Sam Brotherton: When my students are working out these types of problems to really pay attention to the different methods that are out there and focus on how I can sequence that, so that every student in the class can understand it from whatever level they’re at or wherever they’re at in terms of that particular problem, or let that work be a-

Jon Orr: You’re listening to Sam Brotherton, a teacher from St. Louis, Missouri. Sam’s been teaching for five years. We chat with Sam on this Math Mentoring Moment episode of the Making Math Moments That Matter Podcast. Sam’s main reason for contacting us is that he’s struggling with the idea of where assessments fit in his routines. Sam has been teaching with three-act math lessons, Desmos activities, and a variety of other great resources he’s gathered from the math community, but he’s noticing much like we did early on that when it comes to standardized tests or those word problems from the textbook, his students aren’t performing any better.

Kyle Pearce: Listen in as we chat with Sam about these assessment struggles and how the conversation pivots to talk about how to make effective use of the five practices for orchestrating productive mathematical discussions and how purposeful practice fits into our lessons. This is another Math Mentoring Moment. Hit it.

Kyle Pearce: Welcome to the Making Math Moments That Matter Podcast. I’m Kyle Pearce.

Jon Orr: And I’m Jon Orr. We are two math teachers who, together-

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

Jon Orr: … fuel learning-

Kyle Pearce: … and ignite teacher action. Welcome to episode number 13: Where Assessment & Practice fit in Curiosity Sparked Lessons: A Math Mentoring Moment with Sam Brotherton.

Jon Orr: Before we get to our interview with Sam, we want to remind you that you, too, can join us for a Math Mentoring Moment on this podcast. In these episodes, we talked with teachers about real issues that arise in the classroom, and together, we’ve worked through possible solutions. We know that our listeners, math educators like you, will get a lot of value by listening in on this conversation.

Kyle Pearce: If you have a struggle or issue and want to chat with us about it, head over to makemathmoments.com/mentor, and fill out the form. We won’t be able to talk to everyone who fills out the form, there have been a ton so far, but we still make every effort to hear a variety of voices and classroom issues. Again, head over to makemathmoments.com/mentor to apply.

Jon Orr: Before we get to the Math Mentoring Moment, one of our favorite books is How We Learn: The Surprising Truth About When, Where, and Why It Happens by Benedict Carey. Believe it or not, both of us actually listened to this book in audio format while driving, running or relaxing. Now, you, too, can get it for free because Amazon’s Audible platform is offering two free books by going to makemathmoments.com/freebook. That’s makemathmoments.com/freebook. If you like podcasts, then two free audio books with Audible is the way to go.

Kyle Pearce: Awesome stuff, Jon. Let’s get into the mentoring moment with Sam.

Kyle Pearce: Welcome to the podcast, Sam.

Sam Brotherton: Hey!

Jon Orr: Sam, can you just tell us a little bit about yourself? What’s your teaching story so far? Where do you live? All that kind of stuff, fill us in on some details.

Kyle Pearce: What’s your favorite color?

Sam Brotherton: I don’t really have a favorite color. I always tell my students that I like bright colors because you get that question every year as a teacher.

Kyle Pearce: Yeah, for sure.

Sam Brotherton: So I’m from St. Louis. I come from a family of teachers. Ever since I was little, I’ve always just loved math. Sometimes when you’re young kids are kind of out of control or they won’t sit still, so you put them on the iPod or something like that. Back when I was younger, my parents would give me a pencil and a sheet of paper during church, and I would just sit there and work out math problems. So, I think, it’s always been in my blood if that makes sense. So, I’m in my fifth year now. This is my second year in the Mehlville School District at Oakville Middle School. I teach sixth grade math. I absolutely love it. I can’t get enough of it.

Jon Orr: Awesome, awesome stuff.

Kyle Pearce: Very cool. Is your schedule an all-math schedule? So, you’ve got a bunch of different groups of grade six that’s coming through?

Sam Brotherton: I am all-math this year. Sometimes I’ll get like one social studies section. But this year, I have six sections of math to accelerated classes, not just four standard sixth grade classes.

Kyle Pearce: Very cool. So, we were going to ask you why you wanted to become a math teacher, but it sounds like you’ve hit on that. First of all, family of teachers. Were your parents both teachers as well or math teachers, I should say?

Sam Brotherton: No, they’re really not. Well, I don’t like to use the term math people, but they wouldn’t call themselves math people. No, they both taught elementary. Then my dad moved on to administration. He just retired as a superintendent last year. My mom is Director of Outreach at a school in St. Louis area.

Kyle Pearce: What something that I find really interesting is that they might not have maybe considered themselves to be so math centric, if maybe that’s the good term, but they managed to raise their son to find a love of math. That’s something that I would definitely tip my hat to because oftentimes, if a parent dislikes math, they tend to rub off on the children not because it’s genetic by any means, but just because of their attitude towards it. So clearly, they managed to refrain from spreading that math phobia your way which is really nice.

Sam Brotherton: Definitely, and I appreciate them for that.

Kyle Pearce: So tell us a little bit more about your teaching story. How have the first five years of teaching gone for you? Maybe some successes, maybe some challenges.

Sam Brotherton: So, I think I have to jump back to when I was an undergraduate at the University of Missouri. So I was sitting in one of my education classes. Our teacher showed us a video of some guy pouring water into an octagon tank. So I think you guys probably know who I’m referring to with that Dan Meyer task. So, I watched that video. I’m like, “This is cool.” I always understood formulas and those kind of things. But after watching that video, I’m like, “This is cool. This is how I want to teach. I don’t know what this is or how to do it.”

Sam Brotherton: So I think my first two to three years, I really struggled with how do you do Three-Act tasks and how do you do problem-based learning. So I tried a bunch of stuff, hoping that things would stick, like what you guys talked about in your first few episodes. So, it’s nice that a lot of you guys out there paved the way for people like me. But now I’m at the point where I’m facing some new challenges and some new successes as I start to grow as a teacher. One thing that’s been awesome is just watching how excited kids get with different types of problems. So one problem that I refer back to, that’s a big success for me, is at my old school, the kids really like to eat the Frooties. You know what those are?

Jon Orr: Frooties? No.

Kyle Pearce: Yeah, [crosstalk 00:07:13].

Sam Brotherton: Like the flavored Tootsie Rolls right.

Jon Orr: Okay. No, I don’t think… We don’t get those here in Canada.

Kyle Pearce: I have something different in mind. Yeah, I had something different in mind.

Sam Brotherton: I don’t know how common they are here either. But anyway, the kids really liked them. So there was one kid who was selling them, and obviously you’re not supposed to. So principal confiscated them. I was like, “Hey, can I have those? I’m going to make a math lesson out of it.” So, I think it’s Dan Meyer’s lesson with the meatballs and the spaghetti sauce.

Jon Orr: Yeah, that’s his. Great task.

Sam Brotherton: Yeah. So, I took that idea, but I took the Frooties. I had recorded myself rolling them up into a ball, so they start off at cylinders, and then had myself recorded, rolling it up into a sphere and fast forward it, and made a three-act task out of that. That was like a moment where I was like, “This is clicking for me.” I’ve taken a task, and I’ve made a connection that my kids can relate to, and they’re excited about. They can figure out the volume of one of these Frooties, and then use it to make an estimation on the volume of the sphere, and all that kind of stuff so.

Kyle Pearce: Nice. Now, I’m wondering, if we could go back for a second. I’m curious about this Frooties task as well. I want to go back, and I feel like maybe I might have jumped the gun a little bit on asking about a challenge. I’m wondering, it sounds like you bumped into Dan Meyer’s. It sounds like probably his TED Talk, where he shows the octagon-based fish tank or cooler or whatever it is, that [inaudible 00:08:35] water. I’m wondering, before you hit that point, because I remember that very vividly, when I ran into that TED talk, and then had an opportunity to see Dan speak, my mind was blown. What did math class look like for you? So, we always like to ask, what do you remember from math class as a student, or maybe even as a teacher prior to seeing Dan’s TED Talk?

Sam Brotherton: So, there’s a lot of traditional math for me. I liked it because I get just math focus. So, when I’m sitting there working out a formula on paper, I have a tough word problem. That got my brain going. But there were a few instances that I can remember that stick out that were different than just your traditional sense. My high school calculus teacher, Mr. White, he did a lesson with us one time where we had to bring in a random-shaped item, and then we used Plato to show how you can integrate, take the integral of the item. Obviously, I don’t remember the math side of that, perfectly. But I remember I was like, “Okay, so this is engaging. This is cool.” So, there were some moments where math looked like it could be different than your traditional paper pencil, but those were your outliers. So, most of the time, it’s word problems, calculations, those kind of things.

Kyle Pearce: You know what? I think Jon and I both relate when it comes to we typically didn’t struggle with doing the procedural math like pen to paper, let’s do the formula, let’s find some answers. I don’t know if I fully understood it, but I knew how to manipulate algebra. Luckily, I had that ability to recognize patterns and take them. That was helpful, but I definitely was missing some other pieces. It sounds like this calculus teacher created a math moment that mattered for you. That might have been that hinge point or that stick point for you to connect it to the math. Obviously, it’s likely been a while since you’ve done some calculus. That would probably explain why. Maybe the math hasn’t stuck for you right now, but it’s interesting that you do remember that actual experience. So that’s really cool to hear. Thanks for sharing that.

Sam Brotherton: Yeah. There’s probably some high school calculus teachers listening that want to give me a tutoring session.

Jon Orr: Sam, you talked about hearing Dan and implementing your own three-act task, which sounds amazing. That sounds like it’s a great classroom action. We’re wondering what’s on your mind lately. What struggles and challenges have you had implementing those? What do you want to chat with us about today?

Sam Brotherton: I mean I got a whole list for you guys. So, as long as I can keep you here, I’ll keep firing things at you. But I want to just start off by saying I have an awesome group of kids. They stay with me through all this stuff that I throw at them. It makes a lot of these challenges a lot easier. But one of the biggest things I’m struggling with right now is assessments.

Sam Brotherton: So my district is moving towards standards-based grading. I’ve been playing around with it in my classroom. What I’m starting to find is that we’ll do these three-act tasks and we’ll have some great classroom discussions, and we’ll be using manipulatives and using expo markers to show different types of mathematical modeling and everything. I’m looking at all this. I’m like, “This is great.” Patting myself on the back and everything. Then it comes to assessments, state testing, or just summative assessments, word problems. The kids are having a lot of trouble bridging that gap of what I would call authentic math, the things that are happening in class. Not that it’s not authentic, but your traditional world problems that it’s still a part of what we do every day as math teachers and math learners.

Jon Orr: Right. So, if I could repeat back. You’re saying like, you’re implementing the structure of three-act math task, and you’re getting a lot of engagement in your students, a lot of good discussions coming out of this. But when it’s time to turn to say textbook problems, or those standard problems we may have started with in a traditional method, the kids are flopping on those or having a hard time showing that understanding that they’ve already showed in the classroom lesson during that task time. Does that summarize a little bit of that struggle that you’re having?

Sam Brotherton: Yes. So, I could give you an example, if that would help us.

Jon Orr: Sure.

Sam Brotherton: So, sixth grade, dividing fractions is a big one. So, when I introduced that with the kids who started… I think is a Max Ray problem.

Kyle Pearce: Okay, yeah. I’m pretty sure I’m of it. We’ll try to put it in the show notes.

Sam Brotherton: Seven cups of dog food. If your dog eats two thirds of a cup per meal or something like that, how many meals can you feed the dog? So, building a real world example where you might think about dividing fractions, not that anybody starts to measure out all their dog food. So, I’ve got whiteboards all around my classroom. So pretty much a 360 whiteboard room. I just throw that scenario at them. So, you’ll see kids… Initially, they just start drawing models. They draw seven boxes. Some of them will even break it down in sections of three, and they start calling it in. There’s all different things happening. That’s maybe a one- to two-day lesson.

Sam Brotherton: We do some practice problems with that, and we start to show how when you break down those boxes. That’s how you’re multiplying, and that’s why you’re actually multiplying by a reciprocal and all that good stuff. Then on the assessment, it’s like, Little Johnny has a 10-foot board, and he’s making two third foot cuts. How many cuts is he going to make on the board? I get kids raising their hand immediately calling me over like, “I don’t get this. I don’t know.”

Kyle Pearce: So, it sounds like something that… I want to tip my hat to you, drawing models. So it sounds like they’re utilizing some form of mathematical models such as… It sounded like with seven boxes, maybe an area model. So, they’re actually really trying to build a visual piece to this, which is great, and not just rushing to an algorithm, at least, early on. So, between that lesson, so let’s say this Max Ray problem with dividing fractions with the dog food, what might some of the work in between a lesson like that look like and the test down the road like? What does that sound like?

Sam Brotherton: So, how do you follow up those types of lessons is what we’re getting at here.

Kyle Pearce: Right. That purposeful practice piece to make sure that students really have grasp what came out in that lesson, and we’ve consolidated it, and then what goes on after that.

Sam Brotherton: So, I, pretty much this year especially, have started to make all of my lessons through test most. There’s a few lessons on there where you can have the kids manipulating on the computer with fractions. So, it’s generally, for me, most of the time, it will be that jumping off lesson, maybe some sort of three-act task or problem-based learning, and then it’s usually a day or two of some type of Desmos activity. But on those Desmos activities, towards the end, I usually throw in your traditional questions. I can see percentages right then and there of how my students are doing. I think when we’re in class and we’re talking about things, their confidence is a little bit higher and the scores look a little bit better. So to me, I see those happening in class. I also do like Quizlet Live quizzes. We just do a lot of work on whiteboards. So just practice problems on whiteboards, those types of things.

Jon Orr: You got definitely a lot going on in there. I think there’s a lot of good stuff, especially, on Desmos. I definitely use a lot of Desmos in my classes. I agree with you on the idea that they have so many great lessons to get that understanding out into the open or get them to practice and make those connections between graphs or, say, your fractions and the big ideas around those. So I would also be utilizing Desmos to do a lot of things.

Jon Orr: Kyle and I, we’ve had many discussions about this because we were in a very similar situation where when we first started using three-act math task-type lessons, where we would get the kids to talk and get all that out in the open, and they get to try solutions. We realized we had that same problem you did. When it came down to doing our tests or our standardized tests, our scores were worse than they were before, initially. We went back and thought like, “What’s going on here?”

Jon Orr: We realized that we were doing all this great work to get the thinking out into the open, but we were missing a couple things. One of those things that we were missing was, one, we didn’t consolidate the learning explicitly through direct instruction when needed. We also didn’t build in practice time during our units or our lessons that came to those boring textbook-style questions. That were really the test questions anyway. So, they struggled with the test questions because we didn’t practice them. Every day, we were doing these great Desmos activities. We were doing these great three-act math activities like you’ve just said, but we failed to show them the types of questions that they’re going to have to experience or solve.

Jon Orr: Kyle and I will always say that you might be really good at solving this one type of problem, but if you can’t solve this other type of problem that uses the same skill or same context, are we really that good? One thing that we needed to do was, first, we needed to consolidate that lesson directly. Sometimes, for us, that would take the form of seeing what the students are showing us from around the room. If it’s not hitting the learning goal that we picked out for that day, and the skills that we want to bring out that day, that’s the time where we step in and we teach those skills. We show them the skills. We talk about the algorithms. We make it formal. Sometimes, for me, that looked like a standard lesson at the end of class instead of at the beginning of class. Then right after that, we definitely built in practice and more practice and more practice. Then later on, we can talk about some structures around practice.

Jon Orr: Kyle, do you want to add anything to this because I think we definitely have a lot we can build on here?

Kyle Pearce: Yeah, exactly. The reason why we thought it would be excellent to bring you on the show is because we definitely felt this way very recently, or up to very recently, I should say. What I wanted to also mention about that consolidation piece is that there’s going to be many days where there’s enough students in the room who are making headway towards that new learning goal. So, it’s great that students have created student-generated algorithms and solutions along the way. Then oftentimes, when we do that more direct instruction piece, first of all, we can give kids the voice and we can gather them around and bring them in, so that we can do this what we call the most important part of the lesson and ask students to articulate their solution strategy.

Kyle Pearce: So, this will help students at all ends of the spectrum, especially, if we do sequence from most accessible, meaning like a solution strategy that is fairly obvious to see on its surface, so that everyone can… been head nodding going like, “Yeah, I see what’s going on there.” Then all the way to getting closer to that place we’re hoping to get to today, which sometimes, a student will have it, and maybe they’ve had exposure to that learning goal in the past, and they’ve brought it out, or other times, maybe it’s just taking a solution strategy, and it’s extending it. So, taking that one students model that you’ve seen. It’s like, “Wow, this is pretty close to where I want to get to. It makes it so much easier for me to talk less, but have some more meaningful things to say,” if that makes sense.

Kyle Pearce: Like I look back to my typical lessons in the past. It was me talking most of the time, and kids probably zoned in out after five to 10 minutes. Now, in my consolidation, I might be talking for maybe that 10-minute time slot instead. That’s to make those connections. Then the other piece that I think is really low-hanging fruit is taking that context and trying to do some extension problems like Dan Meyer would call them like sequels, but not sequels in the sense that you have to go find another picture or a video, but to ride the context and head towards the word problem. So, I might have done a really cool three-act math task, but then now the extension, after we’ve consolidated the learning, might be another question with the same context.

Kyle Pearce: So it might still be the dog food, but now it’s somebody else’s dog. It’s a different ratio or a different set of fractions that you’re going to have to divide. Now, you’ve helped them by, first of all, they already understand the context, and now it’s just reading the problem and building from there, and then slowly, you can move away. I’m almost suggesting scaffolding for that particular learning goal. Scaffolding away from the three-act math task. While sometimes, that’s not going to be the most exciting thing. What we tend to do is we bring back some of these practice-type problems as something near the end of a class or something even to start off a class.

Kyle Pearce: So, it might just be like a warm-up-type problem. It might be related to a topic from last week. So it really tries to keep it in the forefront, but also get students in this place where they can actually read a word problem and start building the context in their own mind because that’s like a form of abstraction that we’re trying to build for our students. We start by showing them a video because it makes it very accessible for everyone, but then we want them to have the skill to be able to read a word problem and build the context in their mind.

Kyle Pearce: I picture it just like a child who’s learning how to read. You start with picture books, and then the next step is pictures with one word on each page. Then, series of books or the next stage of books would be like pictures and a few words. Then eventually, you get to this place where there’s no picture at all. That takes a long time. So, this might not happen within a day or two days or even a week, if I’m able to mix up my topics a little bit. I can do this over a longer period of time. Does any of that make sense? Or is there anything might be unclear because I feel like we just threw out a ton of ideas, but they were just coming at us pretty hot and heavy there?

Sam Brotherton: Yeah. I think I follow. So I’m breaking down what you guys are saying in two ways. One way, I’m hearing you guys say when my students are working out these types of problems to really pay attention to the different methods that are out there and focus on how I can sequence that, so that every student in the class can understand it from whatever level they’re at, or wherever they’re at in terms of that particular problem, or let that work be a jumping-off point for where our discussion goes.

Jon Orr: Something that really helped us when we first started doing Three-Act math problems and opening the doors to students, attempting solutions before we show them because one of the big mistakes with Three-Act Math problems we made early on was we would present the what do you notice? What do you wonder? Our kids would generate those. We get all this curiosity and engagement, and then I would turn and just show the kids how to do the problem. Then, we lost all that great thinking that could come out. One way that, I would say, saved time is, I guess, it is a time saver, is let the kids show us what they know. It sounds like you’re doing that. What helped us was when we first did this, we didn’t really know how to tie all those together. The resource that really helped us was the book, the 5 Practices for Orchestrating Productive Mathematical Discussions. Have you heard of this book or read this book?

Sam Brotherton: Yes, I have. I’m very familiar. So, that’s why when you guys were talking about sequencing work, I was thinking of that book and just had that in mind. This is actually number two on my list of struggles. They are starting to overlap the student discussion. So picture, you come into my classroom, and we’re working out a problem, whether it’s a Three-Act task or whether it’s just a word problem. There’s a lot of times where the entire room is covered in whiteboards, and there’s 10 or 12 different methods up there on how to do this problem. Usually, you’ll get one or two that maybe have the algorithm that you’re looking for. You might have one or two that have a picture. You might have one or two that are doing repeated addition instead of multiplication. You might have one or two that are totally off.

Sam Brotherton: So as far as my problems with student discussion goes, if I’m like, “All right, Kevin, explain what you did on that board.” Kevin starts to explain it, and then I’d try to have my students respond to him. “Do you agree with that? Do you disagree with that? Can we poke some holes in that? Can we ask him some questions?” After Kevin goes and Sally goes, then I’ve pretty much lost my kids at that point, if that makes sense.

Jon Orr: It totally makes sense. They’re all standing around, and one student relates their strategy. They get Nancy. Then you move on to somebody else. You’re all like, “Let’s all go look at so and so and so’s board.” You go over there and they’re like, “Okay. When is this over?” That’s definitely happened to me. I think I have a couple of suggestions here. But what have you tried so far to help mitigate that problem?

Sam Brotherton: Yeah, a couple of things I’ve tried. So, I mean the goal of it is to put the ball in the student’s hands, right? So I’m trying to select certain students in a certain order on who’s going to talk when and explain what. That’s my way of sequencing it. So, definitely at the very beginning of the years, day one, you have to start building your routines and procedures, I give those sentence starters at the very beginning of the year. I agree because, I disagree because. Can you explain? Or we use, can you speak with conviction I’m having trouble hearing you? I have a lot of six [inaudible 00:25:35] to talk. They’re like, “Quiet.”

Jon Orr: Yeah, I have a daughter like that.

Sam Brotherton: All right. So that helps. I have a pink pointer finger that sometimes I’ll just randomly hand to a student and say, “All right, you’re the discussion facilitator.” So instead of me being like, “All right, who’s going to go?” We do that. Or sometimes I just say, “Hey, you’re the teacher now.” Or sometimes I will just say like, I picked two methods and say, “All right. Let’s decide who’s got a better method going on here?” So little things. I don’t know that I have anything. I’m looking for more.

Kyle Pearce: I love the ideas that you shared. So, you have selected certain students and in a certain order. So obviously, putting some of the book to the five practices to work for you, which is great. I like this idea of you really trying to empower the students, giving students a bit of a voice in this process. I can’t say I’ve done this before, because I haven’t. I did not have a pink pointer finger. But with that in mind, what I’m picturing is like, possibly trying to do a little bit of that, and maybe giving that student or a group of students, a little bit of freedom to, at least, start the consolidation for you, or allow them to go around the room and select maybe two solutions or two groups that you’d like to share out some of their thinking.

Kyle Pearce: Then, for you to take over after that and have maybe your one, two, depending on the scenario, whatever really, really intentional selections ready to go, and then also being very clear with the groups, what you want them to share. This is something that was, for me, a big game changer. It saved a lot of time. It also allowed students to stay more interested in the discussion because it wasn’t a student starting from top to bottom. I think students think when we have them share their thinking, that they have to start from the very beginning, and go all the way to the bottom. I feel like I did expect that from my students for many years. After a while, you realized that, wait a second. What I really want to know from this group is this part right here. You take that pointer with the pink finger on the end, and you really specifically say, “This area right here has got me really curious. I want to know what you were thinking here.” Then helping them to frame the start and end point for them, might help trim up some of that, so that you’re not losing as many students.

Kyle Pearce: This isn’t a guarantee that you’re not going to lose some. But then when that consolidation is over, it’s like you’re doing, you’re thinking, you already know what that learning goal is that you had in mind. That’s when it’s like, “All right, here’s what we need to do in order to connect what we just did to the new learning.” You can be very explicit with students about that, that “Okay, so there’s so much great thinking here. I want to push our thinking just a little bit further. Here’s how we’re going to do it.” Students might come gather around you or however you choose to do that, and then that’s where you get that direct instruction time to try to tie those loose ends and then having in mind, what’s the launch point after you’re done doing that consolidation, what’s the next thing you want them to do. That might be that little push down the line to encourage them.

Kyle Pearce: So for example, if I want the consolidation to be focused on helping students to try something on a double number line, I might then give them a modification or an extension of that problem. Now, I might challenge them to all use that model. So they just had an opportunity to use any model they wanted, any representation, any strategy they wanted. Now, we’re trying to help them, nudge them in a direction to stretch their thinking a little bit because we don’t want students always doing the same thing. We want them to have multiple representations and different modeling and solution strategies that they can use.

Jon Orr: Kyle, just to jump in here, that’s also a good strategy when you’re monitoring and you’re walking from group to group, and you’re noticing, because sometimes this happens, and I would say this happens a good chunk of time, especially, when you’re giving students problems they don’t know how to solve. They’re trying things. If you’re noticing, they are going either down past, that they probably shouldn’t instead of having individual discussions with all of those groups. You can just pause the class, and you can come over to one board, and ask if that’s on the path that you might have that model that you’re saying like, “We could go this way.” So, you can pause the class there, bring everybody to that one more, ask them that very specific question of like, “Why did you choose this model?” Then you talk about it.

Jon Orr: Sometimes, what I do to keep the class going with the discussion is sometimes I’ll ask them that starting question, and then I’ll say thank you, and then I sometimes alleviate some of their pressure because students in my class, for sure, in high school are super nervous to explain that strategy to the class. They’re like, “Don’t pick me, don’t pick me.” They’re worried. I usually let them off a hook a little bit because I know that I’ll go back to them later, and will bring it back at a different time. But I’ll say, “Start here. Tell us how you got started on this.” Then, when they tell us that, I pick up the loose end and continue with that model a little bit myself.

Jon Orr: Then like you said, Kyle, we send them all back to the boards and go, “Now, you try it on this model, too.” So, it could be a great strategy to steer everybody to a particular learning goal you’re looking for. If you notice that they’re doing stuff that could cause them to have mass confusion later on, I definitely make that call on the fly. That’s definitely something that we need to prepare for. We got to prepare for that case coming up.

Kyle Pearce: One thing that I want to just clarify, too, I’m picturing, you might do that if there’s maybe a misunderstanding or there’s some sort of miscommunication. If students are going down a path, they’re following a strategy, and I’m watching it and maybe it’s not making any sense to me, or maybe lots of students are doing things, and I’m not quite certain, I really want to push, allowing them to do that work, engage in that productive struggle. So, if there’s mass confusion, meaning students are uncertain of what the question is asking them or something like that definitely, but at the same time, we want to make sure that they do take risks.

Kyle Pearce: So, if I do shut down, let’s say, certain strategies because I’m like, “They’re not heading anywhere near where I was hoping they would be heading,” I just have to be really cautious because we want to make sure that students do believe that, “Hey, my teacher is going to allow me to do a little bit of tinkering here, and I don’t have to worry about doing it ‘the right way or the efficient way’ yet.” We want to try to help them find more efficient ways or more efficient strategies, but we want to do that over time and definitely not too early in the process. So, just wanted to clarify that just for people listening.

Jon Orr: Thank you, Kyle, for clarifying that. It’s definitely not something that you want to step in because I think you can go down the path there yourself by saying, “They’re not going the way I want it,” we better pause and everyone, basically, you now teach them the way to do it, but that’s not the way [crosstalk 00:32:42].

Kyle Pearce: Yeah. They go, “I’ll just wait next [crosstalk 00:32:43] I’ll wait because Mr. Pearson is going to just cut me off anyway and tell me to do it this other way.” So, it’s hard. Teaching in general is really hard. I’d say math, it’s a complex beast. This is what it’s all about for me. This is what makes me love teaching math is that it is so “it depends.” So many things are variable depending on the situation, which is why we like math. There’s so many interesting things going on and things changing. So, these challenges are just… I just get so excited for them because I want to be there tomorrow when you try some of these ideas and tinker with things just to see what kids do.

Jon Orr: [crosstalk 00:33:25]

Kyle Pearce: Sometimes, it blows up in our face, and then we go back to the drawing board. Other times, it’s like, “Hey, look at that. It was right under my nose.” So, I think-

Sam Brotherton: Yeah, I think what you guys are… Sorry, go ahead.

Kyle Pearce: No, not at all. I was just going to say just anytime you can get three people together, it doesn’t matter who they are. If you can find two colleagues in your building or even one colleague in your building, just to informally throw these ideas around. It’s so interesting how many times I’ll be so challenged by something, and then I share it with one person. It’s like in two seconds, they’re like, “Did you try this?” I’m like, “Oh, no! I didn’t try that at all.” So, I better go do that.

Sam Brotherton: Yeah. I’ve been filmed before. We’d sit there and watch me teach in PLC. It was maybe my first year teaching. My colleagues are watching me. That had this cool lesson, I don’t even remember what it was. But they just asked me the question like, “Do the kids know what your expectations are? Did you ever set your expectations? Do you want them seated? Do you want?” So, I was like, “Oh, I never really did tell them that. It is confusing.” You don’t see that on your own.

Kyle Pearce: No, totally. So many things are so obvious to you, right?

Sam Brotherton: Yeah, right. Going back to what you guys were talking about earlier when you’re talking about stopping kids, when they’re getting to those points. One example that really resonated with me when you guys were talking about that was sixth grade, we do ratios. The way that I do it is I teach ratios, and then we move into unit rate. So, the first time we start doing unit rate, you’ll always get one student that starts to make a ratio table, where they’re putting in 20 spots on a table instead of realizing that you can divide. So, I’ll let those students make their 20 spots and somebody right next to them, divide it. I let them see, “Okay, you know what? There is a faster way to do this” instead of stopping them. Well, maybe you shouldn’t do that table, right? But if I see some kids that are maybe totally off, that’s when I jump in.

Kyle Pearce: Yeah. There’s always that fine line right. You want to give them freedom, but not to a point where a student, at the end of the school year, is still tinkering where they were in the first week of school. It makes it hard. Everything becomes a judgment call. I love Kathy Fasano always says, she challenges students to see if they can do it more clever, if they can find a more clever way to do it.

Kyle Pearce: So, if a student does a ratio table, it’s like let them do it. All right, awesome. I wonder if there’s a more clever way to get to this same solution, and you could walk away and come back a couple of minutes later, see if they fit. Then if they haven’t, then it’s challenging them to try something different. But I think the big part is allowing them to and pushing them, help them see through their original solution or strategy, and then trying to push them to try another. If we can do that, then that makes the consolidation process so rich because if you do have a student who’s behind, and I’ve never met a teacher who’s like, “No, no. All the kids in my class are at the grade six level.”

Kyle Pearce: So, you always have someone in the room that’s going to benefit from seeing that ratio table anyway. So, it’s a nice jumping point to be able to bring that up as a solution strategy for students to see that, “Hey, look, you know ratios and rates. They’re really connected.” It gives you another opportunity to even bring that back to the forefront, like what is the difference? When are we using ratio reasoning? When are we using rate reasoning? Many of these things I never even considered earlier in my career. I would just race through rates. If it was rates today, that’s how we’re solving it. That’s all there is to it.

Jon Orr: I used to be the teacher that if they didn’t solve it with rates on that day, I’d be like, “No, you should solve with rates today” even if they were solving with the ratios.

Kyle Pearce: Something I wanted to come back to this assessment piece because I know like assessment is a beast, especially since, not only like assessment in general, where it’s very, very challenging based on just what society believes school should be from an assessment and evaluation perspective, but then also just the varying approaches from different districts, state standards or provincial standards, depending where you are. Then also, within your own schools and departments. Sometimes, there’s different rules and things in place. So, that makes things very, very thorny, very hairy.

Kyle Pearce: But if I could zoom out on the school year, one of the big challenges that I hear in my own district and in other districts when we travel around and do workshops, is teachers who are stuck between this point of like, “Well, I’m doing these interesting things like a Three-Act Math task or a rich math task of some shape or form.” Then the standardized test at the end is all these word problems. It’s like they don’t seem to jive. In my mind, I now picture the intent and the intention of doing the Three-Act Math task, obviously, to spark curiosity for sure and to fuel sense making.

Kyle Pearce: But if I’m able to fuel sense making in my students enough, then students will be able to buy that standardized tests, be able to take those word problems and build that, call it, like the Three-Act Math task in their mind, if that makes sense. I don’t know if that’s helpful or not, but for me that’s how I see it now, is that, that standardized test is supposed to be at least near the end of that school year. So, near the end of all those standards that were laid out before me in that grade level. That would suggest that we’ve engaged in so much interesting math, and we’ve had so many opportunities to come back and dive deeper and deeper into these ideas that now students don’t need me to show them a video clip in order to understand what’s happening in the context, if that makes sense.

Sam Brotherton: Yeah, I think that does make sense. I guess, one question that I have is… So, you have your kids at a point where they don’t need a video. They don’t necessarily need a rich task, they need that practice. What are the types of things you guys do to keep them curious and hooked and excited while you’re going through your practice, those types of things?

Jon Orr: So, once we’ve got that learning open through the task, and we’ve consolidated, and we’ve decided on an efficient method or the learning goal, and now, we want to say practice, say, the ratios or the unit rates or the fractions. I used lots of different structures to practice to keep that engagement going. A lot of times, I was like, “Okay, let’s open the textbook and just do these questions and sit down.” I used to do it that way, but obviously, you’re suggesting is everything is just going to drop and kids aren’t going to get that practice in. So, what I do now is I set up a lot of structures. I’ve got five different structures that I use on a regular basis. The one that I go to a lot lately is it’s like an appointment clock structure. I don’t know if you’re familiar with this.

Sam Brotherton: You remind me I think I’ve seen something like it. You scheduled times with different students and rotate around.

Jon Orr: Yeah, that’s exactly right. It’s like everyone gets up, you have an appointment clock, which is really just a circle with some lines at different times on around the clock, like 9:00, 12:00, those blank lines for… Then, they go and write each other’s names. They make appointments with each other. Then, what we do after that is, let’s say, we’re factoring expressions. So then after that, everyone gets what I do anyway, is everyone will get a [inaudible 00:40:41] factor, using any of the techniques, and the models that we’ve learned. They’ll own that factoring question. So then, they’ll do that solution on a piece of paper.

Jon Orr: Usually, the questions on a separate slip of paper. Then so I’ll call out a 9:00 appointment, and then everyone goes and finds their 9:00 appointment in the room somewhere, or at the boards they meet. Then, they exchange factoring questions. The nice thing about that is, is that now they’re going to do a new question, so a brand new one. The solution holder of that, at the back of the book, is their peer, the person that they’re sitting across from or standing beside. They’ve got the full solution already to compare against. If one of them doesn’t get it, then they can help each other to get the right answer together. Then, we call out another one.

Jon Orr: So, calling out different appointments, they’re really doing this to practice, from what a textbook might look like, but they’re up, they’re moving, they’re self checking, they’re peer evaluating, they’re having conversations. So, there’s lots of good thinking going on in there and also, like I said, this good discussion.

Sam Brotherton: They’ve got that safety net, their partner, too. So, they know that they have the answer at the end of it. They’re not going to be left without knowing.

Jon Orr: Exactly. So, that’s one I use on a regular basis. That’s really just like, grab some pretty straightforward questions to practice. Another one that I use on a regular basis now is I used to call it two truths and a lie. It’s like that party game, where you’d say two truths about yourself and a lie, and then everyone has to guess which one is the truth and the lie. Well, what I do in my class now is I’ll put up an expression or a math problem or a table or an equation, and I’ll write three statements. I won’t tell them which one’s true or which one’s a lie, and it’s mixed.

Jon Orr: So, as a class, we’ll do one or two together, and often decide which one’s the lie. Lately, I don’t call it two truths and a lie. Why? I just call it truths and lies because they can all be true or they can all be a lie, or a mix and match. So I took out the two and the one condition. We’ll do a couple like that together. Then what I have them do is I give them again like the appointment clock, but I give them a math problem, or everyone gets a different one, and then they create their own two truths and a lie. They don’t tell anyone which are the truths or the lies. Then the great part is they post it around the room, they post their math problem with their statements, and now you’ve got this gallery walk, great math questions and great deep thinking math questions because now you have to verify all three statements for that one math question. They come up with some great stuff because I will say, “Make some easy and make some hard on your peers.”

Jon Orr: Again, they have the solutions on the back of the page. So like the appointment clock, they don’t come to you to check the answers. If one answer differs from the answer that a student’s getting, they go to the person who created it, and they have that conversation with them. So that’s another one. I got five, and I don’t think we can go into all of them.

Kyle Pearce: Yeah, I just jotted down a couple of notes to make sure we put the appointment clock and two truths and a lie, some links in the show notes. So, one of the takeaways that I got from listening to those two practice structures, I guess I’ll call them, is this idea of making practice something that isn’t so long, tedious, boring and maybe seemingly pointless to students. You can have… I heard you mentioned like, so it’s not all rich tasks. I would argue that even the practice there, by taking some of these practice problems that I used to do and I used to just print out the page, 24 of them and fire them off, I could take those same 24 problems, and I can make the process more rich.

Kyle Pearce: The task might not be super rich, but I hope that what I’m having them practice has some richness to it. I’d like to think that a rich task doesn’t necessarily mean a Three-Act Math task or anything like that. We even have a section in our online workshop about taking textbook problems and making them more curious and more rich, and by just making simple changes to how we deliver them or how we introduce them, or how we structure our practice, which Jon’s done a great job of highlighting. So, these are things to keep in mind.

Kyle Pearce: Then the last piece I want to throw in there is the idea that we know that research, cognitive science tells us that space practice is much better than mass practice, which is really just this idea that if we were to take the same amount of practice questions and do them one after the other, for the rest of the class, versus taking those same number and maybe spreading them out over a series of days. It might even be over a week or whatever it might be, you’re actually going to get more benefit in the long run from that and actually longer retention. So we’ll put a link to some spiraling math class resources in the show notes as well.

Kyle Pearce: But at this point, what we’re wondering, Sam, is maybe getting closer to wrapping up, we’re wondering, is there any takeaway that sticks out in your mind from this conversation, maybe something that you might put into action in your class or reflect more about as you head into 2019 and the rest of the school year?

Sam Brotherton: Yeah, definitely. My mind is racing right now. I think for me, my comfort zone… It goes back to the very beginning, I’ve always thought about math. So my comfort zone is the content. You can give me pretty much any topic, sixth, seventh or eighth grade, and I can be like, “Okay. We could make a problem out of it, doing this, this and this.” Then as far as classroom routines and making the practice engaging, that’s where I struggle. Sometimes I don’t step out of that comfort zone. So I think some of these strategies like appointment clock or two truths and a lie, those are all great things that I can start running with. So, it’s good stuff.

Jon Orr: Sam, we want to thank you for joining us here on the podcast. We don’t want to take up too much of your time, but we have one more question. We had a great conversation. We talked about so many great things here. We are just wondering, what’s one big takeaway from this conversation that you got out of our talk tonight?

Sam Brotherton: I think this talk has really helped me maybe take a step back and slow down and say, you know what? Not every single day has to be this crazy video or Three-Act task or whatever it is. How you can take those cool lessons and those great lessons and use them more tactfully start to take away some of the pieces of those lessons, so that students have a better understanding of building their procedural fluency. But procedural fluency doesn’t have to be boring book problems. It can still be engaging. The students can still take away as much, if not more, from those types of lessons than maybe a Three-Act task per se.

Jon Orr: Awesome. That sounds like some good insights there for you. Definitely, it reminds me of some of the things that I got to keep doing in my class, too, and keep thinking about. So, thanks for saying those things. If this is okay with you, we’d hoping to bring you back in six to nine months. We want to hear about the things you’re trying and the things you’ve changed, successes you’ve had. Would you be willing to come back on and chat with us again?

Sam Brotherton: Yeah, absolutely. That would be awesome.

Kyle Pearce: Beautiful. Well, Sam, we’ve learned a ton, as we always do in every conversation. It’s great. Our minds are racing. I’m glad to hear that yours is, too. Your brain’s on fire as much as ours is. We hope that people listening have gained as much from this conversation that we have. So, thank you so much for taking the time to join us tonight on the Making Math Moments That Matter Podcast. We hope to stay in touch with you before we hear from you six to nine months from now, definitely, through social media and email. But in the meantime, we hope you have an amazing remainder of the school year, and we’ll be in touch soon.

Sam Brotherton: Thank you guys so much. It’s been awesome. I appreciate all the help.

Jon Orr: Well, there you have it, Sam Brotherton from St. Louis. We know you’re listening to this Sam, and we want to say one more time, thanks. We appreciate you, and we look forward to having you back on the show in the future to see how you’re progressing on your journey.

Kyle Pearce: This was another Math Mentoring Moment episode, with many more to come where we will have a conversation with a member of the Making Math Moments That Matter community like you, who’s working through a challenge. Together, we will brainstorm ideas and next steps to help overcome it. If you want to join us on the podcast for an upcoming Math Mentoring Moment episode, where you can share a big math class struggle, apply over at makemathmoments.com/mentor. Again, that’s makemathmoments.com/mentor.

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, or simply searching, or use these quick links. For iTunes, go to makemathmoments.com/iTunes.

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Kyle Pearce: Quick Links will work for most other popular podcasting platforms as well. 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. It really does help the show make it to new teachers’ ears and new listeners. So, take a moment right now and hit the Write a Review button and leave us that five-star review. Every single one makes our day.

Jon Orr: Show notes and links to resources from this episode can be found at makemathmoments.com/Episode 13. Again that’s makemathmoments.com/Episode13.

Kyle Pearce: You can also find Make Math Moments on all social media platforms and seek out our free private Facebook group, Math Moment Makers K-12. Well, until next time, I’m Kyle Pearce.

Jon Orr: And I’m Jon Orr.

Kyle Pearce: High fives for us-

Jon Orr: … and high fives for you.