[00:00:00.780] - Pam
Hey! Fellow math-ers! Oh, my gosh. I feel like I just totally screamed, "Hey." Hey! Hey. Hey. Hey. Hey! Hey. Hey! Hey!

[00:00:08.650] - Kim
Hey.

[00:00:09.340] - Pam
Okay, that sounds super weird now. How do I... Hey, fellow math-ers! Is that normal? Does that sound normal? I can't even. What's normal? You say it. Can you say that part?

[00:00:18.440] - Kim
Hey, fellow math-ers.

[00:00:20.330] - Pam
Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris, who can't say, "Hey," without feeling really strange, a former mimicker turned math-er. Sorry.

[00:00:29.760] - Kim
And I'm Kim, a reasoner who now knows how to share her thinking with others. At Math is Figure-Out-Able, we are on a mission to improve math teaching.

[00:00:39.800] - Pam
Because we know that algorithms are super cool human achievements. But we're asking you to consider that they're terrible teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning, meaning they never develop the more sophisticated reasoning, than the problems are intended to develop.

[00:00:58.660] - Kim
In this podcast, we help you teach math-ing, building relationships with your students, and grappling with mathematical relationships.

[00:01:05.340] - Pam
We invite you to join us to make math more Figure-Out-Able.

[00:01:09.590] - Kim
What is happening today?

[00:01:11.840] - Pam
Hey! Oh gosh. Do you ever say something and all of a sudden it just doesn't feel right? You're like, "That just... I... Wow, okay."

[00:01:19.160] - Kim
Sometimes. But harder words than

[00:01:21.300] - Pam and Kim
"Hey".

[00:01:23.630] - Kim
Oh, my God. I'm crying.

[00:01:25.730] - Pam
Okay, but I'm just going to note that in our notes you have, "Hey!" You have the word "hey" written there.

[00:01:30.380] - Kim
Yeah. Hey.

[00:01:31.260] - Pam
Hey. Hey.

[00:01:32.120] - Kim
How's it going?

[00:01:33.650] - Pam
It's fantastic.

[00:01:35.490] - Kim
Shoot.

[00:01:36.320] - Pam
It is fantastic. Hey, we're recording on a Friday. That's good. I don't know if we should tell people we record not on the day it drops? Surely they know that.

[00:01:45.280] - Kim
Yeah, yeah, yeah.

[00:01:46.010] - Pam
Do they know that?

[00:01:47.530] - Kim
Surely.

[00:01:47.930] - Pam
Yeah, surely.

[00:01:48.890] - Kim
Okay, we need to move on.

[00:01:50.640] - Pam
Or Kimberly. Shirley or Kimberly. One of the two. Alright, y'all.

[00:01:54.450] - Kim
Alright, last week.

[00:01:54.790] - Pam
Oh. Sorry, go. You say it.

[00:01:56.920] - Kim
So, last week we... Listen, we talked about seeking for clever.

[00:02:01.270] - Pam
Yeah.

[00:02:01.840] - Kim
And that was a really interesting conversation. It was a good conversation because it's an interesting idea that your first inclination isn't always the best you can do.

[00:02:15.660] - Pam
Mmhm.

[00:02:17.980] - Kim
And you had this happen recently.

[00:02:21.020] - Pam
Yeah. And this idea of what does it mean that we kind of have this cultural thing happening where a lot of teachers are asking for more than one way, and demanding it, and grading it. And so, there's another part of this that kind of comes up and that we want to talk about. So, I was recently several places. So, no one's going to know who, where, or what this was. And I'm not going to be negative here, but I think it's, it's good that we just keep general. I actually had a really good time. Fantastic time. It was a great group of educators. Wonderful. We did some really cool work that day. I got to do some learning that day. A different presenter was presenting. I was part of the learning. And they asked us to kind of consider a different way of looking at solving a specific kind of problems than maybe we were traditionally taught. And I was a little skeptical of what we were doing, but not as... I was skeptical that this was something that I was going to turn around and promote that we teach. But I was more than willing to dive in and just sort of start playing with the relationships, see how it works, kind of see if I could get good at using the relationships in the way they were suggesting. And we did some kind of, you know, preliminary kind of problems that I think it made a little bit more sense. And then we kind of said, "Hey, would it apply here?" And I think we got some variables involved. And one of the participants, very great, great person, raises their hand and says, "But... I see what you're doing. But can't I just solve it this way?" And, to that, the presenter said. Kind of took a breath and then... You know, what if you were presenting and you're trying to get people to, you know, consider something and think about something and somebody's like, "Yeah, but can't I just do it..." whatever way. And I don't even know that we knew what way she did. But she got an answer, and she's like, "Can I just do it this way?" And the presenter said, "But that's not what we're doing right now."

[00:04:15.590] - Kim
Hmm, hmm.

[00:04:16.690] - Pam
That's not a horrible thing to say. It's not like a, you know. But why did you go, "Hmm"? Kim, when I say that, what message does it send? Subtly. I think subtly.

[00:04:29.340] - Kim
Well, who? Who's speaking? Because there's a message for both people, I think.

[00:04:33.700] - Pam
Let's start with when the presenter hears, "Can't I do it this way?" And the presenter says, "But we're not doing that right now. That's not what we're doing right now." What's a possible message that the rest of the class could hear when the presenter says that?

[00:04:49.950] - Kim
I think the presenter could be sending the message that they want you to try something, but what they're saying is, and the message that probably is getting received, is what you want to do is not okay right now. That I want you to do what we're doing because this is the right thing to do in this moment. Like, we're doing it my way.

[00:05:14.210] - Pam
Sort of set aside. Instead of solve it your way, and then let's connect. Or solve it your way, and let's consider. It's like, yes, set aside.

[00:05:26.780] - Kim
It's show me my work. And so, it's interesting because you said imagine if you were a presenter and somebody said that. And in my head, I was like, that happens. When we present, it has happened. And I was trying to think about like what would we say or what do we say?

[00:05:42.780] - Pam
Yeah, how would you reply if you're doing a Problem String or something and like, "But can't I just regroup? Can't I just cross-cancel? Can't I just use long division algorithm? Can't I just...

[00:05:54.880] - Kim
Yeah. And I'd say absolutely.

[00:05:56.560] - Pam
type it in my calculator."

[00:05:58.020] - Kim
Yeah, sure, you can. Because we honor people's thinking. Like, we honor students' thinking in the classroom. We honor teachers' thinking in a presentation. So, absolutely, they truly can solve it however they want. That we let them solve it however they want. I mean, we're not giving them permission. They can do what they want. They're adults and humans. But then we even sometimes go over and help them, you know, if they get stuck on the way that they're thinking about it. But then we craft the conversation that is had among the group about where we're headed. And then, we invite them to consider what we are talking about, even if they didn't do it that way. And that feels different to me than somebody saying, a presenter saying, "No, this is what we're doing right now."

[00:06:46.630] - Pam
Yeah. And I think that's super subtle. Would you agree with this? It's kind of subtle. So, please, by any means. We're not suggesting that's what the presenter meant. I'm just kind of pulling a line out because we thought we could kind of talk about what we've seen in a lot of places.

[00:07:02.300] - Kim
Yeah.

[00:07:03.100] - Pam
We see in a lot of places. Like, in fact, I heard Graham Fletcher say something the other day about this is not about... I can't remember his words exactly, but the message I got was it's not about memorizing a strategy on Monday, and a strategy on Tuesday, and a strategy on Wednesday, and I've taught strategies.

[00:07:18.180] - Kim
Right.

[00:07:18.420] - Pam
Because in that mode, you're saying don't solve it any way you want. You are doing this strategy on Monday. Nope, now stop. Even though we learned that yesterday. Now, today you're doing this strategy on Tuesday. Now, stop that. Like, right? Am I? Is that what you're... Is that part of the message that you're saying could be sent if you say, "We're not doing that right now"?

[00:07:37.780] - Kim
Yeah. Yeah. Yeah. It's, "I'm teaching you this thing, and I want you to do it this way.

[00:07:44.280] - Pam
I did it.

[00:07:44.580] - Kim
because that's what we're doing today."

[00:07:45.870] - Pam
Yeah. I did it. We just did it. Now, you do it. We're suggesting that's not the way we math. It's not the way we teach math. When you said just a minute ago. I just want to dive just a little bit when you said we honor kids' thinking. Just to put a fine point on it, we honor kids' thinking by letting them solve the problem however they do. And, like you said, often we will help them if they get stuck in their thinking. Even if it's not the direction we're going, we might... You know like, they're like, "Ah, I'm not sure how. I'm just off here." And we're like, "Oh, yeah." We honor their thinking in that moment. That doesn't then mean that we dishonor their thinking when we then carefully craft the conversation to go a certain direction to move the math forward, that we have a specific goal for the day, and so then we're going to... In fact, you told me this a long time ago. You're like, it's what you celebrate. So, as you go. You know, everybody solved, and as you go to the board, and you're like, "Okay, did anybody think about this relationship?" Or you might say, "So-and-so. I saw what you were doing. Can you share that with everybody?" In that way, you are crafting the direction of the conversation and kind of focusing. And like, oh, I love when you said and inviting everyone into... Can we think about this? Can we make sense of these relationships?

[00:09:03.670] - Kim
Mmhm. It's so funny that you said inviting, because I was just thinking as you were talking like it's my job to extend an invitation to these students that is appealing, and interesting, and intriguing, and they will go, "Huh. Okay. That's interesting." Like, I'm looking up from what I did and I want to be a part of this conversation and see what's going on here.

[00:09:28.750] - Pam
Yeah, and we want them to feel invited, not commanded. And we want them to be naturally intrigued. Mathematics is naturally intriguing. It is naturally interesting. It's naturally patterns. And we are hardwired to look for patterns, and appreciate patterns, and use patterns. We've just gotten away from that as we've been falsely taught that math is to mimic. What were you going to say?

[00:09:53.310] - Kim
Well, it's funny that you know I was going to say something. I like sit forward. Can you hear me move forward? I like lean forward into the microphone.

[00:09:58.870] - Pam
No, I hear you breathe in.

[00:09:59.710] - Kim
Oh, sorry.

[00:10:00.790] - Pam
No, it's okay. I just hear you take a breath. You take a breath, and I'm like, "Oh, Kim's about to talk."

[00:10:04.790] - Kim
That's so funny because we can't even see each other. You know, it's interesting because what I'm thinking about right now is any human conversation that happens. This happens to you and I. We'll start to have a conversation. We're both thinking about something. And we'll say things like, "Can I interrupt? Are you thinking?" Or I'll start to say something, and you'll realize that I'm taking it somewhere else, and you'll go, "Hang on a second," because you need to finish the way you're solving it, or you need to finish the thought you're having, or the conversation, or the point you're trying to make. And then when you've gotten it out, then we're ready to turn and go somewhere else. And that is absolutely true in students in our classroom. Like, if we say, "Stop what you're doing right now. Don't do it that way. This is what we're doing."

[00:10:54.750] - Pam
Turn 180. Forget your brain was going there.

[00:10:58.430] - Kim
Yeah, the chances that they're going to run with you at that point feel pretty small to me.

[00:11:05.250] - Pam
It doesn't feel like you've honored their thinking in that way, right? It doesn't feel like you're helping them connect what they were thinking about to where we're going today. Or that you, you know, when you say, "Solve it any way you want. Stop. Do it this way." That. It's subtle. This is a little bit. We're talking about some subtle things here. Why, because we're on podcast episode 300-something, so we can be subtle. If you guys are like, "This is too subtle." Well, go back and listen to some of the prior ones. Maybe that would be. Yeah. Kim, I feel like you were going to say something about...

[00:11:40.390] - Kim
Yeah, I think when we talked about this and we were talking about the idea of being clever, that sometimes when you're solving a problem, even if you understand the problem, even if you have already solved the problem, then you can engage in the behavior that we talked about last week, that you can engage in still seeking for something clever that you hadn't seen before. And so, you know, it is true that we might have students who say, "I've solved it, and I'm not interested." But, very often, very, very often, we have students who like take a deep breath, and they go, "Okay," and they're willing to dive back in and look for other pathways.

[00:12:28.240] - Pam
And listen to their classmates to understand. And yeah, and that is, in a big way, what I was feeling that day in the particular presentation I was talking about. That I think I said as we were doing the problems, I was thinking to myself... How do I say this? I was weighing out. You know, it was a bit of a way of thinking about a problem that I hadn't really considered making that general. And so, I was curious. I was like, "Will this be generalizable?" And as I'm asking myself that, I'm also curious, "Do I think this is going to be something I promote after this?" I was a little dubious as I was doing it. But even through all that, I was seeking for clever. I don't know if that's the right way to say it in this case. I was seeking to understand. When the presenter was saying, "Can we think about it this way?" And then they put us up at whiteboards, and I was working with a group, we were willing to like math. We were math-ing. We were like, "This is interesting. Let's see if this... Does it connect here?" Or, in fact, part of what I was doing was trying to find a non-example. I was like, "Okay, I see the ones you've done so far all divide evenly. Hmm, I wonder if that works then when they don't divide evenly." I will give them credit. It does. So, that was interesting, right? That kind of thought of like, "Ooh, let me dig into this and see how these..." Most of the people in the room were very willing to kind of let math be figure-out-able and wonder how we could figure that thing out.

[00:13:51.687] - Kim
Mmhm.

[00:13:52.650] - Pam
Yeah. So, there's another part to this story. And I think you kind of said that a minute ago. Like ,when I said, "What message does that send?" You're like, "Well, for whom?" So, I'm curious. When you hear a participant say back to you. Because you've said you've heard it lots. When a participant or a student says back to you, "But can't I just solve it this way?" I want to kind of get under that question.

[00:14:15.300] - Kim
Yeah.

[00:14:16.210] - Pam
I wonder. And I don't know. I haven't met this person before. I'd love to chat about it. In fact, if you're listening to the podcast, I'd love to chat. Send me an email, pam@mathisfigureoutable.com. I'd love to chat about what you were thinking about. But can't I just solve it this way? I wonder if it's possible that somebody who says that might... I'm not describing thinking. But might have, "Well, if I've got this answer, aren't I done? Can I be done? I got the answer. I got the answer this way, so why jump through these hoops up there if I've already got the answer?" Could that? Is that a possible? Do you get that feeling ever, Kim, when you're working with people that maybe they could be thinking...

[00:14:49.620] - Kim
Yeah, I mean, I certainly think it's possible that they could be focused on getting an answer and just like completing the task. "I'm done. I've got it." I also wonder...

[00:15:02.460] - Pam
Well, before you... Are you going to change tacks? Can I stay there for just a second?

[00:15:04.930] - Kim
Yeah, yeah.

[00:15:05.570] - Pam
Like, that's the way we were taught.

[00:15:07.360] - Kim
Sure.

[00:15:07.860] - Pam
Most of us were taught our job in math class was to get an answer, do it the same way every time, so we didn't have to think. And, you know, yeah. And so, we don't blame anybody for being in that mindset. We do want to kind of encourage people to consider there's another mindset. Okay, don't forget what you're going to say. Go ahead. What was your next?

[00:15:25.960] - Kim
Well, and I think especially because you're talking about something that it sounds like it was something maybe new. A new model, new type of problem. I don't know what the new was.

[00:15:33.420] - Pam
It was a little convoluted. Yeah, it was a little bit like, "If you can think about this in this sort of strange way you've never thought about it, can you then think about this in that sort of..." Yeah, I don't know if that helps.

[00:15:42.180] - Kim
So, I wonder if maybe it has something to do with the answer, but also something to do with comfort. Like, "I'm not really sure about what you're doing, but I'm over here safe doing what I know. And that worked." And so, it's geared still towards getting an answer, but like, "I don't need to go where you're going with this stuff because I'm good over here."

[00:16:06.850] - Pam
"I've got a way. I've got a way." Yeah, "I'm kind of comfortable over here. That's kind of odd. Do we have to make it that..." I'll say it. "Do we have to make it that convoluted?" And, Kim, you and I would agree that if we ever feel like something's too convoluted, we step back.

[00:16:22.050] - Kim
Yeah.

[00:16:22.470] - Pam
And we might recognize that maybe we haven't built some reasoning. Maybe we don't have some understanding that would be helpful, so that the thing isn't actually that convoluted. Like, I think that's why I was willing to dive in that day and, you know, "Well, can I make this part of my repertoire?" And then once I own it, decide, hmm, not for me. You know, or not something I'm going to. But yeah. I think absolutely it could. I think we're inviting everybody to consider when you hit something that you're like, "But I have a way." Could that mean that you're leaning back, you're pulling on your experience from when you were taught. "But I've solved it. I'm comfortable. I've done it. Can we just do that? Do we have to get this..." And that could be one way. We're inviting you, what would it be like to see if you can own those relationships, and then make the decision whether you want it as part of your repertoire or not? Is that fair? Is that a fair way?

[00:17:20.020] - Kim
Yeah. Yeah, this makes me think a lot about last week's episode about clever and about do we make people solve things two different ways all the time. And I think it's important as educators that we recognize that our job is to craft experiences for teachers. I mean, for students. And then as teacher educators, for teachers as well. That help them realize what they know and that they can use things that they didn't realize they knew and maybe go at problems a different way. And so, we can help them make connections that they didn't realize that they already knew were there.

[00:17:58.450] - Pam
Yeah, let's talk maybe about an example of that because the first time you and I talked about this, I was like I think we need kind of a concrete. Like, what do you mean by we can help people realize they can use something they didn't even realize they knew?

[00:18:16.390] - Kim
Okay, so like for an example when we have done some MathStratChat problems, where we have something like 26% of 50.

[00:18:26.070] - Pam
Okay. What is 26% of 50? Mmhm.

[00:18:30.710] - Kim
Yeah. So, maybe people can pause and solve that if they like. But 26% of 50 often will have people say, "Well, I know 10%, and 10%, and 5%, and 1%, and they'll gather some percents together. Or maybe they'll say 20% and 6%.

[00:18:47.880] - Pam
Or 25%.

[00:18:49.970] - Kim
Sure, yeah.

[00:18:50.680] - Pam
A quarter.

[00:18:51.010] - Kim
So, lots of piecing their way to getting 50%.

[00:18:55.780] - Pam
Or 20%.

[00:18:56.710] - Kim
Or to get 26% of 50. But then often what will happen is somebody will say, "Well, I thought about 50% of 26."

[00:19:07.550] - Pam
Wait, I thought the problem was 26% of 50.

[00:19:10.360] - Kim
Yeah. Yeah. But often, people will go, "Oh, my gosh!" Like, "Uh, of course. Of course. It's just multiplication. And I can think about what I know with multiplication. And 26% of 50 is equivalent to 50% of 26." And so, they have this moment where they're like, "Uh, duh," but it hadn't occurred to them.

[00:19:35.250] - Pam
Mmhm.

[00:19:35.310] - Kim
And so, they get an idea from somebody else that connects to something that they already knew. And it's that somebody says something that is clever, they connect it to what they already know, and it's a moment where we're making connections for each other and with each other along the way.

[00:19:58.280] - Pam
Nice. And I know just a minute ago you were like, "Ah, duh." But they could also be like, "Ooh, nice!" Yeah. Like, you don't mean that as a negative, "Oh, duh." You mean that like, "Oh, I know that. I know that, and I could use that. How did I not see that before? Okay, but I see it now."

[00:20:13.070] - Kim
Yeah.

[00:20:13.310] - Pam
Yeah.

[00:20:13.870] - Kim
But can I add, though, that seeing it one time is not enough? This is why we love Problem Strings, right, that we're building...

[00:20:20.850] - Pam
Mental pathways.

[00:20:22.540] - Kim
Yep, we're building strategies. We're building those pathways. We're giving them high-dose experiences. And so, it's not surprising to me that somebody who has made sense of that before, but then has done nothing with it, then sees maybe another problem in MathStratChat, uses, you know, some other strategy, sees a comment that somebody made and went, Oh, yeah. I did know that." And they just... You know, we've got to continue to make these strategies available to ourselves and seek for them, so that they continue to be at our fingertips.

[00:20:55.200] - Pam
Nice. Let me just give one more quick example.

[00:20:57.490] - Kim
Okay.

[00:20:58.050] - Pam
Often, when I do a Problem String where there's some subtraction involved, it doesn't even have to be a subtraction Problem String, but there is some subtraction solving in the problem. And maybe we're subtracting 27 from something. And somebody will say, "Well, I subtracted 20, and then I subtracted 7" that's often a good strategy. And then I'll say, "Did anybody subtract not 27? Ooh, Icky. Did anybody subtract 30?" And I'll pause, pause. And then I'll say, "No? Okay, could you?" And in that moment, I will often see that look that you just were kind of trying to describe with those sounds was like, "Oh, yeah. Oh, yeah, I could use. Yeah, I could use 30." And then I'll go like, "Well, tell us about that. Like, tell us about how you..." And then they'll go, "Well, you know, I could subtract 30, but I've subtracted too much." And then I'll ascribe that strategy to that person. Like, "Well, did everybody hear what David just said?" Or Jane, or whoever. You're like, "Oh." And then they're like, "Well, that wasn't what I did." And I'm like, "Well, you just did." Like, "Well, yeah. That's what I didn't do first." I'm like, "Yeah, but do you now?" And they're like, "Oh, yeah, okay, that's Jane's strategy." So, that's just an example of where... Let me give you one more. I do a Problem String where we end up with a table of values. And we've dealt with 1 to 27 and 2 to 54. So, if I double 27, I get 54. So, 1 to 27, 2 to 54. And then I give them completely unrelated data. And I'm very careful to say it's unrelated. And I'll say 1 to 28. Or 2 to... Well, and I should have said we graphed the points 1 over 1 up 27, over 2 up 54, and we could think about the line that contains those two points. And then I say, "How about the points (1, 28) and (2, 55)?" And then I'll kind of say, "Does anybody see 27s?" And people will go, "Oh, yeah. There's like in 1 to 28 and 2 to 55. It's just like the table before, but you shifted both of the y values up 1." And I don't know. Kim, I'm standing here. I just put my arm where I had the line that contained those 2 points, and I just shifted that line up 1. So, it's an example of where it might not have occurred to you to see 27s in 1 to 28 and 2 to 55. But when somebody goes, "Did anybody see 27s?" Like, "Oh yeah." It's almost like a bunch of multiples of 27. It's just, they're all shifted off 1.

[00:23:14.000] - Kim
Mmhm. And sometimes, you know, in a Problem String, it doesn't even take lobbing out what the strategy might be. Like, you just said, "Did anyone think about the 30?" Sometimes it can be as little of a nudge to say, "I wonder if there's anything up here that might be useful to you."

[00:23:28.990] - Pam
Oh, yeah, yeah.

[00:23:30.260] - Kim
And, you know, we have so much video in classrooms, and it's hilarious how when we say those words, kids instantly look up at the board, and they're like, they stare. And then several, several times, you know, we see them then go down and start to work because it's the nudge.

[00:23:47.900] - Pam
I can use that one. I can use that. Yeah.

[00:23:50.560] - Kim
Yeah, yeah.

[00:23:51.590] - Pam
Um.

[00:23:52.170] - Kim
So.

[00:23:52.650] - Pam
Go ahead.

[00:23:53.510] - Kim
Go ahead. We are all over the place.

[00:23:55.690] - Pam
I was going to move on. Do you want to say anything else?

[00:23:57.850] - Kim
Well, I just think it's important for people to recognize that you have access to things that even if it didn't occur to you right in that moment, you have access, and somebody can help you realize that you have access. And that is an important thing that we do as educators is help students understand that it is within their, you know, zone of proximal development to make this leap, and we're helping them get there.

[00:24:26.550] - Pam
Let's say it this way. That Math is Figure-Out-Able, and they can use what they know to figure out things they don't know yet. And, y'all, you might be hearing some people out there say that you can't do this higher math, or this kind of math, or somesomething until you've memorized the basics. You got to really help kids memorize the basics, so they can then actually do some math. I'm just going to invite you to consider that that perspective fundamentally misunderstands this entire episode and what we're talking about. That we do not believe that students are an empty vessel that we pour math into. We believe they come to us with some. Now, obviously some have less experiences, some have more experiences. We find out where they are, what they can do, and we build on that.

[00:25:09.740] - Kim
Yeah, we're seeking for the "ooh" factor in our teaching.

[00:25:12.410] - Pam
Yes.

[00:25:13.050] - Kim
And adults can have these moments as well, and we seek for those in our professional learning.

[00:25:17.850] - Pam
Absolutely. Alright, nice. Kim, thanks. 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 let's keep spreading the word that Math is Figure-Out-Able.