Christina Michaels: So then with the hot chocolate activity, instead of modeling them, I wanted them to just experience. Okay? So you know the answer's 24. Can you pick two ways to show me? Two of those representations? And so it was a little bit more messy with that, which is understandable if it's their first time using it. I don't expect them to get it right away. So that's where that went. And I just crosstalk-

Kyle Pearce: Today, we speak with Christina Michaels, a teacher for over 24 years from Tucson, Arizona. Christina has just taken our course, The Concept Holding Your Students Back: Unlocking Key Understandings In Proportional Reasoning and is implementing many of the strategies she's learned. The pebble in Christina's shoe is around the question, "When is the right time to show my students a model that they can use to represent their thinking?" What a great question.

Jon Orr: This is another Math Mentoring Moment episode, where we talk with a member of the Math Moment Maker community, someone just like you who is working through struggles. And together, we brainstorm possible next steps and strategies to overcome them.

Kyle Pearce: Let's do this. Welcome to the Making Math Moments That Matter Podcast. I'm Kyle Pearce.

Jon Orr: And I'm John Orr. We are from Make Math Moments. And together-

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

Jon Orr: Fuel sense-making-

Kyle Pearce: And ignite your teacher moves. John, we are diving into another Math Mentoring Moment, and we get to do it with Christina, who we had the opportunity to dive into some learning alongside her in our course, The Concept Holding Your Students Back. I'm really excited to dive into her current pebble in her shoe because I think most of us, we're trying to transform our thinking and our teaching approach, often struggling with this idea, right, moving pre-teaching and not doing that upfront. When is the right time? And I think we're going to have a great conversation here today.

Jon Orr: Yeah. I think this is the big struggle for many, like you just said, Kyle, big struggle for many teachers who are transforming and then they're like, "Am I teaching too much? When should I inject crosstalk-

Kyle Pearce: Am I not teaching enough?

Jon Orr: ... model and the strategy?" Right. Like, "Oh, they're here, they're at this place in their learning." Or, "I see these strategies. Is it okay for me to actually talk about a strategy and bring that in? Or do I have to always allow the students to develop there? Am I allowed to actually do some direct teaching?" So that's a very common question we hear a lot and I'm really excited to talk about it here with Christina and share that conversation with you.

Kyle Pearce: Awesome stuff. Hey, let's not waste any more time, my friends. We're going to dive in right now. Here it is with Christina.
Hey. Hey there, Christina. Thanks for joining us here on the Making Math Moments That Matter Podcast. How are things on your side of the world here today? I guess you're not all the way on the other side of the world, but at least the other side of the continent from John and I. How are you doing?

Christina Michaels: I'm doing very well. Thank you very much. And thank you for having me.

Jon Orr: Awesome stuff. No problem. We are excited to chat with you and also to get to know you a little bit better, but let's take a minute here. Christina, tell us a little bit about yourself. Where are you coming to us from, how long you've been teaching, and what's your current teaching role?

Christina Michaels: My name is Christina Michaels. I'm coming to you from Tucson, Arizona. This is my 21st year teaching, but my fourth year teaching purely middle school math. So right now, I am intervention math teacher for sixth, seventh, and eighth grades. So that means the students have a regular gen ed math class. And then as one of their electives, they have my math class to kind of help them through any struggles that they're having in their regular class.

Kyle Pearce: Fantastic. Fantastic. And as we dive a little bit deeper here, let's roll even further back into your past. And let's talk a little bit about a math moment that you remember from your experience as a math learner. Do you have a math memory that pops into your mind that you wouldn't mind sharing with the Math Moment Maker community?

Christina Michaels: I have two of them. So the first one happened when I was in first grade. We were learning addition and then I was absent. And when I came back, they had learned subtraction, and I had no idea what they were doing about.
So I just very distinctly remember the worksheet that my teacher gave me. And I brought it up to her desk to have her check it, and I got every single one of them wrong, and just that horrible, sinking feeling. She explained it to me and I was able to correct everything, but just that feeling of getting them all wrong and being so confused and not knowing what was happening really kind of stuck with me.
And then another one I had is in fifth grade. My teacher had these little math cards, these laminated cards in the back of the room that when you finished your assignment, you could go pull a math card and do them. And I was the only student to make it all the way through all of the math cards. I made it through the whole deck and just feeling really proud, but getting so much recognition, like all the other students like, "Ph my gosh, she's a math wiz," and the teacher heaping lots of praise on me for being done early and finishing everything.
So those are two that really kind of stand out to me.

Jon Orr: Awesome. Thanks for sharing those and being vulnerable there to share that memory with us. And sounds like you had one memory that was one that reminded you of not being successful, and then you had another memory of having some success. And I'm always curious, Christina, is to... These things stand out in our minds, I think, for a reason. If we dig deep enough, there's something going on there that might have effect on our current selves. And actually, I'm curious into knowing how teachers' math moments have had an effect on how they actually teach mathematics. And I'm wondering if that's the case here with you. You could dig deep here and think about your moments and how does that influence how you teach in your classroom?

Christina Michaels: I absolutely think it affects the way I teach, that first negative experience. Because I'm teaching in a poverty-high turnover school, my students are coming to me without a lot of parent support. Their parents weren't very educated or high in education. And so by the time they're coming to me in sixth, seventh, and eighth grade, they've had six, seven and eight years of that horrible, negative feeling. "I'm getting this all wrong. I don't know how to do it." So I really find that a lot of my job is kind of trying to be their cheerleader and trying to give them those proud moments of, "You overcame this. You figured this out. You solved this. Doesn't that feel good? Don't you feel proud of yourself?" So I think I see both of those moments really affecting my attitude with my students and how I try to present math to them.

Kyle Pearce: Yeah. And I bet too, the fact that you shared two memories, again, as John mentioned, one that didn't make you feel so good and one that made you feel really good, I think it sort of gives you kind of both perspectives, right? And I think that's an area where it's really easy for us, especially people who are trained as math educators.
And what I mean by that is typically, especially high school teachers, for example, who went through and got, let's say math degrees, because they've always felt that feeling of success. Sometimes that can be really hard to realize that guess what? Not everyone has that same experience. And I would say it's a bit of a blessing for you to have kind of both sides, both memories so that you can build off of that. So fantastic so far here as we're diving in and learning a little bit more about you, Christina. I'm wondering, if you were to think top of mind, do you have any recent teaching successes that you might be able to share with the group here? And then we're going to try to dive into maybe that pebble in your shoe. So anything that pops in your mind as a recent teaching success that made you feel good after walking out of class?

Christina Michaels: I do. I had a goosebumps moment. My eighth graders right now are learning about equations and linear equations. They're getting ready to figure out the whole slope thing. So they're starting off with solving equations with variables, and there was just no real understanding behind that. They don't understand what that X means. If they see 2X, they just have no idea what's going on. So I spent my Thanksgiving weekend cutting out about 100 algebra tiles sets for them because we can't share any materials that I made every kid their own set. And we just spent from pretty much November till Christmas just modeling these problems so that when they see 2X, they're seeing, "Oh, it's because there are two Xs and then I'm adding something to it."
And then being able to manipulate the equation by adding negatives, make zero pairs. They're literally putting a positive and a negative together to cancel that out. And so it was very new for them to experience it that way and then to draw them into being able to do it. Some of them were ready to put aside the manipulatives and understand the steps in the equation.
And I've had one student. She's in eighth grade now. I've had her since sixth grade, and she's just really struggled and kind of just does not see herself as a mathematician, but this clicked for her. And so she had a test in her other math class on these and she did great. And she came in so proud and she said, "I just understood it. It all made sense." And I just got goosebumps. It was wonderful.

Jon Orr: Awesome. I'm actually having goosebumps for you only because I just echo your enthusiasm and success with algebra tiles and getting students to either solve equations or model equations, or even when we get into the algebra on expressions. And I just can't rave enough about how it can open eyes for students into... especially the zero pairs. When I was solving equations, we got pretty complex solving equations in my ninth grade class with order of operations and multivariables on both sides of the equal sign and brackets and fractions. And I had students representing them with tiles. And then at no time did I ever use this phrase, but this phrase came out and I was just like, "Yes, this is perfect." A student was like, "I got to zero out that term right there," instead of saying, "I got to move it from one side to the other," which is what I was trained to say as a student.
I actually have zero memory of in-class learning on why I would move something from one side to the other of an equation. I would just say, "Switch the sign." That was what I learned. I never actually was taught or learned. I was actually adding things to both sides to zero it out. But when that term zeroing it out came out from my students, I was like, "Yes." I had also goosebump a moment. So thanks for sharing that. I think if teachers listening who have not used algebra tiles, I think you've just heard from Christina here. You should give it a go.
But Christina, let's dive now into what Kyle always calls a pebble in your shoe. What is a current struggle or challenge you're having right now in your class that we can kind of dig into here together?

Christina Michaels: Okay. Well, I have been doing lots of the activities and three act tasks, curiosity tasks from The Concept Holding Your Students Back from your guys' website, Making Math Moments, and I absolutely love it. I have been definitely approaching it so much differently because I think this is one of those ideas where as somebody who knows math and does math, I feel like the relationships between proportional ratios are... It's obvious. It's right there. Three-fourths equals six-eighths. They both got doubled. But not really understanding that my students aren't making those same connections.
So we've been using lots and lots of models, lots of tiles, color tiles, building things, just really laying the groundwork that in a ratio, the two quantities are completely linked together. So you can't do something to one of the quantities without also doing the same thing to the other. So I feel we're really successful in that, but I feel like my struggle comes from I really want the students to discover it. So I'm very much trying to be hands-off.
So instead of saying, "You got that question right," or, "You got that question wrong," even, just, "Tell me more about your thinking. Why do you think that's your answer? How did you get there and what would that look like with a model?" And the kids are really responding to it, but I feel like me as their teacher, knowing the end game is that they need to be able to look at a ratio table. They need to be able to read a bar diagram, be able to create and use, interpret double number lines. I just feel like, do I just show them these things and say, "This is why it works," or wait for them to kind of organically come up with those ideas on their own?

Kyle Pearce: Yeah. First of all, I've been watching some of your progress in the course The Concept Holding Your Students Back, and so fantastic. And I'm loving the discussions that are coming out in the discussion area. So super awesome. I learn something new every day when I'm going through reading some of the discussions that are going on there. So happy that you're enjoying that process.
And it sounds like you're grappling with a common challenge. It sounds like there's a little bit of a wonder as to like, "How much is inquiry? How much should be me telling?" I'm wondering, can we dive a little bit deeper here? Where are the hiccups starting? And I'm wondering maybe if you can even give us maybe a little bit of a glimpse into what a task might look like, sound like, or a lesson, a typical lesson, so that you can paint us a bit of a picture. I'm making some assumptions based on what I know is inside the course, but just for those who are listening, so they can get a sense as to how you're approaching things. And maybe we can dive a little deeper here to try to help streamline or make some connections as to how we might approach this with our students.

Christina Michaels: Okay. So I started off with the illustrative math task. I inaudible. Jennifer's purple paint, where she makes crosstalk paint.

Kyle Pearce: Okay. Yeah. I'm going to look that up while you're talking and we'll add it into the show notes. I know the question.

Christina Michaels: I always get the colors mixed up. I think she makes two cups of red and three cups of blue to get her perfect recipe, but she needs 20 cups all together. So I just presented that task, gave them some tiles and a piece of paper, and I just wanted to see what they would do, paused in between to like, "That's very interesting. Share thoughts." And a lot of kind of the typical responses came through. There were some kids who were able to recognize, "Oh, five cups in one batch." So they have to do that four times.
So they were able to very easily model it with the tiles, but then coming up with their answer. "Well, how many cups of blue does that make? How many cups of red does that make?" And then more importantly, "How did you know she followed the same recipe?" So that's where they struggled a little bit with that. And then of course, some kids immediately came up with 10 and 10. So my idea for that was to get them to recognize the iteration that's happening, that three and two, you repeat that, then you get six and four, and if you repeated it again, or however the students approached it.
And by the end of it, they were all successful in being able to get to the 12 cups of blue and the eight cups of red. But the modeling of it, using the tiles is what really was key for them and really helped them see it. Even the kids who came up with 10 and 10, when we brought them back to the original recipe, I'm like, "Well, what would happen if she made two batches of that?" That was all I'd have to ask them and they'd be able to see, "Oh, it's not going to be 10 and 10." And they were able to come up with a correct answer.
And then from that, so I just kind of wanted them to experience the idea and the building with the model and putting however they did it with their tiles, putting that on paper and how they knew they got the correct answer. And then from that, we went into the hot chocolate task. And my idea in using that was the numbers were very friendly, extremely friendly for middle school. One cup of cocoa to three scoops of hot chocolate. How many scoops would you need for eight cups?
And so a lot of them, even though they're struggling math students, were right away able to come up with 24, but I put the focus on, "Okay, but how can you represent that? How can you really truly prove on paper that that is the correct answer?" And so that's where I wanted to introduce those ideas of a double number line or a ratio table, or even a bar diagram to kind of help them. We haven't gotten into the graphing yet, but that's the next obvious step that I would take with them.

Jon Orr: So I'm hearing you've tried these tasks, and when we were looking at the perfect purple paint task, we're seeing that there's lots of different solutions outlined actually on the illustrative website and they show you what a ratio table looks like. They're going to show you what a solution on a number line might look like. And so what I'm just going to reflect, you said that you gave the kids manipulatives like tiles or counters or what did you give the kids?

Christina Michaels: I gave them colored tiles.

Jon Orr: Oh yeah. Okay. And so then you said that they solved those problems. They felt pretty confident. And then hot chocolate, similarly solve that problem. And you were looking for... What was your learning goal for these, this series of tasks? Or I think you said you were looking for these things, but maybe can you restate what the learning goal for the group of tasks might've been?

Christina Michaels: The learning task was definitely the idea of what is a ratio? That is, two quantities, but the two quantities are linked together. And then also, how to represent that ratio thinking.

Jon Orr: Okay. And then you propose these tasks. You work through the tasks. You allowed them to kind of come up with their solutions. And then what did you do once they came up with their answers to these problems, they showing that with 20 cups, the ratio works out and it's equivalent to the original ratio? What was your step to kind of tie that to the learning goal?

Christina Michaels: Well, what I did then is I ended up modeling the different representations for them. So we went through a double number line. And so I made anchor charts with them so that when we're looking at a double number line and there's red, but really instead of just saying, "These are the numbers," I'm having them help me create it. So if I'm going to use a double number line for my red, what numbers am I going to count by? How much red are we starting with every single time? So if we're starting with two cups, and then if we double that, it's going to get to four cups, and then... So they're kind of guiding them through the, "Oh, the red number line is counting by twos because we started with two cups of red. And then how is the blue number line going to progress? What number are we going to start with the blue?"
So I kind of made these models for them or pictures on a chart for them to use as an example. So that's where the number line looks like with very explicit directions with it. Why are those numbers increasing that way? And what does it look like, and what does it mean?
So then with the hot chocolate activity, instead of the modeling them, I wanted them to just experience, "Okay. So you know the answer is 24. Can you pick two ways to show me two of those representations?" And so it was a little bit more messy with that, which is totally understandable if it's their first time using it. I don't expect them to get it right away. So that's where that went. And I just felt really iffy of like, I don't want this to be, "Here's this math. You just plug in the numbers."

Jon Orr: And because you're emerging the model afterwards, are you saying that you're struggling with the idea of showing them the model after they've kind of built some thinking and then you're trying to connect it to that thinking, or is it that you want them to come up with the model in the first place?

Christina Michaels: I was hoping that they would come up with the model. So at the time that I asked this question, I had to go ahead and proceed with the activity. And so that's what I ended up doing. It was just pure, "Let's create these models together," instead of, "How could you show me?" and leaving that open-ended and seeing what they came up with.

Kyle Pearce: To me, it sounds like you've got a good consolidation going on. Of course, when it's described, there's interpretation. There's all kinds of things that could be lost in translation here. And just to clarify, are you taking students' solutions and trying to help emerge that model, or do you feel like is there a disconnect when you're like, "I really want to get the double number line going here," but are you feeling like no students are showing any signs of leveraging that thinking, or I just want to dive just a bit deeper into that, just to sort of get right at where you're kind of struggling with this idea of feeling like maybe you're showing or, "When should I show?" is sort of the questioning that we're hearing here.

Christina Michaels: So the idea with the hot chocolate that I really wanted them to represent their thinking in multiple ways, so I figured if they've had these examples, if they're looking at these charts, if I'm constantly bringing you back to, "Remember when we doubled the red paint and we doubled the blue paint? What's happening now?" So when I gave them their choice of, "How would you want to show this?" they're picking certain things and they're purposely avoiding some... I don't know why, but I've heard of quite a few students say, "I don't like number lines." So they just kind of shy away from that.

Kyle Pearce: And that is so common. And I can imagine if I really, really feel comfortable doing something a certain way, as soon as somebody suggests I do it a little differently, there's almost this natural pushback, right? Because it's like, "Now I have to think again." And oftentimes, even if the amount of work a student has to do... And I go all the way back to, let's say counting. One-to-one counting is the easiest way to count. It requires very little thinking. It might take a long time, but I don't have to actually do any thinking. It's very proceduralized.
So when a teacher tries to nudge students towards skip counting, there's a reluctance, an automatic reluctance, because now it's like, "I got to actually think about the numbers that I'm saying out loud, unlike when I'm just counting one-to-one." And then eventually, after I do it enough, it feels a little more natural. And then students are able to skip count by twos, and then you to get them to skip count by threes. And we're at the same sort of barrier again where it's like, "Now this is hard again and I don't want to do that work." I'm wondering though, as you start emerging these models... And I don't know if this is helpful or not, but something that I really like...
And I sort of taken this from a colleague, Yvette Lehman. She's actually been helping us design some of our Make Math Moments units and tasks. And something she does... and she does this a lot in a number talk and also in a consolidation... is she asks students if it's okay if she represents their thinking a certain way. So for example, if let's say a student in your class has done some skip counting, and maybe it's a little disorganized on the page and you're going, "You know what? This is a great opportunity for me to emerge the double number line," I can ask and say, "You know what? Muhammad, is it okay if I represent your thinking on this number line like this? You tell me. Does this do justice to what you did?" And then I can show those jumps of two or four or whatever it is if they're thinking additively or even if they're thinking multiplicatively along that number line.
Now the challenge though, and what I'm hearing from you a little bit is that you're now modeling on this double number line, but you're not seeing students utilize that model. Are we hearing you correctly on that?

Christina Michaels: Yes.

Kyle Pearce: Yeah. So that's something that definitely takes time, as you can imagine, the first time someone sees something. And this is a challenge in math. And especially when we think about how math is often taught and pre-taught where oftentimes, a textbook, for example, we'll say Section 4.3, Proportions. And then it's like, we do a day of proportions and then it's almost expected that students are going to be able to do that work. And the reality is it actually takes a lot longer, right? You've got to slowly, slowly be introducing these things and coming back to them.
And then also, I don't think that there's anything wrong either when let's say after you've done the hot chocolate unit, that on the third day or whatever it is after you've done this, that you now explicitly give them a problem that's similar to the hot chocolate task. And you say, "I'd like you to represent this using a double number line and any other method you'd like." So it kind of keeps that door open to go, "Okay, you can still use your go-to. Rock and roll. But then I want to push your thinking a little bit to get you to try this other model as well."
So I'm going to pause there, and I want to sort of get your thoughts on that based on what we've been discussing so far and see if we can keep chewing on this.

Christina Michaels: Yeah. I think that that's kind of what I wanted to know. Is it okay to make them do this instead of that choice and that they're trying it because they want to try something new? I think some of them do, but I think you hit it on the head when you're like, "They want to do what they're comfortable with and I'm making them uncomfortable by wanting them to stretch and do something new." So it's okay to say, "We're going to do this and you can do that." Still giving them that option. So yes, I really do appreciate that.

Jon Orr: It reminds me of when I was feeding my daughter when they were babies, and it was like, you give them those mashed-up peas. They're going to spit it out. And it's like, "Okay. They don't like peas." And then I remember doing some reading on feeding toddlers or one-year-olds or two... I can't even remember; it's been so long... of how many exposures it takes for them to become comfortable with eating that particular food. Because it was true for all new foods. It was spit out immediately. And it was like after the 10th time, I think. There's a number of times you expose a child to a food before it becomes comfortable enough to eat.
And I view that similarly in mathematics when you emerge or expose a new model to students. And I still do it at high school, in grade nine and other grades, of modeling, using the area model for multiplication. We still use the double number line when we're modeling ratios and solving proportions at the double number line. It's every time or when you're solving a proportion or solving these problems, when it's time to, say connect strategies together, if you model it the way you want them to model it, they're just seeing it again and again and again, and it will become more comfortable. Like Kyle said, it's going to take exposure before it becomes commonplace to see it. And I definitely want to reiterate what Kyle's suggestion is saying. Definitely have questions where like, "Hey Kate, show me using..." Sometimes I'll say, "Hey, I'm going to show you a technique a student did last year. And can I show you that take on Yvette's suggestion? I'm going to show you your thinking using last year's student model."
And then the next question is like, "Hey, I want you to try this next one, but use the model that that student last year did." And it's not like all of a sudden, I know that you're lying to your students right now by saying it's last year's student because you're rephrasing it. But for some reason, that will allow kids to be like, "Oh. Well, another student has done it that way. Let me give it a shot. See if I can do it too." And I think that strategic placement of where those models come and then how you ask them to repeat and practice, they'll see the power in it, right? That's when the will happen is when they see that that model is more powerful than what they were doing in the past or maybe more efficient.

Christina Michaels: Right.

Kyle Pearce: And something else too. As John's talking, that just kind of pops into my mind as well, is thinking as well, this is sort of like a bit of a, now that you're like, "Okay." The reality is on their own, John's daughter probably wasn't going to go grab the peas. Probably go for something that tastes better. Same thing for students. And then I guess this is some of the... We talk about teaching math as a bit of part science and part art. And part of the art of it, I think, is now going, "Okay. So I've slowly helped." And I like to consider it being like, "This student's thinking has helped us emerge this double number line or this ratio table," or whatever model it is that you're trying to emerge. Let's pretend it's the double number line. And you've now used it in consolidation.
My question now becomes, what problem can I give them next? And it might not be today. Maybe it is. Maybe it's not. Maybe it's tomorrow, where the problem is going to be significantly harder to do using their current strategy versus this new model. So what we mean by that is it's like, if you can push the complexity, not so far where students are like, "Man, I'm going to give up on this problem," but enough where they go, "I think I know how I can do this, but my current strategy is not working out so well." That is where you get that sweet spot where you almost... As much as before you had said almost like, "Can I make them do this?" And it's like, you don't have to make them explicitly. It's almost let them make themselves do it.
And now that's a hard thing. It's not something that you just go, "Oh yeah, I'll do that tomorrow and it's all going to be great." Requires a lot of thinking and reflection as to, "How do I take today's problem..." Which for those who know the hot chocolate unit, it's a six-day unit that we have up on the Make Math Moments site. And it's a great task and it's a great multi-day unit. And you'll notice that after day one, it starts to head into compensation strategies where students are either adding on or taking off a little and still using this proportional reasoning and proportionality characteristics. But that task is very low floor, which means that it leaves the door wide open for very, very low accessible strategies. Now I have to think, "How do I now ask them a question that's going to up that a little bit so that those strategies they were using might not be as friendly or as helpful?" So that's something that pops into my mind.
And I'm going to turn it back to you and give you a little bit of time here to think and reflect. How are you feeling so far in the conversation and where's your state of mind now that we've made it thus far into this pebble in our shoes?

Christina Michaels: I'm feeling really, really positive. I think you guys are kind of reiterating a lot of the ideas that I've been thinking, especially with the over and over again. I feel like I fell into that trap quite a bit like, "Okay, we covered it yesterday. Why aren't you guys getting it?" Right? Really realizing they need many exposures and many experiences, not just problems on a workbook page. As their math intervention teacher, I feel like it's my responsibility to give them those experiences because they're getting the workbook problems in their other class. So I have to tell myself that all the time. "They did great yesterday. We're going to almost start over again today." So instead of expecting them to, "Well, we did it yesterday. Let's move on."
So I really appreciate kind of hearing that idea. And then also just the idea of like, I can be more heavy-handed in my guiding. Instead of just having it be so completely open-ended, I can say, "Well, this is the strategy we're going to practice, and let's look at how can we use this in multiple ways." And I absolutely love the idea of introducing those strategies that they're kind of shying away with, but using their thinking like, "I notice you use these numbers. Can I show it to you in a different way?" I do the student thing all the time, especially because I have so many periods in a day.
A lot of times, somebody will come up with something brilliant in my first period. And I'm curious to see, do other classes come up with that as well. And sometimes they do. And sometimes they don't. And when they don't, it's really easy to just say, "One of my other students today came up with this. What do you think about that?" So it's a strategy I'm already using, and now I can just kind of more directly apply it to this kind of thinking.

Jon Orr: I think something else to just keep in mind too is that learning goal piece. It's like, "What is the particular learning goal that I want to be very clear at the end of this lesson?" And then sometimes what I do is pair that learning goal with a particular strategy. And actually that even might be the learning goal, right, is this strategy. And then what model am I using to show that strategy? And so you might see lots of strategies and models around the room, but which one do you want to make sure is very clear and connect to the strategies from around the room.
So it sounds like you've got lots of great takeaways to a head away here with. I'm wondering, just before we say our goodbyes here, is because we're on holidays, what would be your very next step after you get back from holiday?

Christina Michaels: I think my next step is going to be to find some of those problems that Kyle was talking about that's going to kind of push the complexity a little bit that are kind of going to guide them towards using that thinking. So the idea is our standards are that not just that they can create these representations of ratio, reasoning, but that they can actually use them to solve problems so that they're actually using the double number line or table as a tool to help them come up with answers.
So that's going to be my next task because it's trying to kind of find some of those low floor, high ceiling tasks that are going to guide us to that. And then in addition to continuing to model that thinking with, "This is what a double number line looks like, and this is what a table looks like, and this is what a graph looks like." So kind of hand in hand making those representations, and then using them as well.

Kyle Pearce: I love it. I love it. Well, Christina, listening and reflecting on this conversation, it fills our heart. I know John is likely feeling the same way right now to know that you had been doing some learning with us in our The Concept Holding Your Students Back course, which is all about unpacking proportional relationships and the whole, what we call the roadmap from spatial comparison all the way up to proportional relationships, and that you've really taken on these ideas of models and strategies, and really trying to hone in on how we can emerge them in our classrooms. So that makes our hearts feel so full.
You're doing amazing work. We can tell just through the discussion of what you're trying to pull from your students, what you're trying to nudge them to work on. And it can be mighty, mighty hard, but it sounds like you've got a great next step, and my friend, we're wondering, is it okay if we check in with you down the road, say nine to 12 months from now, to see how things are progressing along this journey?

Christina Michaels: Oh, yeah. Absolutely.

Jon Orr: Awesome. Awesome stuff. Christina, we want to just thank you one more time for joining us here on the Making Math Moments That Matter Podcast. And I hope you enjoy your holiday, and good luck on your next step.

Christina Michaels: Thank you so much for having me. Merry Christmas.

Kyle Pearce: Thank you. Thanks for coming on. And we will talk to you soon.
As always, both John and I learned so much from these Math Mentoring Moment episodes, and I'm hoping that you had something that resonated with you during this particular episode.
You got to make sure you do something to hang onto that learning. How are you going to reflect? I know something I enjoy doing is heading into one of our Facebook groups like the Math Moment Makers K-12, or I head into the Math Moments Academy and share some of my reflections there. Make sure you're doing something to ensure that you've got a plan to take action on something you've learned, whether it's big or really small.

Jon Orr: Hey, are you interested in joining us for an upcoming Math Mentoring Moment episode just like Christina did here? You can fill out a form and let us know your current pebble in your shoe. And we choose the people in these episodes from that list. So head on over to makemathmoments.com/mentor. That's makemathmoments.com/mentor to apply.

Kyle Pearce: Awesome stuff. And friends, if you want to ensure you don't miss out on the next episode like this one here with Christina, make sure you smash that subscribe button on your favorite podcast platform and do us a solid. Take a moment, pause, leave us a rating and review. You don't know how helpful that is for the whole community to ensure that we can reach a wider audience.

Jon Orr: Yeah. And show notes and links to resources and complete transcripts from this episode can be found at makemathmoments.com/episode135. Again, that's makemathmoments.com/episode135.

Kyle Pearce: Well, until next time, Math Moment Makers, I'm Kyle Pearce.

Jon Orr: And I'm John Orr.

Kyle Pearce: High fives for us.

Jon Orr: And a high five for you.