Chris Shore: There's a lot of little things that I do with those lap boards that I'm not aware of and so, there are some things to do with that. So, there's the formative assessment piece. There's the giving kids type. The biggest thing is giving kids rigor, putting rigorous problems in front of the kids. We usually, I call them the Set C problems in the textbook. Most teachers don't do those because, "Oh, I don't have enough smart kids to do those problems, so we're just going to do the rote. Now, those are just some-

Kyle Pearce: Hey, there, Math Moment Makers. In this episode, we speak with a colleague and friend, Chris Shore, all about changing the world and why Clothesline math is the master number sense maker.

Jon Orr: Yeah. We're excited to bring you this episode. Chris has spent 33 years in education, helping students understand their mathematical identities. And he's here with us and you to give you resources and ideas around a great teaching tool you need to start implementing in your room.

Kyle Pearce: Here we go. Welcome to the Making Math Moments That Matter podcast. I'm Kyle Pierce.

Jon Orr: And I'm John Orr. We are from MakingMathMoments.com and together with you-

Kyle Pearce: The community of Math Moment Makers worldwide who want to build and deliver problem-based math lessons, that spark curiosity-

Jon Orr: fuel sense-making-

Kyle Pearce: And ignite your teacher moves. Friends, we are super excited to finally get Chris Shore on the podcast. We had some scheduling conflicts. We're trying to get Chris Shore as a part of the Virtual Summit in past years, due to some of the other conferences that he's involved in November of every year, but it sounds like this year he is in. But we've also fumbled the ball a few times trying to get something scheduled for the podcast, so it's great that we finally get a chance to catch up on the podcast because every time John and I meet with Chris, usually at face-to-face conferences or bump into him along the math trail, he's always got great stories, great insight, and just an all-round great person. And you're going to hear that come out through his conversation here with us.

Jon Orr: Yeah. We're excited to share his story. He's got an interesting story that he dives into early in his career and where it led him to now, being a math teacher and then now, in a district leader role, so we're excited to share that. And all about the tool that he's been using while he does teaching mathematics to build number sense in his classroom, the closed line in your work math class, too. So let's get to it. Here we go.

Kyle Pearce: See you on the other side.
Hey, hey there, Chris. Thanks a ton for joining us here on the Making Math Moments That Matter podcast, it's been a long time coming. I know we've chatted a number of times, both in-person back before this thing called COVID happened.

Chris Shore: Before the apocalypse, yep.

Kyle Pearce: Yes. And I remember last summer we were chatting about the Virtual Summit and there was a scheduling conflict and you were doing CMC, I think at the time.

Chris Shore: Right.

Kyle Pearce: And you were on the committee, maybe the Chair. And we said, "You got to get on the podcast." We finally got an opportunity here to get you on. How are things going in your world right now?

Chris Shore: Well, super. My son just got married, so now, two kids...

Jon Orr: Wow, congratulations, yeah.

Chris Shore: ... two married. And it was an epic wedding. They lived down in Mexico and it was...

Kyle Pearce: Oh, wow.

Chris Shore: ... down in the wine country of Mexico.

Kyle Pearce: Love it.

Chris Shore: It was fantastic. So, my wife and I are true empty nesters now and we're out of... well, I'm cautious to say this, but COVID's done and we get to relaunch another year and hope, like we did last year.

Kyle Pearce: Right. Yeah. Uninterrupted.

Chris Shore: This better be COVID-free. Uninterrupted and then that didn't happen, so...

Kyle Pearce: Exactly.

Chris Shore: ... we're hoping this year it does.

Kyle Pearce: I'm like you, I don't want to jinx it, but I feel like we're close to saying, yeah, this is going to be a fresh start and it's going to feel more normal than any of the last two years have felt. So, I'm looking forward to that September as well. Chris, tell our listeners a little bit about yourself in the sense of what's your current teaching or educational role. And then give us a backstory of how you got there, what led you into that role? A little bit of a backstory on your teaching journey.

Chris Shore: Well, right now I'm nearing the end of my official public education journey. I've been in the gigs since for 33 years.

Jon Orr: Nice.

Chris Shore: I started teaching in 1989 and I always joke that the great technological advance my first year teaching was the Expo Marker because I did my student teaching with chalk. And then I got my first job and they replaced the blackboards with the whiteboard and then, we got the Expo Marker. So, my first classroom had no phone, no computer.

Jon Orr: Right.

Chris Shore: And it wasn't until my fifth or sixth year teaching that they started putting these blue cables, blue wires up above the ceiling tiles and Katie Kirk is trying to explain to us what the internet is. So, that's how far back I go. But I started there and taught high school math for 29 years and moved from there to a District Math TOSA position. And then now, I am a Coordinator of Curriculum Instruction, Secondary, so I oversee eight middle schools, a little over 300 teachers. The interesting piece is I spent three decades doing math and now I'm working with Math, Science, Social Studies, and English.

Jon Orr: Wow.

Chris Shore: So, I oversee the four core subjects at four middle schools and four high schools.

Kyle Pearce: Wow.

Chris Shore: It's a wild gig, but I had interesting and I'll say very fortunate journey. I've had a lot of mentors lay a really good path for me, a more innovative path. I wasn't planning on being a teacher. So, originally I wanted to be a comic book writer. I remember when my dad first asked me, "What do you want to do when you grow up?" I said, "A comic book writer." It turns out I have some writing skills, but my drawing skills are amateuristic, not pro. So, then I wanted to be an astronaut and then I threw up on the teacup right at Disneyland. So, I thought, "Okay, that vocation inaudible..."

Jon Orr: V-forces isn't your thing.

Chris Shore: ... that's out the door.

Kyle Pearce: You can work towards it and you stuck with it. You just got to just...

Chris Shore: Yeah, get with it.

Kyle Pearce: ... get to the next level and keep going, yeah.

Chris Shore: So, I go to college, wanting to be either a lawyer or an architect. And then, I decided six months in, I didn't want to wear a tie to work every day because these are the very intelligent decisions that 18 year olds make about their lifelong careers.

Jon Orr: You got to put your foot down somewhere, right?

Chris Shore: Yes, that's the line. No tie. But I had some buddies that were Computer Science majors and remember, this was the '80s, so nobody had these at home yet like we do now. And then of course, the mathematical scientific bent in me said, "Hey, that would be cool." And then at the end of that career, so this applies to what we'll be talking about today here is at the tail end of that career, I've met two very good buddies that I'm still in contact with. In fact, one of them was at my son's wedding and we took a gaming class together.
So, I actually took a class on creating games and part of the assignment was you had to partner up. So, yes, it was a group project in college and we had to create a game together. And apparently, we did a good enough job that the department lead pulled us in and said, "We think after you guys graduate, you need to go market this thing."

Jon Orr: Wow.

Chris Shore: It was an educational game. So, we did that, but as we went out, we had to support ourselves, so we each got to various day jobs that took us away from the task. But I was coaching with the coach. I played water polo in high school and so, over the summers I was going back and I was coaching. And so, I was doing that and I was looking, "What else can I do while we're doing this thing?" And I need some flexibility, and there was a school district that was giving out emergency credentials for substitute teaching. They were so desperate. All you needed was degree. You didn't need the credential.
So, I get this and on a daily basis, I'm going there and then I'm showing up on the pool deck afterwards. I'm like, "Hey, this substitute teaching is fun." And the coach is looking at me like, "Dude, substituting sucks, man. If you that, you need to be a teacher." And so, I said, "Okay." And enrolled in credentialing classes at night, kept doing the substituting during the day, we sold off the game and we all went our separate business ways and that's how I got into teaching.

Jon Orr: Wow.

Kyle Pearce: Wow. What a story. It's really interesting. Because even going all the way back, and first of all, I wanted to comment on the Expo Marker and just the transformative changes in the classroom that happened once the Expo Marker showed up. Not so much, but it definitely looked brighter and shinier and nicer. But all the way through, something I picked up, was oftentimes you'll have someone who says, "I knew from a young age, I always wanted to be a teacher." We've heard the story of educators. And I think John, I don't know, we'll have to count, but I'm thinking more so female educators so far on the show have painted this picture of all the teddy bears in their room, being their class and teaching the class of teddy bears.
And yet, Chris, you're on this other end of the spectrum where you've sampled all the ideas, from comic books to space, to engineering, all over the map, and then finding your home in education, which I think is a really cool journey and a fascinating journey. And I'm sure gave you an opportunity to get into that role, feel good about it, and then also, know that it wasn't like you didn't consider other pathways or other options, which I think is pretty fascinating. So, what a cool, cool story.
Now, I want to know though, so you had all these different ideas, you had all these different ideas of what you might want to do with your life. Some of them, the decisions were forced on you like the teacup ride said no for that role. Others was because of the tie, the wear that you weren't willing to do. So, some of them were personal choices. I'm wondering from your math experience, what's that math moment that if you were to think now, you're like, "Wow, I ended up in education. I've been in this game for 33 years." You've done so many things in mathematics education in particular. Is there a math moment that pops out at you when you think about your own experience in school as a student that you'd be willing to share with the audience?

Chris Shore: Well, sure. And there's the math moment and then also the teaching moment, but you asked about the math moment, so I'll go with that because this influences a lot of what I do, because I pondered this a lot. You ask all the people on your podcast, of course...

Jon Orr: We do.

Chris Shore: ... this is question and it's interesting, I think everybody should ponder this question because it revealed a lot. And one of the things that I always go back to is I was in college. Well, I had two, actually. The first one was in high school. Now, remember I went to high school in the '80s. We didn't have AP classes and dual enrollment and all this other stuff and my school started the first honors class. And so, it was Algebra 2 honors. So, part of that story is I can tell you what it did for my psyche to know that I was identified as a kid that could do math.
The other side, now that I'm adult, is why aren't we giving that to all kids? I'm not saying stick them all in honors classes, but why aren't we all giving them that identity? But I remember watching, being in that class and that's when everything in math started coming together for me like conic sections. So, take two cones, put them point to point, you cut it with a plane one way and it's a circle. You tip it a little bit and it's in an ellipse, you tip it a little bit more, you got a parabola, and you tip it all the way up and you got a hyperbole.
Even at that age, I'm thinking, "Why is that a thing? You just cut these things up and you get something useful?" And then when it started showing like, "Oh, and there's structure to it. We can build these equations and we can figure stuff out." I'm like, "Why is that a thing?"
And so then I go off to college, I guess I was having too much fun because I remember missing the first day that the professor talked about derivatives, because this is first semester of Calculus. I show up on the second day, so the first day they proved this thing and the second day I'm walking in and then of course, I was having too much fun because then now that I'm thinking about this story, I was late on the second day because there was already something up on the board.

Kyle Pearce: Oh, jeez.

Chris Shore: And I'm looking at this, but I'm able to walk into the class on the second day of derivatives and look at the board, this professor is writing on, and I'm seeing what they're doing. So, you multiply the exponent times the coefficient and you decrement the exponent. So, you got three X squared, two times three that's six, the squared becomes one and now, you got six X to the first power. And again, I'm thinking, "Why is that a thing?"

Kyle Pearce: Why is that? Yeah.

Chris Shore: Why don't you add them and then just, raise it to the power and change the X to a Y, like all these things. And then why is that a thing and then why is it useful? And so, I missed the proof the day before, which is okay because we, as kids in high school, we're not often taught that, just do the algorithm. And then you move on to things of like, "Well, why is this useful? Wait a minute. We can actually talk instantaneous velocity. How? How is this working?"

Kyle Pearce: What a blessing that you didn't see the proof ahead of time. Talk about a curiosity killer. You came in and you're like, "Huh," looking at this. Everybody else is like, "Well, you already let the cat out of the bag and we know why this works and probably still don't know." I remember seeing the proof first and still going, "Why is this a thing? I don't even know what's going on here?" But I'm sensing this natural curiosity you had with the mathematics.

Chris Shore: So my point to here is that those math moments really impact what I do in the classroom. And this idea, I talk a lot about the concepts, procedures, application progression, because that's where I was. "Why is this a thing? And then how is it useful?" As well as, "Here's how we do the thing," so I got these bookends on it. And so, I really pushed that in my classes. I pushed that in my writing, into my teacher training, so those are my math moments.

Kyle Pearce: Love it.

Jon Orr: We've heard so many math moments from folks over the last three or plus years now doing this podcast and a lot of them are what you're describing is like this, this moment where it's like, "I had this ah. Why does this work? And that's what drew me in," or sometimes, it's the reverse because people are saying, "I wasn't shown why it works and I didn't have that natural curiosity. And then that's part of why I do what I do now is because I want to give kids that natural curiosity or help understand why these works."
And I'm wondering Chris like when you said you use that in your teaching, in your role now, I wonder if you could give us an example in a sense, you're in a district role now. Because I know that we've got lots of teachers who listen to this podcast, but we also have lots of school leaders and administrators listening in. So, I wonder how do you use that curiosity, that drive towards understanding these ideas in your role now? Because I know that you're not just a math leader, you've got a full bundle of Math, English, Social Studies and Science. So, would you be able to enlighten us just for a moment on how that helps in your role now?

Chris Shore: I do have relationships. The math team, I'm in a different district than I spent 10 years in one district, 20 in another, and now, I'm here, but I'm in the same Valley. I'm at now, actually, in the district in which I live and I knew some of the people. So, there's a connection with the math community and there's mutual respect there, so that's good when I go into talk with math teachers. I have to be a little more gingerly when I talk to English teachers, like, "What does the numbers guy have to say to, what can he tell us?" And I don't try to.
So, with everybody, we build the professional learning community, because it's the only way you can reach 300 plus teachers on a daily basis is building systems in which they can learn. And in fact, you try to build systems in which they do learn. It doesn't become an option. You show up at a work every day and you go home a better teacher. That's the school culture we want to build. And we're in the process of doing that. And it's going well and the teachers are receptive to it. Of course, there's resistance wherever you go, but for the most part it's there.
The big thing is to focus on results, and also the other piece is something that a principal taught me is, "The answer in the mirror, not out the window." So, it's not about the parents, it's not about the income, it's not about what's going on outside and all these things that you cannot. Don't be talking a lot about the things you can't control. Look in the mirror, "What can you control?" And every bit of research says, yes, all those things matter, parents degrees, income, trauma, those kinds of things.
But the number one factor of the quality of education that students receive, decisions that a teacher makes on a daily basis. And so, once you start owning that, that doesn't mean you take everything from the kids, so we got to be careful. The child needs to be empowered for their own learning. You're not stealing that away from them. But you show up every day knowing, "Hey, I can change the world. With this job, I can change the world."
Quick story on why I say that. So, I first moved out here, we moved out to this area because I grew up on the coast and when I got married and my wife stayed home with the kids, we weren't affording a house on the coast. So, we came inland a little bit, left friends behind and it was just before 2000, the Y2K. The world's going to come to end.

Kyle Pearce: Yeah, it's going to crash.

Chris Shore: Everything's going to crash. Planes are going to drop out of the sky. People are going to loot my house. And I'm here getting these postcards from all my Computer Science buddies who graduated and went into the field 10 years ago. And I'm getting these Christmas cards of their houses on the golf course because I could have been out there making bank on this panic that was going on. And, "Hi, I teach high school algebra." And then my wife and I go out on a date and we go into Applebee's. I don't know if you have an Applebee's Café.

Jon Orr: We do. We sure do.

Chris Shore: All right. So, we go to Applebee's, we're sitting at the bar, waiting for the table and my wife goes to the bathroom. And this guy walks in and sits down next to me and he says, "Round on me." And I've seen this in movies. But never have I been there where a guy walks in and buys a drink for everybody at the bar, but he's sitting right next to me. So, I go, "Dude, what's the occasion?" And he says, "I just made $50,000 on a day trade."
And I said, "Oh, you're one of these guys that sits at home on the computer on their underwear and just makes loads of money?" He goes, "Oh no, no, this is just a hobby of mine." He goes, "I'm an engineer." And I said, "Well, we're kindred spirits. I teach math." And he looks at me. Now, remember the world is about to come to an end. We can't find enough people that know math. He goes, "You have a math degree?" I said, "No, actually I have a computer science degree and I teach math." He goes, "Wait a minute. You know math and computers." I said, "Yeah."
True story. Takes out his business card on the back of the business card, writes down a dollar figure that is twice my salary at that time. Now, I got two kids. My wife is staying at home, we just bought a house. And this guy writes this on the back of the card, puts his finger on the card, on the bar, slides it over to me and says, "Interview over. If you want the job, call me on Monday."

Jon Orr: Whoa.

Chris Shore: That's it. You know math and computers.

Jon Orr: Yeah. You're like, "What's the job involved?" I don't inaudible.

Chris Shore: Well, I asked-

Kyle Pearce: It does sound like a movie where it's like you meet this strange character at a bar. They hand you this mysterious card.

Jon Orr: Serious.

Kyle Pearce: You show up and it's like you've been transported to another realm.

Chris Shore: And my whole family is running through a house with hatchets and things like that.

Kyle Pearce: Yeah, all that. It's like The Firm.

Chris Shore: So, I did, I asked him what it was and it was building missiles.

Kyle Pearce: Oh.

Chris Shore: So, the point to the story is I tell every class, ever since that day, I tell them that story, "Kids, you go out there and study anything in STEM, Math, Science Technology, and you could go out there and make bank. We really can change the world. We got kids that are in low income situations that can turn it around in a generation if you just pay attention to what at the gifts I'm going to give you."

Jon Orr: I love it.

Chris Shore: So, yeah, it's all about going out there and change the world. By the way, my wife came back to the bar. The little buzzer in the pocket says, "Your table is ready." And for the first time in our marriage, I sat at the table with nothing to say. She goes, "What happened while I was gone?"

Kyle Pearce: I'm thinking. Let me think for a while. I love it. You are at Applebee's on a date night. Is that from a song? Is that line from a song? And a guy buys around of drinks. I'm not picturing Applebee's being the place where that celebration happens for someone.

Chris Shore: Guy walks into a bar, offers a stranger a job. It's like that joke.

Kyle Pearce: Yeah. Exactly. I love it. I love it. That's great. That's going to be a memory that sticks with me about this conversation for sure. Chris, you've given us all kinds of amazing nuggets. I also want to just highlight this piece about us as educators, trying to get rid of the noise. You were talking about essentially sphere of influence, sphere of concern, rolling back to that idea and really thinking about what can we do? What can we influence? What we can we impact? And there's a lot that we can impact.
And I think one of the other things, the takeaway I got from what you said as well, is that it's not about bringing all students to the same place. It's about helping all students move forward. And for a student who might have some of these other challenges in their life, socioeconomic issues or maybe some family issues or other issues that might be going on in their life, if we can help them and intentionally help that student put themself in a better position than they are today, then we've achieved some level of success, which I think is so important.

Chris Shore: Absolutely. The whole growth mindset thing, people are thinking that somehow we were going to turn everybody into rocket scientists. The whole growth mindset is exactly what you said. No, the kid is going to grow. They're going to be better at math when they're done with your class today than they were when they arrived. That's the goal.
But to finish up your question, some of the things we tell teachers, there's some basic ones, is first off, we're going to teach math. We're not going to teach a textbook. So, that's one of the big ones. Stop talking to me about teaching Section 2.3. What are we trying to get out of the day with 2.3. And the predominant model of instruction in this country, and I say this country, because I've looked a lot at the international studies, is basically we just tell kids what to memorize the night before the test. We spend three weeks telling them on the Thursday night memorize, Friday remember, Monday forget. And we just repeat that cycle.
For teachers, what we said is stop doing that. Like I just said, "Okay, stop teaching from the textbook and teach mathematics." "Okay. If I don't teach from the textbook, how do I teach?" Or, "I don't know because I haven't seen any models. No one has ever done that for me, so I'm just going to go back to the textbook because that's all I know." Which is why the resources you all offer are fantastic, because they need that. But what they also need, not just the resources, is how do you actually do it.
And when I was a math coach and I said, "How may I best serve you? Top two requests? How do you do class discourse and how do you use group work without them tearing apart the room?" And then I would get questions, "I'm going to do a model lesson with you. What would you to see? I really want to see how you use lap boards." Write your answer down, hold it over. In my mind, I'm thinking, "Oh, it's just write it down and hold it over your head." And then I realized when I actually did it and was cognizant of it, there's a lot of little things that I do with those lap boards that I'm not aware of. And so, there are some things to do with that.
So, there's the formative assessment piece. There's the giving kids type. The biggest thing is giving kids rigor. Putting rigorous problems in front of the kids. We usually, I call them, Set C problems in the textbook. Most teachers don't do those because, "Oh, I don't have enough smart kids to do those problems, so we're just going to do the rote." And those are just some of the hits on the list of things that we try to move teachers forward with.

Kyle Pearce: And I appreciate you sharing that because I think you're right. There's this movement in the last 10 years or so to say teaching from the textbook is bad, but you're almost shamed if you were like, "I taught 2.3" or "I taught, I did it my lesson today, but I used the textbook." Teachers are saying, "But I did it this way." And it's because there's so much that teachers have to do today. But telling them, teaching them the textbook is bad, but then not giving them anything. I think that's where we get, "No, I'm not supposed to do it, but I don't have anything else to use right now."

Chris Shore: What's the alternatives. That's right.

Kyle Pearce: I could throw it out the window, but then now I'm going to spend all my time searching for new lessons to replace what I was doing. In a lot of cases that time search sucks up to, "Let me just grab the first thing I see and it might not even be any better." So, what do we do to replace that? And I appreciate you giving some ideas and resources that you've used to supplement and help teachers realize that they can do it differently.
Now, I think there's also some other ideas that you've shared along the way that can help in this way as well, models that come to mind. And I know that you've done a lot of work with Clothesline math in developing some resources around that. We wanted to chat with you today about that. Can you fill us, our listeners on, if they have not heard about Clothesline math as a model to teach mathematics, what that is and how that works? And we'll get into some deeper stuff about it as well.

Chris Shore: Sure. It dovetails into what we were just talking about here. So, well, the journey to it starts with the math projects journal. So, I had a student teacher early on in my career and like I said, the internet came in and then we got computers. And we were part of, when I say we, the teachers at my site, we were part of this very special training program called CQ, out of Cal State Fullerton. And it was called CQ because it was, I'm trying to remember if it was concepts or content, but it was basically about math, collaboration and confidence. So, teachers knowing their stuff, working together and then having the confidence to do more innovative teaching.
In the '90s, innovative teaching was group work. We called it cooperative learning back then. Manipulatives. Oh, and here, before the internet, graphing calculators. So, this is the first time any of us saw TI. And so, we were trained in all of this and then we went out and we actually implemented those things. And then we gave what's here a test in or an old test here in California, Golden State Exam. It was reserved for your college-bound kids, but we wanted to use it to get some results, get some data, it was the only thing available. We didn't have the states test like we do now, the high states testing.
And so, we gave it to all our kids. And then just like in the movie, Stand and Deliver, where all of a sudden you're on the phone with the State Department going, "You know what? Your kids did really, really well, unusually well." And then when they saw that we gave it to all the kids, even the flunkies. It wasn't just the A kids that were going to college. We gave it to every single Algebra kid and every single Geometry kid. And we outperformed the state four to one and giving it to all kids. And that was compared to the upper population that was given the test. So all of a sudden, "Okay, we're onto something. This stuff actually works." We had some data to bolster it.
So then, my student teacher who was very tech savvy, we created a website and we did something called the Math Project Journal. It was going to be an online newsletter, now we call those blogs, but back in the day. And so, he and I started this and we just started sharing out some of those projects. Princess Dido was one of them and that led to the Presidential Award for me back in 2001. And where the story of Princess Dido is that she encompassed enough land by cutting up the skin of an ox that you could actually surround with now. It's the legend of the origin story of the City of Carthage. Could you really cut up an ox skin and surround a land big enough to put a city?
So, we take a king size bedsheet and we do all the math that's involved in this. We make the predictions. And we end up building a circle that's almost as big as the football field at the school. So, that was the early trajectory and I've been sharing those kinds of lessons at mathprojects.com. So, when I see an opportunity for something, to go out there and see something innovative, I'm on it. And my friend was doing a presentation and she said her presentation was they're going to put variables on a number line. Okay, we'll go check this out. We support her. Shout out to Kelly Wise.
We go there. We, friend of mine, watching this. And I had seen it. I had seen fractions on the number line, percentages on the number line. Now, she's doing an Algebra on the number line and our minds were blown. So, my buddy turns to me at the end of the session, he says, "Clothesline is the master number sense maker." And because it's a very visual tool, so I should explain to you. Basically the Clothesline is literally a Clothesline, hanging in front of the kids.

Jon Orr: Strung out in the room.

Chris Shore: String. And imagine a piece of paper folded over like a tent and then it hangs on the Clothesline and you put numbers and variables on that. But the thing about it is it's manipulable. So, when you're on paper and you put this number or you set the benchmarks for your number line, it's static. The Clothesline is dynamic. You can change the ratios, the proportions, the benchmarks, and it's very active. Kids are getting up out of the seats, going up to the front of the room, posting things in groups. Other kids are challenging what's up there. It's very visual. It's very communal. It just blew me away.
So, I want to be very clear as we get there. I didn't create Clothesline. It came out of Cal State San Diego, but they were doing a lot of it with numbers in middle school. And then Kelly said, "What about algebra?" I said, "Well, Kelly, if you think it's good with algebra, I teach geometry and I also teach some trig." So, I started coming up with it and I posted it in the website, and I started a hashtag. So, that's when I realized, "Hey, I'm just as solid as the millennials." Because I started a hashtag Clothesline math and it actually started running.
And then I go to the NCTM conference and I have people coming up to me now that found the website, found the hashtag and said, "When are you going to do some lessons on your website for elementary?" And I said, "Well, elementary doesn't need it because you guys already do number lines, right?" And they're like, "No, we don't." And then they said, "And we don't know how to do number lines, like Clothesline."
So, then, I was talking with a publisher about doing a book and I went to the dinner for all the authors of this publisher and I sat down with the editor and I said, "We're going to do a different book. We're going to do a Clothesline book." She goes, "I was waiting for you to tell me that." And that was San Antonio, 2017. And in one year, 2018, at Washington DC in CTM, the Clothesline book was launched. And it was K, basically it started from a kindergarten lesson in Chapter 1, all the way up through the unit circle. So, we took the number line, Clothesline, all the way up through all the grade levels.

Jon Orr: I love it, I love it.

Chris Shore: In that one book.

Kyle Pearce: That is so fantastic. And I have to be honest, I'm so happy that you shared that about elementary. John and I both coming from the secondary classroom as well. I'm in a K through 12 role right now, but when I had assumptions about what was happening in elementary as well, and you start to realize that we're all in this math place. And I would argue in secondary, we tend to be those students who were considered good at math because we could cram the night before, but we didn't necessarily have those reasoning skills. We didn't have the models and the strategies necessarily that help us to become true problem solvers outside of algebra.
So, we knew the rules of algebra and as long as you follow those, you would eventually, out would come a number and most of the time it would be correct. And these number lines make it so we can actually reason. And you wanted to add something, so I want you to hop in.

Chris Shore: No, I was just going to back up what you were going to say is I've never met an elementary school teacher that was arrogant about math. I've never worked with an elementary school teacher that said, "Nope, I don't need your help teaching math. I got this." What you have is like you said, and this is my tribe, I was one of them. I was high school. I got the math. I had one teacher actually say this to me, "Chris, why are they, the administrators, always putting this on us? When are the parents going to take their kids out behind the wood shed? Just smack your kid into doing more homework and that's the answer for them getting better at math." So, that's a tougher nut to crack getting there.
But for example, so on the Clothesline, so we were talking algebra. Let's say you're going to solve a system of equations with variables on both sides. So, 3X - 9 = 2X + 6. So, one of the things we do on the Clothesline, and it's not just a clothesline like rope, you also have clothespins. We're going to pin a card that has 3X - 9 and we're going to pin it on the same spot, so they're vertical. Same spot with 2X + 6 and they're equivalent. If they're equivalent, they're on the same spot on the number line, but there's no other values on this clothesline.
So, where are we going to go from there? And then we say, "Okay, this is 2X + 6, so where does the 2X go?" And almost all kids will put the 2X greater than the 2X + 6 because it says 2X + 6, so I'm going to go six in that direction. Guys, this is where the 2X is after you add the six. This is 2X + 6, so where was 2X before you added the six? See, they're doing the same thing on the clothesline that they do on their paper. When you see plus six and you want them to subtract from both sides and they add to both sides, they're doing the same thing. And then when they do that, the 3X and the 2X, guys, this is positive here, oh, there you go.

Jon Orr: Yeah, yeah. Just find that.

Chris Shore: Just like that. They can-

Kyle Pearce: Model it for those on YouTube watching here. I'm going to do my best to number talk and facilitate.

Chris Shore: Fantastic.

Kyle Pearce: So, you keep talking and I'll do my best to help.

Chris Shore: All right. So, where does the 2X go? So, this is 2X + 6, so where was 2X before we added the six? And the kids are going to want to go to the right because plus six means to go that direction. And if somebody does, then when I ask the class, "Does anybody want to challenge?" Because this is up hanging on a clothesline and you can imagine it in the front of the room.

Kyle Pearce: So, can I mention right here though, the part I love about this and you already alluded to it is right now that I've written this out and of course, drawing number lines is great when you're fluent and flexible with it, but it makes it way harder for me to be able to allow that student's response to be heard. So, I want to put the 2X over here, but I'm not going to today because I can't erase it. Whereas your Clothesline math, I could place it there and then I can still adjust it later.

Chris Shore: And then you ask for a challenger. So, I always say, "Does anybody want to challenge this?" And a kid will come up and they'll move it to the other side. And then I go back to the first group and I say, "Do you want to defend that?" And they have to justify. I'm not banding on one of your previous podcasts, Your Becauses.

Jon Orr: Yeah. Show your becauses.

Chris Shore: Because why, why are choosing your becauses. And so, we'll talk about it. If we're going to add six, we've got to get there, so there's got to be six going the other way. So now, I want you to imagine that on a clothesline up front, we don't have those little hops. All you see is that distance and this is critical because then when you have 3X - 9 and you say, okay, now the 3X is going to go the other side, but how far? And it should be, if that's six, then we want that and a little bit more. In fact, we want 50% more because that distance there that Kyle did is two 3s, and we want three 3s.
And two 3. Yeah, you got it. And then you need three more for the bingo. Now, here's the beauty. And this is what I did. I would have my clothesline on magnetic hooks on the whiteboard at the front of the room, so I could annotate the lesson as we're going, so we can have both going on there. All right. Kyle, since you're my student.

Kyle Pearce: Well, one thing that I will say when I look at this, this again is highlighting the benefit of being able to adjust because I look at this and I go, well, if there's a difference between here and here of 1X. Basically, if I go in this direction, that's 1X bigger, that means 1X smaller is here and we can see that the proportion is off.

Chris Shore: Awesome.

Kyle Pearce: So, this is where that adjustment comes in, that you're describing. So, I almost have to redraw it now that I know what I know whereas Clothesline math allows me...

Jon Orr: You can shift it, yeah.

Kyle Pearce: ... to shift everything around.

Chris Shore: Exactly. And Dr. Eric Milou, who you've had on your podcast, also. He and I were having this conversation. He said, Chris, "I don't get why you're calling it Clothesline. Why don't you just call it number line?" I said, "Number line is a mathematical tool. Clothesline is a teaching tool. I can manipulate the Clothesline."
So, just as you said, that's 1X, but 1X on that diagram exactly has to be halfway between zero and 2X. So, and I tell the kids, you can slide things, you can squash things, you can expand things, but the proportionality has to be there. So, as we build it, we move it and kids will call that out exactly as. I don't bring it up. The kids start looking at it and they say, "Mr. Shore, but if that's 1X, then what's going on there?"
Now, here's the beauty. I'm going to go back to this because Kyle is my student. Kyle, you already pointed out that the distance between 2X and 3X is 1X and I'll put my hands, I'll go up on the board and I'll say, "This far is X." Good. You ready? What else is this?

Kyle Pearce: Right.

Jon Orr: I love it.

Kyle Pearce: That's a nice visual because you've got 6 one way, 9 the other way. Hey, what's the total units? And what I love about that, Chris is seeing that's 15 units, this absolute value happening in here. We're looking at distance. When you say it's a master number sense maker, I think you're getting so much with number lines themselves, but also with the Clothesline. So much fluency can happen here that you can't get out of, "Hey, let me just show you how to solve this two-step equation or this multistep equation by going let's do opposite operations." Hey, we got to memorize these steps, but if we become fluent using the Clothesline or our double number line or our number line, you can go lots of different ways with this number line.
You looked at it going, "Hey, let me go where 2Xs, let me go where 3Xs and let me find the difference." But another student might go, "Hey, let me look at finding where 2X is and then what's equivalent above the 2X? I could go six down with my - 9 and now, I'm looking at that. And then I'm going, "Okay, well, let me make this 2X disappear. If I take 2X off here, where does that go on my number line?" There's lots of flexibility, which I love about this model and this teaching tool.

Chris Shore: Well, let's jump on that world real quick. So, why is number sense important? So, you're saying about this and understanding that number sense is not memorizing algorithms. Number sense implies flexibility. It means if you look at a problem that you've never seen before, your brain is flexible enough to solve a problem with numbers, many different ways until you can get at that. If I'm fluent in a language, it doesn't mean I've memorized all the responses. I can go down to Mexico now and I can ask where the bathroom is.

Kyle Pearce: Exactly.

Chris Shore: But then someone starts talking really fast to answer, I'm not fluent enough to follow that conversation. I only know how to respond to "where's the bathroom" kind of thing. So, there's this flexibility to it. And we find that the correlation between kids that have number sense can succeed in mathematics than kids that don't have number sense and fail in their mathematics classrooms. It's a hugely strong correlation.
So, I always ask teachers, "If we know that one of the greatest indicators of success in math, the causation, not just the correlation is teaching them number sense, what would be the best thing we could do as math teachers? Teach kids number sense." But now, we have to offer something where it builds in that flexibility, as you said, and that's the beauty of the clothesline.

Kyle Pearce: I love it. I love it. And it's had such an influence and impact both models and strategies in particular, the number line being one of them. The clotheslines is like the tool to access the number line, which I think is really awesome. And then obviously, once students are really comfortable with that idea and using that when they're introduced to new ideas, drawing the number line can also be helpful for them as well because it forced them to maybe think a step ahead where they go, "Wait a second, before I write this down, where does it make sense for me to put down my 2X?"

Chris Shore: Yes.

Kyle Pearce: As we discussed. So, there's almost this progression. So, I look at that Clothesline as an amazing manipulative to investigate, to reason, to work through, to build that comfort with the number line. And the reality is, as you mentioned, you can leverage the number line and clothesline math throughout the grades. It's so, so important.

Chris Shore: All the way.

Kyle Pearce: and something you had said. And I'll just bring it up here for those who are watching, I'll share the screen for just one second, on our website, we have this unit that is called shot put. And it's all about solving equations using linear models and then also, works its way towards essentially solving a system of equations as well. So, you'll see on Day 1, the shot put thrower. We look at the distance, the length we're giving them this visual. I have to move my head here so you can see it, but you essentially have an equation given to you, even though you don't realize symbolically that's an equation. First, we don't tell them how long each stick is, but you have 6X + 8 is equal to some distance. And then we use substitution first and then we work our way through and extend to more complex scenarios.
Giving them here is another one, 4 measuring sticks plus 12 feet is the same as 76 feet. What's that equation look like and sound like? And then you have this it's created proportional for you, so they start with this proportional template and then work their way through. So, lots of love for the number line for sure, which is why we definitely wanted to hit on that idea with you here today.

Chris Shore: Well, and that goes back to Dr. Milou's challenge is we don't want to introduce the kids to the Clothesline and leave them there. The Clothesline is the door to understanding this abstraction of a number line. And like you said, I have to think before I set the benchmarks on this number line, before I do it on paper. The Clothesline is just the doorway to the mathematical tool. That's where we want them to end up.

Kyle Pearce: I love it. I love it reminds me of algebra tiles is great tool. And then eventually, I want to visualize this happening in my mind.

Chris Shore: Correct.

Kyle Pearce: And then I can start representing it symbolically and working through it as well.

Chris Shore: And then last thing on that is you and I have spoken a lot about, you do a lot of work with double number lines, so, well, let's not hold back. We could even go triple. And so when do we use the multiple clothesline? A couple of things as I was pondering, that was I tell teachers that I'll do the double number line. We could have done that system of equations that you just drew on two different lines. We could have the 2X + 6, 3X - 9. I prefer in that case to have them pinned together, because it reinforces the idea that they're on the same spot on the number line, but you could do it separately with that system.
But the better use of the double number line is when you have different units. So, if we were doing a problem with, hey, five ounces of yogurt cost 12 bucks, I can put ounces on one number line, dollars on the other number line, or actually I would do it the other way cause I would do dollars per ounce. And then how much would 10 ounces be? How much would 12.5 ounces be? So, if there's separate units, I would have two different lines.
If we have different forms or different models. So, one of my favorites is three clotheslines. And on one we're doing linear, Y = 2X, another one we're doing exponential, Y = 2X and another one quadratic, Y = X2. And it's one thing on a coordinate plane to see how a straight line, an exponential curve and a quadratic curve parabola. Look, it's a whole another thing to see how these numbers play out on these number lines. It's fantastic.

Kyle Pearce: I love it. I love it. And my excitement over here is hard to contain. You gave me an idea of this triple number line. I'm on just the math as visual site. And one of our math talks is about a recipe for lemonade and we saw that, take that idea and run with it as well.

Chris Shore: There you go.

Kyle Pearce: So, there's so many things we can do with it. And then something that we've found as a door opener as well, when we start to reason through these problems, what you start to realize is the importance of partitive and quotative division. I feel like, John, I'm going to eventually get banned from saying those two words on our podcast because we speak of it so often. Because how we use the model, how we set up the model actually will look a little different depending on whether we're solving for the rate or whether we're solving for the number of groups in a relationship, which again, takes you on this whole other journey that, again, builds towards sense making.
Some people say, "Doesn't that overcomplicate things?" And I just argue, "No, this is complex work and complex work is what we want to be doing in a math class anyway." And these models are what allow us to actually access them and make sense of them, so that students can truly going back to your point, Chris, about students building number sense, number fluency. That is the key. And if they can reason through problems, they build that confidence and they truly will, going all the way back to the beginning, they'll be able to identify as a math person because they feel confident that they can solve problems.

Chris Shore: Going back to that. One thing to just wrap it all up here is how do you use it in the class? People ask me, "Are you actually using it to teach solving of equations?" And these kinds of things. One of the best uses for warmups. And I always say this, "Kids don't flunk current content, they flunk prior content," and most teachers will say that. They're not failing your freshman algebra class. They're sitting in your freshman algebra class failing 6th grade math because they don't know the things in middle school that's required for that. So, rather than just constantly complaining about what the kids don't know, how about we use clothesline to bolster these skills.
So, one example, I was working with the teacher and we were talking about this and they were going to do exponential graphs and/or exponential decay. And so, it was 6 x 1/2X. And I said, "Well, how are you going to have him do it?" "Well, first, I'm going to have them build a chart and plot the graph. You can see the shape and all that." "What do you think the chances are that the kids could actually accurately do the order of operations and produce the table before they even plot the points?" "Oh, some of these kids can't even do that." I said, "Well, then maybe if that's going to be your first step. Maybe we should go back a step and make sure they can do that."
So, we built these cards and we asked the kids to do 6 x 1/20, 6 x 1/21. We're starting with six. We're cutting in half once. We're cutting in half twice. Where will this go? On the number line. I had to clean up so many misconceptions, but the best part was 6 x 1/2100. And a kid came up and took the card and went all the way to the right, left the Clothesline, went over to the side of the room and hugging on the box to the right side of the board. And I said, "Why are you putting it over there?" He says, "Well, anything in hundredth power is almost infinity."
So, this is where the misconceptions come out. By the way, the whole class is along with him. Yeah, yeah. That one is-

Kyle Pearce: Yeah, for sure.

Jon Orr: Makes sense.

Chris Shore: Wait a minute. I thought we were cutting it in half a hundred times? How do you get something that big? And then another kid goes, "Oh, I know, I know." So, he takes it.

Jon Orr: Goes, yeah.

Chris Shore: Goes to the other side of the class, hangs it on the teacher's laptop on their desk and says, "Well, it's going to be negative infinity if you do it that many times."

Jon Orr: Ooh.

Chris Shore: And so, finally somebody raises their hand, picks it up and says, if you take six and you cut it up and half a hundred times and she put it overlapping on the zero on the number and you hear this audible, "Oh." Talk about math moments, baby, right there. Boom.

Jon Orr: I love it. I love it. Yeah.

Kyle Pearce: That's a great visual and I think a great way to leave our listeners here off on going, "Hey, now I need to go and learn about this." Chris, as we wrap here, where can people go right now to get started and learn a little bit more as their next step on Clothesline Math.

Chris Shore: Clotheslinemath.com.

Kyle Pearce: Boom.

Chris Shore: Right there. Easy. Videos are there on some lessons. They can go and print out some of the cards that you can hang on there. Also, go to Walmart, buy clothesline and clothespins for five bucks. Go to home Depot and buy the little magnetic hooks for your whiteboard for another five bucks. And for a 10-spot, you have the master number sense maker ready to rock.

Kyle Pearce: Awesome.

Chris Shore: That and the website and you're good to go.

Kyle Pearce: Awesome stuff, Chris, we want to thank you so much for joining us here in this great session. We talked all about things, getting student identity about changing the world even. And then also, talking about Clothesline Math. So, we want to thank you for joining us here and Making Math Moment podcast. And I'm hoping, I'm hoping in the future, we can bring you back on to talk more.

Chris Shore: All right. Well, thanks for having me. I'm a big fan of your podcast, guys. Keep up the great work.

Jon Orr: Thank you, sir. Awesome to finally connect. Have fun.

Chris Shore: Will do. Bye-bye.

Kyle Pearce: As always friends, both John and I learn so much from these episodes. So, our wonder to you is what is your big takeaway? I know for me personally, a connection I made and I've made with other tools, but it happened through this episode. I have used the clothesline as a tool in the classroom, but I made this direct connection that it's a great place to introduce. It's like starting. It's a tool that really is a great manipulative because you can manipulate it so easily.
And as we learned in this episode, when I was doing Chris's example there 3X - 9 = 2X + 6, it can be really tricky, especially early on trying to space and partition a number line on paper to actually make sense proportionally and that could be really difficult for students to reason through. Whereas if it's able and nice and easy to move on those cards on a number line, on a clothesline that makes that process so much easier as we help students work towards this more visual representation or pictorial representation where students are actually drawing out the number line. So, that was a little mini connection that I made. How about you, John, what was the takeaway that you had here?

Jon Orr: Yeah, echo that, but I also like this, that visual-ness of seeing when we were doing that specific example is that the distance. The distance allows that flexibility and fluency to connect the math that we're working on. So, in this case, it was solving equations to so many other math concepts and build, build that number sense in there. Looking at the distance between the variables had to be the same as the distance in the values, which allows that flexibility to come out. Where normally, if you're solving equations using opposite operations, you can't make that connection about distance without using that number line and clothesline

Kyle Pearce: And something else as well, just reiterate, throughout that episode, we were discussing how we leverage number lines quite a bit throughout our problem based math lessons and throughout our curriculum all the way down when students are actually multiplying and dividing, we're using number lines. We also use the array of course, through the area model. But something that I think is really important for people to note here is that we started with solving a system of equations, but that's not where we want to start with number lines.
Chris giving us an example, we're all geeking out on a math podcast, so it's probably fine and dandy. But we really want to be using this model way earlier so that students are building fluency and flexibility. We don't want students to only see Clothesline Math or a number line for the first time when we're solving multi-step equations. So, definitely if you're looking for ways in order to leverage the number line, the linear model, we do it through our operating of multiplying, dividing, whether it's with whole numbers, our fraction units. We have over 50 units in our problem-based math task section of the website, so definitely head on over there to makemathmoments.com/tasks and that will get you access to all of those units. Day 1 of every unit is wide open for you.
And then we also have parts of number talks and parts of other activities throughout the unit that you can leverage as well, if you are not a member of the academy. So, there's still lots of love, lots of math fun to be had, and really, lots of reasoning that can take place. So, if you're looking for a place to start with linear models or Clothesline Math, I would definitely head on over. And of course, all of Chris's websites, which will be in the show notes here today, also, great, great resources for you to dive into.

Jon Orr: Yeah. And those show notes and resources, and the transcript from this episode can be found over at makemathmoments.com/episode191. Again, that's make mathmoments.com/episode191.

Kyle Pearce: Well, my Math Moment Maker friends until next time. I'm Kyle Pierce.

Jon Orr: And I'm John Orr.

Kyle Pearce: High fives for us.

Jon Orr: And high five for you.