Pam Harris  00:05

Hey fellow mathematicians. Welcome to the podcast where Math is Figureoutable.  I'm Pam Harris. 

Kim Montague  00:13

And I'm Kim Montague.

Pam Harris  00:14

And we're here to suggest that mathematizing is not about mimicking, or rote memorizing, but it's about thinking and reasoning about creating and using mental relationships, empowering teachers and students. We answer the question, if not algorithms, then what? So we will do some math, and we'll talk about issues in math teaching.

Kim Montague  00:39

 Today, we thought we'd tell you a little bit about the first time we met as a way to introduce this podcast, its purpose, who it's for, and why you might want to join us each week. 

Pam Harris  00:48

So you may know me as a numeracy expert. I wrote Building Powerful Numeracy, I teach at the university, I was a high school math teacher. I do a lot of professional development with teachers. I've written some books. However, you might not know sort of where I've come from, and the first time that Kim and I met, we'd like to tell you a little bit about today. But first, so about 18 years ago, I had four kids and I was fascinated. I was enamored, by the way my kids were mathematizing. The way that they were thinking and reasoning about math was so incredible. And yet, they were bored at school. And that was really interesting to me. And so I dove into the research about... I wondered as a high school math teacher if there was a better way to teach elementary math. So Kim, do you remember the first time we met? 

Kim Montague  01:36

Yes. 

Pam Harris  01:38

I would just gotten, the school had gotten a grant and I had gotten asked to do one of my very first elementary workshops and Kim and I hadn't met yet. So I'm in front of elementary teachers. Again, I'm a high school math teacher at this point. And it was the first time I kind of ever tried to teach elementary teachers anything. And so I'd gotten a hold of a really good game that brought out relationships about adding and addition and subtraction, it's called close to 100. We'll probably talk more about that game in a future episode. But I launched that game and we played a little bit and then I gave a sample problem to teachers. I said, "What if I saw that as you're playing this game, you had a problem like 48 plus 39?" I'm like, "How are you guys thinking about solving that?" And I kind of anticipated the teachers were going to talk to me about the algorithm, the traditional way that we've learned to add, and when I asked that question, "How are you guys thinking about it?" It was kind of a little crazy. All of the people in the room kind of turned and sort of poked fun a little bit at Kim. And Kim you got a little bit red because teachers were like pointing at you. Tell me a little bit about that.

Kim Montague  02:40

Sure. I remember being embarrassed because all the teachers were pointing at me. But to be honest with you, I was really excited because we were talking about numbers. And it was the first time that somebody had ever asked me what I was thinking. I looked at numbers a little bit differently, but I'd never had an opportunity or reason really to share that with anyone.

Pam Harris  03:00

So somehow the teachers knew that you kind of did some stuff in your head.

Kim Montague  03:03

Yeah. 

Pam Harris  03:03

Like, is that, was that what you were teaching your kids?

Kim Montague  03:05

Oh, gosh, no, no, no, no, I was teaching standard algorithms. And like I said, it had never occurred to me to even talk about it. In fact, I thought that a lot of people mess with numbers the way I did, but we were all doing the quote, unquote, right way on our paper. So my paper would have look just like everyone else's, 

Pam Harris  03:23

Even though you were really using relationships in your head. And so the teachers were poking fun a little bit at you, because they kind of knew that you did this stuff. But you were like, "Hey, I teach the right way. I just kind of do this stuff in my head." And at the same time, you were kind of excited to actually have us be talking about what you were doing. Let's give everybody a little bit of an example of maybe what you're doing. So like if I ask the question 48 plus 39. Tell us what was happening in your head. 

Kim Montague  03:49

So back then I was doing a lot of adding from left to right. So 48 and 39, might have looked like 40 and 30 to get 70. And then the eight and nine to get 17, and then put those together. 70 and 17 to get 87. Now, I feel like we've worked together for so long that I have a variety of strategies, I might do a little over strategy, and add 48 plus 40, and then back up one, to make it be 48 plus 39. Or what I might do is take one from the 48 to make the problem 47 plus 40, to get 87.

Pam Harris  04:28

A little give and take and so now you have a repertoire of strategies that you can choose from. So do I, we've worked together, we've sort of developed these strategies together. And that's actually one of the thing that sort of sets my work apart from a lot of good people is that we've chosen the strategies we feel like are the most important strategies for kids to learn. We'll talk about that a little bit more later. So at that point, after that workshop, tell us a little bit about, you went back to your classroom and...

Kim Montague  04:55

Yeah, I went back to my classroom being really on fire. I just started asking kids what they were thinking about. I remember playing games with them. And I remember putting up problems on the board and saying, "Hey, does anyone want to tell me how they think about this particular problem?" And I was amazed how many kids, once I said that it was a thing they could do. They started talking about numbers in a different way. And, and I, to be honest with you, it was the first time that I really had a lot of fun with math. It wasn't about practicing. It wasn't about me telling them how to think. It was about us just talking about numbers together. And we had a really good time that year, there was no pressure. It's just all about enjoyment. 

Pam Harris  05:42

And I kind of heard you say before that was, it was actually fun, like playing around with the numbers and kind of hearing what the kids were doing. Not all the kids, right. It's, some of the kids were a little bit like, "Wait, I thought we were just supposed to do this thing." 

Kim Montague  05:54

No. But you know, what was really cool was that even if there were students who weren't coming up with their own strategies, What I found interesting was that there were students in my class who could hear what some kids were doing. And it made sense to them. One kid might share their strategy and the other ones go, "Oh, I hadn't thought about that. But it makes sense what you're talking about."

Pam Harris  06:16

And so I hear you saying that that year, we kind of set you free, kind of gave you permission a little bit to sort of play with numbers with your students.

Kim Montague  06:23

Yeah, that year, when you started working with me and the rest of my colleagues. And we worked in our school and you pushed us to examine how we were teaching, what math was, or it is and put student thinking at the center.

Pam Harris  06:37

And at that point, you were a third grade teacher. Is that right? 

Kim Montague  06:39

Yes. Mm hmm. 

Pam Harris  06:40

So you've taught third grade, fourth grade, fifth grade. It was a little weird for me coming in as a high school math teacher. I think some of you guys were looking at me a little bit crazy. But once we kind of got that same atmosphere going with teachers, that it wasn't about right answers, it wasn't about mimicking some procedure, doing a bunch of steps, but it really is about thinking and reasoning. Well, you and I sort of started a journey together, of working together for about the past 18 years. We're sort of in and out of, I worked at the school for several years, my own personal kids went there, I was lucky enough for a couple of my kids to have you as their third, fourth and fifth grade teacher. Until you left *laughing* I think I've forgiven you. But that'll probably come up later in a future podcast when Kim left to go open a new school with the principal and left my daughter without her as teacher. It's all good. It's all good. And so we started this sort of journey, working together. And I thought we'd share with everybody on the podcast today, two primary ways that you and I kind of enjoy chatting together with each other. One way is that we actually like to do math together.

Kim Montague  07:43

Yeah, having a K-5 background. Sometimes I'll say to Pam, "Hey, can we just do some math?"

Pam Harris  07:49

Like for example, one time, we were doing a professional development workshop. We had traveled to the place. We went to the store to get some snacks and if you guys want to know a little bit more about Kim and me, so Kim's in the gummy section, sweet, sticky stuff. And I'm of course in the real snacks, like the chocolate, right? I'm going for chocolate! Anyway, we totally disagree on snacks, which is totally, fine. And she literally just turned to me and said, "Let's do some math." And so I've been playing around with some Pythagorean theorem relationships. And since you're K-5, it was a little bit out of your zone of expertise, and we had a lot of fun in the hotel room talking about math, and now you have a sixth grade son.

Kim Montague  08:24

Yes. And one of my favorite things to do is to see what he brings home. And to really examine it with a different eye. Right? Things that I maybe would have brought out from memory years ago. Now I can dive into and I love calling Pam and saying, "Hey, let's talk about this math today."

Pam Harris  08:42

Yeah, and then we get to kind of look at the way Luke is thinking and the way you're thinking. And because I have sort of this higher math perspective, I can kind of make sure that he's not doing a strategy that will only works for these numbers or that you know, he's doing something that's going to be generalizable enough. And then I can kind of poke a little bit and like, throw some other things for you guys to think about. It's always so fun because you both get so excited. And then we continue to play a little bit with the math. So that's really fun. So one of the ways that you and I enjoy chatting with each other is literally the math. And the y'all to be really clear as a high school math teacher, that was not true for me. When I was just a high school math teacher, before I dove into the research about how we could really build numeracy with young students. Before I did that, math was just something like I sort of did with kids. It was about mimicking procedures and memorizing a bunch of disconnected stuff. And I tried to make it as fun as I could when I was teaching. But it wasn't ever anything I did in my spare time. It wasn't, there wasn't anything fun about mimicking or memorizing or just doing steps. But now that I understand, math is completely different, now that I come at it from this completely different perspective of what mathematizing means. That it really means using relationships and connections you own to solve problems. That's fun to talk about. It's fun to say, "Kim, how would you have done that?" And then she can share. And then she'll share what Luke did. And we're both blown away because he's amazing. And we really can enjoy that. Another thing that we do is sometimes, so we've worked together now. You're on my team and we do some push in coaching with some schools. And when I know you've been in school and I see your name come up on my phone, I always want to take that call. Tell us a little more a little bit about what that call usually entails. 

Kim Montague  10:21

Oh, it's always fun. We, I'll call you on the car ride home, often to just describe what I saw, what I noticed. And just to beat out with you about, you know, how would we coach teachers next, what kinds of noticings do we have, that can further push teachers along and just help build them?

Pam Harris  10:40

Yeah, in such a way that we help each other with that, because what we don't want to do is demoralize anybody. We don't wanna tell anybody they're doing anything wrong. What we want to do is to notice the things that they're doing well, and then that one or two things that maybe they could work on and that's difficult. That's really, that's not trivial. And so you and I really enjoy talking that out with each other. Like for example, now this is one day where we actually were together. Kim and  I have very similar sort of views on education and math and how we really want to build a community of learners. Every once in a while we'll differ just a little bit. So we were in a classroom together and I'm on one side of the room. And I remember looking out with a big grin on my face, and I'm like, "Oh my gosh, Kim, check this out." And I looked at you and you have this like frown, and your like shaking your head. And so we finally met up in the middle of the room. And I said, "Kim, like, check out like these kids are doing this great strategy." And you said,

Kim Montague  11:34

But they're all doing the same strategy, exactly the same.

Pam Harris  11:38

I said but... so but it's a such a cool strategy.

Kim Montague  11:44

But for such a rich problem, we should have seen, right, a variety of strategies, lots of different ways to think about the problem, not the exact same. 

Pam Harris  11:53

Yeah, for sure, because the numbers in that particular problem really could have been solved with several different of the majors strategies. But I was so excited. As a high school teacher, I'm so used to only seeing one way, and only one way. And you were able to share with me that for a rich problem like that, obviously, the teacher had been too direct. The teacher had sort of said, "This is what you must do today." And the kids were dutifully, like sort of doing the step. This is one of the most made mistakes we sort of see around the country, is that teachers go, "Oh, there's more than one way to solve a problem." And so then they think their job is to then help students memorize these different strategies. And that's not what we're promoting at all. That's not what mathematicians do. Mathematicians don't memorize alternative strategies. Mathematicians create relationships in their head between numbers and between using properties. And then when they hit a problem, then they say to themselves, "What do I know?" And because they have this interconnected network of understanding, they kind of let the numbers influence the way they choose strategy. I remember Cathy Fosnot, I'll quote her, saying that the numbers should dictate the way that you solve the problem. And that's what mathematicians do. What we saw that day was, because with a rich problem, we saw students all doing the same strategy, you clued in that the teacher was probably, we hadn't seen that part of the teaching, but you were able to sort of say, "Ah, the teacher must have been being too direct." Now, if it was a problem, like, I don't know, 49 times 99, well then, really anything times 99, then we wouldn't expect a variety of strategies. We would expect students to say, "Well, if it's 99 times anything, then really can you just think about 100 of those things? And then if we only need 99 of them, just get rid of one of them." We would expect students to use that sort of Over strategy to solve that problem. But in the case, when we were in that classroom, we saw something much more like 15 times 18, where really that's a rich problem. And we would expect students to use a variety of strategies, that kind of the relationships that ping them that day to solve the problem. And because we only saw the one, I think we saw kids using five is half of 10, which is a fine strategy for that problem. But every single third grader in the classroom was using it. You were really clear that the teacher had probably been a bit too direct. So that's interesting, right? That one of the things we find interesting is to talk to each other about things like that. And then we can kind of beat out, Ooh, based on what we saw, then you and I talked. What would we say to the teacher next? What can be a next move? So, y'all in this podcast, it's one of the things that we kind of would like to do. We noticed that as we work with teachers in professional learning situations, the more often we worked with teachers, I can think of a specific colleague of ours, the more often we worked with her, she kind of stopped this and went to everything that I did, and went and did all of our online workshops, and then talked us on Twitter. And the more that she did with us, the more she was able to sort of be clear about the way that she had been taught, and how that was different than the way that she wants to teach. And because it kind of takes some exposure, it takes some doing more problems over time, and getting used to what that looks like in different contexts. Like, what does it look like in multiplication to think and reason? And what does it look like in fractions to think and reason? This particular teacher was secondary. What does it look like to think and reason finding the equation of a line? Like the more experience that she had, the more she kind of was able to be more clear around, "Oh, this is what it means to teach kids to become mathematicians. To help kids mathematize and really develop." So we thought maybe a podcast could be a good venue to share some of those things that we've learned. That we could kind of give everybody an opportunity to kind of learn along with us.

Kim Montague  15:37

So Pam, who is this for?

Pam Harris  15:40

So we think this podcast is a way for us to talk to teachers, maybe parents. I don't know about students, but for sure, math teachers about my work. So let me tell you a little bit about how my work differs from maybe some other really good work that's out there. Just In brief, we'll go into this more in future episodes. There's a lot of really good teachers out there. A lot of really good professional development people, university people who their work centers around helping teachers and students really understand and get better at estimation and using properties and things. But toward the goal of getting good at the algorithm. Like their goal is always, "We're going to like learn some stuff. And then we'll teach you the algorithm and now get good at it. And that's, you're going to use that algorithm to solve problems from now on. It's the most efficient way." I disagree with that. My work has been about finding what are the major strategies for each operation, and we can help students to develop relationships and connections, then those strategies become natural outcomes. And we might study the algorithm. The standards call for the algorithm, so we'll study it as a study in place value, but not as our end goal, not as a thing to do. The thing to do is what mathematicians do, which is let the numbers dictate, let the numbers influence how you might use relationships to solve problems. So y'all if you're more like Kim, who has always kind of played around with numbers, we think in this podcast, we can help you help your students become more like you. And like Kim, we can help you learn even more relationships and connections and more strategies. Kim had some great ones to start with. But boy, we've enlarged that. If you're more like me and you thought math was about memorizing and using a series of steps and rules and choosing which rule to use when, we can help build some math just like I did. I came from a person who sometimes I joke had negative numeracy, it was so bad, to now I'm sort of considered an expert in numeracy. If I can do that, then we can help everybody become experts in mathematizing. And we can help us all learn more relationships and connections. So who's this podcast for? I think it's for everybody, wherever you come from, whatever perspective you come from, if you're interested in helping students become mathematicians, if you're interested in helping students become confident self problem solvers, where what's interesting to them is to talk to each other and you about how they're thinking and reasoning. Then The Math is Figure-Out-Able Podcast is for you.  So, y'all, if you could  do us a favor, like the podcast, give us a review on your favorite podcast hosting site, so more people can see it, that would really help more people find it. If you can like us, and give us a review. You can check out the website MathisFigureOutAble.com. And join us on Twitter, Instagram and Facebook for Math Strat Chat, where we chat about math strategies. I throw out a problem to the world and people throw in their strategies and you can see the way people around the world are really using relationships and connections to solve problems. We'd love to have you join us there. So again, if you're interested to learn more math, we want to help students become mathematicians then the Math is Figure-Out-Able Podcast is for you. Because Math is Figure-Out-Able.