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

[00:00:09.810] - 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 thinking.

[00:00:17.260] - Pam
We know that algorithms are amazing historic achievements, but they are terrible teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than we could be, should be developing.

[00:00:30.550] - Kim
In this podcast, we help you teach math-ing, building relationships with your students, and grappling with mathematical relationships.

[00:00:37.000] - Pam
Y'all, thanks for joining us to make math more figure-out-able. Alright, Kim. You have been getting... Well, a minute ago you were getting ready for school to start. So, it was a while ago. And you were telling me about some things that you were doing with your son.

[00:00:53.670] - Kim
Mmhm, mmhm.

[00:00:53.930] - Pam
Can you say more about that?

[00:00:55.060] - Kim
Well, so, first can we talk about a problem that he did, once school started, and then I'll tell you about what happened?

[00:01:03.470] - Pam
Yep, yep.

[00:01:04.050] - Kim
Okay, so Cooper is in Algebra 2.

[00:01:07.930] - Pam
Go Cooper!

[00:01:08.890] - Kim
Yeah, and I called you and said, "Let me tell you about something," right? So, should we give people a chance to think about the problem that he did?

[00:01:18.960] - Pam
Yeah, so is this the x^2 one?

[00:01:20.930] - Kim
Yeah.

[00:01:21.650] - Pam
Okay, so x^2 plus 14x plus 45 equals 0. And the instructions were to solve for x.

[00:01:30.310] - Kim
Mmhm.

[00:01:30.970] - Pam
Okay.

[00:01:31.430] - Kim
Yeah.

[00:01:32.570] - Pam
So, pause the podcast.

[00:01:34.940] - Kim
Yeah. So...

[00:01:36.240] - Pam
Well, yeah. Like, solve it.

[00:01:38.230] - Kim
Pause.

[00:01:38.360] - Pam
Yeah, and then come on back. Okay, go.

[00:01:39.670] - Kim
Okay, so to be fair to me, I had worked during the day. And I was cooking dinner, and he was working. And I saw Cooper write 2 sets of parentheses and equals 0.

[00:01:55.980] - Pam
Okay, so he's got to factor using the zero product property.

[00:01:59.281] - Pam and Kim
Mmhm

[00:02:00.370] - Kim
And I leaned over his shoulder as he wrote -x. And that's all he had written was -x.

[00:02:07.770] - Pam
Inside one of those parentheses.

[00:02:08.910] - Kim
Inside one of the parentheses.

[00:02:10.570] - Pam
Yep.

[00:02:10.710] - Kim
And I went, "Um..." And I started to put my pencil. And he looked at me, and he put his hand up and said, "Wait."

[00:02:19.720] - Pam
Ooh.

[00:02:20.440] - Kim
And I was like, "Okay." And I went back to stirring my dinner.

[00:02:24.310] - Pam
Bam.

[00:02:25.500] - Kim
Feisty. And so, when I came back over to look at his paper, he had written        -x minus 9, close parentheses, times -x minus 5 equals 0.

[00:02:41.800] - Pam
Look at him.

[00:02:42.840] - Kim
I was like, "What?" It wasn't what I was expecting.

[00:02:49.100] - Pam
Yeah, me either. Yeah.

[00:02:50.950] - Kim
And it just... Cooper's always been the kid, God love him, where he just... He does things maybe a little bit not what you expect. And it's been interesting because along his mathematical journey, he said, "Oh, we're talking about the quadrant last time." He's had teachers who maybe math-ed less and had more direct teach classrooms. And so, you know, Cooper's not wrong. But it's not maybe what is expected. And so, he has gotten a lot of like, "That's wrong," or "You can't do that," or whatever it is.

[00:03:33.180] - Pam
"Do it my way." Yep. "I don't know what you're doing over there. I'm not going to actually even dive into what you're doing. I'm just going to tell you, do it this way."

[00:03:39.100] - Kim
Yeah. And I said to him, "Why in the world would you do that?" And all he said was, "It was the first thing I thought of." And I was like, "Okay, not what I thought of first, but okay."

[00:03:54.840] - Pam
Nice. And so, there's a couple of -1s sort of floating around in there that does make what he did equivalent to what would typically happen where you'd factor into x plus 9 times the quantity x plus 5.

[00:04:06.350] - Kim
Yeah.

[00:04:06.920] - Pam
So, they are equivalent. I had to think about that for a hot second. I'm not sure I've ever seen anybody do that either. So, that's interesting, right? You and he were messing around kind of refreshing stuff, getting ready for Algebra 2. And go, Cooper. I'm just, yeah, that's really fun.

[00:04:25.830] - Kim
Well, it's funny because this happened when he was given an assignment. But you're right that we had done some work in the summer. We were doing some... I knew he was going to Algebra 2. He took geometry last year, so I was like, "It's been a minute since he thought, you know, about anything algebra, so let me get out this fabulous book that I know." Or series of books. And so, I pulled out the Algebra Problem Strings book and the Advanced Algebra Problem Strings book, and I was just like messing around with like what...

[00:04:53.180] - Pam
Thanks for calling my books fabulous.

[00:04:54.580] - Kim
Yeah, they are fabulous. I mean, I flagged all kinds of stuff. And so, I just said, "Hey, before school starts, let's tinker a little bit. Let's just remember what factoring looks like. Let's remember, you know, anything." So, he and I had done a Problem String, and we had done a couple over the course of a week. And he, at one point at the end, I think we were doing something with like systems of equations, I don't know. And he said, "Mom, why don't more people teach like you and Miss Harris talk about?"

[00:05:25.880] - Pam
Hmm.

[00:05:26.380] - Kim
And I was like, "Um..." And I said, "That is a good question. Do you have any thoughts about that?"

[00:05:35.100] - Pam
Oh, what a good parenting move!

[00:05:37.520] - Kim
He's had a lot of really fantastic teachers. Really fantastic. Some were taught like me, and so they haven't experienced, you know, what I've had the opportunity to experience, and so... And some have. And so, he said... This is what I thought was really interesting. He said, "I think it's partly about them and partly about us." And I was like, "Ooh! Like, what do you mean? Tell me more about that."

[00:06:04.850] - Pam
Interesting.

[00:06:06.040] - Kim
And, you know, he's so thoughtful. He said, "I think it's about different types of teachers." And I was like, "Okay." And he said, "I think there might be 3... No, maybe 4 categories." And I was like, "I can't wait to hear what your categories are." And so he said, "I think there are teachers who hear what you guys say, and they believe you, and they want to do that stuff, and so they try." So, for whatever reason, they learn about math-ing or whatever. And then they say, "Yes," and they try it. So, that was one.

[00:06:39.140] - Pam
Math is actually figure-out-able, and they're going to dive in and try to teach that way.

[00:06:42.020] - Kim
And then he said, "I think there are teachers who learn about it. But they think that they can't teach that way because it's too hard for them." Meaning the teachers. "And so, they think that they can't learn how to do it, and so they don't."

[00:06:58.020] - Pam
Like, the teachers are saying. Like, they're listening to us say, "Math is Figure-Out-Able. We can teach it this way." And they're like, "Oof, that made my brain hurt. That stretched me."

[00:07:08.000] - Kim
Yeah. Like, "I don't know how to teach that. I believe you, but I don't know how to teach that, so I'm just going to do what I've always done." And then his third category was teachers who hear about Math is Figure-Out-Able. Or, you know, some whatever. And they think that the students can't do it, and so they have to do what they've always done. So, maybe that's your group where it's like, "Oh, that was hard for me, so my students really can't do it." So.

[00:07:34.790] - Pam
Yeah. "If I really had to think about that, there's no way my students are going to be able to."

[00:07:38.660] - Kim
And so, that group also doesn't, you know, give Math is Figure-Out-Able, or, you know, give numeracy, or whatever a go. And then he said, "I think there's a fourth category, and that's the category of teachers who hear what you say, and they don't believe that what you're doing is best for kids." And you and I, you know, we've encountered some of those teachers as well.

[00:08:03.030] - Pam
Those are the staunch, "Nope, I've got it. I've got to teach kids (unclear)..."

[00:08:06.070] - Kim
Disservice to them, mmhm.

[00:08:06.750] - Pam
"...Drill them on the algorithms. It's a disservice if I don't do that. They're going to need it later. I know for sure that..." You know, yeah. And they, so they just flat out reject.

[00:08:16.220] - Kim
So, then he said, kind of quietly. He's like, "So, only one-fourth of the teachers actually try." Isn't that interesting? 100% unprompted. I was like, "Wow, you've thought about this a lot." And, you know, listen, my kid's grown up in a house where we talk about math. You know like, he's aware of what we do, but it was the first time that I ever heard him really talk about being thoughtful about classrooms. You know, and so, you know, his wrap-up was like, "So, only a fourth of the teachers try."

[00:08:50.910] - Pam
I love the fact that he used a fraction. Like, only 25% of... Because right because he said, "3? Oh, no. 4. Yeah, 4. So, one of those 4 groups."

[00:08:59.120] - Kim
Isn't that interesting?

[00:08:59.520] - Pam
That's a fourth. Yeah. Wow. That is so... I kind of want to unpack those 4. What do you think about his 4 categories?

[00:09:07.790] - Kim
Well, I thought about it a lot.

[00:09:09.010] - Kim
And it came to mind again when we were talking about the 4 quadrants for math-ing and not math-ing.

[00:09:18.930] - Pam
Yeah, that we talked about?

[00:09:19.790] - Kim
Yeah.

[00:09:19.950] - Pam
That we talked about last week.

[00:09:20.850] - Kim
(unclear). Because this happened a bit ago. And so, you know, I don't know that there's a direct overlap or whatever. But it's interesting to me. What I think about here is more about all of these teachers. Oh, dropped my pencil. All of these teachers. All these classrooms.

[00:09:38.270] - Pam
I heard that.

[00:09:38.790] - Kim
Dink, dink, in the closet. All of these teachers have Cooper in their classroom. They have a Cooper. There is a...

[00:09:47.580] - Pam
Or several Coopers.

[00:09:48.320] - Kim
Yeah, there are kids who are doing funky, like put a -1 in your factor. They just... Do they recognize? Do all of them maybe recognize that there is a Cooper-type kid and is that Cooper-type kid allowed to emerge? You know, Cooper gets that fostered at home, but of course not everyone is. So.

[00:10:14.560] - Pam
That reminds me a lot of my daughter as well. Yeah, my sons... Not to get too deep into it. But my older son had very traditional teaching because I worked with his teachers as he had them. Since I was working with him as he had them, my next 2 sons had those teachers, and they had wonderful experiences up to, you know, so high, and then I didn't work with all their teachers. But they had enough Math is Figure-Out-Able that they just continued to make sense of math. And then we did a lot of math at home. But not near as much as my daughter because I stopped working with the district for various reasons around her 3rd grade. 2nd grade, 3rd grade. And so, then she had traditional teaching after that and she, like Cooper, would try to make sense of stuff her own way. And I don't know how vocal Cooper is in class. My daughter was super vocal and just let people have it. Like, yeah, you could do your stuff over there, but I don't get it, and so I'm thinking about it this way, and I want you to dive into my world with me. And very few, very few of her teachers would be willing to dive into what she did, so she'd come home, and then we would.

[00:11:18.560] - Kim
I think Cooper has had some really cool teachers who have said, like, "Off you go, sir. Like, do the thing that you..." As long as he gets the answers right, and then when he doesn't get it right because he's off by a little something, then it's like, "Come back, and you have to do it this way." He's had some who said, "I'm not sure, but off you go." And then he's had some who are like, "I don't... That doesn't make sense. That doesn't work. You got to do it this way." And I think that's not a commentary on like our district or whatever. I think that's happening everywhere. But again, I think the point is there are Abbys and Coopers and, and other kids like them who are thinking and reasoning in each of these types of classrooms. And so, it's interesting because the way that they might be responded to could be different based on where they land.

[00:12:11.370] - Pam
Let's do that. So, if we went in the order that we said them.

[00:12:16.200] - Kim
Okay, so, if it's teachers who hear what you say, and they believe you, and they give it a go, it kind of maybe depends on, at least for me, what I'm thinking about, it depends on where teachers are in their development. If they,  are further along, they might dive in with Cooper and say like, "Ooh, tell me more about it." And like, "What are you thinking? And like, "Ooh, I wonder why that works. And how is that equivalent to how your table partner, you know, got? Why are those both appropriate answers? Or appropriate ways to get started?" If it's somebody newer. Yeah, yeah.

[00:12:48.450] - Pam
Can we stay there for a second? So, that person in a Problem String could have sort of the x plus 9 times the quantity x plus 5 shared, and then while everybody's thinking, could walk over to Cooper and say, "Are those equivalent? And how do you know? And which of those..." And then maybe even I could see a world where that teacher would have Cooper share too and ask the class, "Are these equivalent?" and then have a conversation. If you're walking down the street, and you have to factor this quadratic, solve this quadratic equation, which of these do you find more efficient? So, it's not that Cooper's right or wrong, or better or worse, or whatever. It's a conversation about relationships and a conversation about what do you want your brain to be inclined to do next time? Yeah.

[00:13:33.720] - Kim
So, if it's somebody who is maybe just starting their journey, but they believe that Math is Figure-Out-Able, maybe at the very least they're going to sit down and say, "Tell me what you're thinking." Like, they're not going to maybe push it further, or make a comparison, or talk. You know, bring it forth to the class. But maybe, at the very least, they're going to say, "Ooh, that's interesting. Tell me about that. Like, how do you know? What were you thinking about?" So, I think that's the first category. Second.

[00:14:00.400] - Pam
Okay.

[00:14:01.030] - Kim
Maybe the teacher who is learning about it, but they think that they can't teach that way because it's too hard for them, too hard for the teacher. This is an interesting category. So, if you have a Cooper in your room, maybe they leave Cooper alone. Maybe they're like, "That's good for you." In fact, he had a teacher like this who the teacher and I had a conversation. She was like, "Oh, you know, I know what you guys do. And but..." Actually she might've been more like the third, but she was like, "I can't teach that. I can't teach that, and so I'm going to let him be. And, you know, he's solid. He can do what he wants."

[00:14:39.010] - Pam
So, he kind of doesn't get supported, but he also doesn't get stymied.

[00:14:42.690] - Kim
Right.

[00:14:43.570] - Pam
Okay, okay. Because the teacher, for whatever reason, it seems too hard to teach.

[00:14:49.930] - Kim
Maybe. Yeah, maybe they're not sure how to make that happen, for whatever reason. The teacher that I was actually thinking about would be more like category 3. And what she actually said.

[00:14:59.710] - Pam
Remind us what category that is.

[00:15:02.120] - Kim
Teachers who like learn about what we do. And they might like it, but they say  "The kids can't handle it. The kids can't do it," and so they're going to teach pretty traditionally. So this was a teacher that Cooper actually had who said, "Love it. Love the stuff. Cooper can do what he wants. You know, good for him. But the kids, they can't handle more than one way, and so I'm not going to."

[00:15:28.470] - Pam
Cooper's brilliant, so he can do his thing. But...

[00:15:31.050] - Kim
Just confuse them.

[00:15:31.330] - Pam
...The other kids, they're not brilliant. So.

[00:15:32.960] - Kim
Yeah.

[00:15:34.020] - Pam
Yeah. Yeah, they need one and only one way because they're just not capable.

[00:15:37.780] - Kim
Listen, she was a fabulous person, you know.

[00:15:41.950] - Pam
Oh, and I forgot we're talking about a person, so I was kind of characterizing.

[00:15:43.490] - Kim
No, no, no, I think that's fair. I think it's fair. You know, this particular teacher. And, you know, but I think she could represent several. Like, you know, she has her students' best interests at heart. She didn't want to overload them. But I think... Yeah, but I think it was less about... It was more like Cooper can handle it and the others can't. And less of Cooper's making sense of his own thinking, and that's why he's doing what he's doing. And maybe if we gave kids the opportunity to make sense of thinking, then maybe you would see that they all could as well.

[00:16:21.170] - Pam
Yeah, more and more kids.

[00:16:23.250] - Kim
Go ahead.

[00:16:23.770] - Pam
I mean...

[00:16:24.800] - Kim
Yeah.

[00:16:25.220] - Pam
Are you going to go on to the fourth? Because I want to hang on. I want to hang for just a second and say we can only do... When you know better, you do better. So, we can only do as much as we're sort of. You know, if we don't have our own personal experience math-ing or our own personal experience being helped to math. And we might be in a... You know, you might be listening to this podcast. And you're like, "Well, I really like what you guys are saying, but how do you do that? Like, how do you do that with students?" And yeah, that's why we built our Building Powerful Mathematics workshops and the other professional learning that we do in our Math is Figure-Out-Able Solution when we work with schools and districts. Like, we know. Like, there's no blame here, right? There's this honest conversation about you and I both came from different places and had to have experiences. And then combine that with our belief in the inherent brilliance and capability of kids. Super important what you just said. To think, and then think about their thinking and try to make sense of things. No matter where they are in the landscape, everybody can do that. Like, everybody, where they are on the landscape of mathematics learning, can say, "Let me make sense of this, and let me think deeply about this, and let me try to connect it to what else I know," and therefore make those connections stronger and continue to work up the landscape of learning. That's not trivial if you yourself haven't had the experience, which is, hey, one reason why we have this podcast.

[00:17:58.800] - Kim
And I'll say it's so satisfying for us personally and professionally to work with the teachers in group 2 and 3 because we see a shift in their belief that they can and that students can. You know, they might come to a workshop or they might take a workshop or whatever. They might sit in a conference session. And they're like, "Oh, my gosh. I loved that. And then we get to help them make more sense for themselves and make more sense for the kids. So, that's...

[00:18:28.740] - Pam
Super quick. In our Building Powerful Proportional Reasoning workshop, Melanie was a participant in the first day who said, "My kids can't do this. Nope. My kids can't do this." And then she clearly said, "I'm a special education teacher. You've got to treat these kids better. You've got to..." And she staunchly stood up for them. In the middle of the second day, I heard this gasp, and everyone in the room turns, and she goes, "Oh! I just did that! Like, I could teach this way! Like, I think my kids could do this!" And she, in our, Developing Mathematical Reasoning: Avoiding the Trap of Algorithms book, we quote her. Like, I contacted her a couple of years later, and I was like, "Would you be willing for me to put that in the book?" She's like, "Yes! I still teach Math is Figure-Out-Able and all kids can." Yeah, you're right. Like, we love working with teachers that just have this gut feeling they can reach more students. Yeah, we can help with that. (unclear).

[00:19:22.130] - Kim
Fourth category. Teachers who hear what you say... This is Cooper's category. Teachers who hear what you say and don't believe that what you're saying is best for kids. And that's a tricky place to be. I spent some time as a coach on a campus working with a teacher who, you know, I think we've mentioned her before, said, "I vehemently disagree."

[00:19:45.760] - Pam
"I understand what you're doing, and I say to your face, this is not right for kids." Yeah.

[00:19:50.980] - Kim
And it's unfortunate for a Cooper to be in that classroom. For the kids who are, it's an unfortunate situation.

[00:20:00.060] - Pam
It was unfortunate for Jordan. So, also in the Developing Mathematical Reasoning: Avoiding the Trap of Algorithms, K-12 book, we told the story of Jordan. And that was his 5th grade teacher who said, "Nope, you're going to rote memorize and mimic." And Jordan had been thinking and reasoning in your 3rd and 4th grade classrooms. And luckily, by the grace of God, we grabbed him and helped him realize that just like math was figure-out-able in 3rd and 4th grade, it's still figure-out-able in 5th grade. He went on to do amazing things knowing that math is figure-out-able, owns his own business today.

[00:20:31.350] - Kim
Yeah.

[00:20:33.110] - Pam
Yeah, it is a tough place. We never expect people to go against what they, you know, inherently believe. We just keep trying to give teachers experiences so they can see the brilliance of kids, and see and experience math-ing the way mathy people actually math. Yeah. Kim, tell Cooper he's brilliant, and we appreciate him giving us a great idea on the podcast. Everybody, we'd love to hear what you guys are thinking. Thanks for joining us on the podcast. 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. Keep spreading the word that Math is Figure-Out-Able!