Episode #110: Show Your Becauses – An Interview with Nat Banting

Jan 4, 2021 | Podcast | 0 comments


Today we speak with Nat Banting, from Saskatoon Saskatchewan!  Nat is a high school math teacher turned pre-service instructor who’s created amazing free resources like Fractiontalks.com and MenuMath. Nat was also the winner of 2019 Rosenthal Prize for Innovation and Inspiration in Math Teaching awarded by the National Museum of Mathematics (@MoMath1) in New York. 

In our rich discussion you’ll learn how creating limitations and constraints can drive good learning; how you can spark discussions and deep thinking around fractions; and how to spark curiosity and fuel sense-making with abstract concepts.

You’ll Learn

  • How creating limitations and constraints can drive good learning.
  • How to spark discussions around fractions. 
  • How to spark curiosity & fuel sense making with abstract concepts.
Start your school year off right by downloading the guide that you can save and print to share with colleagues during your next staff meeting, professional learning community meeting or just for your own reference!


Nat Banting: The purpose of the fraction talk isn't to solve a different problem. Reasoning about the image is the problem. And that's what I like about them . It's the self contained nature of the goal isn't to arrive anywhere, it's just to stay a while and think about what's going on. And I think that's where you get all this sort of... It's very inviting in that way where a lot of students they will perform a fraction addition incorrectly-

Kyle Pearce: Today we speak with Nat Banting from Saskatoon, Saskatchewan. Nat is a high school math teacher turned pre-service instructor who's created amazing free pre-services like fractiontalks.com and Menu Math. Nat was also the winner of the 2019 Rosenthal Prize for innovation and inspiration in math teaching awarded by the National Museum of Mathematics in New York.

Kyle Pearce: In this unique discussion with Nat, you'll learn how creating limitations and constraints can drive good learning. How you can spark discussions and deep thinking around fractions. And how to spark curiosity and fuel sense making with abstract concepts. Let's do this.

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

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

Kyle Pearce: ... with you the community of math moment makers worldwide who want to build and deliver math lessons that spark curiosity...

Jon Orr: ... fuel sense making...

Kyle Pearce: ... and ignite your teacher moves. Are you ready, Jon, to dive in with this great, great discussion that we're engaging in with our friend Nat?

Jon Orr: Of course Kyle. Of course as always we're honored to have our guest Nat here with us. Just before we get that, we do want to talk about some upcoming info. But it seems these days that our communities are getting more and more divided. It seems on many fronts, there is an us versed them mentality. And, Kyle, you know the math community is not an exception to this worrying condition.

Kyle Pearce: Yeah, you're right there Jon. When exploring new approaches to teaching and learning mathematics, we're often confronted with very absolute views that seem to pit one extreme against the other. Ideas such as direct instruction verse inquiry lessons or grades verse gradeless assessment practices or we even hear things about homework verse no homework or maybe hands on, collaborative tasks verse independent worksheet practice. It seems regardless of pedagogical approaches we wish to explore, it's a sure bet that you'll find someone out there who believes that it's poor practice.

Jon Orr: So true. And folks feel very strongly on both sides. So join us in our new webinar, The Tortoise And The Hare, How Math Class Missed The Moral and How to fix it. As we explore the two systems for thinking in the brain, and how mathematics education often only serves one.

Kyle Pearce: Yeah, that's right Jon. Now, we're going to be sharing practical classroom lessons and routines that not only help you find the right balance in your mathematics program, but will also help your students to define an identity that they value in your math class.

Jon Orr: Register for this free webinar at makemathmoments.com/webinar. Our webinar will only run once on January 12th, 2021 at 04:00 PM Eastern Time. So stop what you're doing right now and register. By registering, we'll end you a confirmation email that includes the How to Make Math Moments guidebook, plus we'll ensure you get access to the limited replay. Fantastic, yes. So this is the only reminder we're going to send out via the podcast. So pause and head on over to makemathmoments.com/webinar so that you can get all the details for this one time webinar happening on January 12th, 20121 at 04:00 PM Eastern Time. Again, like Jon said, get yourself signed up and you'll get that replay email so that you can hop in and watch it even if you can't be there live. So head on over makemathmomnets.com/webinar. All right. Enough form us. Let's get on with this fantastic conversation with Nat. Hey there Nat, welcome to the Making Math Moments That Matter Podcast. We are pumped to chat with you today. How are you doing over in Saskatchewan?

Nat Banting: Doing pretty well. I know I'm not the first guest from Saskatchewan. So it's becoming a province that's having a little bit more of a footprint on the podcast, but steady as she goes. That's what we're like out West. Just take things as they come and maybe a slower pace kind of thing.

Kyle Pearce: Yeah. You know what? I was thinking this morning when I woke up. I'm like, darn, I used my Saskatchewan shirt that Kiersten Dick gave.

Jon Orr: Oh yeah, I got one too.

Kyle Pearce: Yeah. She sent one to Jon and I. And how awesome. I know that you two cats are fairly familiar with one another. Maybe we'll get into that a little later. Before we do on the connection between you and Kiersten who's also been on, I think, two episodes on the podcast so far, tell us a little bit about yourself. We know you. We've heard you speak at OAME. What's your role in education and what landed you into the teaching space in mathematics?

Nat Banting: Okay. So, role in education. If you were to ask me that five years ago, you would have got a pretty standard and predictable answer in that regard. Now in the last few years, I've been bouncing around as the opportunity has presented itself. And I've learned just to take every year at a time. And when that possibility actualizes in front of me, I'm learning to take it, and then seeing where I go.

Nat Banting: I guess at the beginning of all this, I'm a classroom mathematics teacher. So, I've done that for about 10 years. And then with some other roles sprinkled in there. So there was one year where I was a consultant for my division or my district, I think, or my school board. Depending on where you're from, you call that a different thing. And so working with other math teachers while maintaining my math practice myself. So that was something O really value. So I said I would do the coaching or consulting role as long as half time I got to stay with my learners.

Nat Banting: If you're gong to ask me what the core of what makes me passionate about teaching math, it's being with learners. And that hasn't changed. Yeah. So that was the first 10 years. This year, there was an opportunity that arose at the local university here in Saskatoon, the University of Saskatchewan where the maths Ed methods professor went on sabbatical. And so I had known him for years. And so I applied and got that position for the one year. And so I've been teaching post secondary undergraduate teachers to be. And that has been really fun as well, totally different. But there are moments where I would sit back in class and it's supposed to be a 300 level lecture and I'm like, this is exactly what would happen in grade nine. You would be surprised.

Nat Banting: I found that teaching the younger grades, the seven, eights and nines really prepared me more for the undergrads than maybe the 10s and 11s when school math get really serious. So yeah, it's been a interesting ride. How I got here? Yeah, that's a really good question. I've always been attracted to teaching and mathematics. In general, as long as I can remember, I've had a way of seeing the world. I'd like to mathematize my surroundings. And that's maybe a word that came later through my study of education. But, I can remember ways. When a group of people is working on a task, I'm always obsessed, as a young child, and maybe even to this day, if you ask my wife, I'm sure she'd say so, over what is the most efficient way? What job could I be doing now and what can you be doing so there's no dead space? These sort of ways of mathematizing existence. And so I think that brought me into the profession.

Nat Banting: Now with the work that I'm doing, I'm recognizing that yeah, even though I have these ways of thinking mathematically and making my world a mathematical one. Probably, I also gravitated towards teaching mathematics because it pulled me in. It gave this power that I didn't have to ask for, just by who I was. And maybe I gravitated towards the math because it had this explanatory power. But at least partially, I think I gravitated towards the math because it granted me power.

Nat Banting: So that's a thing that I'm thinking about now is I always painted myself as I chose to be a teacher. I chose to work hard at mathematics, in school mathematics. I chose to see the world in this way. But there's a lot of things that probably happened in my past that brought me in because of what benefit me and not necessarily what I was offering to the profession so to speak.

Kyle Pearce: I'm dying to see what cooking dinner in your house would be. You're going back into that group scenario where you're trying to think of how to be more most efficient. No, you put the rice in this container because it's going to fit more. It's going to fit blah, blah, blah. I don't know why that's what I pictured. I was like oh, I see you in the kitchen is like no, no, we're going start this now and we're going to work on the pasta in two minutes because then it's going to be perfect two minutes from now.

Nat Banting: Yeah. And if I can pull myself under that for a moment. So that's me as a mathematizer maybe, but the math teacher in me wants to actually create a different scenario. I have different goals. So, my goal isn't to be the most efficient off the bat. And that robs the opportunity of maybe my young son and daughter producing that efficiency themselves. So probably I'm a little bit more disruptive now. I'll be like instead of saying, no, you do this and then you do this. I'm not the task driver that task driver that I am. But as an educator, I'm probably the one that's now taking away the third of a cup and being like, oh man, now we have to do this. Because that's a more inaudible opportunity.

Kyle Pearce: Let's really throw this for a whirl.

Nat Banting: Exactly, right?

Jon Orr: I totally understand that because you said you were drawn to math because of the power and the mathematizing. For me, it was the structure of math because I was a product and taught very traditionally. I was a product of that traditional here, this is a very structural place that you're going to be in. And I think I really enjoyed the structure of hey, I'm going to get a lesson today. I'm going to follow these procedures and then I'm going to reproduce them. For me, back in high school, that was I think why I liked math. Even though today, it's a whole different ball game for me on the idea of the creativity that math had that I didn't experience. And this disruptor is exactly where we've become or that you're describing, it's completely changed. And I want to explore that with you just a little bit. What did early teacher look like for Nat? When you first came out of teacher's college and pre-service teaching, did it look like the teaching that you had had? As a teacher, almost all of us have. We kind of mimic what we know. How did that progression change? Can you imagine where it happened and what was the moment? Or maybe you were teaching phenomenally right out of the gate.

Nat Banting: Not to brag, but...

Jon Orr: I'm waiting for someone to say, I was pretty awesome right out of the gate.

Nat Banting: That's when you go sleuthing and find eight or nine ex students to bring on to the podcast and just blow that theory out of the water. I don't know. You ever go back through a home video. Now we have them all in our phones and finally decide, oh, I'm out of space. I have to go back and be a little bit more selective of which videos I keep. It's an uncomfortable thing as teachers. I find curating a blog for as long as I have now. To go back to those early posts. Goodness sake is it uncomfortable. And then I'm thinking, this is public. People are reading this maybe. Maybe not.

Nat Banting: And so this question. Yeah, it's an interesting thing. I came out of teacher prep ready to go. I've always been energetic in my class. I like it to be bustling. I like to be working in the moment. I think that hasn't changed. I would describe how I went through school, fairly, do I want to say standard? This idea of yeah, I looked forward to maybe it was the structure and it was a familiarity of mathematics. I also always knew that I would get it. And that's just part of who math was built for. It's just built for me. I've never had the doubt that I didn't belong there.

Nat Banting: And so when you enter the world of education, that's a tough thing to come to grips to. And that continued through post secondary. I just figured if just sat down. My memory was good enough. I had a work ethic. And this is what I believed to be mathematical. It's to do with these things. It sounds fairly similar to what your experience would be.

Nat Banting: How's that changed? And this is what I told my undergrads this years is I finished my math degree concurrently with my Ed degree. But I think I got, I managed to get through at university an undergraduate degree without doing any mathematics. I just don't think I ever thought mathematically. I was good enough at playing the game that I could just memorize any situation that was thrown my way or maybe notice an anomaly and connect it back to another piece very well. Maybe that's a mathematical act. I knew a lot of mathematics. And so right now almost all of that is gone. I got amazing grades. Don't get me wrong. The school validated me at every turn, but I don't know how mathematically I was actually being until education. And then you start to think about well, what's the purpose here? And I think you pointed at that as like we like it because of structure. And the purpose as a student is to take advantage of that structure.

Nat Banting: And then for me, teaching at the beginning, I think, was the same. I was trying to harness that structure maybe with more student choice, maybe more creatively. And I think I've always had that then since though. And I came along in teaching in 2010. So that very much was the climate of education, was this turn to towards that. But I think it was still my math meaning if I think about it. I think I was still trying to project how I mathematized. And now I've taken a heavy lean towards observing and listening to students. And I think I was just really bad at that. I think if you walked into my room right now, you would be much more likely to see me looking puzzled attentively listening to a student, rather than at the beginning I think I would be like almost waiting in the weeds, waiting for that strategy to come out so I could be like, see. Look, this is how math works. I planned that it happened and then it happened. And try to do that creatively. But now I'm much more comfortable with not knowing and really just listening with students. And I think that's what I truly enjoy now about the craft of math teaching.

Kyle Pearce: Like Jon said, that's exactly my experience in university. I went through. I did okay. I had one or two courses I didn't do so well in, but ironically during that time, during that experience, I was finding ways to blame other people, blame the prof, blame my schedule, all of these things. I did a lot of blaming. But in reality it was because I actually wasn't doing mathematics, I was doing a lot of memorizing. And like you said, I had been validated all the way through because I could maintain those high grades. And it took a long time for me as a teacher even to reflect and say, wow, maybe there's more to this than what I thought. What popped into my mind as you were sharing was just this challenge in math education. I find as we go up the grades, it gets even ore difficult to try to shift beliefs. And this can be educator beliefs, because I think so many of us especially in the older grades who went through and got your math degree and then became a teacher, we've been validated all the way through. And maybe it just hasn't hit a lot of people that oh, it worked well for me that way, but maybe that's not really all there is to it.

Kyle Pearce: And that's definitely something that's really challenging for us in this space to try to help the entire community of math educators shift that way. I'm hearing it in your voice. I know for myself, I'm so happy that I hit that point where it went ah, maybe it's not all about... Hard work is important, but it wasn't just because I worked hard and memorized. Now I'm seeing wow, there's so much more to this. And there's so many different more perspectives that I had no idea about. So I love this idea of you sitting there. I'm picturing you in your classroom that puzzled look you had, which sometimes is probably like a fake puzzled look where you're like ah, tell me more about that? And you're sitting and observing more than just doing all the preaching.

Nat Banting: That's also something I commented on Twitter with a couple friends. They were saying oh, Nat is so good at observing and really listening to students. And it's like he doesn't even know what the answer, and I responded, well, it's really easy to pretend you don't know the answer when you don't know the answer. This humility is there. Yeah, I just think learning now for me is such an intensely personal thing. Making meaning is a really personal thing. And I jus think that I was privileged going through mathematics because the way I made that meaning personally was also the way that was historically validated as mathematics.

Nat Banting: And so I'm not saying that maybe what I was doing wasn't mathematics, I just thing it was the brand of mathematics, maybe capital M mathematics that is just not questioned, it's taken as true. Now I feel like I went through years of teaching 13, 14 year olds and really maybe not observing their meaning making. And I think that's something that we're missing as teachers. I don't think we should miss that opportunity.

Kyle Pearce: For sure. And I agree. It's like there's a lot that we were doing. And in our experience, that were very important parts of mathematics as well. And just seeing that other side. And I love the idea of when you were saying being humble and often times we don't know the answer when we're telling kids I don't know. The beauty is, if you're saying you don't know all the time, kids start to think that you're just faking it all the time. And it almost puncture tires even more. So then the time you're really confused, it's like oh well, I got let of the hook there, which is nice. So there's some fun ways to approach that for sure. But, you know what, before we get deeper into this conversation, we can not overlook this particular question.

Kyle Pearce: We ask every guest to describe a memorable math moment from their past. It could be when you were a student or when you were a teacher. But when you're thinking about math class, is there a moment, maybe it was a defining moment, that got you into education. Maybe it was a moment that made you feel empowered or maybe it was a moment that maybe didn't make you feel so great about yourself in the math classroom. What pops into your mind when we say math moment for you?

Nat Banting: Right. I think one of the nice benefits of being comfortable teaching with a degree of almost like volatility, I like to give students a lot of control, is there's a lot of memorable moments. Because it's maybe not as scripted as maybe it was when I started, but one thing is always common in all of these is that you're genuinely surprised. I think that's just such a nice thing to happen in the class room when you were caught off guard by brilliance. And you just have to take a moment to absorb it. And that can be like an epiphany that comes to you or you're the epiphany of a student. I remember one in particular when I thought about this question having listened before to the podcast.

Nat Banting: I was in grade 10 I believe and my math teacher gave me and one other student the problem to trisect an angle. We were doing some geometry. And so we were fairly precocious. He asked us, please trisect this angle. And we were like yeah, we got this. So we worked together for so long. This is barely on the edge of you just Google it. I think now kids Google it and it ruins the problem. But we went at it. We knew how to bisect an angle. We had all our postulates at our hands. And we were going through it. And at several times, he was also the principle, we went to him with our proof to show him that this was trisected. We had done it three or four different times. So it just wasn't just [inaudible 00:19:18]. And we were going through it.

Nat Banting: And then I remember as I was entering teacher education, I was volunteering in my high school when I moved back, and I was in a math class and they were learning the tool called deductive proof that's taken over geometry. And at one point I said, well, let's try this problem. I remembered it, trisect the angle. And then we were going through it. And someone was like, let's just halve it until it becomes a third. So we went through all these things that in the moment as I was doing this I was like, there's no way this list will ever be split in thirds because this is not a geometry problem, this is a prime factorization problem. And I just looked like [inaudible 00:19:50]. In that moment I was just like, half of me is like Dude, you are such a moron. How could you not see this as a 16 year old. But also I was like, this is gorgeous. Because in a problem that was posed to as one of geometry and I attacked exclusively as geometric. The problem the whole time was one of number and number theory. I just thought that was such a beautiful moment to come into this epiphany of not epiphany, you're brilliant but epiphany, wow math just works really well. This is very nice.

Jon Orr: It's amazing when things click. And there's so many moments. I think math class, now I'm being biased because I've only been in math classes for the last 17 years, but I feel like math class has those moments. And not too many other classes have that for kids to say when that happens, you can see it on students faces and teachers faces too. And it's pretty awesome. I think segue into your moment into a couple of the resources that you actively share and you create in the Twitter verse or blogosphere. And there's a couple we that we want to chat about.

Jon Orr: But we definitely want to chat about fractiontalks.com. And then the other one we want to get into is Menu Math. So, Fraction Talks is a great site to promote discussions around fractions and number theory. And there's so much going on there. What inspired you to put Fractiontalks.com together and how do you envision educators and maybe even parents using the resources for students?

Nat Banting: Okay. So the first thing off the hop that we have to know about this is it was a complete accident. It wasn't like I sat down to-

Jon Orr: Like all good things, right?

Nat Banting: Yeah, right. So what happened was... I'll give you the back story. I'll tell you how it's evolved and then you can choose to believe that this was intentional. It totally wasn't. So, I was sitting actually at a conference presentation by [Lanny Horne 00:21:36]. And it was in 2015. And it wasn't a bad talk. I actually really enjoyed it. But one of her slides had a picture of a area model that was cut with a couple of diagonal lines. And to the moment, I couldn't believe it. I had never seen that. And I was like okay, I've been teaching for five years at that point. I was like, oh, that's curious. And so I started sKetching in my... I have a notebook I bring to all the conferences and jot down my ideas as they come so I don't lose them. And I do five or six other ones and I'm like, oh, this is curious. Which one of these would be the hardest, yada, yada, yada. And then from that, I think this is interesting.

Nat Banting: But I don't have a lot of chance to talk about these things because I'm secondary math teacher. So by the time the students get to me, as I'm sure is the case with all secondary teachers, students completely understand all about fractions.[crosstalk 00:22:20] some utopia where everyone understands fractions. So I started digging in with my relationship to elementary school teachers and working with our grade nines. They do some fraction operations there. And then lower in the grades to see how these things were sponsoring this activity. And the big shift for me of why I started to really inaudible and then I would tweak them and people would make them. I hardly make them at all. I just take the problem, ask them for permission, I just throw it out there and then retweet when people are using them. There's been a really cool wave of people during quarantine doing them on sidewalks, [crosstalk 00:22:56]. It's an interesting way that we've been creative with these things. I'm just happy that it's there.

Nat Banting: So I went through it. And usually when I'd seen a fraction model, they were always predone. There was no decisions left to be made. If you were talking about quarters, it was presectioned in the quarters. They were already lined up. They were usually rectangles. The lines were only vertical ever. And so I was like oh, that's a lot of the work. We're talking about mathematical modeling and the model is there and complete. And that type of model is used to solve a different problem. So it's used to solve the exercise. I say one half plus a quarter. You then realize that halves should be quarters. So they're added together. And you use the models and the fraction strips or whatever and it's there.

Nat Banting: The purpose of the fraction talk isn't to solve a different problem. Reasoning about the image is the problem. And that's what I like about them. It's the self contained nature of the goal isn't to arrive anywhere, it's to stay a while, think about what's going on. And I think that's where you get all this sort of... it's very inviting in that way where a lot of students they will perform a fraction addition incorrectly. They will not make a common denominator and I'm so frustrated. But with the image, they're like, well, those aren't the same size. So I'll just make them the same-

Kyle Pearce: Yeah, you can't do that. These are different. I can't count those.

Nat Banting: And they're welcome. They would question. And I know we have these students. Before they do anything, they're like, well, can I, and they ask permission, where they don't feel that power or agency to do it. With the fraction talk, oh, I'll just cut it. It's so inviting. They're like, I'll just slice it. It's just so natural. Whereas with the notion, hey, should I find a common denominator? As if I-

Kyle Pearce: Did I do this right?

Nat Banting: Yeah, go ahead. I am the gatekeeper and I say you may use the common denominator. And so that's what I loved about them. The coolest part is just watching people use them. I think that's the nicest part of any resources that takes off. It's just in my limited sphere of influence. I use them with students as I am creatively bound to do so. But watching other people and their context has been really, really fun, particularly with the really young kids. I have a sun now going to grade one next year. And so as he grows up, I'm paying more attention to how the primary grade teachers have used fraction talks. People like Simon Gregg on Twitter who's just masterful. I want to somehow get over to Europe so my son can be in his class. I feel like after what I've seen, it would be phenomenal.

Nat Banting: That's what it is. They're sitting there at franctiontalks.com. And I've done my best to present not only the images but also the how to kind of spot where there's some teacher prompts and the rational is described. And then there's a blog that goes with it where teachers who are using them, I try to amplify what they've done. And sometimes they have permission to post some videos. There's a teacher, also a Canadian teacher, who has made a fraction talk war game, which is really, really cool. Very creatively sectioned. That's [Carla Dawson 00:25:39], I think, in British Columbia, I believe. Her stuff is all linked there.

Nat Banting: And so the goal is community building. Really I have to go back and credit Dr. Horne for her influence on it, even though I don't know if was on purpose. And these things aren't new. I found them in text books early as the 1930s when we went through and dusted off all these old [inaudible 00:25:57]. They were dividing squares like this for a long time, but it's the context in which that they're divided, this idea that these are a tool for conversation and sense making and reasoning, and not a tool to get at a solution for an exercise. And so yeah, it's been fun.

Kyle Pearce: I'm looking at those fraction cards right now and they look slick. It's like you crosstalk

Jon Orr: ... toss a link in the resources there.

Kyle Pearce: Yeah, I will for sure. And it's like you print them out. And then it's like throw this down. And who's got the bigger one? That's pretty cool.

Nat Banting: Mm-hmm (affirmative). You can put any game around them once you feel that agency as a teacher.

Kyle Pearce: And so many things come to mind as you're chatting there, Nat, backing all the way up to the presentation where you're looking going like, huh, why is that surprising to me and why is that interesting to me? When we're young, we see what's in the text book and like you had said, already partitioned area models, and into very predictable like a Pizza. Oh okay, that's one of the benefits of a circle area model for fractions. That's one of the benefits. There's also some negatives as well early on.

Kyle Pearce: But when you look at that and you're like okay, I can see that's a fourth without even seeing the whole. I can use my spatial reasoning. We see these different things. When you think back and all the things we do as children, we see so much of the same thing that we forget that there's other options. You take a square and then you actually show the same square but show it on a 45. And now the kids call it a diamond. And you're like, but no, that's just a square. It's just sitting on a point and that's it. So, the same thing with fractions, it's like there's so many different ways that we can partition. And something else that kind of hit me as you were talking is, I love the fraction talks is a way to build from that concrete to, I know it visual on the web, but you could be doing this concretely like actual have the kids cut them out and play with them and partition their own fractions.

Kyle Pearce: But then as we start to work into more in depth equality of fractions and actual operating with fractions, being able to take some of these images and these models and having kids create their own so that they have a clear connection between the visual of the fraction and what they're doing so that they don't feel like that permission asking like you had said. When we teach procedures first, kids are constantly asking us, is this right? Because they have no sense of reasoning. Whereas when you go back and they actually have a model to work with and they're looking at it and their going wow, this piece isn't the same size as that piece, I can't add those together, you can then play in this land of well, what about these two pieces which maybe they are the same area but they're a different shape.

Kyle Pearce: There's so much that you can do there and I behalf the math community, I just want to thank you for putting those out there because you've left it with this open playing field for people to utilize them. But hopefully people are seeing that you can do so much with them even beyond the warm up math talks sort of approach, you can do that. But you can also bring them into your actual lessons so that kids can actually make more meaning. So I thought that was super cool.

Nat Banting: Yeah. I think the productive thing if anything it's all product of culture. So I think if we produce a culture even around these images even though there's potential richness sitting in them, we can dissipate that quite quickly by creating a culture of, hey, this is how. Even like you say, this is just like a term from Jennifer Holm who's a professor in math education, Canada. This is just a procedure with pictures. It's the same right. Here's how. I've really tried to have this attitude to get students to say, well maybe here's why. I think this is like, here's something that I'm thinking. Well, here's why I think that. Or often times I'll say, and the secondary students hate this, but I'll say, show your [bercuses 00:29:47]. They'll tell me something. The younger kids love that. And my son at five and six, he liked that whole show your bercuses. I don't know why you think that. But when you tell an 18 year old and they're like well I think the log rhythm dah, dah, dah. Show your bercuses. And I don't quite take it the same way.

Nat Banting: But I think that's an important piece here is these things live on the internet and that's great. But they only hold the potential for richness and a lot of that is the job of the teaching. It's an activating and amplifying and accentuating that potential.

Jon Orr: Let's move into one of the other resources that you've been talking about and sharing, which is Menu Math. Can you give us a little elevator pitch or an elevator chat about what is all about Menu Math? Are people picking off a menu? Help a listener understand what it is and how and why, I guess, we should be using it, because I definitely think we should be using it in our classes.

Nat Banting: Those are kind of basic. Menu Math was a much more intentional design than Fraction Talks. So Fraction Talks is a happenstance occurrence. So I teach secondary, again, to belabor that point, but a lot of the secondary mathematics content that I am responsible for creating opportunities with is very algebraic and very abstract in nature, and that's one of the number one complaints for secondary teachers like, ah, this is really great. But there are rules when you get high enough, which is not true. The meaning is obscured by the abstractness I think. So I started to look for ways to continue to have students make decisions with algebraic models. It's particularly functions is where this started. So I started with quadratics. And what I noticed as I zoomed out on teaching quadratics as a whole is that generally, we ask to think in one way, and that is here is the model, please describe it. So they would say, here's the model. Oh, here's the X intercept. Here's the direction of opening. Here's it's width. We're describing the model, but the model's given to you. And all I did is, what would happen if I just flipped it on its head and I said well, here's some characteristics. Can you please build me a model that satisfies them. That was the whole goal. So, that's the design behind it.

Nat Banting: And so I did that for a while and I really liked it. Students were creating if we thinking blooms at the very top of that pyramid. They were creating some very good solutions. And then I thought, what if I could put this on steroids? And so then what I did is I had these cards that I would flip over. And so in the moment, we'd be in our grade 11 class that we'd be dealing with quadratics. I'd lick my finger and flip over three of these playing cards. And each of them had one of these characteristics on it. So, no X intercepts. Never enters the first quadrant, what ever. And everyone was responsible for building me a parabola or proving that it was not possible. And so I thought, you know what? So I put eight of them on a list in one day instead of saying please build me one, I said you can build me as many as you want as long as each of these is satisfied at lease once. And there's eight of them. Or you can build me eight different separate parabolas and then you're still creating a lot of decisions.

Nat Banting: But the kicker with Menu Math that makes it really, really nice is you say can you please build as few as possible. So can you think at that next level where you're not just thinking about what affects that characteristic, but which of those constraints actually pair really nicely together like which ones can I check off the box with one function for sure? Which ones are impossible? Can I do it at one, two or three. There's all these kind of questions that come in. So they're modeling, they're making those decisions. But also there's this next level of okay, now I'm looking at the synergy between all my creations. So I started with functions. And then like the community does, they just took it and ran with it. So there's things from second grade numbers. There's shapes, looking at polygons. There's sets of data. There's calculus.it's all over the map. I just got one on graph theory. I have to go back and relearn graph theory to see if I can do this Menu Math.

Nat Banting: So it's fun to see people, again, submit and be like, hey, this is what I used. And the power of the structure is that it creates this opportunity to make decisions, and decisions happening at multiple levels. And so those are all on my website as well. And again, I'm just building a table of everyone else's work. So there's less credit sure the idea I put together this structure that I've called Menu Math, but most of the creating and innovating is happening outside myself. I'm running around like a mad man trying to get all the pieces up in a nice place for teachers.

Kyle Pearce: But you know what? I think one of the big pieces is the idea. When you give somebody an idea like this, it's like they now have a tool that of course collecting them and sharing them that makes life even easier so that they don't have to spend nearly as much time like digging and doing that deep thinking, even though I think everyone should be doing that thinking as well. But it's nice to have a place to start. But now that you have this idea, you go huh. It's almost like it starts coming out of everything that you do. You're like huh, I could see how I could take what I used to do with this lesson and I could just make the way I present it or the way I can just tweak it a little bit. And then all of a sudden now I have something that's a little bit more of a routine that I can build in. I see some similarities between this and some of the open middle idea. Even though open middle's different, but it does force this more of that critical thinking. All of that goodness that we're hoping that we can put out of our math class. I think it's fantastic.

Nat Banting: You do see inaudible too like when you have to make one decision to use one of those numerals, you're also making the opposite decision, I'm not going to use it the next. And so you do see that give and take, that real critical density that happens with these types of activities that just give students choice over the creation of the model.

Jon Orr: Yeah. I love this idea of taking something that we would normally do and then turning it inside out and seeing how that in the constraints can happen. There's another routine that's out there, that is two truths and a lie and it's a similar thing that kids have to figure that out themselves if they're going to create two truth statements or any kind of statements from this equation. But they got to build that equation to make it true and they create the actual menu that goes with it. I just always loved this flipping things around and seeing how kids can attempt to pull information from, and then you just sit back and listen and provide guidance. And I think this whole idea of constraints is really a winner for kids.

Jon Orr: And I know that your OAME in 2019 talk was all about constraints, and constraints to instigate mathematical thinking. Do you mind giving our listeners a little bit of a calls notes of that presentation?

Nat Banting: Yeah. I think the more I've thought about it, the more I've gone through the academic side of the profession as well as the practicing teacher angle that I've lived. This notion has been more powerful than anything else in creating how I think in the moment about teaching mathematics and also how I design mathematical experiences. And so I put it all together for one of the first times around that late 2018, and then for OAME 2019 is this idea of, what if teachers felt like they could have autonomy over the problem. So what if we could have a way of thinking where in the moment teachers felt actually in control of the problem rather than having the problem run it's course and you be out of control of it. Because I think that's a very common impression when you teach differently, oh it's going to be a fiasco. So I sat down again. I'm a big fan of sitting down and really trying to generalize themes of practice. I think I like to redefine mathematics teaching to the pint where I'm like oh, yeah, now I can bring this back into context.

Nat Banting: So the one thing that I noticed right away is we notice difference. We notice when things are different. And I realized that when I was reading a lot of Gregory Bateson who talks about this idea that all information is actually just a difference that makes a difference. So, we notice things, it pops into our awareness when something is different. And one of the examples of that is there's all those studies of the baby where you have three dots and if you take the dots away and then there's only two, they'll dwell on it longer. So from a very early age, we have this biological predisposition to notice difference. And unfortunately I don't think there was ever... You mentioned it already in this chat, math class is just horrendously predictable. And so to do that, if that's the unit of information, something needs to change.

Nat Banting: The other thing is that teachers are collectors. And I think this is fascinating. And this started an internship when one of my friends came in, biggest smile on his face. And it was like mission accomplished. And then I was like, what's up? And he goes, I got the binder. And it was like this big moment. On student teaching like, what happened? I did it. I landed the jackpot. It's almost like, I got the whale. All the lessons.[inaudible 00:37:50]-

Jon Orr: I don't have to make any anymore.

Nat Banting: Yeah. And he'd spent his whole day in the school that day photocopying the binder, and now we have the USB or hey, I got permission to the Google drive or whatever we have here. This is the same analogous thing in our teaching day and age. But I think teachers, we think that things teach for us. So that creates the impression that we're scared to change them like oh, here's a rich task. I got it off this very reputable website from this very reputable author. It's perfect because it was there. And so I don't want to change it. So there's this notion that we're not teaching. The prompts. If only I had the best prompts, I'd be a great teacher.

Nat Banting: So, taking these things, we often think that we're bad teachers if we take opportunities away from students. But I actually started just perfectly thinking instead of instructing students, obstructing students. So putting this purposefully in their way to create these ideas of difference so then we can keep thinking going. And so that's where this idea is born from is this notion of obstruction. S when we talked about cooking way, way back at the beginning here, taking away that third measuring cup. That could be seen as a bad teaching move for a lot of people because you're limiting what the students can do. But actually through the creative limitations and constraints on students, we actually see a lot of creativity and mathematical thinking come out. So that's what I was playing at there.

Nat Banting: And I think it just grants teachers a lot of license. And so that's one of my favorite moves to do in the classroom is the I forgot teacher key sort of thing, or you're not going to believe this, I didn't tell you the right problem. I left out a piece of the problem. crosstalk you got to be kidding me. But the beautiful thing is they don't blame you and they don't blame themselves. They just keep thinking because it's no one's fault. And so the analogy I think I told at OAME is the idea about the fishing story. So, in Saskatchewan almost everyone fishes. It's what people do. I don't-

Jon Orr: You guys have lakes there?

Nat Banting: Oh, tons of lakes. Anyway, I don't frequent them. But my son went fishing actually with my father in law. He got to fish. I have caught one in my life. So he's ahead of me. I go to the staff room and everyone will be talking about their fishing stories. And there's a fishing story, everyone says I caught the biggest fish. That's the classic fishing story. And the next most popular fishing story is I had this giant fish. I was doing really well and I almost got the fish in the boat, but then it got away. The line broke. And that's still a great story. And I sat there listening. That's a terrible story because your story is like, I'm not good at fishing. But that's not the message there. So when you change the problem on a student, they don't get the impression that their bad at mathematics, they get the impression that the line broke. Oh, okay. Now I have to reset. And then they go through this very valuable process of, okay, what have I done that's still valuable in my new context? What have I done that no longer applies? Which is really important piece of mathematizing.

Nat Banting: And so I just started playing around with it. And I enjoy being whimsical and off the wall in the room. And that's very much fits my emo. But I've seen teachers do this with amazing precision. Where as leaving something out conveniently can actually have huge ramifications for maths thinking down the road that are just beautiful to watch how students adjust to a changing context, how they make a difference because of these differences.

Nat Banting: Sometimes I plan them. I pre orchestrate when I'm going to slowly release these new constraints. And sometimes, it just hits [inaudible 00:40:57]. What if that wasn't true? And that what if not sort of thing is a nod. And maybe we can out this in the show notes as well. A book I read very early on studying to be a teacher which is The Art of Problem Posing has been very influential. So they have this process called the what if not process. So you just list everything about a problem and then ask yourself if that wasn't true. And that's the same thing. You're playing with the different constraints on a problem and it really breeds the creativity out of it.

Kyle Pearce: I love it. So many things to pull at there, and backing all the way up to you had mentioned the rich task idea. It's perfect. I found it. Or here's the text book or here's the notes from the teacher who's taught this course for 13 years straight, all of these things. I'm going back and thinking as well that Jon and I talk about trying to keep questions open at the beginning. Make them low floor, high ceiling, but sometimes what people can hear from that is oh, every has to be wide open. And what I love about your message here is what you're saying is actually, there's so many different approaches to questioning and good question and thinking through. And I think intentionality is so important in order to get out what you're after. And I love all of these tips. I'm hoping people ae taking them away and saying, huh. now I think what you need to think about. And I say you like the we, all of us thinking, is going okay, I want to use constraints. Now the question is, when?

Kyle Pearce: When do I incorporate them? When's the moment you drop that constraint or open it up even more or what ever it might be. But I think it's such a valuable tool to add to to your tool belt. And I love the part of you said sometimes it's very intentional. And then other times what I kind of heard, I'm going to go ahead and tell myself this story that was you were saying is when you come up with one of the fly, it's almost like curiosity. Your own curiosity is taking over a little bit. And you're like, I am so curious about this. And Jon and I speak about sparking curiosity all the time. And there's so many different ways to do it. And I think kids eat that up when they can look you in the yes and see that you truly are curious, and that maybe wow, he's not actually setting up for this one. You want to know the answer too?

Nat Banting: And I think so. As far was when, that's a tough thing. So my first thing I would say against that is there's no right when. There's no answer to that. But the caveat being that we always have a purpose from the teaching and that is to avoid inactivity. Inactivity is distinction. The second, activity stops, thinking stops. So my goal at all costs is to disrupt enough, provide enough disruption, enough coherence to keep thinking going. And so whether it might be right or wrong, I want to throw out this idea of optimal, optimal times, optimal response, but really shift again, lean heavily into what was effect. What was the impact of my interaction with them.

Nat Banting: And you can always just take it back. That's so disruptive. Like oh, forget it. Don't worry. Let's go back here. That's fine. There's no right or wrong way to do that. And posing of the problem is nice when you prepare it. Whether its' rich or not. Whether it came from a site that is very reputable or not or a text book or the teacher who's been teaching for ions. All those things have in common. Again, what do all those mindsets have in common is that all those the questions live in the past tense. So we're not worried about what's happening mathematically, we're worried about how our students' are aligning with what we thought would happen. And so we're actually orienting ourselves towards the past, towards our preparation. Where I think when we're really free, when we're really teaching for understanding and really responding to our learners, we actually are teaching in the present tense. This idea that when that thing hist us, why not? What is the downside of trying to sponsor that student understanding.

Nat Banting: You can always take it back. This is not binding in anyway shape of form. I think it's actually this very nice liberating thing for teachers to have control over their own classrooms is a fundamental to teacher change. I think that's a fundamental aspect.

Jon Orr: And I think a lot of teachers, we're getting at the heart of thinking and fueling sense making for students. So many of our teachers are always asking us. How do you engage kids. And I think people get hung up on the I got to have that real world math problem or I got to do that vide problem or I got to have this really cool thing from a website. But I think what you're providing here is such rich discussion that it creates that engagement automatically. And people often ask us well, how do you make it engaging to do these abstract math concepts as we get into higher grade levels. People think, oh, I can't do a good activity with my university bound grade 11 or grade 12 because it's too abstract. But these frameworks that you're putting together provide really great fueling sense making moments. But also great engagement moments. I got to thank you for talking about them here and making them clear for everybody. And also talking about the benefits of why we can't not do them.

Jon Orr: So I want to thank you so much. We're looking at the time here, Nat. We don't want to take too much more of your time. But thanks so much for joining us on this conversation. And if you could leave everybody with one last statement. One last thought before exit here. What would you say?

Nat Banting: Wow, that is tough [crosstalk 00:46:01]... all mathematics education. I think that now teaching post secondary, teaching undergraduates and going back to teaching students, the fundamental piece, and we're learning this more and more as we've gone through this COVID error is that teaching is about wellbeing. And that's mathematical wellbeing or otherwise. And so I think if we can not learn to develop a sincere empathy for those we are charged to instruct, we can to do that job well. And so I would say start there.so that means observing the knowing as if you were knowing as them. Really try to step beside them empathically but also as human beings. And so I think if we can guide our practice with that, I think we will find ourselves coming out on top.

Kyle Pearce: Oh, that's fantastic. Well said there Nat. It's been a pleasure chatting with you here. And again like Jon said, we want to thank you so much. Natbanting.com has a bunch of goodies, Fractiontalks.com. Anywhere else anyone can get in touch with you before we say our final goodbyes?

Nat Banting: The natbanting.com will get you there. Usually I'm fairly connected. There is a contact form and things. I'm on Twitter a lot @natbanting as well if you wanted to interact there. If you're ever bumming around Western Canada, maybe we'll see each other in person or maybe on the website sometimes there's talks that I'm giving around Canada. Maybe we'll end up in the room some day. So say hi. I love to chat about math education and other thing obviously. Not as much as math education, but you get the point.

Kyle Pearce: Fishing in the lake I'm sure.

Nat Banting: Yeah, fishing. I'd love to talk about it.

Kyle Pearce: Thanks so much Nat. We hope you have an awesome day. And I'm sure we will bump into you sometimes soon once these face to face conferences get back up and running.

Nat Banting: Hopefully, yeah.

Kyle Pearce: Take care my friend.

Nat Banting: You guys too.

Kyle Pearce: We'd like to thank Nat for spending some time with us here today. We felt like it was such a deep and worthwhile discussion. Both Jon and I walked away with awesome take aways and I hope you'd did too. We want to just give you a quick little reminder that registration is open right for our webinar The Tortoise And The Hare, How Math Class Missed the moral and How To Fix It.

Jon Orr: In this webinar, we'll be learning what awe need to do to cultivate slow and fast thinkers in math class and how to do that. Why math class rewards extroverts. And how to structure our math class to engage all students on an introvert, extrovert spectrum. How to teach problem based lessons that are accessible to all learners and how to help students conceptually understand their basic math packs.

Kyle Pearce: Yes my friends. Lots of awesomeness going on in this one time webinar. And again, this is the only reminder on the podcast. This episode is the only time you're going to hear about this webinar in particular. So head on over to makemathmoments.com/webinar to register for that one time offering on January 12th, 2021. It's happening at 04:00 PM Eastern. But again, if you register, you're going to get the replay link as well. So head on over there. Make sure you don't mess it out on it. We are going to be really looking forward to this once.

Jon Orr: That's maemathmoments.com/webinar.

Kyle Pearce: In order to ensure you don't miss out on new episodes as they come out each week, be sure to head over to subscribe on your favorite podcasting platform. And do us a huge solid by helping us grow the podcast by heading over to Apple Podcast, hitting that subscribe button. Leave us a short rating and review. And my friends you don't know how much that means to us for us to get that constant feedback in order to keep us going and pushing this podcast to new heights.

Jon Orr: Show notes and links to resources and full transcripts from this episode can be found at makemathmoments.com/espisode110. Again, that's makemathmoments.com/episode110.

Kyle Pearce: Well, until next time my friends, I'm Kyle Pearce.

Jon Orr: And I'm Jon Orr.

Kyle Pearce: High fives for us.

Jon Orr: And a big high five for you.

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