Pam Harris  00:01

Hey fellow mathematicians. Welcome to the podcast where Math is Figure-Out-Able. (pause) Oh shoot! What is wrong with me? (laughs)

Kim Montague  00:13

Your Pam and I'm Kim.

Pam Harris  00:19

And we make the case that mathematizing is not about mimicking steps or rote memorizing facts, but it's about thinking and reasoning - about creating and using mental relationships, we take the strong stance that not only are algorithms not particularly helpful in teaching, but that mimicking algorithms actually keeps students from being the mathematicians they can be. We answer the question, if not algorithms and step by step procedures, then what?

Kim Montague  00:46

So in this episode, we thought we'd continue the conversation that we've been having for a couple of weeks. We've been talking about algorithms versus relationships. And now we want to talk about correct answers versus relationships. So this actually came up because we were talking about an email that we got from one of our journey members, so your membership site, and she said, "I have to share with you what my husband said during a car ride this past weekend." He was describing doing some math, he handles the finances. And so it's probably about taxes or something. And he said, "So I thought to myself, What would Katrien do?" And I said, "What?" This is her talking. He said, "I've heard those math webinars that you've listened to where that woman shows those other strategies. So I thought to myself, hmm, I have to subtract 75 from something crunchy. And so I thought about subtracting 80 and then adding back five. So I've been trying that for a while. Every time I have a math problem, to see if I could do it. And it's really cool." Isn't that interesting? She says, "A wholly unexpected influence. You're rewiring the brains of 55 year old dads, the over strategy for the win." Isn't that great?

Pam Harris  01:59

Oh my gosh, when we got that email, I was like, sweet. So Katrien, thank you for sending that story. And that's fabulous. I'm so glad. Like, that's amazing. Um, I'll be honest, I, how was my husband? 58? Like, we've got a 58 year old guy over here. And he's sort of rewiring the same way that we've been rewiring him for a while. And I mean, of all the people. Rewiring me, like, I was the one who just mimicked procedures and just did what the teacher told me to do. I tried to make sense of it. I wanted to know why and how. I did the best I could to figure out which step went next. But now I have a slightly different perspective. That I'm not just a mimicker. I'm not just the button pushing monkey. I can mathematize. And I love that. I love how that has set me free.

Kim Montague  02:49

Yeah. So this question about correct answers versus relationships actually comes up pretty often because people ask you all the time where you stand. Right? And so I am going to ask you to share a little bit today. One of the things I've heard you talk about before that relates to this is about Phil Daro. And I think one of the first time you ever shared his research and what he had to say about this topic really blew my mind. So I am going to sit back and listen a little bit today, and let you share with our listeners.

Pam Harris  03:20

Yeah, thanks for asking. So I've actually gotten to meet Phil a couple times, he won't remember the first two or three, because I was just in awe. And so I just sat there quietly. But then we did some work in New York at the same time, and I was able to chat with him a little bit. I've met him at conferences. This is a very wise, smart guy. And it's cute, because he's kind of self deprecating. And he tells the story. And I'll give you the link where if you want to watch it from his lips, I'm going to repeat sort of what he says, so if you want to hear it straight from him. We'll give you the link in the show notes. But he tells this experience, and he kind of what kids himself a little bit. You know, he's like, "I can't believe it took me this long to figure it out." But, y'all, he's the only one who's figured this out. And I think it's really important and amazing. So when I share this with you today that I think this might help all of us take a little bit of a different stance or a little bit of a different way of looking at why Kim and I are so strongly trying to change the way we view math teaching. And how it can help with that. So Phil Daro was involved in the TIMMS study, T, I, M, M, S, the TIMMS study. So it's an older study. My understanding of study is it's worldwide and it was all about comparing nations to each other. Like let's take a look at teaching around the world and how that's happening. And so they would take camera crews and they would land in eighth grade classrooms. "Surprise, we're gonna video you today." So no warning. It's not like you could prepare your best Chinese lesson, right? They just would show up and like, "Here we are, go." And then video the lesson. And they collected these lessons around the world and countries after countries. And Phil's great. He says, "I noticed something, I began to notice this thing and it troubled me." And he began to notice that when he compared, now remembers all around the world, but when he specifically when he compared classrooms in the United States and classrooms in Japan, that he said, "I noticed this thing that teachers could range the gamut in both countries." Like he could find sort of less sophisticated teachers to pedagogical geniuses, he called them both in both countries. And when he looked at both countries, teachers that range the gamut, from really good teachers to poor teachers, the teachers in Japan got really good results. And teachers in the United States, even the pedagogical geniuses in the in the United States, got mediocre results. Let me say that, again, teachers in the United States, even the pedagogical geniuses in the United States, got mediocre results. But teachers in Japan, no matter where they were on that spectrum, got excellent results. That was troubling, y'all that's troubling. That is a note. Now, it's not like we want Japan to not do well or anything. It's not that. It's like we want our pedagogical geniuses, no, no, no, we want teachers that range the gamut. We want them also to get excellent results. What are the teachers in Japan doing to get excellent results? And again, he's like, "I can't believe it took me so long to figure it out." But here's what he's figured out. If you take the problem of the day, in either country, same problem of the day, if you take that problem of the day, teachers in the United States tend to say, "Okay, I gotta get my kids like, here's the problem of the day. What is my lesson going to be today so that my kids can get answers to these kinds of problems? So I'm going to teach today to help my students get answers to these kinds of problems." Right?" Everyone's nodding. They're like, "Yeah, that's what we do. That's teaching math, Pam." Okay. Alright. So we've got that settled. Here's the problem. Today, my job is to help students get answers to those kinds of questions. Alright, set. However, teachers in Japan would look at the exact same problem in the day. And they would say, "Hmm, how can I help my students? Learn the math? How can I help my students create mental relationships? Connections? How can I help my students develop their brains in such a way, so that they can get answers to these kinds of problems?" Let me say that, again, United States teachers, "How can I help my students get answers to these problems?" Japanese teachers, "How can I help my students think and reason, so they can work these kinds of problems, so they can answer these kinds of problems, they can think and reason through these kinds of problems." That is a distinctly different goal. Consider that if, and I'm saying to Phil, so I'm still kind of paraphrasing, but he says consider if you are teaching, if you're if you say to yourself, how can I help my kids solve this proportion? Well, humm, in fact, I wish I would have gotten an example yet. Let me stay general for a second. "How can I help my students get answers to this question, whatever this is question of the day? Well, let's see, what do they already know how to do this, and they're gonna do that. Oh, I can use those things. I can use things they already know how to do. And if they put them in this order, then they can solve the problems. That's good. That's fantastic. Because they already know those things. If I got these procedures, these rules or whatever, they can use those. It's going to be very reliable. I'm gonna be able to do that nobody's gonna feel stupid. Nobody's going to be slower than anybody else. They're all just gonna be able to, like, clop along, and they're gonna get answers to these kinds of questions. Alright, we're doing that now. Got my lesson plan." Versus the teacher in Japan, who says, "Hmm, okay, looking at this problem today, in order to reason through this problem, what kinds of things do my students already know that I can build from? So that now they can reason in a new and more sophisticated way? Because this problem is calling for more sophisticated reasoning than they currently have. So they're currently reasoning this way, what kinds of things could I do if I ask this? If I give him this problem to solve, what could we do to help build the reasoning so that they are actually reasoning through this problem at a sophisticated level. Because they might give them a problem to solve that they could use a less sophisticated thinking or strategy to solve. But that's not what I'm trying to build. I'm not trying to just get an answer. I'm trying to actually build reasoning so that they're solving this problem in a sophisticated way, the way this problem was meant to be solved. That's why we wrote this problem, because it needs this kind of reasoning. So how can I actually help build my students' brains to reason in such a way that ah, I can reason through this at this sophisticated level?" Can you get this flavor for how different that is? How different is to have those two different goals? Now, if I may, I'd like to take a specific example. So what if I'm a middle school teacher, and I say to myself, "I need my students to solve this proportion. Okay, well, my students can already multiply and divide, or at least they should be because they should come to me with that." We all know that might not be true. But they should have learned they've learned that before. Maybe that's the best way to say they have learned before how to multiply and divide. "So I'm just going to say, here's the proportion. I'm going to draw this butterfly, or maybe," Kim, what do we see a bat and a ball? 

Kim Montague  10:20

Yes.

Pam Harris  10:20

What is it? A bat and a ball? There's several other kooky things that you can do. "To help the students go, I'm going to cross multiply and then divide. And if you cross multiply these two numbers and divide that number. Look there we go. We solve, up nice. All you gotta do is memorize this one little bit, cross multiply, divide this bat and ball, this butterfly, whatever this thing is, just memorize that little tiny thing. Get that down, and you're just going to do, use those things that you've learned before. Whoa, we are selling proportions. And by the end of that class period, everybody's successfully solving proportions. And, and everybody's like, very comfortable," may be a word I use. They're all just like, "Okay, we did it. We're done. Move on." Nobody's intrigued. Nobody's like, was challenged, nobody struggled "Woo, did my job. We got this easy thing done. Good. We can move on and do something else." Versus that same problem of day, my kids need to solve this proportion in a Japanese classroom, where the Japanese teacher then says, "Okay, what are the kinds of things I'm going to do to build Proportional Reasoning? To help my students reason proportionately, so they can actually use Multiplicative Reasoning, not just a multiplication algorithm. But reasoning, oh, you know, what I know, I know, there's going to be some students who are going to try to use Additive Reasoning here. That's not gonna work. So I might look for students trying that. And we might bring that out in a very positive way. "Oh, look at the student," as a canary in the mineshaft. "Look at how this student is going down this rabbit hole. Let's check it out. Will that work?" And because it's probably in context, we can reason about how, "Oh, you can't like, if you subtract a pizza, you can't subtract $1. Because a slice of pizza didn't cost $1. We don't know what a slice of pizza cost, but we know it wasn't $1. So I can't just subtract a slice." And I'm using a context that we've used before, where we have four slices of pizza for $5. Kids will sometimes subtract a slice of pizza and subtract $1 In order to solve, say, for three, the price for three slices of pizza. If they use that Additive Reasoning, we know that's going to be a thing that's wasted to been reasoning. And so we know that. We know the sort of the landscape of what's happening. We know that the Development of Mathematical Reasoning. And so we say, "Okay, we're gonna look for that, we're going to highlight it, and we're going to discuss it, we're gonna use context to make sense of it. Oh, now look, now my students are reasoning more sophisticatedly. Nice, how am I going to continue that? How am I going to help them continue to reason more sophisticatedly?" Now in that classroom, at the end of the day, I might not have all students solving that problem correctly. Now, I might, one problem by the end of the whole class period. But in that, we might have some struggle. Okay, we are going to have struggled, if we're doing it well, students are going to be like grappling with these ideas and trying to make sense of how the relationships work differently than they sort of were thinking about them, especially if they were thinking additively. And in that grappling, that's a little uncomfortable. That's a little disconcerting. You're off balance, because it's not all just like, fresh in front of you. And you might be like, "Pam, can we just give it fresh in front of them?" No, it doesn't work. Learning doesn't work that way. I mean, they might solve, if you can see my hands, I'm kind of on this side, the American side, where I just gave him that one little bit to memorize. And by the end of the day, they're all like, "Yeah, okay, we did the thing." What you don't get is satisfied students. You get satisfied students now, maybe not that day. But as they use those relationships, and it starts to actually make sense. Those students are like, "Okay, I can do that. That makes sense to me. I am clear on those relationships. I was intrigued. And now I'm like," but that's not an overnight process. That's not a five minute. Let me, I do. I just did it. Now we're going to do together. Now you go do it. Oh, good luck, how... It's not that, humm, what? Easy?Like it takes, learning takes effort. And that effort is then rewarded, not effort, just repeating stuff 29 times, but effort in making sense of what's happening, because in that making sense of what's happening, I am now a more sophisticated reasoner. And that is the goal of math class.

Kim Montague  14:32

So I love his research. So we have some American teachers listening and maybe maybe some other countries as well, who recognize, "Maybe I'm a little focused on getting answers because that's just what I've been used to." Right?

Pam Harris  14:51

Sure. Sure. 

Kim Montague  14:52

What does this mean for them? What does this mean when they say, "Oh, I recognize that and I'd like to make a shift." 

Pam Harris  14:58

Yeah. So if I may first, no blame here. Like, please, please leave here going, "Wow, that's interesting. Let me try to move forward with that." Please don't go, "Oh, that's terrible. Oh, no bad." Like, there's no blame. We only can do what we know. Most of us are probably teaching the way we were taught. So all I would ask is at this point, now that we know a little bit different, now we can do differently. Now we can begin to shift our practice away from how can I just get kids to use what they've already learned in the easiest way possible in order to get answers, to how can I help my kids actually reason about this thing? And that is not trivial. But that's why we're here. The Math is Figure-Out-Able movement is to help all of us learn to teach more and more math that is Figure-Out-Able. Not rote memorizable. But it's about how can we help each other, create lessons that really helps students dig into the relationships and make those mental connections from what they know, to this new thing, and then use those to solve problems. Because when I say that, making mental connections, that's literally being able to think more sophisticatedly. And that is the goal of math class. So if you're that teacher, please recognize so as I. Like, that's exactly where I was. And we can help you become more and more the teacher where math in your classroom is Figure-Out-Able.

Kim Montague  16:26

And they're already a part of the movement, right? By just listening to this podcast, they're here, they want to know more.

Pam Harris  16:32

You bet. So keep listening. And we will keep helping you keep helping you do that. Let me give you a little bit of a glimpse into what it could look like when you can sort of know that you're on that track. Several years ago, Kim was teaching for reasons I went in, and I videoed a bunch of our kids. It was hugely in the early stages of my research. And what I'm going to tell you about has absolutely nothing to do with the research I was doing at that point. But here's an outcome that I found that I think is fascinating. So one of the questions that I asked students were, in fact I asked two most missed facts. So I think I said seven times eight. And then I said, I think 11 times 12. And then some other things in the interview. And when I asked Kim's students seven times eight, here's what I never heard. I never heard them say, "I don't know." Instantly when I asked them, they never said I don't know. What they said was either 56. Either they had it right off, or they thought about it. They used some relationship. And then they told me 56. Every one of their responses wasn't that it was a know or not knowable thing. Every one of their responses was, "Ah, my job is to tell you what it is. And I have power over that. And so I will figure it out." And they figured it out. And they told me. Now don't get me wrong. Plenty of her students just do seven times eight because it dealt with a lot. And in fact, when I asked the 11 times 12, one of her students said, "132." And I said, "Whoa," it was just so fast. And matter of fact, it's fourth grade kid that's kind of unusual. And I said, "How do you know that?" And she goes, "We deal with twelves a lot in class." I know exactly who you're talking about.  So some, many of the kids just knew the fact that they dealt with them a lot. They had a lot of experience. But those who didn't, if they ran into effect, they didn't know, they were clear it was Figure-Out-Able because that was the atmosphere that was built in that math class. That's what we're after. We are after the sense and this feel, that Math is Figure-Out-Able. And that's what we do here. We figure it out. Not retrive from rote memory, not mimic your steps, but figure it out. So if you want to learn more mathematics and refine your math teaching so that you and students are mathematizing more and more. Enjoying the Math is Figure-Out-Able movement and help us spread the word that Math is Figure-Out-Able