Maia Kingery Gallagher: I think I am personally a rather visual person when it comes to math and I gravitate towards certain representations from students that I'm more familiar with. So there is probably a piece of planning that I need to kind of focus on every time we jump into a new topic or anticipating student responses to a task that if I had more variety that I was looking for and familiar with, I might be able to then focus-

Jon Orr: Today we speak with Maia from the Boston area. Maia is a middle school math teacher and is working hard to make all students feel a sense of belonging in a classroom, just like they felt when they were in grade two. You'll hear about her own math moment that she remembers from math class and how it inspires her to build her own lessons. She'll share the work she's been doing, implementing structures from the routines for reasoning book and how she wants to better help her students gain confidence to share their rough drafts thinking with their peers.

Kyle Pearce: This is another math mentoring moment episode where we talk with a member of the math moment maker community who is working through common problems of practice. We are together. We brainstorm possible next steps and strategies to overcome them. Let's hit it.

Jon Orr: All right.

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

Jon Orr: I'm John 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. Welcome everyone to another math mentoring moment episode.

Jon Orr: Let's get ready for another jam packed episode. But first we wanted to give you a quick heads up that in this episode, we talk about gaps in student learning and the concept that holds students back.

Kyle Pearce: Yes, after talking with so many teachers in these math mentoring moment episodes, just like the one you're about to listen to and from interactions with our own students, we have decided to build a course to help teachers from kindergarten through grade 10, in order to strengthen their understanding at addressing common gaps in student learning, as well as build their teacher confidence in the math classroom.

Jon Orr: The course we're talking about is called the concept holding your students back, unlocking key understandings and proportional relationships to reach all students. In this hands on comprehensive course will not only unfold the fundamental concepts for teaching proportional reasoning so you could close gaps with your students, will also show and give you the lessons and resources to use in your classroom to make it all happen.

Kyle Pearce: Yes, that's right John. Registration for this online and fully self paced course is open only until September 25th, 2020, and we will not be offering it again until next September. So learn more and register at make mathmoments.com/proportions.

Jon Orr: All right, let's jump into our conversation with Maia.

Kyle Pearce: Hey there, Maia, thanks for joining us here on another math mentoring moment episode of the making Math Moments That Matter Podcast. We're super excited to have you on the show today, but first, before we begin, how are you doing and where are you coming from?

Maia Kingery Gallagher: So I am coming from Boston, Massachusetts, where I've been staying in my apartment for the past few months, sort of teaching remotely. Then we had some summer break and now I'm getting ready for the new school year to start soon.

Jon Orr: Awesome stuff. Maia, why don't you tell us a little bit about yourself, your teaching journey. What are you teaching in the classroom? What grade level? All that kind of stuff. Fill us in a little bit of a backstory on teaching for you.

Maia Kingery Gallagher: So I've had a lot of variety in my experiences, but most recently I've been teaching sixth grade at a school in the suburbs of Boston called Clark Middle School.

Kyle Pearce: Awesome. Fantastic. As you ended that little phrase, I was jotting down six, seven, eight, nine, 10, 12. I'm like, wow, she's got a good range.

Jon Orr: She's got range.

Kyle Pearce: It's almost like we want to do composite or opposite here of what you haven't taught. It looks like you've got-

Maia Kingery Gallagher: Just 11th grade.

Kyle Pearce: You've got a good range here which is great. I need to know a little more though. You said eight or five years. I want to know a little bit more about that before we dive in.

Jon Orr: That's got a story behind it for sure.

Kyle Pearce: Yeah, tell us more.

Maia Kingery Gallagher: Sure. I did student teaching in undergrad and worked at a Boston public school. I realized that the end of student teaching, I had my teaching certification, but I really didn't feel like I was ready to take on a whole classroom. So I did a program that was an extended learning day. So we taught in the afternoon and I taught math. I had my math teaching certification, but I taught a small group of students for two years and then I got my master's degree working with the Boston Teacher Residency. So I was very lucky to get to study on under Amy Lucenta and Grace Kelemanik authors of Routines for Reasoning.

Kyle Pearce: Oh, nice.

Maia Kingery Gallagher: So they were in my classroom during that year and really helped shape the math teacher I've become.

Kyle Pearce: Fantastic, and I love to hearing that you had a math teaching certification. So there must've been some sort of interest there to kind of get into the math side of things. Before we dive in any deeper one of our favorite questions we ask guests before we dive into a problem of practice where we can all kind of riff on is your math moment. That math moment that pops into your mind when you hear the words, math class. You think back to your experience in the classroom as a student, what moment pops into your mind?

Maia Kingery Gallagher: Well, I'm a huge fan of the podcast and I listen to this on my way to work. Actually, every time I'm trying to think of a math moment that sticks out to me and I connect with many of the past guests, their experiences with those timed multiplication tests for sure. But I have one that really sticks out in my mind. I remember this one day in high school we were sitting around in math class before it started, because we were nerdy like that and talking about some rumor. Earlier in the day, there had been, I guess somebody had thrown a pen in one of the classes and so we were all gossiping about it.
As we were gossiping, the story got bigger and bigger. The pen like flew close to the principal's head. Our math teacher overheard us talking about this right before class started, and he leaned over and said, "Actually the pen never hit the wall." We all like our mouths dropped because we're like, "Oh no, he heard us gossiping and talking about this." But then we actually went on this kind of tangential conversation about Zeno's paradox and density of rational numbers, and before the bell rang, he had us back graphing inequalities on the number line. It was just sort of this perfect turn from what was really interesting to us, to what was on his math agenda.

Jon Orr: Having that discussion is always like mind-blowing to students. When that idea comes up about, there's an infinite amount of numbers between one and two and thinking about like, it's never actually going to get there and demonstrating that to your students and talking about it. It's definitely awesome, I'm really glad you shared that math moment. I think that's also a math moment of mine. When you hear about that idea, it's just one of the best parts of math class.

Kyle Pearce: John, I'm listening too and I can't help but think, talk about what an amazing way to spark curiosity. I know John, you and I sort of fabricate when we're creating tasks, we use that three part math framework for the make math moments framework. We talk about sparking curiosity and we tend to do it through like a fabrication of giving them video or giving them an image or giving them a provocation, which is helpful ahead of time because you don't know what's going to happen. But for your teacher Maia, to be able to take that scenario and on the fly, be able to kind of craft that experience, that to me is expertise at its finest. So how awesome is that?

Maia Kingery Gallagher: Absolutely. It's definitely something that's kept wanting to be a teacher for so long. Is this idea that I could reach a level of mastery in my practice where I could overhear this thread of conversation in students, and be able to connect it to whatever the day's lesson or target is. I think that that's truly a mark of a teacher with expertise and mastery.

Jon Orr: Yeah. I was just actually going to ask you that Maia is how you think that teacher or that moment has influenced your teaching career or how you view lessons in your classroom. But I think you've answered that, and I'll give you a moment maybe if you want to elaborate on more of that. Many teachers bring up this idea of how do you connect with your students and how do you like on the fly adjust in the moment. I think one answer to teachers who have questions like that is, that it takes practice and experience and comfort in the classroom, comfort with your students. The students feeling comfortable too, that you can kind of pivot and move in directions when you need to, so that they can engage.
Also mastery of material. Knowing the math concepts and knowing how math is super important to being able to feel comfortable in front of students when you're talking about math concepts. So yeah, would you mind elaborating any more on how say that moment or that teacher has influenced your career?

Maia Kingery Gallagher: Sure. Well, I think that when you speak about mastery of your content as being essential in this process and understanding sort of the trajectory of where this math that I'm teaching in this moment is going, I think has been central to my experiences because I have taught so many different grade levels. It's been very challenging during the year to have three different classes or three different grade levels of lessons to be teaching at one time.
But it's also been such amazing experience to see, okay, in sixth grade we're doing ratios, but in seventh grade, that's going to come across when we talk about proportional relationships, and then in eighth grade, we're going to start introducing linear relationships more. So being able to see where their math is going, I think really helps in determining what's the most essential thing for students to take away from this conversation today.

Kyle Pearce: For sure.

Jon Orr: I love that you've articulated this understanding of sort of how the math develops, and I think that is so important. Content knowledge is something that while a lot of people will come and listen to the podcast and they may be in various stages of their own learning and their own development of their own expertise, we often don't push the content knowledge piece right out of the gate because it's kind of scary. But clearly you have that on your radar and you're thinking about, okay, how am I going to kind of build on what students know, where they are and where are they going. This is something really central to what we talk about in make math moments. So that's fantastic.

Maia Kingery Gallagher: One of pieces that I'm actually realizing is a gap of mine is actually knowing where students are coming from. So with the schooling last year being somewhat interrupted and teaching now sixth grade, I'm really at the border between middle school and elementary and there is a bit of a rift there. A lot of times, and I don't really know the progression of ideas throughout elementary school. So that's something that I'm hoping to learn more about in the coming years to sort of improve my practice.

Kyle Pearce: You know what? That does tend to be, or at least for us, I can only speak for John and I and the folks that I've collaborated with a little more closely. We tend to, as educators start with this idea of engagement. You had a great example with your own math moment where it was almost like you just were almost sucked into the math. That is really important, we've got to get their attention. But then once you start to become comfortable in that area and it's never a finished process, then you start to realize that, okay, engagement's great, but it doesn't guarantee everyone's going to be successful in math class. I have to start thinking about that development and you've addressed something that both John and I experienced, we came from the secondary classroom and didn't have the opportunity to teach in sixth, seventh, eighth grade like you have.
The lowest level we taught at in the classroom was grade nine. So when I came into this role, I realized, well, much like you're realizing that, oh my gosh, I do not understand when that student comes into my grade nine class. I, in my mind expected students to be right where I need them to be. That's sort of like this assumption we make. Well, in grade eight, they learn this, that, and the next thing and when they come to me, they're going to be ready to gobble up everything I serve up.
The reality was is it's just not like that. We know that, when we sit there and we reflect on our own classroom and the makeup of our classroom, you realize there's a group of students, and that could be a small group in some classes, but many times it's much larger group where students haven't built that fluency, that flexibility with the concepts, with even just the math facts and how to work with operators, like addition, subtraction, multiplication, division. That forced me to kind of take a deep dive myself back into, how does this stuff all work? Then I started to realize how much I really didn't know.

Maia Kingery Gallagher: Absolutely.

Kyle Pearce: We were going to ask you about a success you've had, but I'm wondering maybe we should riff into this a little more. I'm wondering, do you have any specific struggle or challenge that sort of pops into your mind? Maybe it's a memory in your own class where I guess lack of expertise sounds bad. But what I mean is essentially just not having the experience, having taught in a grade five, four, three, two, one classroom. Did you want to riff on that a little more?

Maia Kingery Gallagher: Sure. I'm trying to think. I had prepared a little bit for some of the other questions, but I can try to think for a minute. So I think that one of the big questions for my teaching is around equity and how do I ensure that every single student in my class, regardless of their race or their gender identity, that they're engaging with math and learning and growing. I think that one of the challenges that I've had in my classroom around that is really more of a sense around that engagement and hook piece that students who feel invited into the conversation are more willing to express confusion about a math topic.
Because I have like a lack of understanding, I was watching a video the other day of some, I think it was a second grade classroom where students were learning to count and they were counting by tens up to 120. So they counted 80, 90, 100, 101, 102. It was like this moment where I was like, I've seen that in the classroom, and I didn't know in the moment how to address it, and yet this second grader sitting next to the kid in the video, very fluently said, "No, we're counting by tens 10, 20, 30, 40." And like show the student how to do it.
Eventually they started mimicking and then they started continuing on the counting with themselves to like 150, 160, 170. That's sort of like coach and fade model, I think was so cool to see in a second grader, they're just able to jump right in and do that. So I think that my experience in the classroom of witnessing students who are struggling with content that's below grade level or things that I'm not as familiar with, the real sort of gem in the rough year is how can I make sure that peer interactions are safe and supportive in a way that they could be in like a second grade classroom.
That's certainly very challenging, in middle school there's a lot of social emotional development, fear, a lot of things that go into that as well as identity issues of, I don't want to be the only kid in the class who looks like me, who's having this problem. So I'm not going to raise my hand and ask about it. I'm certainly not going to turn to my partner who's the most popular girl in the school and ask her to explain to me how to count past 100. So I think that's sort of in my mind where the answer to that dearth of my own knowledge is how can I ensure that kids are able to support and help each other, that collaboration piece.

Jon Orr: I think this is a great insight and a great challenge to work with. I think you've phrased this very, very nicely. I guess I never thought about it in a sense that we often talk about trying to make our classrooms a safe environment for everyone to express ideas and not feel criticized or comfortable enough that you don't have to worry about what's going to happen after you leave the room or what kids are going to say about you. Between middle school and high school, this is often our stumbling block on engagement and collaboration. I don't want to sit with that person because they did this at recess.
I really liked how you compared it to the second grade class because we don't see it happening there. It's this middle school time that changes are happening and social circles are becoming more of an influence on our students than they are as say at those younger grades. But yeah, I would love to be able to figure out how to make my class also the second grade class. Comfortable enough to kind of share that, and that can be a great goal for us.

Kyle Pearce: I think it brings up another idea about you learning a technique that helps in a second grade class and how you can apply it to your classrooms. I think that's an important aspect too and it's tied together with this landscape of learning that we know about the before, the prior knowledge that kids need to know before coming in. All these ideas are just tying together and I think in a perfect world it would be great if each of us had to teach a year at every level in that subject that you're looking at.
I could imagine how much I would learn if I taught grade one grade, two grade, grade three, all the way up to high school and use that understanding and knowledge and tools that you learn along the way at all these different spots. Because I think watching someone in a grade seven class and how they handle their classroom and the tools they use is much different than a grade 12 classroom but you can learn a lot from each other. That's something else it's like, maybe in a perfect world we can't teach all of that because it's going to take us 12 years of teaching just to do that.
But watching lessons from grades one to grade 12 would be super important and super useful for everybody. But if we want to go back to your big struggle and challenge that you were bringing up, I'm wondering before, if I'm just going to reiterate like this challenge of making your classroom safe enough for everybody.

Maia Kingery Gallagher: Absolutely. I think that it was interesting because I wrote about that to you guys in what? December and you sent it to me recently and I realized that despite distance learning and despite the challenges, it's still very much the central focus of my practice, and my challenge is, is around this piece of equity and collaboration. How can we make sure that all the students in my class, whether they are from non-dominant religions or they're English language learners, or they have a learning disability. All of the things that they bring into my class, how can we make sure that those are not getting in the way of them learning through tasks?
And then additionally, this piece of collaboration I think, is essential because like I was saying, my students have been in a fifth grade math classroom far more recently than I have. So it's an untapped resource in many ways, and with distance learning it does become more challenging that collaboration piece simply because there's a lot of teacher to student interaction, but not as much student to student interaction.

Jon Orr: This is something that we often ask teachers when they present a struggle or something they're working on or problem of practice. Maia, what would you say you have done so far just to help address this issues? What does a classroom look like to help some of these issues out? Because where you are in this journey of this problem of practice is different than someone who's listening right now. So we want to paint a picture of where you are on this journey so that people listening go, "Okay, this is where they are. This is where I am." It's a good picture to have moving forward to help kind of provide you with some next steps.

Maia Kingery Gallagher: Right. So in my classroom, when I was in the classroom, students were always working in small groups from the moment they walked in the door. I typically have some sort of math problem posted when they come in and I actually had one student complain. They said, "We spent the whole class on the warmup." Because we tend to have a conversation afterwards. I've taken a lot of the Routines for Reasoning and sort of made them a little bit less structured and more just responsive so that I don't have to necessarily follow the structure that Grace and Amy made, but I sort of modify it based on where the conversation is going with students.
If two ideas come out as really strong, we might jump into the design and defend. If students are struggling more with comparing different strategies, we might do connecting representations. So I plan ahead, I have one in mind myself, but I'm able to change it if needed based on what the students.

Kyle Pearce: How do you find that working for you? Are you feeling in terms of the level of success? Is there some positives that you're seeing? When you think back and say, "Okay, before I started doing this." Or these routines that you've sort of modified from Routines for Reasoning, we'll include the link in the show notes for those who are listening. But what's working and then maybe where are you still feeling like this challenge's poking its head? So we can try to really hone in here.

Maia Kingery Gallagher: I don't know how much of this is just the culture of the school that I'm at, but getting students to a point where they can share rough draft thinking with each other and understand each other's rough draft thinking has been very challenging. I think that a lot of times students want to come forward with their fully finished product to present to other students, to think about, and instead getting them to the point where they are comfortable sharing, sort of the rough draft thinking is sort of a year long process I'm noticing.
It generally starts in September with me sharing my own errors and rough draft thinking sort of intentionally planned mistakes that we analyze. But then also just sort of having a pervasive, constant voice of rough draft thinking, rough draft thinking in the classroom. Narrating when it happens, celebrating when it happens and then looking for it constantly. Always looking for that rough draft thought that could be shared with the whole class.

Jon Orr: The rough draft thinking, I've never used that phrase in the classroom, but I really like it. In English class or in language class often you spend a lot of time on your rough draft before you move into your final draft, and then often math class we don't do that. We're always looking for a final draft and we're always talking about that, but rough draft makes it very clear that you're going to value that. I guess one question I have Maia is that, sounds like you're encouraging rough draft thinking in your classroom.
I'm wondering what let's say, assessment then looks like, because sometimes kids get into this habit of like, yay, we're talking about rough draft, but then I'm only focusing on final draft because that's where I gain my marks, that's where it counts. So yeah, she's talking about rough draft, but I really need to only focus over here. So I'm wondering, what does a snapshot of assessment in your class look like? Because I think sometimes kids pick up on where the value lies.

Maia Kingery Gallagher: Absolutely. It is very challenging because there are district level initiatives and requirements for assessment. So at the end of units, there is a test that students have to submit their first try score on for the district to place students in honors. So that is a conversation that we have to have that every other assessment that we take is something that students have to revise and they have the opportunity to retake it as many times as they would like.
I have actually the entire class retake assessments, even if you got 100 and I sort of modify all the questions and we talk about the difference between performance and retention. That's sort of my justification there for why, "I got 100 Ms. Gallagher, why do I have to take it again?" It's like, "Well, we want to make sure that you've retained it." And also I want to create a classroom where the norm is, we're all working on this, it isn't extra work for somebody who is struggling the first time.

Kyle Pearce: To me that sounds amazing. It sounds to me just from that experience that you're really trying to, and John used the word, what do you value? We always say that, when we say certain things, but then our actions sort of show students something else, and a lot of times that is through assessment and evaluation procedures. We say, the example I use a lot is growth mindset. A lot of people have growth mindset messaging on the wall, but then when it comes to it, the assessment process sort of says, "Well, sorry, it's one and done." So here, I'm hearing that in your voice that you're actually kind of walking the walk here, which is fantastic. I want to roll back to this idea.

Maia Kingery Gallagher: It does mean I have doubled the grading, and I have to design two tests every time I have a test. So it isn't a piece of practice that goes without effort.

Kyle Pearce: For sure.

Jon Orr: For sure. I think any, would say effective or really well thought through teaching oftentimes does bring some of those things in. While over time, we might be able to try to find ways where we can save a little bit of that time.

Kyle Pearce: Try to streamline it.

Jon Orr: Yeah, and streamline it. It sounds like you've got a lot of really good things going on. So I wanted to come back to this idea of the rough draft thinking, as John mentioned, this idea that in math class, it feels like, and students have sort of been trained that math class, there isn't rough, it's sort of like lining everything up and the equal sign always has to be lined up along the margin. We sort of get kids thinking that whatever we write down on paper has to be already thought through. We already have to know the direction we're going in.
When I think back to this, it sounds like there's a little bit of confidence issues where students being comfortable to share with their peers. This probably has nothing to do with your class in particular, just that they've been going on through years. So at least five years plus kindergarten of how math class has been shared with them. So that takes a lot for us to bring them into our classroom and to try to shift their mindset around what math is. So I want to dig a little deeper here, and you went and talked about some routines for reasoning that you've modified and you mentioned something about connections and representation.
So my wonder now, as we try to hone in on this, getting kids confident to share their thoughts and share their process and really kind of be open with their problem solving process with their peers. So that they can feel like it's at grade two classroom, that everyone belongs. My wonder is how do you feel your incorporation of tools and representations to show our model mathematics? How do you feel there? Because I know you had already mentioned earlier about this having not taught prior to grade six, that, that was sort of something that you wanted to focus in on. I'm wondering about the tools and representations, like math models. What does that look like, sound like? What's your comfort level feel like in that area in your classroom?

Maia Kingery Gallagher: I think I am personally a rather visual person when it comes to math and I gravitate towards certain representations from students that I'm more familiar with. So there is probably a piece of planning that I need to kind of focus on every time we like jump into a new topic or anticipating student responses to a task that if I had more variety that I was looking for and familiar with.
I might be able to then focus in and validate different students who are coming up with these more unique ones that I might just walk by in the classroom and not register like, "Oh, they're on the right track." Or, "Oh, they've got something that would be a useful misconception for everyone to think about." Or, "Oh, this is a unique representation that I don't see in other students that I would like everyone to start using." So I do think that there's a piece of anticipating student representations and models that I could improve on for sure.

Kyle Pearce: How often are you thinking about that now? Or is that something you're like, "Oh, I got to do that." You were saying that you can improve on that, but I'm wondering beforehand were you considering that? Or is that something that's like, "Oh, I got to do it now."

Maia Kingery Gallagher: I think it's more like an afterthought of like quickly, oh, I picked out the task, I've sequenced the objectives for this unit. I've written the two unit tests that I need and then I get around to anticipating student responses. So maybe flip-flopping that order there. I wonder just sort of prioritizing for a unit or two that anticipating of student responses.

Kyle Pearce: We were chatting before we hit record today. I said it and I meant it, that when we have these episodes, the reason why we love them so much is because we feel like our brain's burning as we're thinking through. In a way it's kind of like planning a math lesson because you're describing your classroom and right now what's going on in my mind is I'm trying to anticipate some of what's happening and the in-betweens and all of those things and I think it's amazing.
I think back to you had already referenced that you want students to feel like it's a grade two classroom. I also want to say too, that I feel like you're probably really well on your way with that because you sound so reflective and intentional about what you have been doing that of course we can all get better. We can always continue to get better, but I want to make sure that we do make you realize that from what we're hearing, that I'm feeling like students are already feeling pretty comfortable in your classroom.
When we go and we think back to not only how do we help them feel comfortable as if they're in a grade two class, but this might be helpful for our anticipatory activities here. We talk about the five practices for orchestrating productive mathematics discussions, and that resource has been so instrumental in what John and I do in the three part framework. But I'm almost thinking as well, beyond getting them to feel comfortable like they're in a grade two class. I'm wondering if you try to help yourself think like a grade two student as you work through some of those solutions strategies. Kind of thinking like, I always like to pick a student or two that I've taught.
I know right now we're recording this in summer. So maybe it's a student from last year where you're thinking, you're trying to think like that student and you go, "Okay, I know this student has these tools in their back pocket." An example I'll use just to keep it very broad is I know that they're able to skip count by twos, very easily fives, very easily, tens, very easily. Maybe three's not so bad. Maybe some of the others not so great, but what might help that student? What sort of math model might help that student if the problem is more proportional in nature?
You had mentioned this idea of ratios and then going into proportional relationships. I'm picturing for that student is a number line a helpful tool for them? Is the double number line. Are they ready for that tool? Or do we have to go back even further? And maybe it's just grouping like sets. Or maybe they're ready for a ratio table where they can actually do some skip counting in the ratio table and we try to help them take jumps where they start to do things like doubling and so forth.
So these are all just random ideas that are popping into my mind, but it's almost like you've got this idea about helping students feel comfortable in your class like during grade two. Maybe we have to start thinking about how do we think like we're in grade two? So that, that one student in your class or that two, or maybe there's five students in your class that are not quite ready for the concepts that we're working with here, how do we present the task or the problem in a way where they can access it and feel comfortable accessing it from where they are through the appropriate tool or representation.
And then on top of that, I'm thinking about helping the other students in your classroom better understand that developmental progression as well. So that they almost inside and out understand what they're doing because there's oftentimes where you'll see two students who are at varying levels and we tend to not want to put them together because we think, uh, they're too far apart.
But imagine if that student who's using a proportion to solve the problem actually understands that that's just two lines in a ratio table and that student's able to maybe help meet that student where they are, the other student who's performing a couple grade levels below so that they can kind of work as a team, and in the process they're both getting a deeper conceptual understanding. By one being able to sort of practice that thinking and for the other to be able to see that, "Oh my gosh, my thinking isn't all that different from this student who I thought was at a completely different place than I was."
So I'm going to stop talking here because I just went on kind of like a little tangent and I want to pause and sort of put it back to you, Maia and get your thoughts based on what we shared. If there's anything that pops out that seems relevant, or maybe it's completely off base.

Maia Kingery Gallagher: Actually, as you were speaking, you reminded me of a success from this year and I'm wondering if it was successful because of exactly what you're saying. I shared actually with students the Graham Fletcher video that shows the progression of division through elementary school because we were trying to introduce the traditional algorithm. My students were really engaged with this high speed video that was showing division and with lots of different models and very different progressing throughout the grade levels.
Every single one of them said, "That's not the order I learned it." Or, "That's not when I learned how to do that." Or, "That's not the way I did it." So we had this great moment where they were able to themselves see the progression of these ideas, compare their own thinking to this progression. Then I think it did influence their group work after that, where they were able to connect back to, "Oh, that's like the model when we saw with the box." "Oh, that's like when you have a whole bunch of partial potions in a list." It definitely did facilitate group work for them to have sort of this broader view of the progression of ideas themselves.

Jon Orr: I think that strategy and I think when you were wondering whether that was a success or not, and as a result and I think totally it is in the sense that like what Kyle said, I think it's really important for students to see that map. We've used this kind of analogy before in a game that we play in our live sessions, the game of Nim. The game of Nim is this game where you start with 21 stones in a pile or sticks or things in a pile, and each of you take out between one of three at a time on your turn and whoever takes the last one wins.
We play this game in our live sessions because if one person knows there's a winning strategy here, it's actually one of the basic games of game theory. If somebody knows the winning strategy, they can't lose. If the other person doesn't know the winning strategy, they're going to lose, and there's nothing they can actually do about it. There's no move they can actually make to try to win if the other person knows the winning strategy. So we play this because it's like, we often sometimes think about moves that we make in teachers like progressions or the way a unit looks or the way the course is going to unfold for our students.
We teachers, if we've done all of our homework and we've done all of our planning, where we've done all of our anticipation, we know how this is going to unfold from start to end, especially if you've taught the course many times. But we rarely communicate that to our students upfront or in the moment, or where does this lesson fit in, in the grand scheme of the unit or the grand scheme of the course?
I can imagine students sitting there in class going, "I'm wondering how this fits in with what we're doing." They might appear a loss. It's like, this is where kids get the question of like, "Why are we doing this?" Like, "Why do I have to learn this?" Because they're just seeing this large picture of where these important lessons fit in with the grand scheme of things, and like showing progressions, I think is so important. So it's no doubt in my mind that that helped your class do that.
I often do this with senior level students, like go back. When we're talking about, say, here's an example where in grade 12, we're talking about how to factor polynomials of degree three or higher. We go back to learning how division was taught in grades three and four and five and show that progression of we've actually been dividing things for a long time and showing what that looks like, just like a Graham Fletcher kind of video. When kids can see that it falls together nicely.
So I think stressing the importance of where lessons lie in progressions, like that landscape of learning that Kathy Fasno would say is so important, not just for the teacher, but also for the kids. So I think there's no doubt in my mind there that you nailed that with that success.

Maia Kingery Gallagher: Yeah. So now I have to figure out how to replicate that, not just with long division and our numbers unit early in the year, but sort of maintain that throughout. Because I think it's interesting that I reached out to you in December time, there's sort of a trajectory within the year. You have, in September there's sort of the setting everything up, and then there's a little bit of the honeymoon with students where they're trying to please, and then we get a little bit into that chaos range where they're starting to test boundaries. We've been together for a while. We're waiting for breaks to come.

Kyle Pearce: What can I get away with, yeah?

Maia Kingery Gallagher: Right. So how to really maintain it throughout that time of the year, it's going to be an important thing to learn as I continue doing this hopefully for another 30 years or so.

Kyle Pearce: Absolutely. You bring up, I know John's referenced it and you've referenced it. There are so many great resources out there to help. I know Graham actually has a couple different progression videos. He has multiplication, he has a fractions video as well. I'll also include in the show notes, I have a progression of division blog posts that this conversation sort of reminded me about. I'm going to have to go back and modify it a little bit before this goes live because as we mentioned, expertise is this thing. It's you're never a true master.
So when I did that progression of division posts, I think two, three, maybe four years ago, I've learned so much about it over time that I'm going to have to go and actually update it and sort of change my thinking a little bit that I originally shared. When you're thinking about what you might do to continue building that content knowledge, John's mentioned Kathy Fasno as a great reference. She was on our podcast earlier. I have to double check the episode number, we'll put that in the show notes. But one of the other tool-

Maia Kingery Gallagher: You had a great show, I listened to it.

Kyle Pearce: Oh, she's so great. She's just so fun to listen to and just so knowledgeable. So her work has really influenced my own personal understanding of mathematics. And then also John Van de Walle, his elementary and middle school mathematics book, Teaching Developmentally. I will put that in the show notes. That book in particular had a great, great influence on my own understanding, but then also our course, the concept holding your students back. We have a nine module course that we offer. It'll actually be coming out in September. We're going to be opening it up and getting folks into that course.
It is from start to finish a progression along the proportional reasoning or proportional relationship roadmap that we've sort of pieced together from all of our thinking around John Van de Walle's work, some Alex Lawson's work and Kathy Fasno. So definitely something to consider. There's tons of things you can piece together from the internet. Like we said, from Graham picking up some of these books is great. And then, like I said, this course, I tried to take these big ideas and really we tried to plan it so that we can chop it up. Use it with the three part framework and really try to make it as useful as possible.
So hopefully there's some takeaways there that give you some next steps in those who are listening, we are actually going to give you access to that, the concept holding your students back course. I just sort of said that on the fly. So John, that's what we're going to do.

Jon Orr: Yes we are.

Maia Kingery Gallagher: Awesome.

Kyle Pearce: We didn't plan this based on the conversation. So we'll send that along after the episode. For those who are interested listening, they can go to make mathmoments.com/proportions, and that will bring you to more details about that course. But we want to turn it back to you, for those who are listening and in particular for you, we're wondering what might be a big takeaway for you and how are you feeling after having this call?

Maia Kingery Gallagher: Well, I think one of the things that has prevented me from really diving into the anticipating of student ideas and this idea of knowing sort of the progression in elementary school, is that it seems like extra work. What I'm hearing from both of you is that it doesn't have to be extra outside work. It can be work that I do that I directly share with students, which I think can make it feel so much more relevant and purposeful that it's not just anticipating move, it's also a planning and a sharing with students this progression. I think they would really benefit from thinking about how the math ideas develop.

Jon Orr: Awesome. Thanks for that, big takeaway and I think you've got some next steps for yourself along the way coming into your new school year.

Kyle Pearce: I know that the new school year is going to look very different for everybody, but we're really looking forward to hearing back from you. We would love to follow up with you in let's say like six months, like after you've kind of got your feet wet, you tried some of these ideas out, maybe in a blended learning model or a digital. That's just making me curious. What is your classroom going to look like come the fall?

Maia Kingery Gallagher: At the moment we're doing a blended hybrid model. Some students at home, some in the classroom, but it's likely going to change to fully remote at some point.

Jon Orr: Gotcha. We would love to kind of follow up with you in about six months. So I'll be back on kind of talk about where you are, what that look like. So would that be okay with you if we kind of invited you back on?

Maia Kingery Gallagher: Yeah, that would be amazing. Thank you.

Kyle Pearce: All right, Maia. Well, it sounds like we've had a really productive conversation. I have really thoroughly enjoyed myself. As I mentioned earlier, my brain's burning with all kinds of ideas to take with me and to reflect on. So I think that's it for us. I'm Kyle Pierce.

Jon Orr: And I'm John Orr.

Maia Kingery Gallagher: And I'm Maia Kingery Gallagher.

Kyle Pearce: High fives for us.

Maia Kingery Gallagher: High fives for you.

Kyle Pearce: As always both John and I learn so much from these math mentoring moment episodes, but in order to ensure we hang onto this new learning so it doesn't wash away like footprints in the sand. We must reflect on what we've learned. An excellent way to ensure that this learning sticks is to reflect and create yourself a plan to take action on something that you picked up here today.

Jon Orr: A great way to hold yourself accountable is always to write it down or share with a colleague or a friend from the math moment maker community. You can head to our free private Facebook group, Math Moment makers K-12. Tag us at make math moments on all social media and we'll get back to you.

Kyle Pearce: Yes, that's right. But before you head out thanks again for listening all the way through this episode, and just like you heard, we were chatting with Maia about enrolling in our course, the concept holding your students back, unlocking key understandings and proportional relationships to reach all students.

Jon Orr: In his hands-on comprehensive course we'll not only unfold the fundamental concepts for teaching proportional reasoning so you can close those gaps with students, we'll also show you and give you all the lessons and resources you need to make it all happen in your classroom.

Kyle Pearce: Yes, that's right. Remember registration for this fully self paced, all online course is open only until Friday, September 25th, 2020. So learn more and register at makemathmoments.com/proportions.

Jon Orr: That's makemathmoments.com/proportions.

Kyle Pearce: Are you interested in joining us for an upcoming math mentoring moment episode where you share a big math class struggle? Apply over at makemathmoments.com/mentor. That's makemathmoments.com/mentor.

Jon Orr: In order to ensure you don't miss out on new episodes as they come out each Monday morning, subscribe on your favorite podcast platform.

Kyle Pearce: Show notes, links to resources and full transcripts from each and every episode. But in particular, this episode can be found at makemathmoments.com/episode94. That's makemathmoments.com/episode94. Well, my friends until next time I'm Kyle Pearce.

Jon Orr: And I'm John Orr.

Kyle Pearce: High fives for us.

Jon Orr: And high fives for you.