Pam  00:01

Hey, fellow mathers. Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris, a former mimicker turned mather.

Kim  00:09

And I'm Kim Montague, 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 teaching.

Pam  00:17

We know that algorithms are amazing human achievements. But, ya'll, they are not good teaching tools because mimicking step-by-step procedures can actually trap students into using less sophisticated reasoning than the problems are intended to develop.

Kim  00:30

In this podcast, we hope you teach mathing, building relationships with your students and grappling with mathematical relationships.

Pam  00:38

We invite you to join us to make math more figure-out-able. Whoo! 

Kim  00:43

Hi there.

Pam  00:43

Hi, there. How's it going? 

Kim  00:45

It's good. How are you?

Pam  00:46

Crazy, thanks.

Kim  00:49

Um, yeah! 

Pam  00:49

Wait, wait, wait. My life is full of many good things.

Kim  00:54

Wonderful things.

Pam  00:54

And I am open to... Yeah, whatever. 

Kim  00:58

Alright.

Pam  01:01

I got to hug my grandson twice this weekend, so... 

Kim  01:03

That's exciting. 

Pam  01:04

That made things good, yep. 

Kim  01:05

Super fun. 

Pam  01:06

Yeah, he's so cute. He will... How do I even say this? He will join me in doing things for longer than I think any of my kids ever did. Like, he and I made cookies last night, and he'll just, you know like, grab the... He's 15 months old, ya'll. He'll grab the measuring cup, and we'll dip it in the flour, and we'll pour it in. And then we'll dip it in the flour, and we'll pour it in. And last night, I lost track of him, and all of a sudden, I looked at him, and he had his hand in the flour, and he had it all over his face. I was like, "Oh, hey." Anyway. Oh, he's just brilliant. 

Kim  01:37

I think I forget how old he is, and that he's doing stuff like that. I picture him still tiny, bitty.

Pam  01:43

Oh, he's so cute.

Kim  01:44

So cute, so cute. 

Pam  01:45

And both of them are, I just see one a little bit more often.

Kim  01:47

Yeah. I actually see your other one more on Zoom. 

Pam  01:51

Because you see him virtually, yeah. 

Kim  01:52

That's hilarious. Okay. Alright, well, I have a super, super great review. Jeremy Prior says, "All the right things! I am absolutely loving all that I'm learning from Pam Harris. As a teacher, it doesn't feel like we have extra time." No kidding. "But I genuinely look forward to owning a new idea, strategy, or concept that I can easily turn around in my classroom to help build and strengthen students thinking. Pam makes even the most complex ideas comprehensible and purposeful, so that then we can be prepared to help our students find that math is actually figure-out-able. Highly recommend." Oh, Jeremy! Nice! Wow! (unclear) The thing that I'm like attaching to right now is that I can easily turn around in my classroom. How many times do we learn something and we like feel like it might resonate, but it doesn't translate to...

Pam  02:43

What do we do? 

Kim  02:44

Like, yeah.

Pam  02:44

Yeah, where do we go. What now? Yeah.

Kim  02:47

That's super great, Jeremy.

Pam  02:48

Yeah, thanks. Ya'll, we super appreciate the ratings and reviews that you give. It helps more people find the podcast. It's not just to make us feel better.

Kim  02:59

It is fun! 

Pam  03:00

That is nice. But it really does. The algorithms use it to send the podcast out to more people. So, we appreciate that. Ya'll, if you get a chance today, in all your spare time because we know you're busy teachers, we'd love it if you'd rate, review the podcast. That would be fantastic.

Kim  03:14

Okay, so big congratulations to you Miss Pamela Harris because your brand new book is out any day. I'm so pumped for this book. It is such a great read and a really, really helpful, necessary addition to the math world. I know a lot of people have had a chance to preview and have said amazingly wonderful things. Which is very sweet, but also it's welcomed and it's needed. And I cannot wait for people to get their hands on it. Yeah, I know it's been a long journey. 

Pam  03:49

It has, yes. You know, they say writing a book is like having a baby, and in a lot of ways it is. It's a long, long journey. And you think it's almost over, and then labor takes forever. And this particular last set of edits, wow. I was like, "Seriously, can we get the font right? Anyway, so it's been brilliant. And yeah, I'm super excited about it. I don't think I could be happier about what we've been able to say. And yeah, just really want to encourage everybody. When you get a chance, I think it's your new... What? Book study book.

Kim  04:27

Yeah, for sure. (unclear). 

Pam  04:29

K-12. Yeah, I'm just really excited about the message. And like, I've really taken what I've been thinking, and working on, and researching for a very long time and tried to put it together and give it to people. And like you just said, a really turn-around-able make-sense-able kind of teacher useful way. Mmhm.

Kim  04:30

Yeah. Mmhm. Yeah,

Pam  04:33

so I think that there's a section that we should talk about in this book that... Ooh, I can't wait to hear. Which one? 

Kim  04:50

Yeah, so in this book, you have a section about math distortions.

Pam  05:03

Ah.

Kim  05:04

And I think... You know, I've heard bits of this over time. 

Pam  05:09

Mmhm.

Kim  05:09

But I think that it will really resonate with people like never before. And the first time that I had a chance to hear you speak about these distortions in this particular way was at NCSM. And typically, I know like what your presentation is going to be. And no disrespect, but I kind of like half listen because I've heard.

Pam  05:33

I mean, you help me create,. You know like I run it by you usually, make sure. Yep, mmhm. 

Kim  05:37

But listen, I don't know why. I don't know why I didn't know what you were going to say this time, but I remember like... I remember you starting to talk, and I heard you say something that I was like, "Okay, I know what she's going to do here. Butyou described these distortions differently than I ever had. And I just got to tell you. Like, when you started, I kind of looked up, and I just looked at you, and I like sat up. And I remember sitting up and leaning in. And I can't remember who I was sitting next to, but I looked at them like, "This is... Like, wait, I'm hearing something new I haven't hear before. And the people in the room heard it so well as well. I remember like just feeling the silence in the room because it was such like the story, and the connection, and the the way that you described what I'm going to ask you to share today just... It just made sense in a way that other things, I think, like people maybe have kind of heard. And so, I hope that you'll share today about the math distortions. 

Pam  05:51

You know,

Kim  06:03

I remember that particular NCSM presentation because in that silence that you're talking about... Yeah.

Pam  06:53

...somebody went, "Huh." 

Kim  06:55

Yeah. 

Pam  06:56

I kind of zeroed in, and I was like.. You know, I heard the noise, and I was like, "What?" And that person goes, "It's just so good!"

Kim  07:04

Yeah, I think because you have kind of talked about these distortions in a different way, and this time, it just hit differently. It just hit differently for me. It hit differently for the people in the room. So, I think we should talk about that today.

Pam  07:19

Cool. So, maybe I'll just mention. The book is called Developing Mathematical Reasoning: Avoiding the Trap of Algorithms. It has been a labor of love. And I'm going to give Cameron, my son, some credit. He had heard me talk about kind of, I think, the way that you had heard me talk about these distortions before. And I would talk about them as perspectives. In fact, we may have a podcast or two where I have talked about perspective. 

Kim  07:44

Mmhm, yeah. 

Pam  07:45

Yeah. And he said, "I don't think you should call these perspectives because that makes it sound like valid, and good, and we should, you know like, I don't know, honor that people have this perspective kind of on life." And he goes, "But these aren't perspectives. These are distortions."

Kim  08:00

Mmhm.

Pam  08:00

And I was like, "That's so good! Yeah!" So, yeah. In the book, just overall, I've poured a ton of learning and thinking to help math teachers and leaders really realize that algorithms can trap students. I know we sort of start the podcast that way.

Kim  08:14

Yeah.

Pam  08:14

But we really dive into details and kind of help teachers go, "Oh, that algorithm. I can see how that does." We outline some traps that kind of hit all algorithms. We've said before. Algorithms are amazing human achievements, but they are terrible teaching tools. 

Kim  08:30

Yeah.

Pam  08:31

Yeah, so when we were at NCSM, and I've been starting to talk about these distortions a little bit in a different way, I showed an image that I think, if I remember right, you don't really like. So, ya'll, listeners, do you remember in the 1990s, and you've seen them since probably, but there's these images called Magic Eyes? Or the technical name is an autostereogram. 

Kim  09:01

Yeah.

Pam  09:02

If you can picture it, yeah. So, it's like this colorful, or can be black and white, but it's kind of a bunch of dots. 

Kim  09:09

Mmhm.

Pam  09:09

And so, it's like you're looking at a rectangle that's just like this field of dots that are of different hues and different, you know, dark and light, and they're just kind of scattered everywhere.

Kim  09:19

Mmhm.

Pam  09:19

You can sometimes kind of see some patterns and whatever, but it really just... 

Kim  09:23

Oh, no, no, Pam. You do not see patterns if you're Kim Montague. 

Pam  09:26

You don't see any patterns at all?

Kim  09:28

No! You put that up there, and it's like static on a TV to me. I remember. I remember you put it up there, and I didn't even bother. 

Pam  09:36

Just like, bleh.

Kim  09:37

Yeah. 

Pam  09:38

Really? 

Kim  09:38

Nope.

Pam  09:38

You didn't even try?

Kim  09:39

No, because I have seen those in my youth. At the mall, they have them like on these stands. And I did not bother. I was like, "I know what's happening here, and I'm not going to see it." 

Pam  09:49

Well, so let's talk about that. So, ya'll, in case you've been under a rock and you've never seen a Magic Eye, or you're not sure what I'm talking about, you could Google "Magic Eye". And just sort of look at these kind of images, where these random patterns of dot. Well, maybe. I don't know. Some people see patterns.

Kim  10:03

Yeah.

Pam  10:04

Of dots. And the idea is, if I understand the science correctly, that they've taken these series of dots, they've taken an image, and they've overlaid these dots on the image, and then they shift the dots in such a way that if you're able to look at the image but focus behind the image. So, when your eyes focus, your two eyes have to, you know, they converge, and they see the image, and they send the information to your brain. But instead of focusing on the image where it is, if you can focus behind the image, it kind of shifts the dots in a way, then a 3D image can appear.

Kim  10:43

Mmhm. 

Pam  10:43

And it's Magic Eye. It's this sort of weird way of kind of taking advantage of how our eyes work and our brains work. There is actually... What do I say? A medical condition that there are people who cannot see it. So, maybe you have that, Kim. I don't know. Have you ever been able to make one work? No?

Kim  11:01

The random occasion. Like, I could. 

Pam  11:04

Okay. 

Kim  11:04

Maybe. But I remember you putting it up there. And...

Pam  11:07

Yeah.

Kim  11:08

...we all knew we were supposed to do. Like, several people like looked up. They leaned forward. They tilted their head.

Pam  11:13

Oh yeah, the entire room was like. Yeah, tilt the head, kind of squint. Kind of like. And there like leaning forward and backward.

Kim  11:19

Yeah.

Pam  11:19

Yeah. 

Kim  11:20

And...

Pam  11:20

It's a... Go ahead.

Kim  11:21

Well, I was going to say. After a little bit, I remember people like murmuring that they saw. (unclear).

Pam  11:28

A few, yeah. I've actually found that with that particular image, if I showed on a big screen, like we were in a big ballroom, that it's actually harder for people to see. There's something about that focal length. There are usually a few people that can make it work. I found that if they take a picture of it on their phone, and then look at it on their phone, they're able to sort of change the focal length. More people can see it that way. I've also found that if I've done it in a virtual setting, that a lot of people. If people can make a Magic Eye work, you know, they typically, can sort of... Yeah, make it work. That on a zoom when it's in front of their... Basically, I think it's when you can adjust the focal length, then they can make it work. So, it's funny. I've done it enough times now that... Yeah, so as I throw that image up, people are trying. You know maybe they've been successful in the past. Maybe they're like you, and they're like, "Ugh, whatever! I hate these things!" 

Kim  12:24

Yeah, for sure. 

Pam  12:25

So, it's really less important whether you can make it work or not. In fact, I just did a presentation a few days ago. I showed this image, and I said, "Ya'll stop trying," because in the moment, it's less important that you can see the shark. And they look at me. And I'm like, "Oh yeah, because if you can make this work..." The particular image that I was showing, "...A 3D shark will pop out of the middle." Now, when you just look at the particular image I was showing, there's kind of these vertical squiggles.

Kim  12:57

Mmhm.

Pam  12:58

And there's kind of this one horizontal, kind of like wave that goes across sort of the bottom two-thirds of it. So, if you just kind of glance at it like that, you don't try to actually make the autostereogram work, 3D image. You just kind of glance at it. There's definitely a kind of a weird pattern that you can kind of pick out with these squiggles and this wavy line. And I say what's much more important in this moment whether is not if you can see the shark but that you've had the experience of trying to see the shark. 

Kim  13:26

Yeah. 

Pam  13:27

Because that's what I'm trying to draw on. I'm trying to draw on this idea of you've been in this position where you've seen this thing. You know that people are telling you that there's a shark there. And you're like, "Okay, let me see if I can make that work. So, Kim, would you consider that there are people out there who can see the shark a little bit easier? Let's just sort of use this as a metaphor for a second to say that there are people who have natural talents, I think, out there. You know, some people play basketball easier. Some people run. Hint, that's you. Like gazelles. You know like, we all have natural talents, and inclinations, and interests that sort of make some things come a little bit easier for us. That doesn't mean that I can't run. I don't ever be as good of a runner as you, but I can run if I put the time and effort into it. So, that there are some people out there who saw patterns and put them together to make more math. Like, they were out in the world, and they had a low dose of mathematical patterns, and they used them, and they put them together, and then all of a sudden, now, like more math becomes apparent or useful to them, and then they can put those patterns and relationships together, and then more math. And they mess with those relationships. And then one of the things as they were messing with these mathematical relationships is they dug deep and they generalized. They were like, "Oh, check it out! Like, if I line these numbers up, I can actually add these numbers in columns because there's this place value thing happening." But that's after they've really explored, and messed with, and really created mental maps, and connections, and relationships. Then one of the things. Not the end goal, but one of the things they do is that they create algorithms. They create generalizations that could allow people to operate on stuff. Those procedures can help people get answers. And ya'll, historically, that was super important. The algorithms that Algorithmi...I pronounce his name wrong, I'm sure...came up with in the I think 600s 700s A.D. They were super important because they allowed shopkeepers. You didn't have to be trained in the abacus anymore. You could just actually do the sums and the figures yourself to keep track of your finances in your shop. That was super important. But just mimicking those steps of an algorithm to get answers? Again, super important. But that's not really doing the same processes that mathematicians, the same mental actions that those mathematicians were doing as they were creating those mental relationships and creating those generalizations. Would you consider that then teachers and textbooks took those steps, and somehow that's what gets taught?

Kim  13:27

Mmhm. 

Pam  14:18

So, again, if everybody can think of this kind of a Magic Eye image. You're thinking about those dots and everything. Here's the analogy. Be a student in a classroom and be in that moment where we're learning something where students often get kind of stuck. Like, maybe it's long division. Maybe it's fractions. Maybe it's solving proportions. Or like solving equations. Like, pick a moment where students often kind of get stuck in mathematics. Maybe it's memorizing their multiplication facts. You know like, whatever. And could you be a student who is sitting in that class, and the teacher says, "Alright, everybody, ready? Does McDonald sell cheeseburgers? Here are the steps of long division algorithm." And you could be a student who has, for whatever reason, you can just see a few, get a low dose of patterns. And you're like, "Whoa! Like, that's really cool! I could see that shark!"

Kim  17:10

Yeah. 

Pam  17:13

"Like, I see..." And you start playing with the relationships, and you can kind of see what's happening behind the scenes. And you're like, "Whoa, I could see the shark! Why is the teacher making me write down squiggles?" Like, because the teachers like, "Here are the steps that you do. And now, you got to do this." And the kids like, "No, like I can just... I see what's going on here. And I can just see that I..." Pick a problem 4 divided into 364. They're like, "Well, I can just see what's happening. Like, I can think about how many fours are in 360, and then there's only one more 4 left over. Like, what? You're going to make me write all these steps down? The students who, with some low doses see the shark in that moment, they're like, "Why are you making me write down all these squiggles?" Do you have that student in your class where they just come up with an answer? And you're like, "Yeah, you got to show your work." And they're like, "Really? Really I have show my work?" They can just see the shark. And then, let's be clear, they might not even know what worked to like really what their brain was doing because it certainly wasn't the steps that you were just trying to get them to to memorize. So, we might have some students that do that. But then be some other students in the class where the students might go, "There's supposed to be a shark there? I don't... I just see a bunch of squiggles, and dots, and stuff." And the teacher says, "Alright, start doing these steps. Memorize these squiggles." And the students are like, "Oh, that's math. Okay, math is memorizing these squiggles. This is dumb. Like, okay, I'll memorize your squiggles. But wow, this isn't related to my life. And when are we going to use this? Alright, math is completely irrelevant. But here we go. Step, step, step. Does McDonald's sell cheeseburger. Blah, blah, blah, blah, blah,blah, blah." And math, to them, becomes about squiggles, and wavy things, and dots, and they never actually see the shark. They don't ever do the mental actions that mathematicians actually do. They just buy into the myth that math is memorizing a set of disconnected facts and mimicking procedures. Alright, I think there's another. So, these are the distortions, right? The first... Well, I'm going to get distortions in just a second. So, the last group of students. Could you be a student in the class that goes, "There's a shark there? Cool! I want to see the shark!" And the teacher goes, "Alright, so here's the first squiggle and the second squiggle." And the students are like, "Whatever, with your squiggles? No, I want to see the shark! Like, I believe there's a shark there. Can you help me see the shark?" And the teacher is like, "Start squiggling. Like, pick up your pencil."

Kim  19:26

Yeah.

Pam  19:26

"Show me the squiggles." And the students like, "I don't... What? I'm not... I'm never going to memorize those squiggles. I'd like to see... If you'll show me how to see the shark, then I'll be able to actually do this. But as long as you're making me memorize squiggles, I don't memorize well." Like, you almost feel the frustrate. That was my daughter. Like, in such a big way, once she could see the shark, she could reason through anything. But man, trying to get her to memorize squiggles? Mmm, not going to happen. That was just not her. So, here's where the distortions fit in. Be those students now becoming teachers. So, I'm the student. First group. Just saw the shark. That student becomes a teacher because they love seeing the shark, and  they want to help everybody see the shark. And they still like, "Hey, everybody! Check it out! Here's a shark!" And they're like, "Okay, a couple kids like are with me. But man, most of the kids... I don't know why they can't. Ah, that's why my teacher did those squiggles. Alright, I guess that's what we need to do. Guess you all need the squiggles, so here's the squiggles." So, that's a distortion. The distortion is that kids need squiggles. That if you can't just see the shark, if you're just can't magically see it, then the only alternative we have is to memorize squiggle and waves. Not so. That's a distortion. We now know ways to actually help all students actually do the mental actions that mathematicians do. Really see the shark. So, second distortion. Be the student who doesn't see the shark, needs higher doses of patterns, and just would buy into memorizing the squiggles. That student becomes a teacher, and that teacher says, "Alright, everybody, ready? This is a shark. Now, I don't really know shark, but we're just going to sing some songs, so that you remember. Do some raps and rhymes, so you remember that these squiggles means shark. Okay, everybody? I know it doesn't really mean shark, but that's okay. Math, whatever. We're just going to like... We're going to memorize squiggles. That's what math is. They might have kids in their class that see the shark. But boy howdy, that teacher's going to be like, "No, you got to show me squiggles."

Kim  19:33

Mmhm. 

Pam  19:44

"Because math is about squiggles, and you're going to show those steps, and if you don't show those steps, then... You know, because that's what math is." Now, I'm not blaming that teacher. That was me.

Kim  21:26

Yeah.

Pam  21:34

Like, I was the one who believed that's what math was. I wasn't doing the mental actions of mathematicians. I was memorizing and mimicking squiggles. And man, I did it well with my students and didn't really know what my kids were doing that could naturally see the shark. It was this distorted view of what mathematics actually is. I felt like you were going to say something (unclear). 

Kim  21:54

Well, I was going to say as you're describing the students in this distorted view classroom, you've got students who see the shark, but are forced to do squiggles. 

Pam  22:04

Yep.

Kim  22:05

That's frustration for those students. 

Pam  22:07

Sure.

Kim  22:07

You've got students who want to see the shark but aren't getting to see the shark. That's frustration. So, we have classrooms of kids who are not loving math. Who are not wanting to be in the room. Who are always talking about, where does this fit in my life? And...

Pam  22:25

How is this relevant? Why do I have to memorize these steps? Math is stupid (unclear). 

Kim  22:29

And for different reasons. They're not all frustrated for the same reason, but it's super helpful to know like that we can address all those frustrations by helping them see the shark, by talking about the shark.

Pam  22:45

Actually being able to. We now know how to help everyone see the shark. And when we actually get kids sharking.

Kim  22:53

Yeah.

Pam  22:54

And not just memorizing and mimicking, then everybody comes alive. And no longer do we ever hear kids talk about, is this relevant? When am I ever going to use this? Because it's so interesting because they're actually seeing the shark, and thinking, and reasoning mathematically. Some quicker than others. That's okay. Some need more doses of the patterns than others. And that's okay. The distortion is, what is the nature of mathematics? And if we can help everybody actually see the shark, then bam, off we go. We can actually not be seeing life through this weird distortion, but we can actually see that 3D image of real mathematics. So, let's be really clear. I ran into a teacher that I've worked with before in the past where I'd called these perspectives and not distortions. And she said, "Oh, yeah. Like, I'm this one." I don't eve nremember which one she said. And I was like, "Oh, not anymore, though, right? Because that was a distortion you were seeing throught.

Kim  23:48

Right, right.

Pam  23:49

Now you're on Team Real Math. Now, you're actually helping kids really see the shark, and do the mental processes and actions that that mathematicians actually do. And she's like, "Oh, yeah! Oh, I really like how... Oh, that's the new way..." Yeah. So, it's been super good to kind of gain some clarity about how to discuss the fact that those are distortions. We don't want any of those distortions. We want to get real mathing

Kim  24:16

Right. Yeah. I'm thinking about how, you know, in when I was a student. And, you know, I did see the shark. And I wonder if other people in my class would have seen the shark if the teacher had asked how I was thinking about something. And I have to believe that if we were discussing mathematics and not just steps, if more kids would have come alive. I believe that that's true. That if (unclear)... 

Pam  24:45

Absolutely.

Kim  24:46

...about mathing, talking about the shark in a way that other people could really see it and not just memorize, we would have generations of students growing up with more mathing, and with more relevance in their lives, and more opportunities. But it's not too late! We can make those shifts for current students. 

Pam  25:07

And maybe let me make just one fine point in case somebody's kind of getting caught in the metaphor. When we say that, we're not actually suggesting that teachers need to go to their students and go, "Let me tell you about a shark." It's not that. It's that we actually are seeking to help everybody understand what real mathing is, and then discussing that, making students... So, pulling out student thinking, making that thinking visible, and high dosing students with those major patterns. So, it's not talking about anything. It's not random. Like, the standards are there for a reason because they actually do build us towards the important mathematics. But knowing what those major models, and strategies, and big ideas are to focus on those in our discussions, and in what we make visible. That's why Problem Strings are written to high dose kids. Thanks Brendan Scribner for that. I'm loving this high dosing thing. 

Kim  25:43

Mmhm.  Yeah, it's great.

Pam  25:58

So, when you said if your teacher would have asked you to describe your thinking, more kids would have come alive. Totally believe that. If your teacher would have asked you to discuss your thinking and high dosed kids with those patterns, all kids could could have come alive. 

Kim  26:14

Yeah.

Pam  26:14

Because most of us need higher doses than just a little bit. I think number talks kind of got us started toward that, but number talks don't give us a high enough dose.

Kim  26:23

Sure.

Pam  26:23

Of the patterns. We need more of those. Yeah. Whoo!

Kim  26:25

Well, listen. I love. And you've written about this in the new book, and it is going to be out any day now for people to catch their own copy. And I hope that book study groups all over are using this to help understand, help people understand what mathing can really be in classrooms. So, I'm pumped. Congratulations again! 

Pam  26:46

Thank you so much. And, leaders, this can really help you understand where your teachers are coming from, so you can get a chance to help them actually see the shark. Alright, ya'll 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. Let's keep spreading the word that Math is Figure-Out-Able!