Cathy Fosnot: The transformation from using a textbook that is the type of textbook that just basically tells you procedures. Even if it’s a one that tries to explain why the procedures work, it still application. It’s not the math that we’re talking about.

Kyle Pearce: Right. The surprise is gone, right?

Cathy Fosnot: Exactly. There’s no surprise. There’s no generating. There’s no doing of what mathematicians do. Mathematicians raise conjectures, they inquire, they work on arguments, they plan ways to handle the problem. They love the surprise. [crosstalk 00:00:34]

Kyle Pearce: On episode 24 we welcome the Queen of Context for Learning Mathematics, Cathy Fosnot. Cathy is a Professor Emeritus of Education at City College of New York and was the founder of mathematics in the city, a national center for professional development located at the college. She’s authored over 40 books on mathematics education including Young Mathematicians at Work series, The Context for Learning Mathematics series and now her new online platform, New Perspectives.

Jon Orr: We are so excited for our math moment maker community to dig into this great episode with a champion of fueling sense-making in math class as she will share her mathematics journey including some discriminatory barriers that she had to overcome along the way. Listen in as we speak with the wonderful Cathy Fosnot. Play that intro music now.

Kyle Pearce: Welcome to the Making Math Moments That Matter podcast. I’m Kyle Pearce from tapintoteenminds.com.

Jon Orr: I’m Jon Orr from mrorr-isageek.com. We are two math teachers who together …

Kyle Pearce: With you the community of math educators worldwide who want to build and deliver math lessons that spark engagement …

Jon Orr: Fuel learning.

Kyle Pearce: … and ignite teacher action. Welcome everyone to episode number 24; Landscapes of Learning: An interview with Cathy Fosnot.

Jon Orr: Get hooked into an intriguing conversation with a game changer in mathematics education who is constantly helping the mathematics education community to mathematize vertically upwards, Cathy Fosnot.

Kyle Pearce: Before we get into our conversation we want to let you know that we have just opened up the doors to our brand new Make Math Moments Academy and it is now accepting members. Our Academy will help you create memorable math moments daily with access to curiosity rich tasks, in-depth PD video trainings and a system for making these moments a reality.

Jon Orr: At the time of this recording, we already have over 60 founding members so if you’re looking to build a classroom you will love that generates resilient problem solvers, happy students and proud parents check out the Academy and see if it’s the right fit for you. You can learn more at makemathmoments.com/academy. That is makemathmoments.com/academy.

Kyle Pearce: Now let’s get on with the interview with Cathy.

Kyle Pearce: Welcome Cathy. Long time no see. For a while there I felt like I was bumping into you every couple of months but I think we’re long overdue. How are things doing over on the East Coast?

Cathy Fosnot: Actually I am great. I have had the last month off with no travel and I have been writing like mad, so I love that kind of precious time. I have, in the last two weeks, actually finished the Renew units.

Kyle Pearce: Wow.

Cathy Fosnot: I know. Very exciting. I’ve finished one first on a grade four unit it’s called Number Detectives. It’s all about numbers, geometric shapes of numbers, how many factors numbers have, how we can tell, is there any way to predict, all kinds of common factors, multiples … Just a lot of fun playing with number. That’s called Number Detectives and then I also just finished two shorter units on time for Grade 1 and Grade 2, time and money.

Kyle Pearce: Oh, beautiful.

Jon Orr: Amazing, amazing. That’s fantastic.

Kyle Pearce: Yeah, we’re going to dive deeper into that for sure when we get to some of the resources that we wanted to ask you about as well. I have jotted those down and make sure we don’t forget to come back to those.

Jon Orr: Cathy, I think for our listeners that they don’t know who you are, would you be able to just tell us a little bit about yourself, where you’re from, what’s your teaching story, what’s your journey? Can you just fill us in a little bit?

Cathy Fosnot: Oh, sure. Well I’m originally from Norwich, Connecticut and I lived actually not far from my old hometown now because I returned here 10 years ago and my mom and dad were aging. I was able to commute into New York for a while from here and then I ended up settling here so my history.

Cathy Fosnot: I think I honestly always loved math. I can’t say that I didn’t. In the beginning, I did. It was probably my favorite subject I loved. I was one of those people that stayed up with a flashlight under the covers at night reading Nancy Drew mysteries and the Hardy Boys. It was something akin to doing that with mathematics for me, it was about solving problems, cracking the mystery. I was very lucky when I was an elementary student to have an absolutely fabulous principal who also loved mathematics. He tried to, some of the students that he knew really loved math, involved with him and he would always give a test at the end of the year that was all problems. They were all open-ended problems.

Cathy Fosnot: I did, I remember, quite well on one of those and we got the students that did there were about six or seven of us got taken to a Red Sox baseball game in Boston. That was really big for a 10-year old-

Kyle Pearce: That sounds amazing.

Cathy Fosnot: … 11 year-olds, you know?

Kyle Pearce: For sure, or for a 38-year old.

Cathy Fosnot: Yeah. I started off like that which sounds like, “It’s going to be a wonderful journey but now let me paint the other side of what happened.”

Cathy Fosnot: When I got to high school, I thought I was going to take a lot of math courses. My plan was to go to college and to major in mathematics. I was in my high school geometry class and loving that because of course, I was finally being asked to prove things, right? What fun! Not only were you cracking the case you then had to write your argument, the convincing argument. I was doing very well in that course too and I went up to my teacher and I had been placed in … We had tracking in my school. It was a very large high school, 3,000 children.

Cathy Fosnot: We are tracking and I was not in the top, I was in the next to the top and I asked why I couldn’t be in the top math class because I had straight A’s. I could show the teacher more of what I could do I was already doing really well in his class and he looked at me and he said, “Well Cathy, that track is for the boys.”

Kyle Pearce: Ouch.

Cathy Fosnot: “That track is for the boys who were going to be engineers. They’re in that track because they’re going to be taking advanced placement classes at the university.”

Kyle Pearce: That is so it’s heartbreaking and I want to believe that things have changed significantly but the problem is, is that that tracking still does exist and now it’s you might be able to swap out boys for something else, right? That’s for-

Cathy Fosnot: Yeah, that’s right.

Kyle Pearce: [crosstalk 00:07:52]

Cathy Fosnot: That’s right.

Kyle Pearce: Yeah. Yeah, totally. That hurts for sure.

Cathy Fosnot: Yup, yup.

Kyle Pearce: Sorry, carry on.

Cathy Fosnot: Yup. Then I continued with my high school courses and went to college and signed up for the math track anyway and had to take a few courses first because I hadn’t had the advanced classes. I finally got into calculus and oh my God, the calculus teacher was abysmal that I ended up with he was known on campus as [Flunkem Small 00:08:20]. He gave tests and then curbed everything and I did fine. I did fine in his class but I wasn’t learning anything. I mean you could get a 20 on his exams and as long as it was better than other students he used a bell curve, you’d get a C or D or an A or whatever but I wasn’t learning anything and the math I loved wasn’t happening and so what I ended up doing was switching majors. I switched into Elementary Ed. I took math electives because I loved math and I ended up with a self-designed program in math but not a degree in mathematics. I had a degree in Elementary Ed with a specialty in math.

Cathy Fosnot: That’s my history of how I started. Now, I ended up now in the department where I was teaching math to students to teachers and of course, it became my love again. I want to tell you two things that were instrumental in changing my whole career path because when I first got out with my PhD, I also have an art degree because I’m a painter. I had real interest in writing. I love to write and the first position I took at a university teaching was in writing.

Cathy Fosnot: Well, I was going through a divorce and I have joint custody with my children when they were young and so I had to take a position that was very near my ex-husband. The only position I could find right near me so I didn’t have to move was literacy but they also wanted someone to teach cognitive psych and that was also my specialty because on my PhD I had focused on cognitive development, on epistemology. I was really curious about how people come to know, which was probably why I loved the math that I knew when you were actually generating it and proving it was about making.

Cathy Fosnot: What ended up happening was that I taught literacy for about 10 years at the University in Connecticut and I had written a couple of papers on EdTech and on cognitive psych and on constructivism in particular and I had won an award on constructivism and Ed Technology. I was asked to come … Do you know Marty Simon?

Kyle Pearce: No, I don’t.

Jon Orr: Yeah, no I don’t think so.

Cathy Fosnot: A researcher and he’s a professor in math education at NYU. Anyway, he had a grant with NSF at Mount Holyoke College and it was called Summer Math for Teachers. It’s now the Big Idea Project, the Deborah Schifter and a few others are involved with so you may know that. Anyway, he invited me to come and give a talk on constructivism. I watched Marty teaching math to teachers and it was all about inquiry. I said, “Oh my gosh. Somebody knows how to teach math.”

Kyle Pearce: Nice.

Jon Orr: Yeah.

Cathy Fosnot: I took a leave of absence from my university. I said. “I’m going to switch careers.” I joined Marty and Deborah Schifter at Mount Holyoke College for a couple of years and loved it and then switched into math. The second example I want to give is that when I started doing inquiries with my students, teaching in a way that where I was more mentoring rather than expecting them to get my answer. I was trying to get underneath their thinking. They would often go in directions that I was unprepared for. I had my way of solving the problem but my students would often go in different ways our students will do and I was trying to follow their thinking and get underneath what they were doing.

Cathy Fosnot: I was known, people that worked with me in math in the city will tell you, I was known for staying up all night if I had to trying to crack inquiries. The math I now know did not come from the formal preparation of a PhD in Math Ed. It came from working with my college students, K-8 teachers and doing math with them.

Kyle Pearce: Right. Right, and staying up all night likely.

Cathy Fosnot: And staying up all night. I was curious. That’s why math in the first place. I wanted to figure out the increase. I wanted to model the problems in a variety of different ways. I wanted to see where my students were going and how their strategy was different than my strategy. I stayed up because I was hooked. I stayed up all night for geometry in high school because I would do every problem in the book so that the teacher could never throw a problem at me I hadn’t already solved.

Kyle Pearce: But it wasn’t so that you were better prepared to be a problem solver on the test but because you had seen every possibility, right? It was sort of like whatever they could throw at you, you’re ready to take it and you didn’t really have to do that inquiry piece like you just described. You would just know what to do.

Cathy Fosnot: Well, I would do the inquiry at home, that’s why I stayed up all night. I work on too. I’d solved problems that I thought were typical of the type he was going to put on a test. Was I just trying to get the best grade? I don’t know, honestly. I can’t remember back that far. I just know that I wouldn’t have stayed up all night if I also didn’t love what I was doing so it wasn’t just for the grade.

Cathy Fosnot: When I was doing it with my students, it clearly wasn’t for a grade. It was because I loved it. I wanted to solve the problems. I wanted to understand their thinking.

Kyle Pearce: Yeah, I think these past six years, I feel the exact same way like I feel like I’ve learned so much more now and I know that a lot of what I learned when I was younger was obviously foundational and very important otherwise I wouldn’t be thinking the way I am now about certain ideas and math and trying to figure them out. But actually, a colleague of mine, Yvette Lehman, she is a big, big fan of yours as well. She was actually very jealous that I was going to have an opportunity to chat with you tonight again.

Kyle Pearce: Her and I will send pictures of different so strategies, trying out different models and trying to figure out those in-betweens. It sounds a lot like probably what you were tinkering with back then when you have made your switch from the previous University over to the summer math for teachers program.

Cathy Fosnot: Yep. Then from literacy to math feeling like, “Oh my gosh.” You have to be a risk taker because you don’t know what your students are going to do. Then fun of teaching this way is that you’re learning as you’re teaching. You’re not teaching what you know … I mean, of course, in a sense you are. We design problems, we bring them to our students because there’s certain mathematics that we’re working on with them. But you don’t always know when you’re teaching this way what students are going to do with the problem and so you’ve got to be curious. You’ve got to enjoy the process of the doing of math. You’ve got to be willing to take risks and work with them and that’s what a good mentor does. I think the role that we’re all trying to play now is we shift our teaching more to this model is that we’re becoming mentors.

Kyle Pearce: Yeah. There’s a question that we ask all of our guests and all of the teachers that we talk to and I think you’ve answered some of this already but I’d want to give you a chance to add to it but we always ask people, “What is your most memorable math moment?” It could be in school or teaching. You’ve given us a lot already so I just wanted to give you one more opportunity if there was something else that pops into your mind in the history of your math experience as a student or as an educator. Is there anything else you want to add to that memorable math moment?

Cathy Fosnot: Oh goodness. There’s so many so I don’t know if there’s only one. I’m thinking to one day I was working with students and we were doing mental arithmetic and one of the students had the problem 17 times 23 and my colleague Martin Dolk solved it with 20 squared minus three squared. I said, “Oh, now that’s interesting.”

Jon Orr: It is.

Kyle Pearce: I love that.

Jon Orr: Oh, I love it too.

Cathy Fosnot: I was completely hooked. I said, “Why does that work? Wait a second. Wait a second.” I went home and I played with a raise and I cut out pieces first and saw what was happening with the array but then I said, “Okay. There’s got to be a way to also model this algebraically.” Of course, then you see that it’s A plus B times A minus B and all of a sudden the middle terms are dropping out and you end up with the A squared minus B squared. I didn’t know where that was going to go but I loved trying to solve it in a variety of ways. It’s that surprise.

Cathy Fosnot: I think you mentioned you were at the winter Conference, the OAME the leadership.

Kyle Pearce: OAME, yup. Yes, great conference.

Cathy Fosnot: Yes, it was great wasn’t it? Dan and I and-

Kyle Pearce: Graham. Graham.

Cathy Fosnot: Yes, thank you. We’re speaking and the night before me. What I loved about one of the things he said was the surprise. When you hit upon something and that I then looked up one of my first-grade lessons that I’d filmed in Canada and started the next morning with it because I loved the little first graders. They were in shock about-

Kyle Pearce: Oh, they are fabulous.

Jon Orr: Yeah.

Cathy Fosnot: They were fabulous talking about odd and even numbers. It is that surprise, isn’t it?

Jon Orr: Yeah, that part I loved so much was that he had described that very same situation with the adding and on … Any odd number with another odd number will come out even. Everyone in the audience was sort of like, “That doesn’t seem that surprising.” You knocked it out of the park the next day when you brought that those students chatting about like, “These ones have a middle and these ones don’t have a middle,” and I just thought that was so slick after dinner and you said, “Oh, I’ve got to go and get ready for tomorrow.” It all connected the next day. I’m like, “That’s why she had to go get ready for tomorrow because she’s pulling the Audible.”

Cathy Fosnot: I wanted to find the video clip, that’s right.

Kyle Pearce: No, that was so great.

Cathy Fosnot: What’s so wonderful and we can talk about this later in the broadcast, if you prefer. I’ve been working on this online platform that’s full of video and that I’m building PD system … I think actually you’re using it in a couple of schools, aren’t you?

Kyle Pearce: Yes, we have. Yes, we are we have great trial and now we actually have the full license for, I believe, we have it’s 40 teachers involved in it.

Cathy Fosnot: Fabulous.

Kyle Pearce: Yeah, the response has been really positive and. There’s just so much great content people can just gobble up depending on where they are and what they’re after. That’s great. Thank you for that.

Cathy Fosnot: That’s why I was able to get that video clip that night so quickly because it’s on my whole video library, is on that platform and so I know all the different videos that we filmed through the years in the classrooms and what the content is. I knew we’d actually filmed the sequence actually in Canada, in Ontario at the York Board and I had that first-grade lesson where the kids were talking about odd and even numbers. That’s why I went and grabbed it.

Kyle Pearce: Well, while we’re discussing why don’t we take a little bit of a side road here and chat a little bit more about that. The platform is called New Perspectives. Do you want to tell us a little bit more about that platform maybe when you fought to create such a tool and maybe tell us a bit of the story there?

Cathy Fosnot: Well, first I think I’m going to take a little bit of an excursion first and talk about PD in general, Professional Learning in general. Because one of the things that I’ve learned through the years is that this is a very steep journey for all of us. I’ve been learning as I’ve been doing this now for over 40 years yet we also expect the teachers that we work with to get excited and then to be able … They do. They get excited. They’re in a workshop, they get excited. They say, “Great, I can do this.” Then they go back to a classroom.

Cathy Fosnot: The transformation from using a textbook that is the type of textbook that just basically tells you procedures and even if it’s a one that tries to explain why the procedures work, it’s still application. It’s not the math that we’re talking about.

Kyle Pearce: Right. The surprise is gone, right?

Cathy Fosnot: Exactly. There’s no surprise. There’s no generating. There’s no doing of what mathematicians do. Mathematicians raise conjectures, they inquire, they work on arguments. They find ways to model the problem. They love their surprise. In fact, if there’s no surprise or it’s a trivial problem they just say, “Well, that’s a trivial one. Why bother doing it? I don’t know the answer yet but I know what I would do so why bother?”

Kyle Pearce: Where’s the fun in that, right?

Cathy Fosnot: Yeah. If we expect teachers to change their teaching to be more generative and to help learners generate mathematics and to be able to have really good problems that they can do with kids, and to know how to question in powerful ways, that is a very steep, steep learning journey.

Cathy Fosnot: The bulk of my work has always been professional learning. Whether I’m doing workshops in trying to think about what to do with the workshops, whether I’m writing units and trying to put enough information in the unit so it isn’t just a host of activities, it’s got boxes behind the numbers. Why did I choose this when I wrote that string? Or here’s some of the examples of what you’ll likely see your children do when you do this investigation and. If you’re going to confer, here’s some tips on what you might want to try to do you know writing all that kind of stuff.

Cathy Fosnot: Having professional development as the real thrust of my work, I finally reached a point in the last, oh I don’t know, 10 years or so. My colleague Martin Dolk and I, actually it’s more than 40 years ago. We got a big grant from NSF to start filming classrooms of my really fine, fine, fine teachers. Teachers that I’ve worked with the years that you know I knew that they were good examples of this work and building professional modules, professional learning modules, around the footage.

Cathy Fosnot: Our first attempt was, and this will tell you, this was when I tell you the technology. You’ll see the date on this. They were CD ROMs.

Kyle Pearce: It’s better than floppy disks, right? That’s okay.

Cathy Fosnot: You’re right. They’re better than a book. It’s a book that has a CD with it.

Jon Orr: Right. It was probably cutting edge. Cutting edge.

Cathy Fosnot: It was cutting edge back then. Then, of course, from that we went to DVDs because we could host more video on it but then of course, the DVDs are old as well.

Cathy Fosnot: Five, six years ago Martin and I said, “Look, let’s just take those old DVDs off the market. Do a whole lot more filming so we also have a lot of new footage as well and build an environment that isn’t online.” That’s what we started doing. We’ve been filming now for about three or four years. Martin films me talking about the footage. We have all the samples of children’s work so I can use the samples and do online modules with teachers about the work samples. But the other thing we’re doing with it is my curriculum, I call it that in Canada you would call it a program. My program, my resource is now over 40 units. I think in Ontario mostly people know my early units which they tend to call the kids. The kids have about 18 units in them but I now have over 40 units on the market.

Cathy Fosnot: I’m pushing to make it a full program a full core program so that teachers have enough materials to use. It’s not a textbook. It’s individual units but teachers have enough individual units on topics that they can do math workshop every day.

Kyle Pearce: Nice. That’s beautiful.

Cathy Fosnot: Then I filmed everyone and show that back to … I’ll just say this one last thing. I don’t want to-

Kyle Pearce: No, it’s okay. Go ahead.

Cathy Fosnot: … monologue here. On my platform has the filmed footage of all of the units and so it allows teachers that are using the units to just go online and have examples of me and a teacher in a classroom the kids doing that exact unit that they are now about to embark on.

Jon Orr: That’s actually what I was about to ask that Kyle and I’ve been kind of diving into this realm too and we know it’s a lot of work. The one thing that I was thinking about for you is like, “What does it look like from a teacher’s perspective if a teacher logs on and they get to your module what’s the most value that teacher gets out of the tool. It sounds like you alluded to that just a little bit but how does a teacher see that fitting into the regular classroom routine, like their daily planning routine?

Cathy Fosnot: First there’s many different things you can do with the platform. It’s sort of like asking the question, “Well, what do you do on the net?” You say, “Well, it depends on what I’m searching for or what I’m doing that day.”

Jon Orr: You got to Google, of course.

Cathy Fosnot: Go to Google, right? When you go to the platform you have a whole bunch of different things you can do there and you can search and you might search for a unit that you’re about to use. If you do the unit comes up, now you still have choice. If you’re a beginner with my units and you don’t even know what math workshop looks like in a classroom or you’ve never used the unit before. Overview with you just want to know what it’s about then you click on getting started and you have some choices there. Math workshop, what does it look like? Overview of the math in this unit. Overview of the unit itself but if you’re a more experienced teacher you might not and maybe you’ve even already taught that unit once and your interest is not what’s the unit about, your interest is learning to question better or learning to plan math congress’s, which pieces of work do you use or maybe it’s my landscapes and you want to study the landscape then you go there again.

Cathy Fosnot: Again, tip of your finger you take the pathway you want. It’s a multi pathway program.

Jon Orr: I’ve been inside and I’m picturing as you’re talking I’m like, “Yep. Been over there, yeah. Awesome.” It is like a Google for everything mathematics and particularly in the style of that Cathy’s put together which is fantastic.

Jon Orr: You had mentioned two things about Math Workshop and Math Congress. I’m picturing myself if there’s a listener at home right now who’s going, “Okay. I’m feeling stuck.” You’ve described how the textbook tends to teach things procedurally and takes away much of that surprise. Even a textbook that really tries to do it really well, at the bottom of the page the surprise is right there. You have to almost instruct kids to put their arm over that portion of the textbook so they don’t see the surprise if they want it to be so could you help them understand like what would math workshop look like so I’m a brand-new teacher or at least a new teacher to this way of thinking about math pedagogy and teaching.

Jon Orr: What is up math workshop consist of? What does it look like if I was to walk into Cathy’s classroom?

Cathy Fosnot: Well first, it’s not always a three-part lesson. It might be some days but it’s not often a three-part lesson not happening all on the same day. Let me start there because a lot of people think of math workshop as being a three-part lesson where you have the development of the context, then you have kids go off and work and then you bring them back and you have a share.

Cathy Fosnot: Ours is a little bit different. I think of math workshop as basically turning the classroom into a community of mathematicians. There are always mathematicians in the community. Math workshop may only be an hour a day and it may start on one day with a mini lesson with a string of related problems and then the context is developed but the kids may be working on that problem for the rest of the math workshop in pairs and the teachers moving around and conferring, and kids have math journals and they’re writing down what they’re noticing. They’re trying things. They’re modeling it. Then they turn in their journals that night and I read their journals and I comment back, so I’m getting to know my kids better in terms of what they all did because I maybe only got around to confer with three or four pairs of kids and I didn’t get to all the other kids.

Cathy Fosnot: The next day we pick up where we might start where they read my notes that I wrote to them in these dialogue journals where I commented on what they’re trying to do with the problem. I do a little math with them in the journal and then they join peer review groups and in the peer review groups they’re usually to pay partnered now four groups of fours and the kids talk about what they’ve been doing with each other, and then they go back to work and they start … I confer a bit more. Then as they’re working and then they move into writing arguments.

Cathy Fosnot: Now, the conferring shifts a little bit away from where before, on the first day, I might have been conferring in a way that was going to support kids as they’re working on the problem and trying to get underneath their strategies so they make progress, suggesting ways they might model and so forth. But the next day is they start posting. I’m now trying to get under their arguments and so I’m trying to support them to write a cohesive mathematical argument, questioning on what do you want your audience to know? How are you going to convince them? How you going to hook them?

Cathy Fosnot: Then we have a gallery walk where their arguments are posted and we all go around with review pads and we read a few of the arguments. I spread the class out so that everybody does three or four, nobody does them all but everybody gets comments. Then after the gallery walk people read the comments that they got from reviewers and then we convene a math congress. The math congress is where I picked two or three posters that I think are going to foster some deep conversations. Sometimes it looks like a progression from least efficient strategy to more. People think I’m always carving congress’s that way but most often I’m not. That’s more the ban shoe that the Japanese do.

Cathy Fosnot: More often I’m picking posters where I see kernels of big ideas and I know that if I have that poster as a focus of discussion, I’m going to be able to push beyond just solving the problem to a generalization of something big. Freudenthal calls the difference between those two as vertical mathematizing and horizontal mathematizing.

Cathy Fosnot: Horizontal mathematizing is when you just go from problem to problem to problem. Vertical mathematize you push towards generalization and you begin to go vertically with your conferring and you push for almost a mathematizing of the mathematics.

Jon Orr: Almost like following that trajectory, like pushing them up that developmental trajectory?

Cathy Fosnot: Yup. When you’re going horizontally, you’re still grounded in the context but there are moments after kids have modeled and solved the context and generated some powerful mathematics, where you just want to dig into the generalization of the mathematics. Then you leave the problem and you get into conversations where, “You broke this up when you multiplied. Are you’re saying that whenever you multiply, you can break it up?”

Jon Orr: Yeah, does it always work?

Cathy Fosnot: And how are you going to defend it that works. Now you’re moving into proof.

Jon Orr: Yeah, I love that. It’s very clear to me as I’m picturing moving horizontally. It’s almost like when you move to that mathematizing vertically, it’s almost like coming to the realization that that context was there to spark that curiosity and get you on this train. Now that we’ve got you, now we’re going to ask these what-if questions. Like, “Well, what if this? What if that?” That’s when you depart from that activity or from that context is what I’m hearing.

Cathy Fosnot: It’s mentoring moves and it’s back and forth. Back and forth. You’ve got the landscape, you know the landscape if it’s multiplication the kids are working on, a multiplication inquiry of some type, you know that multiplication landscape. It informs the kinds of questions that you’re asking and you go back and forth from horizontal to vertical constantly trying to support movement on that landscape.

Kyle Pearce: Yeah. That’s great. We’ve been diving into that a lot lately and it brings up the idea of how intentionally we choose the tasks or activities or those contexts or even the examples. It’s so key. We need to be clear even with that just ourselves as teachers on what learning goal or progression we’re aiming towards and you know like not just going to grab any old question out of the textbook or Googling fractions and picking a task they found on somebody’s website. We’ve been finding it’s so important to choose the tasks wisely for your students.

Kyle Pearce: I know you have some thoughts on this. I’d love to hear them.

Cathy Fosnot: Yeah, I’m so glad you brought that up too because I don’t know if you could hear me in the background … Oh, yes. We’re agreeing with the points that you were making wholeheartedly. I think what I saw in the U.S. I mean, I remember the work I was doing with Marty Simon and Deborah Schifter. A lot of people were still using textbooks but the big movement that was starting to happen in the U.S. was using rich problems with kids, with low floors and high ceilings and really involving kids in the doing of math.

Cathy Fosnot: What I found however is that at least in the States, people still had a didactic of just doing an open-ended problem and then the next day would be a different problem, and the next day a different problem. The hard thing that I found that teachers were having to grapple with is they would say, “But I can’t just do problems every day because I’m responsible for these year-end objectives and I’ve got to ensure that my kids get there. I can do problem-solving one day a week but I can’t do it every day.” I think that was coming from not having thought and … In the U.S. educators and I don’t mean teachers right now, I’m talking about curriculum designers.

Cathy Fosnot: What I did, I started learning about what the Dutch were doing. I felt like they had a piece of the problem. They were using what they called learning lines. That’s way too linear for me now but back when I was learning about what they were doing, when they would write curriculum they had a line, a developmental line, a hypothetical trajectory of investigations that they thought would build upon each other. They were using context and crafting it very purposefully in order to engender progress. If I may give an example. If you want to develop the ratio table you might start with a grocery problem where kids are having to figure out the cost of a turkey with kilos and they’re developing a chart if it’s one kilo it’s blah, blah, blah. If it’s two, if it’s four, if it’s eight and they’re developing charts.

Cathy Fosnot: Then these charts, you start playing around with the chart and then a series of problems might be charts that you give to kids where part of the information is there but not all of it and then they’re developing strategies to move around the charts. You generalize from that and you’re developing scaling. I saw your lovely talk with your session with Phil last year on scaling and scaling in tandem. If you know that that’s important then you design contexts that are actually going to generate conversations around the scaling.

Cathy Fosnot: That’s what the Dutch were doing. I was fascinated by that because that’s not what we were doing in the U.S. we were just designing low flow, high ceiling. Nice problems, rich, nice. You’d have a really good day with your kids and a lot of nice strategies but then the question was, “All right. What do I do tomorrow and the next day and the next day and so… ” I thought the Dutch were doing some interesting things with context. I took a leave and I studied with them and I also got money from NSF and I brought many of the Dutch people from the front and Thal Institute over to work with me and then when after working with them for several years Martine Dulk and I wrote the young mathematicians at work books there’s. For them out on the market now we publish those in 2001-2002. Bill Jacob actually did the fourth one with me on algebra but ones on early number sans, ones on addition and subtraction, multiplication division fractions and one on algebra.

Cathy Fosnot: What we tried to do in those books is move away from a linear pathway towards a landscape and we try to connect strategies with structures so Piaget talks about structures and as a mathematician you know what I mean when I say structures you would number is a structure, multiplication is a structure and the number system is a structure. Mathematicians are building structures. My notion of big ideas is not the same as, for example, in Canada Marian smalls work is well known and she of course talks a lot about big ideas but I’m defining big ideas differently I see big ideas more as learners building structures so it gets at the heart of the doing of mathematics and it can therefore also be a big idea in mathematics because mathematicians years ago invented that structure but when a learner is inventing a big idea the learner is actually structuring …

Kyle Pearce: Placing bricks here and, “Oh, that doesn’t quite work,” and then moving it over there and …

Cathy Fosnot: That little boy that you saw on the tape with the odd and even numbers were structuring the number system into even and odd numbers and exploring the properties of what he could do with the even and odd numbers. What happened when he used operations with the even and odd numbers? That’s structuring.

Cathy Fosnot: To me the big ideas are akin to the process of structuring and then the generating of a structure, which to me along the way, there are big ideas that are being invented their generalizations about the structure. The distributive property to me and to Marian Small would both be a big idea, because it’s in for me it’s involved in this cheering of number in relation to the operation of multiplication anyway you know what we tried to do in those books in building those landscapes was to build networks because the brain is actually building a cognitive network. As humans, we try strategies. Our strategies are progressive. They move from informal to efficient and more formal strategies. There is a progression. The Dutch call it progressive schematization. But there’s also the structuring and then there’s also the modeling.

Cathy Fosnot: We have to think out which models for a given topic do we know we need to generate for multiplication? There are critical models, the open number line is critical early on when kids are doing repeated addition and it’s also a lovely model for kids to start exploring common factors and common multiples because the number line becomes a double number line but to shift from additive structuring to multiplicative structuring to get at that scaling, we need to ship to a ratio table. Then of course, there’s the array if we want to get to quadratics and partial products. Base 10 blocks won’t get us there they’re too rigid.

Cathy Fosnot: They’re designed for the four partial products to get to the algorithm they’re not designed to flexibly build the partial products that I’m at somebody with good numeracy is going to build. You can’t break the pallet up.

Cathy Fosnot: The models we choose and how we generate the models also becomes really important and that’s why my landscapes are models, strategies and big ideas. I am trying to look at how they cluster, how they’re near each other. It’s not a linear pathway.

Kyle Pearce: Educators it seems like we … I guess it’s a human thing, we want that linearness. We were on and off type people. We want yes or no but unfortunately with mathematics, so many times it depends. I’m picturing your landscapes even though I feel like I know them well, I can see them and how they’re structured and it is very interconnected and like a web as you described but now I’m even looking at them, even from a different light or a different perspective. Just visualizing like certain students that are building these structures as you’ve described building their own understanding and that Tommy heads this direction and then Molly heads this direction but at some point, they’re going to have to meet. They’ll converge at some point.

Kyle Pearce: When you look at the landscape, sometimes you could look and go, “Whoa! There’s a lot there,” but in reality that’s just what it is. It’s hard work. It’s really important for us to know that it’s not going to always go down that perfect path like I used to plan with my long-range plan and I had all my topics and my lessons all set up and by day four of the course, I was like, “Whoa, I’m way off track.” Probably because I was not paying any attention to this idea that my plan is a linear one and unfortunately that’s not how learning works.

Cathy Fosnot: I mean the mind is built around neural pathways and synapses. It depends on the experiences we have the strategies we try the structuring we done and all we can do in a classroom … I think we can plan sequences of big nice juicy problems. In some sense that is a little more linear when you’re building a unit but the problems themselves are open enough. Each investigation has low floor and high ceiling so kids are going to do all kinds of different things. Then when you’re moving around and conferring, it’s really all around the landscape.

Cathy Fosnot: Now actually by the way Kyle, I think people are most familiar with the original three because those are the ones that are in the kits with the four operations but I think I have … I don’t know, 10 or 12 on the market now because I’d done geometry and measurement and-

Kyle Pearce: Wonderful.

Cathy Fosnot: … The last couple of weeks I was working on time.

Kyle Pearce: One of the things teachers often say to us when we present them with ideas and strategies is, “What does that look? How do I get started? They love these ideas.” They’re like, “What I do to begin?” When I think about your when teachers land linearly and it makes so much sense to think of the landscape instead, what do you say to teachers who are scared of this non-linearity? What do you suggest to them? How do they start? What could they focus on? What would be their next action step?

Cathy Fosnot: Yeah. I don’t want this to come out wrong because I’m not trying to push my materials. Please trust that I’m not. If I wanted to make money on my materials I’d charge so much more than I’m charging. Think like this, my units sell for about $30 so the whole program isn’t going to cost very much with 40 units, that’s the whole program, right?

Cathy Fosnot: I don’t want this to come out like I’m trying to push my materials but the reason I finally sat down and wrote units is because the question that you just asked me is exactly what my teachers were constantly asking. If I only give them a problem to start with and I give them the landscape, it’s still overwhelming. I was in classrooms in the early days of math in the city four days a week and I would be designing sequences with teachers, planning problems. I didn’t even have landscapes at that point in time. We were just coming up with nice problems and analyzing children’s work but even that wasn’t enough and that was every day.

Cathy Fosnot: What I tried to do is write units the teachers could just pick up and get started and then there’s design the online platform so they could see it in action, they could study the video, they hear me talking with them. I analyze children’s work on there with them. That’s the only answer I have for your question, honestly. Do I think it’s enough? No.

Kyle Pearce: Yeah, you know what though? It makes complete sense. I’m visualizing myself and the reason I really like the tool is because as a teacher you know just like you’re saying, I can start basically wherever I am. It’s not like I have to go back to the beginning. I could go where I am today and I can step in there and I could watch a very short video clip it’s usually whatever the three minutes and get an understanding as to why that context why this idea where it’s going and the intentionality behind it and get started. In a way, that’s almost like you putting the teacher in the same place that we want to put our students in, which is to construct this new learning. Because we can’t press pause on life and then go and do three years of going through the entire system and feeling 100% confident with everything before I come back to the class. I just need to get started.

Kyle Pearce: That’s the message that I’m hearing from you and I really like and I think the tools design in a way that it can help teachers get started that way.

Cathy Fosnot: Even if people don’t have the platform, the online platform if they just pick up a unit, they’re two weeks long if they get started with a unit. That’s the easiest way to start.

Kyle Pearce: Cathy it’s been an amazing discussion so far. We have just a couple short questions here for you. We like to ask people before we head out on our way, we know how important and I know you know how important reflection is in order to generate new learning. Why don’t you share with the audience what is your professional development focus right now? I know that you’ve been working on those units but what’s the math concept or the math idea, maybe it’s pedagogy what’s going on in your brain right now that you’re trying to work through? Because I almost want to bet that there’s something up there that you’re working on and maybe just tinkering with late at night before bed.

Cathy Fosnot: Well first let me mention my book, Making Moments Matter.

Kyle Pearce: That’s a great title.

Cathy Fosnot: I know. When I realized that you know my book and I didn’t know what you were doing when we both discovered this, we just, on our own, came up with similar titles.

Kyle Pearce: I think we were in the car when I picked you up from the airport back in the summertime and we were driving and we were just chatting. We made that connection like, “What? Oh, well that’s awesome. Well clearly it must have been a good title.”

Cathy Fosnot: Yeah. That book is now two years old. It’s not my brand-new work but it’s a book I’m conferring. I studied. I really wanted to spend time really thinking deeply about conferring moves and how they’re different when you’re developing proof, how they’re different when you’re supporting a learner to go further with the inquiry and so that book is called Conferring with Young Mathematicians: Making Moments Matter. That’s an Amazon book.

Cathy Fosnot: Anyway-

Jon Orr: We’ll add the link to the show notes for sure.

Cathy Fosnot: Oh, yeah. Good. Thanks. Yup, thanks. Oh. By the way, that book also has quick scans in it so you can actually see several conferrals when you’re reading the book with you if you have your cell phone with you and you just download the app-

Kyle Pearce: Like little videos you can watch on your cell phone-

Cathy Fosnot: You can watch on your cellphone as you’re reading.

Kyle Pearce: Nice.

Jon Orr: Nice, nice. I love that.

Cathy Fosnot: My most recent work aside from filling out the unit’s to be a full core, some of the more recent things I’ve been spending time with is thinking in literacy how they do mentor text you Any of the literacy work of Lucy Calkins. She’ll use mentor text where kids will study like the way of writer’s crafting a lead or something. I’m fascinated by how something like that could occur on the writing of proof because we’re faced in mathematics with a different issue. The child, obviously, has to develop a proof. You can’t show a child of proof and adult proofs are too hard for children to eat and understand. God knows I can’t understand most of them and depends on the math.

Cathy Fosnot: I mean, my level is not the level of a professional mathematician. How do we introduce children to the writing of proof? Early on we encouraged them to write a convincing argument and all of that’s good. That’s where we have to start. But somewhere along the line, I don’t know, Grade 3, Grade 4, Grade 5, we need to start introducing some crafted proofs that are simple enough on the mathematics that children have been working on. They would be able to understand and talk about as a mentor text.

Kyle Pearce: That’s so interesting to me because here in Ontario we have our process expectations similar to the practices in the Common Core and one we have is reasoning improving. For each division, we have our primary division, junior, intermediate then senior all the way from basically Grade 1 to Grade 12. You can see this reasoning improving of doing or at least telling us what we’re supposed to do in a similar fashion but without the how so we’re trying to help kids with making conjectures and I heard you mention conjecture earlier like that.

Kyle Pearce: I love that idea of you know for me that’s what inquiry is all about is getting kids to really go like, “I wonder if,” or, “I think that when you do this, that happens.” Then it’s like, “Oh, it worked there, is that just a coincidence? Let’s try some others. Let’s see if we can figure this out,” but I’m not too certain how many of us educators in the classroom are able to really help develop that reasoning improving. That really struck a chord with me, that’s for you.

Cathy Fosnot: Well, I’ll tell you what I tried. If this interests you, I would invite you to join me in trying to explore this because I’m in the very infancy of trying to figure this out with this topic but I’ve tried one thing so far. I was in a third grade classroom, if you go onto the platform someday Kyle, go in and look for it’s an orange unit I filmed it in Grade 4 and it’s called Puzzle Packing. If you go to supporting the investigations, you’ll see several videos there about conferring and there are two where I’m conferring with three boys and I’m conferring on the title of one of the clips is Structure and Regularity and the title of the other clip is building an argument or argumentation, I don’t remember now exactly what I called it. I’m trying there to help the kids craft an argument … Well, it’s long and you probably want to sign off its … We’ve been on for a while. Should I keep going?

Kyle Pearce: I’m actually very curious.

Cathy Fosnot: Okay. The kids-

Kyle Pearce: If other people as sad because unfortunately for them. Sorry everyone.

Cathy Fosnot: If it’s still on, we can cut it.

Kyle Pearce: We could always do that. That’s the beauty of technology right?

Cathy Fosnot: Yeah. Puzzle packing is about puzzles like Rubik’s cubes that come in cubes and we had the kids figure out how many cubes would be in a whole host of different boxes that are structured like the base 10 blocks. The tool that we had available were base 10 blocks but the only thing we told the kids was that the little cube comes in a cube box, one puzzle, and then there’s a long boxes that holds 10 and then there’s a pallet that holds 10 or the 10s. Then there’s the big shipping box where there is 10 pallets that get stack and then there’s the shipping box being taken out to a truck and it gets layered in on the bottom to become a big, big palate and then a giant massive cube.

Cathy Fosnot: The kids have to figure out how many Rubik cubes are actually in that shape. Now one of the kids and then the last part of it which happens on day three or day four afterwards is, so now that you know how many are in there. There is actually a million. Is that truck going to be able to go down the road? How big did that truck have to be?

Kyle Pearce: They’re like, “I thought we were done.”

Cathy Fosnot: Is this even possible?

Kyle Pearce: Yes. Yes.

Cathy Fosnot: Of course it’s not. It’s not possible. The truck would end up being 25 feet on one dimension so it wouldn’t even feet. But anyway, what the kids are talking about on this clip is they have hit upon the fact that the shape changes between the commas and it goes cube long pallet. Cube long pallet. Cube long pallet. It just keeps getting bigger and they also-

Kyle Pearce: That’s fascinating to me that they-

Cathy Fosnot: Isn’t it? I know.

Kyle Pearce: Yeah.

Cathy Fosnot: They also hit upon the fact that when you’re multiplied by 10, of course you can move the number over and put the zero on the right. They figured out why and they realize that because of the commutative property that if you were multiplying by 10 like 10 times 100, you also had a hundred tens and so therefore, the 100 had to move over because you had to a zero down to get 1,000, so that you could then seated on the left, they’re not 110.

Cathy Fosnot: They also hit upon the idea that no matter how many if you were going to take care of us? Multiply a hundreds times a thousand that a hundred was really 10 times 10 and 100 was 10 times 10 times 10 and I introduced at that moment exponents because it was natural since they were talking about it. Then they realized they had 10 to the second times 10 to the third and the answer was going to be 10 to the fifth.

Cathy Fosnot: They know why. These kids were eight years old. Actually, it’s a-

Jon Orr: Right.

Kyle Pearce: In grade four, I want to say that again, correct? Grade 4, third grade.

Cathy Fosnot: The film that you’re going to look at, I filmed in May in Grade 3. I knew the content was going to be hard for a lot of third graders and I decided to make it a grade four unit to align with our Common Core but honestly, you are looking at third graders. They get up in the Congress and two pairs get up and they argue the associative property and then they argue the shape, and then they argue the exponents and then they say, “For example, if you have 10 to the 21st power. We know it’s going to be one with 21 zeroes but we also can tell you what shape it’s going to be because every three there’s a pattern of every three.

Kyle Pearce: This is an amazing example of is it proves the point for all those teachers that say you know like, “I can’t teach this way because I have to cover all the curriculum expectations in my time.” You just unpacked one shop or math workshop lesson where those kids went farther than they would have if I just laid out the lessons in the linear fashion.”

Cathy Fosnot: Exactly when I wrote that unit initially, the draft I had going in did not include exponents. I wasn’t planning on teaching exponents or-

Kyle Pearce: It was those unruly kids that made it happen.

Jon Orr: We have to be ready for it, right?

Cathy Fosnot: I just quickly though want to now tell you what I did with that about mentor text. The kids developed their own proof and it was quite nice and I supported them and I helped confer about their proof. But everybody shared and the congress was over, I had written my own proof using the children’s work and tightened it up so it was a crisp proof and then I got so excited about what you were doing I’ve wrote a proof about exactly what you were talking about too. Your proofs beautiful. Here’s mine. Mathematicians love writing proofs and mathematicians never write the same proof because they’re all writing their own. “I love yours. You got me excited.”

Cathy Fosnot: I’ll share mine now. Mine was Crisper. I wrote it Chris Lee on purpose because I wanted it to be a mentor text so I don’t know. I didn’t hurt ownership at all the kids had already owned their own proof I filmed it but I wasn’t happy with it. It went too long so I didn’t put it on the platform but we do have the footage of it. I put out to my staff this past summer this idea that I want to work on mentor texts and invited anybody that wants to try doing some things in classrooms, I don’t have the time to sit and plan things with people but if you try things and you do something that you really like then send it to me I would really love to see what you’re doing and what success you’re having. I’m starting to try to do it too and I’m just happy to share with you successes I have.

Kyle Pearce: That would be great. I would love to see the actual video of the proof piece there that’s very curious to me I’m geeked out I know the problem but to think of what those students did with that problem I’m sure people are listening and they’re just jaws dropped going like, “You’ve got to be kidding me.” But at the same time, you definitely have that shows that the intentionality behind the work that you’re doing and making sure to those moves and really giving students that … Clearly, there’s a community that you’ve built, a community of mathematicians as you’ve expressed earlier. That’s something that’s so important to build that community.

Kyle Pearce: It will take a while for those listening but give it time and just keep working at it. With the resources that what Cathy’s mentioned, I think it’s definitely doable if we’re willing to give it a shot.

Cathy Fosnot: Yup.

Jon Orr: Well, Kathy we can’t thank you enough for joining us here tonight. I know that I’ve learned so much just in this conversation. I know that all the teachers listening here right now would probably agree with me. We just want to thank you for joining us and we really hope to connect with you in the future. I think both of us are going to jump on that opportunity you just gave us and I’m sure there’s teachers out there to jump on this and construct proofs with our students. Thank you for that and thank you again.

Cathy Fosnot: Thank you. Thanks for having me. I look forward to hearing it when it goes live. Let me know, okay?

Kyle Pearce: Have a great night, Cathy. We’ll talk to you soon.

Cathy Fosnot: Yeah, thanks.

Jon Orr: We just want to thank Cathy one more time for joining us on this episode. Cathy, we know you were listening right now. You have so many insights into learning and your laughs, oh my gosh, it is contagious.

Jon Orr: Kyle, I know you had some takeaways here so let’s hear them.

Kyle Pearce: Oh my gosh, Jon you are so right. That laugh is contagious and I just don’t know where to begin. I can’t get enough of the learning I get out of the many conversations I’ve been lucky enough to have with Cathy over these past couple years.

Kyle Pearce: Jon you and I focus a lot on rich tasks and while this is great for getting kids into the doing of mathematics it can be easy to feel like you’re spinning your wheels and maybe not covering curriculum because you aren’t managing to find a way to uncover those ideas we’ve been mandated to teach. I love how she called out this big issue as many try their best to teach through tasks but often find themselves feeling as though they aren’t getting to where they need to go.

Kyle Pearce: You and I, Jon, often have conversations with teachers who are trying to break free from more traditional teaching practices as well as break away from units to interleaving or spiraling but then find themselves struggling to organize their course where those big ideas come out. It’s great to hear that Cathy recognizes that this is a common challenge for math teachers and that’s what she’s working towards as well as what we’re working towards through this podcast.

Kyle Pearce: How about you Jon, what resonated with you?

Jon Orr: As humans, we try strategies when we want to solve problems and it’s natural for us to move along progressions with those strategies, let’s say from contextual to more efficient strategies and you know I think it’s important for us to show our students how to move along those progressions. I liked especially how she mentioned that well when she was working with a group they were calling them they’d also thought about these learning progressions but they were calling them learning lines. She said she did not want to call them learning lines because she said they’re not linear, like these progressions are not linear. She called them landscapes because they’re like these big ideas, these clusters that live near each other like a landscape. I thought that that was such a simple yet huge idea and I appreciated thinking of it that way.

Kyle Pearce: You’re so right Jon. I love in her resources how she does have landscapes and they actually she builds them out like landscapes, they can be overwhelming when you look at them at first glance but if you really start from the bottom and you just see that you know some students are going to go this way and some might go that way, it’s just really important for us to be aware that it’s not necessarily going to be this than that sort of situation so I can’t agree more.

Kyle Pearce: Jon, we’ve been getting a lot of feedback through email, on Twitter, Facebook, on Instagram. You know what would be great if everyone who’s listening right now, no matter where you are you’re walking the dog on a run at the gym mowing the lawn, driving to work, okay well maybe not if you’re driving don’t do this but wherever you are, we want you to just stop. Just stop and your phone out of your pocket and take a photo, not of you but of where you are, what you are doing. You can be in the photo but we want to see where are people listening to this podcast and we want you to take that photo, we want you to tweet us @makemathmoments on Twitter or on Instagram, @makemathmoments or you can go to Facebook.com. You can go to Facebook.com/makemathmoments and let us know where you are listening to this podcast right now.

Kyle Pearce: Let us know. We can’t wait to see where you are and you know what if you don’t have a social account, you can email us admin@makemathmoments.com.

Jon Orr: Yeah, that would be super cool. Imagine we saw this tweet and you’re like you’ve just you know climbed not like a mountain but you were on a hike and you got to the top of this lookout-

Kyle Pearce: People are feeling bad about themselves they’re just on a walk and yet Jon wants you on a mountain but that’s okay.

Jon Orr: Yeah. If you’re at the top of a lookout and you’re like, “I’m just listening to this episode. Oh, I’m going to take this picture, I think that would be super cool,” or maybe you’re just riding a lawnmower. You could just shoot us a picture. That would be super cool too. I think would be awesome to see where all our math moment makers are. Where are you right now? I can’t wait to see what you send us.

Kyle Pearce: Awesome. Awesome. Yeah, I’m super excited about that I can’t wait. We actually are really looking forward to this episode coming out so awesome stuff. One quick reminder before we exit here if you are interested in becoming one of our founding members in the Make Math Moments Academy, be sure to check out makemathmoments.com/academy. That’s makemathmoments.com/academy. Check it out. See if it’s right for you. We can’t wait to dive into some deep learning with all of you.

Jon Orr: In order to ensure you don’t miss out on new episodes as they come out each week, be sure to subscribe on your favorite podcast platform.

Kyle Pearce: Also if you’re liking what you’re hearing, please share the podcast with a colleague and help us reach a wider audience by leaving us a review on iTunes, Google Play, Spotify or whatever platform you are listening to right now.

Kyle Pearce: Show notes and links to resources from this episode can be found at makemathmoments.com/episode24.

Jon Orr: You can also find Make Math Moments on all social media platforms and seek out our free private Facebook group, Math Moment Makers K through 12.

Kyle Pearce: Well, until next time, I’m Kyle Pearce.

Jon Orr: I’m Jon Orr.

Kyle Pearce: High-fives for us and high-fives for you.