Pam Harris  00:01

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

Kim Montague  00:07

I'm Kim.

Pam Harris  00:08

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 mathematics 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? So, Kim, let's talk about how perspectives influence the way you view the nature of math and how to teach math. We think there are three main perspectives that kind of keep us stuck in teaching fake math. In the last episode, we talked about the X perspective, which represents a worldview, that math is about patterns and relationships, and that you can use those to solve problems. But it also assumes that you think that the way you learned all that Real Math was through the teaching of fake math, through your teacher showing you algorithms to mimic. So an X perspective tends to teach the way you were taught, thinking that people will catch on like you did. 

Kim Montague  01:08

Yeah.

Pam Harris  01:08

 If you want to know more about that X perspective, check out last week's episode.

Kim Montague  01:12

Yeah. So to start off this episode, we're gonna actually read a podcast review that we got, because it's a great introduction to what we're going to talk about. The subject is, "Mind blown." And JCPTX writes, "I didn't know that there were other ways to think about math until I listened to Pam. I thought algorithms were the end all be all answers. It was almost like opening an entirely new universe to learn that I could play with numbers. Thanks for giving us some background, and an opportunity to expand horizons."

Pam Harris  01:45

Bam! Woohoo! Like that's the point right? What we're trying to do is open that new universe of what Real Mathematics actually is. Thanks, tons for the five stars and the review, it's a great way for other people to find the podcast, we appreciate that. But it's also a really great intro to another of the three perspectives that keep us stuck in teaching fake math, and we call it the 'Z perspective'. So just like a variable x, y, z, this is the Z perspective. 

Kim Montague  02:12

And this was you for so long, right? So I'm gonna just sit back on this one and let you share a lot about your journey, about what you used to think math is all about.

Pam Harris  02:22

Yep, totally. Alright. So when my teachers said, "Today, we're going to learn to add big numbers," and she lined them all up and we started to carry or regroup as we now say. I thought that was the definition of addition that that was the one and only way to add.

Kim Montague  02:38

And subtraction was even better. Hey, tell us about the time that you were in calculus. 

Pam Harris  02:42

Oh this is so embarrassing. So y'all, I was in second semester calculus at BYU. I have lots of stories about that class, because it was crazy. While I was in that class, there was a moment where I had to find someone's age. And so it was like 1990. And I had to find somebody's age, and they were born in like, I don't even know 1957 or something. And so I thought to myself, "Well, to find age, one subtracts. That's how you do that problem." And so I wrote down literally, like 1990, and then 1957. And I drew the line. And I could not remember the subtraction algorithm. And I was like, Okay, I know, there's something about crossy-outies and there's something about nines." And I was like, trying to, and I literally retaught myself the subtraction algorithm. It didn't take that long, because I kind of actually understood what was happening with regrouping and everything. But that was my view of math. My view of math was that in order to find that person's age, I had to do the subtraction algorithm. And so since I couldn't remember how to do it, I taught myself again, how to borrow from the different places and I didn't think about - it's so funny now - I didn't even think about like 57 to 60, and then 60 to 90. I mean, it was ridiculous how I wasn't thinking about the numbers and relationships. I literally thought to myself, this is University days, I thought to myself, the definition of subtraction is to do that algorithm. And the way to do that algorithm are those steps. And so I dutifully remembered how to do those steps and then solved the problem to find out how old the person was and I thought in digits that whole time, right, it was digit thinking, because the algorithm is all about thinking digits. So I could have even been off if I'd done one of those digit subtractions wrong and would never have known it unless I like redid the algorithm or something to check my work. And then I would have had two different answers and wouldn't have known which is - I mean, the whole thing.

Kim Montague  04:33

I'm so sad for the former you. 

Pam Harris  04:37

Thanks, thanks, Kim.

Kim Montague  04:39

So you asked me about me as an X student. 

Pam Harris  04:43

As a former X student.

Kim Montague  04:44

Yeah, yeah. Tell us about you as Z perspective student.

Pam Harris  04:48

Okay. So I tried to make sense of what to do when, but it was always about the why of making sense about what I've been shown. How I would know to do that step next or this rule for this problem. It wasn't ever about using relationships to do something that came naturally. It was really more about: Okay, the teacher has shown me this, how am I going to make sense of, that was Tuesday? And so then on Wednesday, when she showed us a new way, how am I going to make sure that I know which one to do when and then when I'm doing those, how do I make sense of which was the next step to do in those. So I was doing a lot of thinking and reasoning and making sense of things. But the things I was making sense of were someone else's procedure, someone else's relationships. And it was always this sort of general procedure, which meant I was always doing every step every time, if you would have asked me to do 2001 subtract 1999. 2001 subtracting 1999, I literally would have done all the steps to do that. It's ridiculous, thinking about it now. But I was also really clear that it was all about practicing over and over and over. And another embarrassing thing, this came out in my teaching, because I would tell students the story I'm about to tell you. And I would suggest to them the way to be successful in here is to practice over and over and over. Because, y'all, that's how I became successful at that. That's how I knew what steps to do when and which rules to do in which case. So I'm gonna tell on myself again in the second semester calculus at BYU. So big class, lots of people taking it. BYU is a huge University, it was a big, big class, you know where you're in the big lecture hall with the professor and then you meet with the TA three times a week for help, oh sorry, the other way around three times a week with a professor in the big hall and then twice a week with the TA. And if you're lucky they speak English as their native language and all the things. So all the reasons that that makes it hard. When I was taking that class I wanted to do well, I was gonna be a math teacher, I wanted to do well. And so I literally, this was the way that I succeeded in that class. I did all the homework. And as I did the homework, if I like really had to stress and fuss and go look up the the procedure, look up the steps or whatever, then I marked that problem. If I didn't, if I just could do it, I didn't mark it. And so I did all the homework that way. And then I went back. And I redid the problems that I had marked. Because I had to fuss about them. And so I redid them. And as I redid the homework, the second time, I marked problems again that I couldn't just do; that if I had to look anything up, or it took me too long, or I really had to like stress about them. I remarked those problems, and literally did them a third time. 

Kim Montague  07:26

Wow. 

Pam Harris  07:27

So by the time I got to a quiz or a test, nothing surprised me. I had done all those problems before, many of them multiple times. However, when I reached a problem that I hadn't seen before, I had nothing. Like if it was at all like the problems that I had done, man, I zipped. I had done them all three times. I could no problem, I could just cram through that test. Not a problem at all. But if there was a problem I hadn't seen, it was really hard for me to do anything. If I could sort of figure out which rule to do, even if I hadn't seen it, if I could figure out which rule to do, then was okay. But if it wasn't one that I could just apply a rule I had seen before - I know, I know - then I had nothing.

Kim Montague  08:13

This reminds me of when I told you that Luke was joining the math club and you were like, "Oh, I've got a math club story." Tell us a little bit about that.

Pam Harris  08:20

Oh, you just are embarrassing me all over the place today. So in seventh grade, I didn't have any friends. I was gonna say I didn't have very many, but I didn't have any friends. I was a bit of a loner in middle school. I still say middle school should be crumpled up and thrown in the trash. It was a disaster time for most of us. And so I said to myself, I'm good at math. And there's a math club. And so I will go hang with the math kids, you know, like, surely they'll accept me, because I was a bit of a nerdy kid. And so you know, that'll work right? So I walked in to the math club, sat down. Now they'd already started. So it was already kind of in process for the year, not just that day, but both - it was kind of in process that day. So I don't remember why I was late, but I sat down. And the math coach who's a nice guy, he's actually my teacher came over and he handed me a piece of paper. It had a few questions on it. And I was like, "Okay, we're gonna do some math, cool." And then I looked at the first question and looked at all the questions, and I didn't know how to do any of them. No problem. I've been here before. I knew I was joining the club late. So okay, obviously, I missed the day when they talked about how to solve these problems. I raised my hand. I said, "Hey, Mr. Choles, Coach Choles, I don't know how to do or I've never been shown how to do this. You know, could you please teach me?" And he said, "Oh, yeah, that's not what we really do here. What we do in math club is we kind of, you know, like, think about the problems and we kind of, you know, work together a little bit about how you - there's not like one set way to do it." Kim I didn't know what to do with that!

Kim Montague  09:38

Yeah.

Pam Harris  09:39

I looked around the room. Everybody was kind of leaned over. A couple of kids are talking to each other, plugging in stuff or whatever. I said, "But you don't understand. I've never seen these before. I don't know how to do this problem." I mean, thanks. You're like feeling for me. And he said, "No, no, really, like just start using what you know, just start messing around with the problem." It was so out of my worldview, I quietly got up, I walked out of the room. I never walked back. I was done.

Kim Montague  10:05

Oh man.

Pam Harris  10:06

That was my brief, brief moment of hoping the math club would be my social savior. And it wasn't. Because it was so out of my perspective, this perspective, we're talking about that math is doing what your teacher tells you to do. They show you how to solve the problem, and then you can solve those problems. And that's not what obviously what they were doing in math club. That's not what your son's doing. He's had a much better experience in his math club, right? 

Kim Montague  10:31

Yeah, yeah.

Pam Harris  10:32

Good. So as a math teacher, educator, and as a former Z, now I know about Real Math, and I know what mathematicians really do with relationships. I work with a lot of Z's, I work with a lot of people who still have that perspective. Probably the majority of teachers I work with K-12 have this Z, rote-memorizing perspective. Now, I don't know that that's true of people. But I do think it's true of teachers, that it's the majority. Not like everybody, but I think the majority of people have this Z perspective. I know it was me, right? I think a lot of us did well, in math, we did well mimicking our teachers, we did well, solving the problems the way we were told to do. And we liked kids, so we went into teaching. In fact, sometimes I'm a little bitter that I could have actually learned all that stuff that I rote memorized. 

Kim Montague  11:22

We hear that a lot in workshops. 

Pam Harris  11:24

Absolutely, absolutely. It's this perspective of why doesn't everyone teach math this way? If this is really what it is, why were we sort of forced to go through all that rote memory stuff. So getting to work with Z teachers, it's really fun, we get to see their Ah-ha, as we get to see them learn what Real Math is. It can also be quite unsettling for teachers with a Z perspective, because they've kind of, you know, sort of based their life's work on one perspective of what math is. And now we're kind of giving them this idea that mathematics is actually something different. So in that moment, I have to quickly make sure I catch them, make sure that they realize that they can absolutely do the Real Math and we usually do. We usually, you know, catch them quickly enough that they're like, "Okay, well, I can do this." But I want to warn you, the folks out there that identify with the Z perspective, don't get so unsettled that you're like, "Wait, what? It's all a sham!"  No, no, no, you're still a brilliant teacher, now we can use all of what you're good at, to now teach Real Math to help kids learn the Real Math. So if you identify with this Z perspective, you might inadvertently be carrying that into your teaching. If math is all about rote memorizing things that are just made up to be that way, then you're focusing - if you that's your perspective - you're probably focusing on helping students memorize and retrieve the right things at the right time, you probably make things super organized and help students remember all the things with memory stuff. So my goal when I work with a teacher with a Z perspective, is to help them understand two major things. First: I want to get you mathematizing. I want you to catch the vision about what Real Math is. And then know that that can be unsettling, right? That could be uncomfortable. And so like Brene Brown suggests, we need to normalize that discomfort, that learning is uncomfortable. And so I want to normalize the fact that it's not going to be quite memorize this thing, do the steps in this order, you'll get a correct answer every time. That it's a little more about playing with numbers, using relationships, and that that can be a little unsettling. So I want to normalize that discomfort as I help you mathematize yourself and catch the vision of what Real Math is. And then secondly: I want to encourage teachers with a Z perspective to watch their teaching and kind of catch where the Z comes back in. In other words, we'll often get teachers who are like, "Okay, alright, I get it, I'm gonna go teach Real Math." And they go teach Real Math, but sometimes the kind of Z perspective sort of slips back in a little bit where all of a sudden they're helping students memorize an alternative strategy. Or they might, "Today guys, we're gonna learn the Doubling and Halving strategy. Step one, step two... " like they're turning the relationships into steps. 

Kim Montague  14:11

Right.

Pam Harris  14:11

And so you kind of want to watch your teaching, you want to kind of be on guard a little bit not to let the Z slip back in. It's not about steps. It's not about rules. It's not about making strategies into rules and learning the steps of the strategies. It's about learning the relationships, creating the mental connections in students' heads so that the strategies become natural outcomes. Now there are ways we do that. We do anchor things, we do make anchor charts, we do solidify things, we do talk about them as what relationships we're using here. How can we make sort of general sense of that? We do make statements about it with that, that helps solidify the relationships. But y'all it helps solidify the relationships, not a bunch of steps and rules. So just sort of the two main things working with a Z so if you are identified that Z perspective join us in mathematizing. Join us in learning more and more Real Math and being comfortable, that might be a little unsettling, that you might be in a little disequilibrium as you learn more Real Math. And then secondly, watch your teaching and make sure that that Z-ness doesn't sneak back in, that you start turning things into steps and things to memorize.

Kim Montague  15:22

So in this perspective, we're talking all about the perspectives that we grew up with about what math is, and what it means to teach math. And really, we're trying to help us all recognize our perspectives, because when we do, when we can see them clearly, then we have choices. We can have a choice to teach the way we actually believe about teaching and learning Real Math. It's not about pigeonholing or scapegoating. There's no blame or shame here. It's also not about any one of these three perspectives being the right one. It's about getting us all to do more and more Real Mathematics, like mathematicians do. 

Pam Harris  15:55

Absolutely. Well said. 

Kim Montague  15:57

If you haven't had a chance yet, we would highly encourage you to go over to www.mathisFigureOutAble.com/XYZ and take the super cool quiz to help you identify maybe a little bit about what your perspective was growing up. You can also check out a couple of blogs that go into more detail about each of these perspectives.

Pam Harris  16:18

Super. So we'd love for you to take that quiz, in part because we're sort of collecting data. And so that would be really helpful for us to hop on over and take that quiz at www.mathisFigureOutAble.com/XYZ. We'll also put that link in the show notes. Remember to join us on #MathStratChat on Facebook, Twitter, or Instagram Wednesday eves as the entire world talks about strategies for solving problems using Real Mathematics not memorized algorithms. So y'all if you're interested to learn more math, and you want to help students develop as mathematicians then the Math is Figure-Out-Able Podcast is for you. Because math is Figure-Out-Able!