[00:00:01.150] - Pam
Hey, fellow math-ers! Welcome to our closet. I mean, to the podcast where Math is Figure-Out-Able. I'm Pam Harris. Oh, gosh. A former mimicker turned math-er. Now, I just almost swallowed the honey drop that I'm... It's going to be a day, y'all. Hang on tight.

[00:00:17.260] - Kim
And I'm Kim Montague, live from my closet. And I'm 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.

[00:00:27.130] - Pam
Y'all, we know that algorithms are super cool achievements, but they're not good teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than we could be, should be developing. Hey, Kim!

[00:00:43.220] - Kim
In this podcast, we help you teach math-ing, building relationships with your students, and grappling with mathematical relationships.

[00:00:50.700] - Pam
We invite you to join us to make math more figure-out-able! And how you doing over there?

[00:00:56.000] - Kim
Oh, my gosh. It's going to be a day. Holy cow. I've already dropped the microphone. I've got no big screen. Probably can't see anything.

[00:01:03.120] - Pam
Ah, yeah.

[00:01:03.540] - Kim
It's going to be great. It's going to be great.

[00:01:05.560] - Pam
So we've shifted location just a little bit. You might not know, I record from my house and Kim records from her house. And our podcast editor, who we love, told us that he was hoping for some better audio from Kim. And so, Kim has wonderfully voluntold. How do you say that? You said you'd be willing, but maybe not so much anymore. To be in a... You have a tall kitchen, right? That's where you used to record.

[00:01:31.060] - Kim
Yeah, dining room. Mmhm.

[00:01:32.890] - Pam
Yeah. So, a little echoey kind of. And he was like, "Stop the echo!" Anyway, so we're going to give this a try and hopefully not drop the microphone again. That's why we were laughing when we got on. Okay. So, Kim, the other day.

[00:01:46.970] - Kim
Yep.

[00:01:47.110] - Pam
We were having a conversation with some friends of ours at Innovamat.

[00:01:50.180] - Kim
Mmhm.

[00:01:50.270] - Pam
They are super thoughtful. Love having conversations with them. Deep thinkers. And some stuff came up that we thought we would continue to beat out on the podcast.

[00:01:59.880] - Kim
Yeah, yeah. And I think it's fair to say this is like fresh, right? This is why we have a podcast because when we're talking about something, and we have a conversation, and we don't want it to stop, we just say like, "Let's keep talking about it on the podcast." So.

[00:02:12.310] - Pam
Take it to the podcast.

[00:02:13.260] - Kim
Not always fully formed, but it's fresh and it's on our minds.

[00:02:17.810] - Pam
And we reserve the right to tweak things as we go and maybe take stuff back.  Because it's new. Yeah, new. Okay. I don't even remember how it started, but it came up. I think somebody said something about 4 quadrants, and we kind of started fleshing that out. We were talking about maybe teachers that we work with or ourselves. And okay.

[00:02:42.210] - Kim
Yeah, I think we were talking about types of teaching styles, right? Like when are you... There are teachers who are more of one kind of teacher versus another. And I think you chimed in and said, "I think there's another thing to consider."

[00:02:59.760] - Pam
Yeah. And it had to do with... So, types of teaching as in maybe you lean towards more direct instruction. You tell kids what to do. Or you lean more towards inquiry. We've just had several podcast episodes where we kind of were ferreting out that we are always intentional when we teach. There are times when we tell kids to write stuff down. There are times, you know, directly tell them what to do. Those are often more in social situations. But really it's all about keeping kids in thinking land.

[00:03:29.390] - Kim
Yeah.

[00:03:29.580] - Pam
So, there are definitely times where it's more inquiry. But, you know, you might lean one. As we talk to people, they might lean one way or the other. You know, not... I don't... More oriented towards direct teaching or more oriented towards inquiry. But I also... Was it me?

[00:03:46.010] - Kim
Yeah.

[00:03:46.220] - Pam
That brought in that there's also this bit about whether you math or less math. How do I say that? Whether... Like, when I started teaching, I wasn't really math-ing. I didn't really know what math-ing meant. I thought math was I wait till the teacher told me what to do, and then I rote memorized and mimicked that stuff. I did a lot of thinking, but it was always about someone else's thinking. It didn't occur to me that I could start with what I know. So, we kind of brought up that maybe you might be out there saying, "Ah, you just got to tell kids what to do." And that, to me, then it kind of depends on when you say that, are you coming from a place of, "Yeah, because I'm going to tell kids these relationships. I math and I use relationships and connections to solve problems, and so I'm just going to tell kids to do that." Or are you coming from a place more like I was as a new teacher? Where I'm telling kids what to do because I think that's the nature of math-ing. Does that make sense?

[00:04:43.810] - Kim
Yeah, yeah.

[00:04:45.080] - Pam
So... Yeah, go ahead.

[00:04:46.520] - Kim
Well, I was going to say, so maybe let's describe the...

[00:04:50.460] - Pam and Kim
Four quadrants.

[00:04:51.500] - Kim
Yeah. And then we'll get into each type.

[00:04:54.490] - Pam
Okay, so the kind of the way that I was thinking about the axis is along the left side is sort of the orientation about the way you're sort of oriented toward teaching. So, it's kind of on a continuum, more direct teach to less. More direct teach towards more inquiry. Does that make sense? You're sort of...

[00:05:15.030] - Kim
Mmhm.

[00:05:15.070] - Pam
And I'm pointing up and down. So, for whatever reason, I put oriented to more direct teach at the top, and then sort of going down towards leaning more towards inquiry.

[00:05:24.920] - Kim
Yeah.

[00:05:25.330] - Pam
Okay. And then the other axis going from left to right is whether you math, you have experience where you use relationships, connections. You know what math-ing is. Going to the right is more like I used to. Not so much math-ing and more mimicking. Rote memorizing and mimicking. So, if those are the two axis, let's describe the four quadrants. I'm going to let you do that.

[00:05:50.590] - Kim
Oh, okay. So, if you tend to lean towards more direct teaching and you math.

[00:05:59.910] - Pam
Mmhm.

[00:06:00.220] - Kim
That's in the kind of upper left quadrant.

[00:06:02.470] - Pam
Yeah, yeah.

[00:06:03.190] - Kim
And towards the right quadrant, the upper right would be where we kind of placed the maybe you have a more direct-oriented classroom and you didn't maybe math a lot. You solved problems in math class, but maybe you weren't math-ing.

[00:06:19.090] - Pam
More mimicking. Mmhm.

[00:06:19.810] - Kim
Yeah. And then bottom left, maybe you're a more inquiry-based teacher and you math. And then in the bottom right would be maybe you have an inquiry-based classroom, but you're not doing a lot of math-ing or you didn't do a lot of math-ing.

[00:06:36.160] - Pam
Yeah, so if you don't mind, let's describe kind of some things that are happening in our heads when we think about these four quadrants.

[00:06:43.250] - Kim
Yeah.

[00:06:43.370] - Pam
Can we start with the lower right? Is that totally backwards from where we just did? It's the last one that you mentioned. So, you kind of lean towards inquiry, but you're... But. And you think that math is more rote memorizing and mimicking.

[00:07:00.520] - Kim
Lean towards inquiry?

[00:07:02.060] - Pam
Lean towards inquiry. Did I say that wrong?

[00:07:04.170] - Kim
Yeah.

[00:07:05.120] - Pam
Oh, sorry.

[00:07:05.680] - Kim
Yes.

[00:07:06.060] - Pam
Yeah, lean towards inquiry but have had less experience math-ing, and so you still think math class is about getting answers, rote memorizing and mimicking.

[00:07:16.670] - Kim
Yeah.

[00:07:16.910] - Pam
So, to me, these are the teachers that I work with that... And this is so cut and dry, so like nobody think that I'm talking about you, right? It's just kind of a way of sort of thinking about these 4 quadrants. These are folks I meet with that go to some PD, and they experience some kind of inquiry something, and it feels really fresh and energetic, and they're like, "Oh, I could do this cool thing in my math class," and so then they turn around and they try it. They're oriented towards inquiry, towards, you know experience. And they find something, and so they go try it in their classroom and they're like, "Ah, it's just the kids aren't figuring out the steps on their own. It was kind of cool. It kind of brought some energy into class." But now either they might think it failed because kids didn't come up with the algorithm and they weren't all doing the same steps by the end. Or they might think, "Okay, now that we've had some kind of fun, and I've got kids kind of energized, now I'm going to give them the steps, and now they're going to mimic those steps." Does that track for you?

[00:08:17.950] - Kim
Yeah. And did I hear you say that that was you early on?

[00:08:22.290] - Pam
It was. Not quite at the beginning. I would say I was more in the upper right-hand corner at the beginning. Okay. So, the upper right-hand corner is where you're not really math-ing. That was me. I wasn't math-ing. I was rote memorizing and mimicking. And I thought that the best way to do it was to just be really clear about the rules.

[00:08:40.180] - Kim
Sure, yeah.

[00:08:40.550] - Pam
This was me when I was a tutor in the math lab at Brigham Young University. So, it was a great job. I just got to wander around. People would raise their hands, and I would go up to them, and I would say, "What are you doing?" And man, it could be from all different levels of math. And they would say, "How do I do this problem?" And I would take a deep breath and I would go, "Okay, so cosine, cosine, sine, sine, different sine," and I would like have some kind of rhyme or, you know, how did I remember the thing? Now, often I tried to make sense of when to do which thing. But I often had some kind of mnemonic or something that would help me remember the steps that belonged to those rules. And like, for example, if somebody was solving a log problem, I would say, "Hey, if the number's in the front, then put it up on the exponent. And if it's in the exponent, put it up in the front." And they're like, why? And I'm like, "Because you just do. Like, it's just that just remember, these are the things to do." So, I wasn't really math-ing, and I was trying to be super clear, super direct, super, "These are the steps, this is what you do." But then I began to experience graphing calculators. And like they would just come out. And my cooperating teacher was using them. And I could play with the math. And I watched her teach. Pam Giles, like I don't even know where you are these days, but hats off to you. I just learned so much from you. I started hanging out with T³ instructors from TI, Teachers Teaching with Technology. And I experienced some inquiry kinds of ways of teaching. And I was like, "Okay, like, this is amazing." I started to lean towards inquiry, but I wasn't math-ing as much yet. Like, again, I don't want to, you know like, caricature anybody as putting anybody too much. So, now I was more like in the lower right-hand corner. I still wasn't math-ing. I was math-ing more. But I had this feel and a sense that inquiry would bring some different energy into class and that there were times where we could do that kind of thing.

[00:10:30.440] - Kim
Yeah. You know, I'm glad that you said continuum when you were talking about the axis and the shift that happened because it isn't that people are like, "This is me in this one quadrant.This is me. I'm stuck here. You're describing who I am."

[00:10:44.900] - Pam
Yeah, yeah, yeah.

[00:10:45.350] - Kim
It's a current state of maybe the kind of teaching, but that I think what's exciting here is that you and I have shifted where we were based on the work that we do and what we've become aware of. So, should we talk about the other quadrants?

[00:11:03.500] - Pam
Yeah, yeah.

[00:11:04.130] - Kim
Okay. So, we went top left into top right. So, we're going to go clockwise. So, let's go now to the bottom left is where we put things. This is maybe you math, maybe you've always math-ed, maybe you math now, and you lean towards an inquiry-based classroom. And this is where you and I are now, but this is where I kind of lived growing up. I was a math-er, and I played with relationships, and connections, and patterns, and all the things, and so my classroom would look very much like exploring and some really guided discussions. And so, that's kind of the work that we talk about now and what we see for our students.

[00:11:51.310] - Pam
Mmhm.

[00:11:52.530] - Kim
But that wasn't my first year of teaching. My first year of teaching, I would've been in that top left quadrant where I math-ed, but I was very direct teach focused. Which I think is an interesting place to be because it seems to me now that if I math, then I would want my kids to math and to build it through inquiry-based experiences. But as a first-year teacher, I was not aware why I math-ed the way that I did and other people didn't.

[00:12:26.320] - Pam
Sure.

[00:12:26.480] - Kim
And so, I kind of lived under this assumption that I somehow made it happen for myself and really hadn't explored why. So, I just assumed that when my teachers taught me very directly, that I made sense of it, and so if I then was a direct teach teacher to my students, they would also become math-ers like me.

[00:12:51.830] - Pam
Sure.

[00:12:52.070] - Kim
And so, my very first year of teaching, maybe my second year of teaching, I did very similar direct teaching to what we were taught. And then I kind of made a shift to when I started becoming aware of, you know, you helped me unpack and Fosnot's work helped me unpack like that I was math-ing maybe differently than what we were taught. Then I probably. I don't know. But probably in some ways went back and direct taught the relationships that I saw. So, I wasn't direct teaching steps anymore, but I was direct teaching, "Don't you see this pattern? Oh, think about it this way." And so, even I made this transition between a direct teach classroom into an inquiry-based classroom as a math-er.

[00:13:40.850] - Pam
Yeah, it's super interesting. And I think that's part of what gets kind of... I think if this continuum... And again, we're sort of fresh and this is a bit new. But if this makes sense, as I talk to teachers, sometimes it's the math-ers who direct teach, who lean towards... They're oriented towards more direct. Like, you just said, I'm going to either, I'm going to... And I think it's an either. I'm in that upper left-hand corner where I math and I lean towards direct teaching. Either I direct teach the steps of the algorithm, like you did when you were a young teacher, because that's what the way it was done to you and it worked for you, and so that's what you do. Or I run into people who math, who say, "Yeah..." who do what you then did where you're like, "Okay, hey, use this relationship. And can't you just see it's a little bit over? And, you know, think about these fractions this way." You know like, "And then here's the rule." And some mathematicians that I've talked to recently who have sort of pushed back on my algorithm stance. They're like, "Pam, algorithms are beautiful. They're so cool to study." And I'm like, "Study?! Let's focus on that!" Yes, they are so cool to study, but what we don't want to do is give them to kids as the way to do it because too many kids take it as this cheap trick that then they just do these steps and they get stuck in less sophisticated thinking. And, Kim, those math-ers who lean towards direct teach look at me in horror. They're like, "Of course, we would never do that!" I'm like, "But look at what you're doing." Like, nobody's doing anything nefarious on purpose, of course, right? Like everybody has a good heart, everybody wants. And so, but I think it's interesting that if you naturally math-ed for whatever reason, would you consider that other people can do that? They just need to know it's a thing.

[00:15:39.180] - Kim
Mmhm.

[00:15:40.010] - Pam
Like, one thing I'm thinking about today is if we're stuck in that upper two quadrants, whether we math or not math, and we treat math as, "Here, do these steps." Whether the steps are an algorithm or the steps are using relationships, if we treat math that way, too many students just perceive math that that's what math is. It's, "Wait until the teacher tells you what to do and rote memorize that stuff and mimic those steps," and they don't ever get a chance to math.

[00:16:14.240] - Kim
Yeah.

[00:16:14.390] - Pam
And we now have ways to help students math. And so that's kind of... Anyway, it's exciting.

[00:16:18.600] - Kim
I'm going to raise one other wonder.

[00:16:21.220] - Pam
Mmhm.

[00:16:22.500] - Kim
That the potential for you math and you kind of direct teach is people who are new to math-ing. And they get excited. And they then want to tell what they've learned. Or I think that was me in the beginning. So, I think part of it is why are you teaching the way you are? I know that my direct teaching relationships early on in that may be first or second year was because I was excited that I had permission to talk to my kids.

[00:16:54.840] - Pam
About what you were thinking, how your brain worked. Mmhm.

[00:16:57.270] - Kim
I sometimes see this with new coaches, who are excited and want to go back and tell their teachers. So, anyway, I think there's some reasons why you might teach directly, but at the heart of it, you know, we believe that giving kids experiences, giving teachers experiences to math for themselves is what's going to make a difference.

[00:17:18.370] - Pam
Yeah, nice.

[00:17:19.330] - Kim
We also have some people who are maybe direct teaching because they think they're supposed to but also because maybe they have pressure from colleagues that it was the right way to teach. Maybe pressure and time pressure. So, it is a lot of reasons to explore. But I think these two axis in conjunction with each other raise some interesting ideas.

[00:17:40.830] - Pam
Yeah. And maybe we'll just throw that out just to, you know, we'd love your feedback. What do you guys think about? Again, like Kim said, this isn't like identity. We're not. It's more about kind of what you lean towards maybe and why. And might you be open to other possibilities that could help us teach math more figure-outable?

[00:18:02.490] - Kim
Yeah.

[00:18:03.210] - Pam
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. Let's keep spreading the word that Math is Figure-Out-Able!