[00:00:01.300] - Pam
Hey, fellow math-ers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris, a former mimicker turned math-er.

[00:00:10.770] - Kim
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.

[00:00:18.930] - Pam
Because we know that algorithms are amazing historic 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 should be, could be developing.

[00:00:33.540] - Kim
In this podcast, we help you teach math-ing, building relationships with your students, and grappling with mathematical relationships.

[00:00:40.910] - Pam
We invite you to join us to make math more figure-out-able!

[00:00:44.990] - Kim
Hello.

[00:00:45.790] - Pam
Hey. So, I kind of have a fun thing that we're going to start with today, if it's okay with you.

[00:00:50.130] - Kim
Yeah.

[00:00:50.710] - Pam
So, little-known fact. My third son, Craig, is actually our podcast editor.

[00:00:55.810] - Kim
Love Craig.

[00:00:56.870] - Pam
He has a... He is awesome. He has a real job. He has a kid and a half. And he likes...

[00:01:02.070] - Kim
Is that true? Okay, hang on a second. Kids aren't half.

[00:01:05.440] - Pam
He has a kid and, uh, 9/10 of a kid.

[00:01:09.290] - Kim
The kid is the kid.

[00:01:10.670] - Pam
I mean, they're due. No, they're due anytime now.

[00:01:13.700] - Kim
I know, but the kid is...

[00:01:15.460] - Pam
Oh, kid's still a kid. Okay, that's true.

[00:01:19.410] - Kim
Alright, alright, alright.

[00:01:21.170] - Pam
He is going to have a baby any moment now. Which, actually, when this podcast airs, I bet he does have. Anyway.

[00:01:27.210] - Kim
That's great.

[00:01:27.900] - Pam
He's amazing. He worked for us for a while when he was at university, and so he knows a lot about. And he likes to edit the podcast because it's a way for him to kind of keep his... He knows what we're doing, right, as we talk about stuff. And anyway, so he came over for dinner with his family and my other kids. And we have Sunday dinner together. It was awesome. Did a new recipe. Thanks, Reba. My sister's name is Reba, and she gave me a new recipe. It was yummy. Totally yummy. And he said, "Hey, would you be interested to know what I thought about the episode that you and Kim did about explicit teaching? Explicit teaching. I think we called it explicit teaching. It was episode 325.

[00:02:05.650] - Kim
Yeah.

[00:02:06.210] - Pam
And I was like, "Yes!" Yeah, I love. Craig and I can just talk about anything all day long. It's amazing. And he said... So, I'm actually going to quote him because we had a chat and I said, "Will you say that?" and he gave me a kind of a funky voice memo. So, I did the transcript, and I'm actually going to quote this voice memo that he sent while we were sitting there stirring stuff Sunday dinner. So, he said, "I really like the term 'thinking land' and how you can use it to guide you into the types of teaching to use and when to be explicit and when not." And I was like, "Thinking land? Tell me more about that."

[00:02:43.780] - Kim
Yeah.

[00:02:44.340] - Pam
And then he gives an example from the episode that we did a few weeks ago, 325, where we used two phrases that you and I kind of joke with each other about because they... Go listen to 325, everybody. But, in brief, they remind us of when we want to be direct when we're teaching and when we want to more help kids grapple and solve problems that are figure-out-able. When something's social versus when something's logical-mathematical. And so, he said... The 2 phrases are, "Just write that down" and "Sure wish I knew something about quarters". And he said we could use those... Or we did. Sorry, in the episode we used those two phrases to talk about when to be direct and when to help create experiences where students can grapple and solve problems as figure-out-able.

[00:03:29.240] - Kim
Yeah.

[00:03:29.650] - Pam
Did I just give enough background for the rest?

[00:03:32.550] - Kim
Yeah, I think so.

[00:03:33.600] - Pam
Okay, cool. So, he said, "For example, with the 'Just write that down', it's not taking students out of thinking land. It's saying, 'Okay, put that thinking that you've done. Put it down, so that we can keep track of it, and we can continue thinking.'"

[00:03:51.640] - Kim
Yeah.

[00:03:52.530] - Pam
"Whereas you could have a teacher..." Still Craig, quoting him. "Whereas you could have a teacher say, 'Okay, so next we're going to do this. Write this down. Write the things that I just wrote down,' and then that triggers the student to get out of thinking land and get into rote memorizing and mimic land and to just follow what the teacher's doing and remember the steps."

[00:04:13.920] - Kim
Oh, you know what? It's making me think that did we describe well enough in 325, that episode? Did we describe that what I said when I said to the student, "Write that down," that it was her thinking. She had been involved in the thinking, and then I felt like she was kind of at capacity. And so I said...

[00:04:34.800] - Pam
To keep it all in her head, right?

[00:04:35.810] - Kim
Right. So, I said, "Write that down. We're freeing that up, so that you can remain in thinking land." Yeah.

[00:04:41.760] - Pam
Exactly that. Yes. But it's interesting that he's like, "Because you could hear somebody say, 'Write that down,' as in I'm going to be direct and tell you what to do, which would then cue a kid to get out of thinking land." That's why he kept saying, "Thinking land can help teachers realize if what I'm about to do will keep kids in thinking land, yes... If what I'm... Like, 'Write that down, so we can keep following your thinking and keep using it to reason.' Or is what I'm about to do, 'Okay, everybody, write this down because you're going to mimic this.' Bam, you've taken them out of thinking land. So, it's not a magical phrase you use. It's why you're using it and what it will trigger for the kid." So, then he goes on, "Similarly, we used, 'Sure, I should know something about quarters.'" Because, Kim, we joke about that because you used it with a group of students who were kind of fussing with some relationships. And the problem included quarters, and so you knew that if... And you'd heard one of the kids said something about quarters. And so, you knew if you threw that out, that would keep them thinking, right?

[00:05:41.300] - Kim
Mmhm.

[00:05:41.800] - Pam
So, Craig says, "Similarly with the, 'Sure wish I knew something about quarters,' that's inviting students to stay in thinking land, to explore in thinking land further to see if there's a relationship that would be helpful."

[00:05:54.540] - Kim
Mmhm.

[00:05:54.670] - Pam
"This is opposed to a teacher saying, 'Okay, remember, for this one we can do quarters.' That takes students out of thinking land and says, 'Right, I need to stop thinking, and I need to stop looking for relationships that I can use. Instead, I need to remember for this problem what I'm supposed to do is use quarters. I don't need to think about it, try to figure it out. I need to remember that this is the approach for this problem.'"

[00:06:19.160] - Kim
Yeah.

[00:06:19.980] - Pam
That's so... He's so good! Way to go, Craig! And then, so he's listening to us right now because he's editing this podcast.

[00:06:26.220] - Kim
Must have had some good teachers and parents.

[00:06:28.360] - Pam
Oh, yeah! That's awesome. Because I'm his parent and you were his teacher. I love it. I love it. So, then he says, "So, I like thinking land, and I think it needs to be part of the conversation." And I think that's brilliant.

[00:06:41.570] - Kim
Agreed, yeah.

[00:06:41.840] - Pam
Brilliant. At one point my daughter-in-law came in. Not his wife. And was like, "What are you guys talking about?" And he's like, "Oh, it's math pedagogy, something we never talk about in this house." And everybody just cracked up.

[00:06:52.460] - Kim
Yeah, for sure.

[00:06:53.330] - Pam
Because we talk about it maybe a little bit.

[00:06:54.930] - Kim
Yeah, I like it.

[00:06:55.770] - Pam
So, Kim, I think that we can... I'll start adding that to my repertoire that, you know, one of the ways that we can help each other realize what to do in this moment, should I be direct? Should I try to create an experience where kids are figuring or knowing that math is figure-out-able using relationships, keeping them in thinking land? How do we keep kids in thinking land? If keeping them in thinking land, we can be direct, do it. If keeping them in thinking land is raising an idea, lobbing it out, so that they stay in thinking land and wonder about using that relationship, and mess with it, and grapple, then bam. So, let's do that, Kim, with one more example Problem String.

[00:07:35.110] - Kim
Okay.

[00:07:35.640] - Pam
We've done this in the last couple episodes. I have a new one for you today.

[00:07:38.610] - Kim
Okey dokey.

[00:07:39.330] - Pam
And I'm just going to try... Help me as we go through. If you hear something direct or more experiential, point it out. Alright, so here's the first problem. 2 times the quantity x minus 5. Close that quantity. So, 2 "parentheses" x minus 5 "parentheses" equals 8. And we're solving for x.

[00:08:00.630] - Kim
Okay. You said 2 times the quantity x minus 5.

[00:08:03.980] - Pam
Yep. And that all equals 8. So, sometimes in a Problem String on the podcast, you kind of answer and stuff. Today, I'm just going to talk you through it. So, I would put this up on the board, and I would say, "Hey, I'm super curious. What value for x would make this true? And I wonder how you're thinking about it." And then I would very purposely walk around or maybe ask kids questions. But I would try to find somebody who does what I think is the typical. So, I think typically kids would do what they've kind of seen done before, and they would multiply through. They would distribute. So, they would say... Oh, go ahead.

[00:08:38.750] - Kim
You mean typically because they've been told that before? Or?

[00:08:42.500] - Pam
Yes.

[00:08:42.930] - Kim
Okay.

[00:08:43.150] - Pam
Yes. Yeah, thank you. Yes. I think we kind of have a tradition.

[00:08:48.280] - Kim
Mmhm.

[00:08:48.410] - Pam
Where we often say when you see 2 times a quantity, then you should distribute that 2 through.

[00:08:53.250] - Kim
Right, okay.

[00:08:54.080] - Pam
Okay. So, you know, and often teachers practice that and whatever. So, I would expect that I would have somebody share that there's like, "Well, 2 times x would be 2x. And 2 times -5 would be minus 10. So, now underneath that I've written 2x minus 10. And that still equals 8." And then they might say, "Well, something minus 10 was 8. And if something minus 10 was 8, and that something is 2x, then what minus 10 is 8? That has to be 18." And so, they would say, "I know that 2x has to be 18. And now 2 times something is 18, so x has to be 9." So I've just kind of written that down. Now, notice I've tried to stay in thinking land while doing that. I tried to be relational about solving that. Not just doing a bunch of steps.

[00:09:36.770] - Kim
So, let me... I'm going to interrupt you here.

[00:09:38.840] - Pam
Mmhm.

[00:09:39.010] - Kim
Because you did say that relationally, and I think many students would say, "And then you add 10 to both sides." And, you know, they would describe the steps that they were told to do. So, I think right off the bat, something that you do that is helping them stay in thinking land is that when you say that back to them, it's very purposeful.

[00:09:59.770] - Pam
Yeah. Yeah, true that. Then, while I was looking for someone to solve that problem. So, I was very intentional about which one I was going to have solve first, and it was to kind of get everybody on the same page. Yeah. Everybody's probably seen this. I guess I'm coming at it from a high school perspective. Kids have done this before, so that's kind of what's in my head.

[00:10:19.510] - Kim
Yeah.

[00:10:20.040] - Pam
Then, I might... I will, specifically, have been looking for a kid with this strategy. If I don't see it, it would be the next problem in the string. But if I see it, then I would say, "Hey, Kim, how would you solve this problem?" And so, Kim, I'm kind of asking you to read my mind a little bit, but I have a feeling you have a different strategy.

[00:10:37.580] - Kim
Yeah. So, if 2 times the quantity x minus 5 is 8, then just x minus 5 would be 4.

[00:10:46.810] - Pam
Why?

[00:10:48.090] - Kim
Because if you have 2 of something is 8, then just one of them is half as much.

[00:10:54.260] - Pam
Nice. Yeah. And so, right next to the equation, I wrote x minus 5 equals 4. So, now I have Something minus 5 is 4. So, now I would say, "What minus 5 is 4?" So, then you also got x equals 9.

[00:11:06.670] - Kim
Yeah.

[00:11:07.490] - Pam
2 different strategies. And I would go, "Huh. But I thought you were supposed to like multiply the 2 through." And I would try to engender a conversation where kids could go, "Well, I guess you could do either of those. You could distribute that 2 through, and then solve the equation. Or you could say, "Well, if 2 times that quantity is 8, then 1 of those quantities would have to be half as much." Just like you said. So, I'm going to be very purposeful about who I call on and very purposeful about how I model those strategies. Like you said, I'm going to try to bring meaning to the steps they might be just mimicking. If nobody said Kim's strategy, the next problem in the string would be, "Hey, what if I've got x minus 5 equals 4? Go ahead and solve that any way you want. Then I'm going to try to create a conversation about are these two related and how? And pull out Kim's thinking.

[00:11:54.410] - Kim
Okay, before you go on.

[00:11:56.090] - Pam
Yep.

[00:11:56.350] - Kim
So, what I heard was you are directly stating, "You can do both of these."

[00:12:00.810] - Pam
Yes.

[00:12:00.900] - Kim
You're acknowledging they are both acceptable ways. But then behind that you wondered, "I wonder if there's... I don't remember exactly what you said. But you...

[00:12:09.950] - Pam
Wonder when it might be helpful to do...

[00:12:11.730] - Kim
Yeah, Kim's strategy.

[00:12:12.930] - Pam
Yeah, yeah. Do we always do Kim's strategy? Only sometimes? I wonder if there's a time where it might be beneficial to just do one or the other. Cool. Okay, next question. Then I'm going to say, "9 equals 3 times the quantity x plus 2. Solve that any way you want." I'm going to let kids solve it. I'm going to walk around. I'm going to find somebody who distributes because I'm 90% sure someone's going to distribute. If they don't, I might say, "Somebody talk us through solving this distributing." Then I'll model that. So, I'll have something like, so 9 equals 3 times the quantity x plus 6. And I'll say, "Okay, what are you thinking about?" Something plus 6 was 9. So, that something, the 3x is the something. So, something plus 6 is 9. That's going to have to be 3. So, now I know that 3 has to equal 3x. 3 times what is 3? And so, x is going to be 1. And I've just written down sort of what you would think typical equation solving things. I've got 9 equals 3 times the quantity x plus 2.
Underneath that, I've got 9 equals 3x plus 6. Underneath that, I've got 3 equals 3x. And underneath that, I've got x equals 1.

[00:13:15.950] - Kim
Mmhm.

[00:13:16.670] - Pam
I could have written 1 equals x. Either way. In fact, I just erased it and wrote 1 equal x. And then I'm going to say, "Did anybody do Kim's strategy?" Or if nobody had done it, but if somebody had said something about it, I might've tried to give them that name. You know, or give them the possession of it. The... Not the...

[00:13:33.850] - Kim
Ownership. Ownership.

[00:13:34.790] - Pam
Ownership. That's the word I was looking for. Thank you. You know, did anybody think that way? And then someone's going to... Well, in fact, do you want to just like. How would you think about this one?

[00:13:44.040] - Kim
If 3 of something is 9, then 1 of something is 3. So, the thing is x plus 2. So x plus 2 is 3.

[00:13:53.160] - Pam
Yeah. And I wrote down 3 equals x plus 2. Yeah. But I could have written x plus 2 is 3. Either one. And then I'm going to say, "So, something plus 2 is 3. What would that be?" And that would also be 1. And x equals 1. Or 1 equals x. Cool. "So, it looks, guys, like we could distribute. But we could also look to see if there's a relationship we could use, and just sort of divide first. In fact, as we look back, it looks like, kind of on the left-hand side, it's like you multiplied by 2 first. Like, you distributed the 2 first. And then you ended up having to divide by 2 in the end."

[00:14:28.310] - Kim
Mmhm.

[00:14:29.650] - Pam
"Versus Kim divided by 2 in the first, and then just had to sort of mess with what was left over. Or if we look at the second problem, you could distribute the 3, but then in the end you were going to have to divide by the 3. Or you could just divide by the 3 first." Now, as soon as that comes out, I'm very explicitly saying that. "Looks like you could do this or that." Very intentionally. Like, just saying it out loud. Almost suggesting that, "Well, it looks like we better just do Kim's method all the time." And I'm going to kind of almost set that up as, "So, do we always want to do that?" Next problem. How about if I were to have 16 equals 4 times the quantity x minus 6? So, I don't know, Kim, if we need to talk about that. But I would let students solve it any way they want. I would ask somebody, "Talk us through distributing first." I would ask somebody to talk us through, in this particular one, since you have 16 equals 4 times a quantity, you could divide by 4 first.

[00:15:22.330] - Kim
Mmhm.

[00:15:23.120] - Pam
I would again very explicitly bring out, "Huh. It looks like we could multiply by the 4, distribute the 4, then we'd have to end up dividing by the 4 at the end. Or we could just divide by the 4 at the beginning, and we have sort of less to do. Wow, seems kind of efficient."

[00:15:36.260] - Kim
Yeah.

[00:15:36.500] - Pam
Then at that point, I might get a little bit more complicated with something like 2 times the quantity x minus 1 plus 8 equals 10. Just a little bit more complicated. Kind of let kids do whatever they want. And then very intentionally, I'm going to pull out both of those strategies. We're going to keep comparing. And then I might throw out something like... And I'm actually making this one up on the spot. Like 3 times the quantity x plus 5 equals 5. Mm, 7. Anything not divisible by 3, really. And then I'm going to go, "Okay, it seems like we better do Kim's strategy here. Like, let's divide first." And let them fuss a little bit. This one's actually not a bad one. Because I'm going to have kids who can 3 times the quantity x plus... In fact, Kim, do you want to run with your strategy?

[00:16:23.970] - Kim
What did you say it was? 3 times?

[00:16:25.730] - Pam
3 times the quantity x plus 5. End that parentheses. Equals 5.

[00:16:31.380] - Kim
Equals 5.

[00:16:32.270] - Pam
Yep.

[00:16:33.250] - Kim
So, then x plus 5 would be equivalent to five-thirds.

[00:16:38.970] - Pam
So, you are going to go ahead and divide even though there's not a common factor. Tricky.

[00:16:43.820] - Kim
Yeah.

[00:16:44.470] - Pam
Now, you've got something plus 5 equals five-thirds.

[00:16:48.670] - Kim
So, five-thirds minus 5. So, yeah.

[00:16:52.790] - Pam
Yeah. And then I can let you sort of simplify that if you want to. And that's going to be x.

[00:16:56.560] - Kim
Yeah.

[00:16:56.870] - Pam
Or. And I just made this one up, so it's not a real even answer. But that's okay. Or we could have distributed first and solved. And now we can have a question. When do you want to distribute first? When don't you? You could have. And we really want to pull out the conversation about common factors. And the kids are saying, "Huh, I could look for common factors first."

[00:17:17.810] - Kim
Yeah.

[00:17:18.370] - Pam
This is a desirable outcome where kids say to you, "Huh, we could look for common factors first." That is an outcome of keeping kids in thinking land where they have come up with a generalization that, "Man, if I've got common factors happening on both sides of the equation, why don't we just divide first? And then I don't have to multiply it all the way through, distribute it through, and then end up dividing at the end. I can just get rid of it." If there's not common factors, eh, I kind of have a choice, you know?

[00:17:47.650] - Kim
Yeah.

[00:17:47.760] - Pam
Like, maybe it's smart to do, maybe it's not. But I could go that direction and kind of see how it works out. Yeah.

[00:17:53.760] - Kim
Yeah. And well, I was going to say, so for this particular one, even though there aren't common factors, it's not as messy. But if you had something like, you know, on both sides you have that you're adding or subtracting something along with, then it gets a little bit messier. Then you might want to do distribute first rather than looking for the common factors.

[00:18:20.960] - Pam
Absolutely.

[00:18:21.860] - Kim
I think the point that you're raising is that you're looking and analyzing and you're almost naming, "We've got a couple of strategies. Look first for patterns." And I think that... "Or look for relationships." I think it's an appropriate direct thing to say. "When you have a mathematical problem and there are a variety of strategies, before you dive in, we're going to look. We're going to analyze the problem."

[00:18:47.050] - Pam
Yeah, it's figure-out-able. And one of the things that's figure-out-able is what would be a good first move?

[00:18:52.520] - Kim
Yes.

[00:18:53.160] - Pam
Yes. And that is a desirable outcome. And we can... Hopefully it's helpful to think about am I being direct here to keep kids thinking? Or am I being direct here to have them now go out of thinking land and just rote memorize or mimic something that they've seen before? I think that's super helpful.

[00:19:11.210] - Kim
Yeah. You know, it reminds me when... You know, we've talked about this, maybe even on the podcast, when I was working with some first grade students and the teachers said, "These kids have some strategies, and I know they do because I've worked with them, but they default to the same thing."

[00:19:28.760] - Pam
Counting by ones.

[00:19:29.730] - Kim
Yeah, right. You know what I'm talking about.

[00:19:32.460] - Pam
Yeah.

[00:19:32.530] - Kim
And so, the conversation that I had with them was, "You know things to do. I want you to stop and I want you to think."

[00:19:41.110] - Pam
You were very direct.

[00:19:42.010] - Kim
I was very direct. I was very direct because I knew that they had relationships and strategies that they could be using, and so at that point, I was directly saying, "And now, it's time to use strategies."

[00:19:54.200] - Pam
Yeah.

[00:19:54.350] - Kim
Because I knew that they had been built in them.

[00:19:56.690] - Pam
And that direction that you gave them was to keep them or get them into thinking land. You knew that they could use those relationships. "You guys, it's time. We're using the relationships." Yeah. That wasn't you saying, "Okay, guys. I had you memorize these strategies, and now you must mimic those strategies we've memorized before." Nope.

[00:20:16.710] - Kim
Yeah.

[00:20:16.930] - Pam
You've developed the relationships. Strategies are becoming natural outcomes. Let's use these now. You're saying, "Use them now." Direct. Nice. Craig, thanks. Thanks for the little heads up about thinking land. I love it. And, 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!