Pam Harris  00:01

Hey fellow mathematicians. Welcome to the podcast where Math is Figure-Out-Able. I'm Pam. 

Kim Montague  00:07

And I'm Kim.

Pam Harris  00:09

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. That math class can be less like it has been for so many of us and more like mathematicians working together. We answer the question, if not algorithms, then what?

Kim Montague  00:32

Last week, we introduced the idea that there are a variety of perspectives we bring to the math classroom, as teachers and learners. And in today's episode, we're going to begin parsing out one of those three perspectives. As you listen guys, we invite you to consider: was this my point of view growing up? Is this my frame of reference today?

Pam Harris  00:54

So both right, we want you to be clear on that. It's what you sort of thought about math as a kid, and then is that still the same way that you're thinking about it today? And to be clear, we're not actually advocating any of these perspectives. 

Kim Montague  01:06

Right.

Pam Harris  01:06

We are advocating that we all recognize what we might have been influenced to think. And now make a choice to teach and do Real Mathematics. So this episode is all about the perspective we call the 'X perspective'. X like the variable x, like the letter x, the X perspective. These are the people that when their teacher said, "Alright, class, today, we're gonna learn how to add big numbers," and so  like 37 plus 99. And then you line them up, and they had you line them up vertically, and they then said, Now, don't pay attention to the 90 and the 30. And that 99 plus 37, just pay attention to the nine and the seven, the sort of the small numbers, and they taught you all the steps. And they just like, do everything with digits and all the things - that when all that was happening, the X-perspective kids, were kind of like, "Okay, I mean, I guess you could do all that work for other problems. But for this problem, you don't need to do that." Right? Like you kind of looked around at everybody, like, we're all in on this joke here, because you could do all those steps for a problem like 99 plus 37. But nobody is right? The X perspective kids were thinking to themselves 99... 100. And so 99 plus 37 became 100 plus what's leftover, 36. And they're like, just kind of chuckling. Of course, the rest of everybody is thinking that too, right? It's almost like you kind of were like yucking it up a little bit like kind of elbowing the person next to you. Right? Ha, ha. Like, you were kinda looking at what the teachers doing and thinking, "I mean, you can do all that work, but not for that problem." And the rest of us who were looking at you like, "Uh no." Like the rest of us were just dutifully following what the teacher did. But the X perspective, that we're going to focus on today, did stuff in your heads to think about that problem. And thought the rest of us were, too. So Kim, you relate to this perspective, right? 

Kim Montague  03:00

I do, yes.

Pam Harris  03:01

So tell us about what you were thinking as a student, go.

Kim Montague  03:04

Oh, go. Okay. So as a student, there were a lot of opportunities in that scenario that you just described, where I would look at the problem and kind of mess around a little bit. Right? Like it depended on the numbers. But I can remember specifically, something that happened a lot for me in like, I don't know, fourth, fifth grade. And I did at the time, a lot of what you and I kind of call, like a rolling accumulation of amounts, maybe kind of place-value-ey. And I would do that a lot. And then I would put all the marks back on my page, because I was supposed to, I'm air quoting, "I was supposed to." And I remember, sometimes my dad and I would talk about it, or he'd sit next to me while I was working on assignments. And he would be so annoyed, because he thought a lot about numbers like I did. And he would say, "Why in the world are you putting that up there? That's not exactly how you were thinking about it." And I just wanted it to look right. And so I did.

Pam Harris  04:12

So tell us more like what what do you mean, you didn't need the marks? Like what marks are you talking about?

Kim Montague  04:17

Okay, so like you gave the problem 99 plus 37. But so maybe like for a problem 56 plus 78. I would a lot of times start from the biggest amount. Right? So from left to right. And if the problem was 56 plus 78, I would think about the 50 and the 70. And I would call that 120. And then I would look at what's left, and I would say, "Okay, I have the six and the eight left." So I would have 120 and then I'd maybe add the eight to get 128 and then I have the six left from the 56. And so I would say, "I have 128 and six is just 130... 134." And I would just keep adding on pieces to the total amount that I have. And I did a lot of that when I was younger. I would add a lot of left to right until I had some more sophisticated strategies. But I wanted my paper to look the way that it was supposed to. And so instead of recording chunks along the way, then I would record the little tally marks at the top of the numbers that would show the way that it was supposed to be shown.

Pam Harris  05:30

The way the teacher showed you. 

Kim Montague  05:32

Yeah. 

Pam Harris  05:33

So you were using that rolling accumulation strategy to add up all the numbers. But then you were a little bit annoyed, or at least your dad was, that you had to put in the carry, or now we call regrouping marks in order to get credit. 

Kim Montague  05:43

Yeah.

Pam Harris  05:44

Kim, can I can't believe you did that!

Kim Montague  05:45

Well, it mattered. Credit mattered to me, right. And so if my teacher said, this is the way - and I loved my teacher, and I wanted to please her - and so if she said, this is the way that it was supposed to be done, then I would do the things in my head, but then my paper would look 'correct'. Yeah. So as a kid math just kind of made sense to me. I did a lot of um... Oh, man, Pam, you should have known me. I counted all the things all the time, like, lots of counting, and lots of generalizations. Like I thought a lot about problems and kind of categorized them. And so my dad and brother also had an X perspective. And it wasn't weird. We talked about math all the time. And really, I assumed that lots and lots and lots of people did that. And we would talk about math all the time in our house. My dad would would constantly be asking us questions. And so it's just kind of a thing in my world.

Pam Harris  06:48

So you guys, your dad, your brother, you saw through the rote procedures.

Kim Montague  06:52

Yeah. 

Pam Harris  06:52

And you used relationships, you played with numbers. And then when the teachers showed you an algorithm, then you tried to make sense of it, or you made sense of the problem, at least. 

Kim Montague  07:03

Yeah. 

Pam Harris  07:03

And then you decided to put the marks back into it to get credit. 

Kim Montague  07:06

I did. I did. Yep. 

Pam Harris  07:07

So X perspective sort of does that. They make sense of what's happening. Let me give you an example from my life. And this is in a huge way, how you and I got to know each other. Because I started on this journey of learning what my kids were doing. 

Kim Montague  07:22

Right. 

Pam Harris  07:22

So my first kid, his name is Cameron. He's our oldest son. He totally has this X perspective. And it was fascinating to me, because it was so beyond what I had ever - I just didn't even know it existed. And so I just watching him kind of do what he did. Let me give you an example. He came home from first grade with a paper and it was subtraction with regrouping. Which means, for everybody who doesn't teach elementary right now, it's what we used to call carrying or borrowing. And so it was problems where you couldn't just subtract by place value, you had to do some borrowing or carrying in order to subtract. And so we came home with this paper: subtraction with regrouping. And I said, "Hey, all I see are answers, like I don't see any crossey-outies, little ones, you know." If they're were a lot of zeros on the paper, you should have some nines showing up, right? And I'm like, "Dude, you know, like, Are you cheating?" And he goes, "No, no. I didn't really understand what my teacher was talking about. So I made up my own way."

Kim Montague  08:15

Nice.

Pam Harris  08:16

Kim! I was like, that is not a thing! What are you doing? I had no idea. I said, "Show me what you're, tell me, what you're talking about." And so he did. Y'all my numeracy was so bad at that moment I went off to the side, did a little algebra to make sure his method would work every time. It would. I was so floored. It was so not my perspective. I had no idea what to do with it. I kind of put it on a shelf. I sort of set it aside. I had four kids, I had lots to do, lots was going on in my life right then. I just kind of ignored it. I was like he's getting grades, whatever, move on. And so he just continued through first grade, first grade ends, second grade comes along. He brings home a paper subtraction with regrouping. I'm like, "Dude, did you tell your teacher how you did the subtraction last year?" And he looks at me straight in the eye. And he goes, "Nah I couldn't really remember what I did last year. So I made up a new way." Again, I was like so floored. I was like, seriously! He told me what he did. I'm like, tell me, explain that. He explained it. I did some algebra off the side to make sure it would work every time. It would, it was a good general solution. So fascinating to me that he would just, A: know that it was a thing that he could make sense of the math and come up with his own way to do it. And B: it was general enough that it would work every time. I was just fascinated with this. And he wasn't bothered at all that um, he wasn't bothered by the marks at all. But he also wasn't bothered that he had to sort of mimic the teacher. He's like, "No, I'm good. As long as I understand what's going on." He was definitely not as much a teacher pleaser, as you were, which is totally okay. That's his personality a little bit. Either way, right? It's not that part that matters. What matters is that you, both have you had this perspective, that math is figure-out-able and that you could use relationships to figure it out. I thought that was so cool. I'm so grateful that God gave him to me. One other little bit I'd like to talk about with this perspective is when I taught in Michigan. So I taught high school, I'm a former high school teacher. I taught in Michigan, had a wonderful experience, worked with some really great people. We had a math hall. So everybody that taught math was in the same hallway. We called it the math hall, we often talked about things, you know, like, well in the math hall, whatever. It was me and eight men, the math department of the school, and it was wonderful. Again, I really respect and like them personally. We had different perspectives on teaching math. And it was fascinating, because as we would talk, they would say things like, "If they don't want to be in this hall, let's just not even make them. Let's just not even make them come in this hall. If they're not interested in math." Like they really thought that they could let these freshmen, sophomore kids like make these decisions. I was like, I don't know, I think they need some math. Like, I would probably make them take some math classes. Not make, not force. But you know, like I think that it -

Kim Montague  11:03

Created a desire in them to want to be there?

Pam Harris  11:06

Create the desire, yeah! Like let's help them be successful. I was aware, even then with my Z perspective, I'm so aware that math is such a gatekeeper. And I wanted kids to have the ability to make choices to do what they wanted to do. So I wanted them to be successful as long as they could in a subject that I enjoyed, right that I liked. I like math. And, and so I wanted to help and it didn't make sense to me why they - they definitely had this perspective, and we really haven't talked about it too much, so let me bring it in now. Part of the X perspective isn't just that you made sense of the math, but part of the X perspective is that you think that others can too, but others will see through the procedures, they'll see through the steps. They'll make sense of it on their own. 

Kim Montague  11:51

Yep. 

Pam Harris  11:51

Or maybe they just don't have the math gene. 

Kim Montague  11:55

Yeah. 

Pam Harris  11:56

That's the part of the X perspective that we don't like. 

Kim Montague  11:58

Right.

Pam Harris  11:58

There is this idea that we're thinking that, "Oh, I'm doing this and so is everyone. Or I guess if you're not, then I don't know why you're not because it worked for me." The teacher showing you the steps sort of, quote unquote, "worked for you." But it didn't really right. But you think it did. And so that part of that X perspective could be that you think, "Oh, I'm just going to teach the way I was taught, because it'll work for kids have the math gene."

Kim Montague  12:25

Yeah, I think I might have had some of those teacher friends of yours in high school. They were your math teachers? Yeah. It was really, um, I wouldn't say easy, it was really a connection that I was able to make on my own for a lot of years. People would say things, and I was able to draw relationships on my own and make connections on my own. So I hit this point in my career where I didn't have a guide, right? Like, I wanted to make sense of things. And I wanted it to be connected to what I already knew. But there wasn't somebody there helping me make those connections any more. And so once it became about just memorizing a bunch of stuff, that's when I decided I was done, it wasn't fun anymore.

Pam Harris  13:11

And it was high school, there were guys, you were distracted. I remember telling me you were like, "And there were other things for me to be interested in like guys."

Kim Montague  13:20

Yeah.

Pam Harris  13:21

Totally cool. So I'm a teacher educator. And as such, I work with a lot of people that relate to this X perspective, especially at the secondary level. These are well meaning teachers who just keep teaching the way they were taught because it worked for them. My goal when I work with people with an X perspective, is to help them understand a few things. So one: help them understand it actually didn't work all that well for them. They could have gone much further, deeper, and faster if someone was actively helping them. 

Kim Montague  13:53

Yeah.

Pam Harris  13:53

And two: that just because others didn't pick up on it - they didn't pick up it on their own, they didn't sort of see behind the scenes, read between the lines - doesn't mean that they can't. It doesn't mean that they don't have the math gene, they absolutely can. The rest of us can mathematize. Just, don't get them lost in all those rules and procedures. Instead, help them build the relationships. Don't expect people to build the relationships on their own, like you said, be a guide to help them connect with what they already know and build those relationships. And then three: a really important way to build relationships to help others make those connections in their brain is to talk about and make visible about what's happening in your brain. 

Kim Montague  14:38

Right. 

Pam Harris  14:38

So sharing with others, your process is so important. But you have to also make that thinking visible with models. And that's a necessity. Those models are a visual way of putting your brain on paper, so that - or on the whiteboard - so that all of us can see what's happening. Now we can all comment on the relationships because we can see them. Because we can make sense of them, we can all sort of parse out what's happening because you're making it visible. The relationships in these models, that's what mathematicians do in their heads. It's fascinating to me when people sometimes will balk, "What is this? What is this thing you're making my kid do with an open number line, or an open array or whatever." But the relationships that we're making visible on those models are actually what mathematicians have going on in their head. That's what's happening for them. We're just helping others go, "Oh, these are the relationships" by making them visible. So if you relate to this X perspective, three things one: it didn't actually work all that well for you, because you could have gone much further and farther and faster. And two: just because others didn't read between the lines doesn't mean that they can't do Real Mathematics. They can, they just need to know it's a thing. And three: talk about and make your thinking visible. You know, I'll tell you it is a blast working with people who relate to that X perspective, that X perspective growing up because now they are lit up, right? They want to talk about all the math things, and they kind of actually get discouraged when nobody can hang with them. I love talking with them because they like creating others who can hang with them, it re-energizes their teaching. Like Kim, I'm reminded about you and that guy named Clay, remember? 

Kim Montague  16:18

Oh, yeah, yeah. So I remember that. We did a workshop, right? And I remember you talked about the XYZ perspectives, and had everybody in the room. You described them all, and you had us go in the room to where we thought we grew up identifying as. 

Pam Harris  16:37

Like a corner of the room for the Xs, one corner for the Zs, and one corner for the Ys, uh huh.

Kim Montague  16:41

And  there were just a few of us in the room that went to the X perspective corner. And we started talking. And I remember just chatting it up. Just on and on and on. And what do you think about it? What do you do? What are your quirks and it was a lot of fun for me. I love finding other X perspectives in that situation, because building relationships with other people who are new to numeracy is great, but it's fun to meet other people who have existing connections, because we can build off each other. And we can wonder about new things. Listen, math is fun to me. It's awesome to find other people who say the same, because we do want to talk about the things that are happening in our head. We do want to talk about strategy. We do want to play different games. And yeah, that was I remember finding Clay in that moment and thinking we should have this like professional relationship where we share ideas about things. 

Pam Harris  17:37

Yeah, that was a cool workshop. And just to be clear, when you said you like to work with people who are beginning their journey.

Kim Montague  17:42

100%. Yeah. 

Pam Harris  17:43

But it's also fun for you to go for where you are to like build on from where you are. So that's not a beginning, that's why it can be fun to talk to people who have already built some things. Yeah, that was cool. So Kim, we all want to be you. Right? 

Kim Montague  17:56

Well, of course, right? No, no. Messing around or playing with relationships is a desirable outcome for everyone. 

Pam Harris  18:06

That part, yes.

Kim Montague  18:08

But thinking that there's like this math gene, and people can or can't is not ideal. The thinking that everyone is already messing around with numbers naturally, just because I did or I do is not ideal. That's not the perspective we want to have for people.

Pam Harris  18:24

Right. We want people to be messing like you did, but not just assume that it's happening with everybody else, we want them to actively promote it. So maybe it's safe to say that we all want to play with numbers and relationships like you do. And we want to actively help it happen in our kids and ourselves. 

Kim Montague  18:41

So if you relate to this X perspective, keep playing. But also talk about your thinking and make your thinking visible to those around you so that they can enjoy math like you do. In the next couple of weeks, we're going to step into the Y and Z perspective and learn a little bit more. If you're not sure what you identify with, you can head over to mathisFigureOutAble.com/XYZ or check out the show notes and take the quiz.

Pam Harris  19:08

Totally fun quiz that we've created that can help you kind of determine which perspective you thought more about math as a kid as growing up. Like which one are you more like an X like Kim was are you more like a Z like I was? Like where do you fit? Take our really cool quiz at mathisFigureOutAble.com/xyz or we'll have the link in the show notes. We also have a blog series about it so you can read more about it, get more examples and all the things. So head on over to there and you can read more and take that super cool quiz.

Kim Montague  19:35

Thanks so much for the five star ratings on Apple podcasts. We love reading your comments and Ah-has that you're having.

Pam Harris  19:41

Yeah, in fact, Kim here is a five star review that we got recently. I love it. The subject line is "an absolute necessity". Bam! And it reads, "This podcast is an absolute necessity to any and all math teachers. This podcast is some of the best PD around!"

Kim Montague  19:58

I love it.

Pam Harris  20:00

Yeah, so that was really cool. Thank you so much for those reviews and ratings that helps others find the podcast. Remember to join us on Wednesday-eves at #MathStratChat where we throw out a problem and the whole world has a number talk together where we explore problems and solutions. So everybody, if you're interested to learn more mathematics and you want to help students develop as mathematicians in the Math is Figure-Out-Able Podcast is for you. Because Math is Figure-Out-Able!