk today. I thought you were going to say something else

Kim  6:37  
No.

Pam  6:37  
to set the stage. No, we're good? Okay, so I'm going to give a problem. And, listeners, we would invite you to solve that problem. Try to use relationships. If you are someone who's built a few strategies, maybe think how many strategies can you come up with? Or how many of them are major? Which ones would you be thinking about pulling out of your students? Blah, blah, blah. Okay, so here is a Number Talk, what we call a Problem Talk, that we're going to do today. And, Kim, that is 15 times 36. "So, class 15 times 36. Solve it any way you want. Try to be clever. Try to be like efficient, clever, slick. Try to like find a slick strategy. Go." I'm going to give that to students. Then, I'm going to circulate. And see, I would suggest that this would not be one of those where there's no paper and pencil. So, kids might have whiteboards. Whatever. I don't care how they're writing. Doesn't need to be formal in a notebook or something. This is kind of scrap paper time. Could be drawing on their desks with whiteboard markers. Whatever. I'm going to circulate, kind of see what's happening. Once enough kids have chewed on that problem well enough. Might have asked a couple kids what they're thinking if I can't kind of tell, they're not representing their thinking very well. I'll come back to the board and I'll say, "Alright, fantastic. Kim.

Kim  8:03  
Yep.

Pam  8:04  
Tell us how you were solving 15 times 36."

Kim  8:06  
Well, I'm really glad that you talked for a minute because I was like racking my brain of all the different ways that I know you're going to ask me for more than one. Yeah, I'm good. Okay, so my first inclination was to think about 15 times 36 as an Over strategy. So, I actually thought about that like 40 times 15. 

Pam  8:25  
Okay.

Kim  8:26  
Which is 600. And then I didn't need forty 15s. I only needed thirty-six 15s. So, then I subtracted 40. I mean, sorry, four 15s. So, then I wrote 40 times 15 minus 4 times 15 is 36 times 15. And I wrote 600 minus 90 is 540.

Pam  8:52  
Minus 90.

Kim  8:54  
Mmhm. Is 540.

Pam  8:55  
Is 4 times 15, 90?

Kim  8:58  
Did I say 90? I don't know. 600. Or 60. Sorry. I literally wrote 90 on my paper. And then, funny enough, I wrote 540.

Pam  9:08  
There you go. So, I think you were thinking 60. For whatever reason, you wrote 90. I think you were thinking 60. Yeah, cool.

Kim  9:12  
That's hilarious. Okay.

Pam  9:15  
So, if a kid were to... First of all, they might not be as clear as you were. If a kid were to say exactly what you just said, I might write on the board 40 times 15 equals 600.

Kim  9:30  
Mmhm.

Pam  9:30  
4 times 15 equals 60. And then draw a line. And then write 36 times 15 equals 540. 

Kim  9:38  
Okay.

Pam  9:39  
But I'm thinking about what I... So, I don't want to do too much teacher talk right now. I think I might also say that, "It feels like Kim was thinking about 15s." So, I asked the question 15 times 36, but I don't think that you were thinking about 36s. I don't think you found 15 of them. So, if you're thinking  about 15s, I could also represent your strategy on a ratio table. And I've just drawn a vertical ratio table where I've got 1, 15. So, 1 to 15. And you found 40 of them was 600. So, now I've got a ratio table that it's vertical. It's got 1 to 15, and 40 to 600, and then 4 to 60, and then 36 to 540. So, it actually looks kind of similar to what I have in the equations as far as the numbers go, but it's sort of in that table next to it. Cool. Yeah, it just got me thinking. I think because of where I think I'm going with this... No. I'm just going to say what I would say to kids. "I'm also going to represent that as a 15 by 36. So, everybody help me draw that rectangle. If this was 15, the 36 should be..." And I'm looking for somebody to say... Well, sorry, Kim. If I've drawn 15 down, what would the 36 across look like?

Kim  11:03  
It's going to be more than twice as long.

Pam  11:06  
More than twice as long. So, I'll kind of there, and then. I've got a rectangle that's 15 down and more than twice across. But you didn't really find the area of a 15 by 36. Not off the top. You found an area of 15 by 40. So, I'm going to draw a little bit. Like, tack on a little bit. I've now stuck the 36 inside the rectangle. So, 15 by 36. There's that little 4 hanging over on the side. You said that. So, now I have a 15 by 40. So, I've written 40 across the top of the long rectangle because I stuck that extra little bit on the side. And you said that that 15 times 40 was 600. So, now I have a caret below the chunk that was the 15 by 36, connecting it to the 15 by 4. You said that total area was 600. The 15 by 4 area was 60. So, the leftover area is 540. So, I now have three models on the board for this one strategy. And I'm sort of curious. I might have students like say to themselves which of those models speaks to your heart and your soul? Which of those helps you kind of understand Kim's strategy better? Cool. Alright.

Kim  12:12  
Okay. 

Pam  12:12  
Now, I might say, "Great." Now, I'm going to call on another kid. "How did you solve this problem?" Is that Kim too? Do we give it a new name? Is that...

Kim  12:22  
Everybody needs more Kims. Yeah, so a second student...

Pam  12:27  
New kid, new kid. Yep. Okay.

Kim  12:29  
A new kid might say, "I broke up the 15 and the 36 into place value parts.

Pam  12:39  
Mmhm.

Kim  12:39  
And so then I did 10 times 30 and 5 times 30 and 10 times 6 and 5 times 6. 

Pam  12:45  
Cool. So, I'm going to redraw that 15 by 36 right underneath the area model, the rectangle, that I had above. And so now I have two 15 by 36s. But I don't have that extra tacked on the end. I just have the 15 by 36. And Kim, you said that you broke up the 15 into 10 and 5?

Kim  13:04  
Mmhm.

Pam  13:05  
So, about two-thirds of the way down, I've now drawn a horizontal line. And in the inside of the rectangle, I've drawn a 10 and a 5. And then you broke the 36 up into 30 and 6. 

Kim  13:14  
Mmhm.

Pam  13:15  
So, way over to the right, a little bit bigger than the 5 across the bottom is a 6. It probably looks kind of the same because 5 and 6 really when you're dealing with. But I'll probably say out loud, "A little bit bigger over here because this is 6." So, now in the inside, I have 30 in that inside rectangle, and then I have 6 over on the edge, that sort of strip that's going down below. So, the two rectangles are right above each other. That one for the 15 by 40 strategy. And right underneath it, I have copied that 15 by 36. But now I've cut it into 4 chunks. And then, Kim, I'll let you keep going. What is 10 times 30?

Kim  13:51  
300.

Pam  13:52  
So, that's that inner, one of those inner rectangle's area. And then 5 by 30?

Kim  13:57  
150.

Pam  13:58  
And then 10 by 6?

Kim  14:00  
60.

Pam  14:00  
And then that last little tiny one down there, 5 by 6?

Kim  14:03  
30.

Pam  14:03  
And then if we add all those up together, did you also get 540?

Kim  14:07  
I did. 

Pam  14:07  
Alright, so if we add all that together, 540. Alright, so Kim number two, nice strategy. I might step back and say between these two strategies, does either one of them feel more efficient? Let everybody think. I might even invite kids to partner talk at this point, "Which of those two strategies feels more efficient?" Cool. Kid number three. What are you thinking about?

Kim  14:34  
I just broke up the 15. So, I did 10 times 36 is 360. And 5 times 36 I knew was half as much as the 10 times 36. So, that was 180.

Pam  14:54  
Cool. So, I've redrawn the 15 by 36 again right below. So, now, I have three 15 by 36s. Bam, bam, bam. Right on top of each other. One of them has that little extra 4 on the end. One of them's cut into 4 chunks. And this one. Can I just redraw? This one looks terrible. Hang on. Wow, I can't draw straight today. Okay, so this one, you broke the 15 into 10 and 5. So, that's going to look very similar to how I cut the rectangle right above it into 10 and 5. But then I'm not cutting it vertically. I'm just leaving the 36 alone. 

Kim  15:33  
Okay.

Pam  15:33  
And then what did you do after that? 

Kim  15:35  
I did 10 times 36 is 360.

Pam  15:38  
Okay, so that's the area of that big top chunk. Mmhm.

Kim  15:40  
Mmhm. And then I knew that the 5 times 36 was going to be half as much as the 10 times 36. That's just 180.

Pam  15:48  
Nice, cool. And then when you added those together, did you also get 540?

Kim  15:52  
Yeah.

Pam  15:52  
Cool.

Kim  15:53  
Yeah.

Pam  15:53  
Now, I might invite a conversation with the whole class. "What do you think about these three strategies?" Then, I could point to them because I have three rectangles on the board. "What do you think about these three strategies? And I want to have a conversation. What do you think about? Do they make sense to you? Might they occur to you? Which one do you want to occur to you next time? Which one feels kind of clever and efficient? You're hoping that occurs to you next time?" Kim, can you envision a kid saying something in that class discussion?

Kim  16:23  
Yeah, I could see one of the kids saying that something they hadn't thought of before might be nice to think about. Like, if I was kid and never thought of Over, they might say that. If a kid broke up both factors, they might like to break up just one.

Pam  16:38  
Mmhm. Cool. I could also envision a kid saying something like, "Well, the middle strategy had four chunks and the other two just had two chunks." So, that might speak towards efficiency. You could also envision a kid saying, "Yeah, and that one subtraction. I don't really like subtraction. I'd rather add." Then a kid might say, "Yeah, but you had to divide the 360 in half. And I'd rather like just use the sort of place value things that were happening with the 600 and the 60."

Kim  17:12  
Mmhm.

Pam  17:08  
So, the subtraction wasn't very bad. So, I can see that this could be a nice conversation. You know, where like, "No, I hate subtraction!" "Really? But 600 minus 60?" "No, it's just easier to add." "yeah, but really, you had to divide." You know, and so, yeah. So, I could imagine that conversation. Cool. Alright, kid number... Let's see. I think four, right? We're

Pam and Kim  17:12  
kid number 4.

Kim  17:21  
Yep. 

Pam  17:23  
Kid number four, how did you solve it?

Kim  17:30  
I could see that a kid would see that it's a multiplication problem and want to add a bunch of 15s together because then they can add all the 10s together, and add all the 5s together. 10s and 5s are pretty simple to chunk.

Pam  17:47  
If you're going to add a bunch of stuff together, you might that. Great. So, if a kid says, "Yeah, I started adding a bunch of 15s, and I was bringing these together," I might say, "Oh, that's interesting. Are we looking for thirty-six 15s? Sure enough." Next kid. So, I'm going to choose, in this moment, to reference, to say, "Are we doing that?" Everybody nods. I'm like, "Yep." And then not represent that repeated addition on the board. So, I'm going to acknowledge it positively. "Yep. We could do that. Nice." Kid number five. What do you got?

Kim  18:20  
Kid number five could say, "I know that 36 is close to 40, so I thought about that like 15 times 40 minus 15 times 4.

Pam  18:35  
And in that moment, I might say, "Kind of like..." and then I would point to the first rectangle that I drew. And I would say, "So, you were thinking about 15 times 40." And I would point to that 15 by 40 that we already have drawn on the board. "And what did you do?"

Kim  18:56  
I subtracted 4.

Pam  18:56  
Hey, and here it is. There's the four 15s. Sorry, I'm interrupting. 

Kim  18:56  
No, go ahead.

Pam  18:56  
Here's the four 15s. So, does this represent what you were thinking? In that moment, depending on how the kid answers, I'm hoping the kid goes, "Yeah, that represents what I was thinking." Then I might go, "Okay, yeah. That is a nice strategy just like Kim number one did. Nice." So, I'm not going to redraw or re-represent the same strategy, even though the kids said it differently.

Kim  19:21  
Mmhm.

Pam  19:21  
If it's using the same relationships. Okay. Kid number... I don't know what number we're on. Are we on six? Kid number six.

Kim  19:28  
I could see another student say, "I'm gonna break up the 15 and call that 3 times 5 and break up this 36 and call that 6 times 6.

Pam  19:45  
So, I've got written on the board right now 3 times 5 in parentheses times 6 times 6 in parentheses. And I should say I rewrote 15 times 36 and I've written... I rewrote 15 times 36 and underneath it I wrote the quantity 3 times 5 times the quantity 6 times 6.

Kim  20:02  
Mmhm.

Pam  20:03  
"Okay, random. What do you..." No, I wouldn't say that to a kid. I might, actually. I might go, "That seems really weird."

Kim  20:08  
Yeah, you will.

Pam  20:09  
Like, "All these kids are breaking up area of rectangles. What are you doing?"

Kim  20:13  
So, I think that I could say I want to do 5 times 6. Put those two together.

Pam  20:19  
This 5 and that 6. The two in the middle. Okay.

Kim  20:19  
I'm going to call that 30. Um... Yeah, 30. And then I'm going to multiply by 3 next.

Pam  20:31  
Okay.

Kim  20:31  
For 90. And then I want to do 90 times 6 is 540 because I know 9 times 6.

Pam  20:38  
Okay. So, just so everybody knows what I've got on the board. So, I rewrote 15 times 36. Underneath it, the quantity 3 times 5 times the quantity 6 times 6. And then underneath that, I kept the numbers in the same place. I wrote 3 times the quantity, parentheses around 5 times 6, times 6. So, it's 3 times parenthesis 5 times 6, end the parentheses, times 6. Underneath that I wrote 3 times that 5 times 6 is 30. So, 3 times 30 times 6. And underneath that, I wrote 90 times 6 equals 540. Nice, flexible factoring. Now, depending on how much time I have, at this point, I might say, "I wonder if anybody could flexibly factor or did anybody flexibly factor differently? I'm looking around the room. I am kind of curious, Kim, do you have a different flexible factoring? Or is that probably the best one? 

Kim  21:37  
Yeah, if you know 30 times 18. Like, if you know 3 times 18 is 54. I mean, I do. I don't know that... 

Pam  21:46  
Yeah. And I think you got that by factoring the 15 into 3 times 5 and the 36 into 2 times 18. So, now you end up with 5 times 2 in the middle times 3 times 18. You're like, eh.

Kim  21:57  
Wait, say that again? No, I did 5 times 6 is 30 and 3 times 6 is 18. So, then I did 30 times 18.

Pam  22:05  
So, you kept it factored the same way, and I factored it slightly differently.

Kim  22:09  
Yeah. 

Pam  22:09  
So, I was thinking about 3 times 5 times 2 times 18.

Kim  22:12  
Mmhm.

Pam  22:13  
And then the 5 times 2 becomes 10, so I end up with 10 times 3 times 18.

Kim  22:19  
Mmhm.

Pam  22:19  
So, then you might be at a place where you're like, "3 times 18? Really?" But you could do that if you kept factoring that 3 times 9. That's 27 times 2 to get the 54 times a 10. Anyway, so some ways that we could play around with flexible factoring. Alright, kid number seven. What do you got?

Kim  22:41  
I could see another kids saying 15 times 36, I'm going to Double Halve and turn that into 15 times 18... Uh, 30 times 18. Sorry.

Pam  22:55  
And so, I'm drawing 15 times 36. I'm drawing that rectangle. But you want to? What part are you

Kim  23:01  
doubling? I'm going to double the 15.

Pam  23:04  
Because that's a nice 30.

Kim  23:06  
Mmhm. And then I'm gonna halve the 36. And that's going to turn it into a 30 times 18.

Pam  23:23  
And now we're stuck back at that 3 times 18.

Kim  23:26  
Mmhm. But I could do it again, and then get 60 times 9.

Pam  23:34  
And I'm drawing. Let's see. I got to cut that in half. And I've run out of room on my paper. 

Kim  23:40  
Me too. I had to move it somewhere else. And then I have 60 times 9, which is 540.

Pam  23:46  
And 60 times 9, which is 540. Nice. Alright, now we're going to have another class conversation. "What do you think? Which strategy do you want your brain to be inclined to do next time? And why? Turn and talk to a partner." And then I'm going to go listen in, and I might call on one or two kids to just chat about, really, at this point, it's not whether a strategy wins. I'm going to try to get kids to talk about their thinking. Yeah. Alright.

Kim  24:15  
Yeah.

Pam  24:16  
There's my Problem Talk.

Kim  24:17  
Okay. 

Pam  24:19  
Whoo!

Kim  24:19  
You know, you and I, I think do them a little bit differently, which is fine. I wonder about... You have me thinking about how much instruction I do in the middle of a Problem Talk or along the way. Like, there were a couple of moments where you said that you would have them reflect, and compare, and like come up with another strategy. Like, if one kid had done a flexible factoring, and then maybe say, "Could we do it another way?"

Pam  24:46  
"Could we think of another flexible factoring way?" Mmhm.

Kim  24:48  
Yeah, it's one of those things where I don't know that we've ever been in the same room as each other doing a Problem Talk. So, that might be a fun thing to do. You know, when we traveled and presented together for a lot of years, we would see each other do something and go, "Huh? Why do you do it that way?" So, this is raising some questions for me about when we would do what. I think when I do them, it's more like information gathering because I want to know what students are thinking about, and so I wonder if mine go a little more quickly than yours. I think there's definitely, for me in my mind, this step back towards the end that you just mentioned. Yeah. I'm thinking. I do have some questions. When you circulate and you're looking at papers, are you purposely calling on someone? Are you... How are you getting? You know, we had Kim 1, 2, 3, 4, 5, 6, 7. How are you deciding who you call on?

Pam  24:49  
So, first of all, good point that I'm definitely deciding who I'm calling on.

Kim  24:49  
Mmhm.

Pam  25:08  
And so, it might depend on the goal of the day. And, for today, I wanted to compare the major strategies that we had developed. 

Kim  25:28  
Mmhm.

Pam  25:28  
And so, I was looking for an example of those major strategies. 

Kim  25:28  
Mmhm.

Pam  25:28  
The kid who talked about repeated addition would have been... I think would have been me thinking the kid was doing something different. So, I don't know that I would have called on the repeated addition kid.

Kim  25:28  
That's good to know.

Pam  26:21  
In this case. So, (unclear). 

Kim  26:24  
Which...

Pam  26:24  
Go ahead.

Kim  26:25  
No, can I interrupt? Because...

Pam  26:26  
Yeah.

Kim  26:27  
...there might be people who push back and say, "Well, why'd you call on, and then not represent their thinking?" So, you didn't necessarily know the things that I was going to say. But if you were seeing, you know you would acknowledge, high five them for thinking about the connection that that's the problem you're solving. But not call them up, and then say, "I'm not representing it."

Pam  26:46  
Yeah. Yeah, it's a little tricky. I definitely want to make sure that the kid is respected for their thinking and that we've honored the fact that they've dug in and they've done fine. You know, they're thinking about thirty-six 15s and they chose to add up 15s instead of add up 36s. I could back up a little bit. If I'm doing this early, and I have a lot of kids that are doing repeated addition or even a handful of kids that are doing repeated addition, then we need to do more work here. In that case, I might have said, "Hey, you who added up the 15s instead of added up the 36s..." so that the kids who were adding up 36s could go, "Oh, look at the numbers before you just start adding. Adding up those 10s and 5s would have been so much easier than all the work I did to add up 36s."

Kim  27:32  
Mmhm.

Pam  27:33  
That could have been a conversation. I chose, having a feeling that you were going to come up with flexible factoring as one of the strategies, that says to me that I don't have a lot of kids that are doing repeated addition. So, how do I say this? I want to honor all kids' thinking by letting them solve the problem any way that they want, interacting with them as I'm circulating. Then I'm going to move the math forward by purposefully choosing specific strategies to compare.

Kim  28:05  
And that's important to note because there are a lot of people who love Problem Talks as a way to have students share what they're thinking and choose to say, "Everybody can share. We're going to put lots and lots of things on the board." And I think what you're saying, for you, Problem Talks are about instructional moves, and they are still about purposely choosing what goes up to move the math forward. And I think that's a big distinction that people need to take away from this, that Problem Talks for you are not just anything goes, put everybody's up as a celebration. They're still instructional.

Pam  28:49  
Very much so. Yeah.

Kim  28:51  
Okay.

Pam  28:52  
Yeah.

Kim  28:52  
I think one more question is we had Kim be a student each time, and so you looked for specific things. How many kids do you have in mind that you would call on? Like, you purposely planned ahead of time what you thought about. You said you were looking for strategies. Is there a number of kids that you think about when you have share?

Pam  29:13  
Yeah, I'm kind of aware that I almost wish we would have done this podcast slightly differently because I might be giving the impression that it's like, "Anybody else? Anybody else?" Where, really, I do think I would have. And it's just kind of the way we set it up today.

Kim  29:27  
Yeah. 

Pam  29:27  
Really, in a classroom, I would have gone to find the strategies that I think will move the math forward. And I don't think I would have called on seven kids. So, depending on what the majority of kids are doing, I want to choose to share something that will help link what they're doing to something more sophisticated. I don't know a different way to say that. So.

Kim  29:50  
Yeah.

Pam  29:51  
Probably four would be around the number. So, maybe three, maybe five. But probably four kids is... Yeah, maybe three sometimes. It's not about getting everything on the board. And I fear that we might. So, we didn't. Maybe we gave that impression today just in the way we sort of set it up in the podcast. When I'm in class, I'm going to be intentional about who I'm choosing. And I think three to four, maybe five is going to be the max that I might have share.

Kim  30:20  
Okay. So, one of the things we didn't mention that is important to mention is that a Problem Talk needs to be a really rich problem.

Pam  30:29  
Mmm, mmhm.

Kim  30:30  
So, that you can get the strategies to come out that you're looking for. So, 15 times 36 was a fantastic problem because there's a lot that you can do with it. 

Pam  30:39  
Mmhm.

Kim  30:39  
What may not make a great Problem Talk is a problem like 47 times 99. Say more about why that's not a good Problem Talk.

Pam  30:50  
Yeah, that's a good choice. If we give the students something like 47 times 99, we're really hoping that they're going to choose an Over strategy because almost everything else is... Okay, everything else is just less efficient.

Kim  31:05  
Cumbersome.

Pam  31:04  
So efficient. And so, it uses place values. We're hoping to develop the intuition in students, so that when they look at a problem, they ask themselves, "What will work well here? What do I know that I can use?"

Kim  31:05  
Yeah.

Pam  31:06  
And we hope we built intuition for that particular strategy, for those numbers that they can choose well. And the only way that we can do that is doing it. You know like, giving kids problems, comparing strategies, and going, "Why? What is it what about these numbers that nudged you that way?" Oh, now we learn, and then hopefully we get better at that. 

Kim  31:12  
Yeah. Okay, I hope that you like what you've put on your paper because we are going to capture that image and share that, so when people listen to this episode, they'll have a place that they can go and look at what your final blog will look like. Sweet.

Pam  31:54  
That'll be fun.  Alright. 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. And thanks for spreading the word that Math is Figure-Out-Able!