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Andrea joins us with a pebble she is dealing with regarding the connectedness of the content (or lack thereof) she is feeling with certain topics at the end of the unit. She even goes as far as to describe this issue with some topics as having all of the sprinkles throughout the unit to engage students and help them achieve, but no real “cupcake” by the end. 

This is another Math Mentoring Moment episode where we chat with a teacher like you who is working through some problems of practice and together we brainstorm ways to overcome them.

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FULL TRANSCRIPT

Andrea:

I just felt like our unit didn’t have any sort of glue. It was just like sprinkles thrown onto this random thing but there was no cupcake. We were just doing fun activities that were good math activities. They were good thinking. I had good resources, but there was no cohesion to it. And my school is an IB school. So, theoretically, we want our assessments to be conceptual-

Kyle:

In this episode, we hop on a call with Andrea, a middle school mathematics educator and Texas native now living and teaching in Germany. Andrea joins us with a pebble that she’s been dealing with regarding the connectedness of the content or maybe the lack thereof, that she’s feeling with certain topics at the end of the unit. She even goes as far to describe the issue with some topics as having all the sprinkles throughout the units to engage our students, and to help them achieve but really having no cupcake ready to go by the end.

Jon:

This is another math mentoring moment episode where we chat with a teacher. Just like you, a math moment maker just like you who is working through some problems of practice those pebbles rattling around in our shoes. And together we brainstorm ways to overcome them. Let’s do this.

Kyle:

Welcome to the Making Math Moments That Matter podcast. I’m Kyle Pierce.

Jon:

And I’m Jon Orr. We’re from makemathmoments.com.

Kyle:

This is the only podcast that coaches you through a six-step plan to grow your mathematics program, whether it’s at the classroom level or at the district level.

Jon:

And we do that by helping you cultivate and foster your mathematics program like a strong, healthy and balanced tree.

Kyle:

The trunk of the tree represents leadership in your organization or the classroom pillars in your math class. The roots of the tree represent mathematics content knowledge and what it means to be mathematically proficient.

Jon:

Like a tree requires soil, water and sunlight, your mathematics program requires a productive educator mindset and the belief that all students can achieve at high levels. Your professional learning structure or your professional learning plan is represented by the limbs of the tree. The branches of the tree represent the development of educator pedagogical content knowledge, including effective teaching and equity-based teaching practices. And the final section, but not the least important is the leaves. And that is the resources that you have in your classrooms, those tools and your classroom environment.

Kyle:

If you master the six parts in an effective mathematics program, the impact of your program will grow far and wide. Every week, you’ll get the insight you need to stop feeling overwhelmed to gain back your confidence and get back to enjoying the planning and facilitating of your mathematics program for the students or educators you serve.

Kyle:

Jon, let’s not waste any time. Here comes Andrea from Germany. Let’s dig right in here. Tell us a little bit more about yourself, Andrea. And where are you coming from? What grades do you teach? What’s your role and any other details that might be helpful here before we dig in?

Andrea:

My name is Andrew Clark. I’m teaching in Munich, Germany, and I just moved last summer. So, this is my first year here, but I’ve been teaching for 12 years. I’m currently teaching fifth grade in a middle school, but it’s self-contained. So, I teach all the core academic subjects. Previously, I was in middle school math teacher teaching fifth to eighth grade, but I’ve taught everything from second to eighth grade starting in elementary. Fifth grade is my wheelhouse or it’s where I have the most experience. So, it’s nice to be back in a familiar place, though I am a little sad that I don’t get to just teach math.

Kyle:

Interesting. Tell me more though, because it sounds like your safe place is fifth grade. But I’m wondering is your safe place Germany or is there more to that story? Tell us more.

Andrea:

Well, we were living in Texas and my husband and I we wanted to-

Kyle:

See the world.

Andrea:

… try living abroad. Yeah. We have young children and apparently, it wasn’t hard enough so we wanted to ratchet up the challenge.

Jon:

I love it.

Andrea:

So, we decided to pick ourselves up and move here and I got a job.

Kyle:

Amazing.

Andrea:

And here we are.

Kyle:

Also, fantastic.

Jon:

I’m so envious. I’ve always wanted to do that. We actually looked at trying to move to New Zealand right before COVID. The stars didn’t align for us, but we were almost there and I know Kyle wanted to do some of that himself.

Kyle:

Yeah. We did apply for spot at the NATO school in the Netherlands. And we unfortunately did not receive an interview. There was tons and tons of applicants, but my wife and I wanted to go do something very similar. So, the three of us have more than math in common here, except one of us was successful. So, good for you.

Andrea:

Some days, it doesn’t feel like success. Some days, it just feels like we’re just making it through.

Jon:

We’re going to dig into that. But Andrea, give us a favor, let us know what your math moment is. So, when we talk about what comes to mind when we say math class, there’s usually a memory that pops into people’s minds from their past that just sticks with them. Sometimes it’s good, sometimes that bad. There’s no judgement here. We just want to hear what comes to mind when we say math class.

Andrea:

So, as a longtime listener, I had a lot of time thinking about this question. And unfortunately, I have a negative moment and I can’t shake it. So, in eighth grade, I was in advanced math. So, I was in algebra, and I was doing fine in it but I wasn’t crushing it. I was not excelling in it. And my teacher had, I think, what he thought was a motivational strategy where every time you took a quiz or a test, he would read the … When he would pass them back, he would read your score if you got an A.

Jon:

Out loud?

Andrea:

Out loud. And if you didn’t get an A, he would just read your name. And he would start with the As. And it wasn’t in descending order. So, it wasn’t like if you were the last name, you got low score. But you know that if he just said your name, you did not get an A.

Kyle:

Yeah. You just did not get that mark.

Andrea:

Yup. And I got only one A- on a quiz. And I only got it because he had marked something wrong that was actually right. So, I didn’t even get the satisfaction of my name being [inaudible 00:06:46].

Kyle:

You’re like, “Can you call it out again please after you correct this, please call it out again, just so I know.” That’s too bad.

Andrea:

Unfortunately, that is not a good one. But when I think of my childhood math experiences, that’s what sticks out in my head.

Kyle:

You know what, for so many people, and actually, Jon, you are always better at remembering the books, Jon and I have probably read 90% of the same books over the years. We talk about that all the time on the podcast. And one of the books, I think it was the Power of Moments actually where they talk about how negative moments are so much more vivid in your mind than positive moments.

Kyle:

And that part of how our brains work, I think is natural when we ask this question. When we ask for your math moment, oftentimes, the things that are positive, they feel good in the moment, but they’re not something that necessarily sticks with you over the long term unless it’s something super significant. But those negative moments, those negative moments, just like etch in that neural path, it just etches that idea in your mind, and it comes back.

Kyle:

And the part that really saddens me about a lot of these math moments that we hear and we’ve heard over the years and yours I think would fall into this category, like you had said, I’m guessing that this teacher thought they were doing a good thing, by trying to motivate people to make that goal or to reach that plateau. And that may actually have worked for some students. And that’s the other part about this.

Kyle:

Sometimes what works for some students in a positive manner has a very different effect on other students. Now, I’m going to guess that for the students who had the positive response, if there were any, that is probably not their math moment. But for you, it’s the negative math moment because it didn’t motivate for you.

Kyle:

So, it just reminds me and this is a big takeaway already in this conversation for myself is just to remind me that the things we say and do, even though we might have good intentions for them, oftentimes, there’s negative impact on some. And that’s where I think we all just have to take a second thought and reflect on the things that we do in our classrooms. Because it’s possible that I do this thing in my classroom that I have no idea might actually turn into that negative math moment for someone down the road.

Andrea:

Oh, no.

Kyle:

I know hopefully that was-

Jon:

You just dropped the bomb on everybody.

Kyle:

Everybody just turned their podcast off. Everyone was like, “Come on, man.” But what would I hope it does do even though it might feel like oh, geez, that’s not so great. But even if we take that time to reflect and if there is something that’s going on there, I just think of how much better it will be over the long term if we knew now. And you’d probably say, “Hey, I’d much rather know now than say never.” So, that’s a reminder for me and I’ll thank you for sharing that moment so that I could do a little bit of my own learning here.

Andrea:

I always try. I want my students to feel accepted in math class, and that’s always sticks in my head every time I need a new group of students and like, “I want to be the teacher that invites everybody in.” And I felt very excluded. And I love school. I’m a teacher. I love school so much I want to spend every day there. I was a hard worker. I was motivated. I was the exact kind of student that should have been motivated by this, but it didn’t work. It wasn’t enough to take me over that hump.

Jon:

So, yeah, and I love that you are using that moment to I think a lot of us use our moments like that to influence how we behave in the classroom, what choices we make in our classroom. And it sounds like you’re using that to fuel your classroom experiences, which is always great to hear. Andrea, what’s on your mind lately, though? What would you say is a current pebble in your shoe that we could chat about here today?

Andrea:

So, at my current school, I’m kind of the math lead. This isn’t an official-

Kyle:

You’re the guru. You’re the guru.

Andrea:

This isn’t the official title, but I’m like, I do the planning for math for the grade. Other teachers in my team-

Kyle:

They look to you.

Andrea:

… do other planning and share lessons. So, in many ways, it’s fun to teach other subjects but math is my favorite. So, it’s fun that I get to spend the most time in there. And I have a ton of freedom. I know what sixth grade does. I have an idea of what fourth grade does. So, I do have a bit of a map. But I have a lot of freedom to do whatever I want. And I have from listening to your podcast, reading lots of books, I love math education. And so, I spend a lot of time collecting resources. I have a ton of good resources. But it all feels a little piecemeal right now.

Andrea:

We just finished this huge fraction unit. Fractions are really big and fifth grade. You add and subtract and multiply, divide. You do a ton of things with fractions. It’s really important. I want my students to have a good conceptual understanding. I don’t want them to leave with what invert and multiply. I want them to really understand what they’re doing. And I just felt like our unit didn’t have any glue. It was just like sprinkles thrown onto this random thing but there was no cupcake. We were just doing fun activities that were good math activities. They were good thinking. I had good resources, but there was no cohesion to it.

Andrea:

And my school is an IB school. So, theoretically, we want our assessments to be conceptual, grounded in a real-life situation, as it’s not just a test where I want students to multiply fractions randomly. Ideally, there’s some context that we’re working in. And I just didn’t really have that for my unit. And it just felt like every week, I was like, “Okay, what activities are we going to do this week? And so, let me create this unit.”

Andrea:

I had the roadmap of the standards, but I didn’t have what was pulling all of it together. I didn’t have something that we could come back, context we could come back to like, remember this activity? How does it fit with this context? Or can we bring this activity in with another one? It was like, “Let’s do this activity, this activity, you get an activity, you get an activity.”

Kyle:

I feel like there’s more than just the three of us nodding right now as they’re listening. I think everyone has felt this. But I want to know for you, in particular, what made you feel that way? You did an excellent job even use the analogy of there were no sprinkles on the cupcake I think you said.

Jon:

No. There were sprinkles. There’s just no cupcake.

Andrea:

There was no cupcake.

Kyle:

I’m sorry. I’m sorry. I got it backwards. I think I was just focusing on what I like best, which is the sprinkles. But anyway, so there were sprinkles on there but no cupcake. That analogy I think is a really good one. So, I guess my wonder for you is why do you think you felt that way? What wasn’t happening for you to feel that way?

Jon:

Yeah. Or why do you think that you’re missing something? What makes you think that that wasn’t successful? I think that’s where usually we get these uneasy feelings. It’s like, I wasn’t successful in this scenario and I want to rethink it, and I want to get some insight on that. But something is making you feel like what you did isn’t successful, I think.

Andrea:

So, because I teach multiple subjects and some subjects, the contexts are a little bit more obvious. I teach English and I teach social studies or history. It’s called different things in different places, and science. And it’s really easy to mix those together. We just did a really great integrated unit on ancient civilizations and English and social studies. The students were doing research and they were reading these nonfiction pieces.

Andrea:

Everything that they were learning, they were learning how to take notes, and how to read nonfiction articles and how to write nonfiction, all these skills that they were learning led up to this really beautiful, final product that they wrote about an ancient civilization. That was the cupcake. And all the things that we were doing were the sprinkles, but they like went on the cupcake at the end. And the students had this great product that really could demonstrate. Look at everything I learned in this unit.

Andrea:

And with math, I would get to the end of a cycle of topics and well, we need to assess. What do we want to assess about? Baking, patterns. It didn’t feel natural. And some subjects are easy. They’re there to be a final product. We just are finishing a data analysis unit. Easy to ground in a connection. Students created a survey, everything we’re doing with probability and mean, median, mode, all of that is grounded in data that they collected and it’s really meaningful for the students.

Andrea:

And then the end, they’re going to have these graphs and these probability statements and all this stuff as they’ve learned throughout the unit. And so, some things lend themselves really well to that, like geometry, data analysis. But some things, fractions are just harder to come up with something that’s meaningful, that can sustain. It doesn’t need to sustain for the whole unit.

Kyle:

Your cupcake is becoming more clear to me in terms of what that cupcake is in your world. So, it sounds like your sprinkles and your icing is, you’re able to maybe day-to-day keep students engaged, keep them interested. I’ll finish my thought, and then I have one more question for you just to clarify. But then at the end of the unit, you feel like they did this one thing here, they did this one thing there, they did this other thing over here.

Kyle:

But then at the end, it’s like in these other subject areas. And in other even topics that sounds like in math are some topics where you feel like you can bring it all together into some meaningful experience or project or assessment opportunity that students are able to go like, “Wow, it’s so makes sense that on day whatever we did this. And on day three, we did that.” And it all loops together.

Kyle:

My wonder is would you say that they’re actually conceptually and maybe even procedurally understanding the different various parts to this, we’ll call it unit of study? Are they successful with the mathematics itself, but you’re just feeling like they’re doing it, and you’re just randomly asking these jaded pieces? Or is there also a struggle there with the actual math learning piece? How would you describe that?

Andrea:

I mean, I have a range of students. So, it seems like the majority of them get the majority of the math and there’s some that get all of it and some that get less of it, but the majority of them seem to be, they are understanding the activities that we can do, they can apply it in a different situation, they can talk about it the next day. There’s some recall with the activities. It just feels like in the end, I’m like, “Okay, sit down here, take a test.” Despite the fact that we did all of this cool visual work.

Jon:

You tell me if this is what you’re feeling then. When you said there are data management, it’s clear that there’s a product at the end, and we all work towards this product. And it makes sense. This is where this math is useful. And right now, it sounds like you’re struggling to go, “Hey, we did a fractions unit.” And I’m struggling to help the kids understand that this was useful in this area or this product that we all came to create at the end. We have this big unified product that says, “This is where you use fractions.” Is that where your head is at? It’s like, “I just wish I had something like that.”

Andrea:

Yes. I mean, so in the United States, baking is a pretty common thing to use because you use cups and tablespoons, and it’s all in fractions. In Europe, they do not use cups and tablespoons, they use milliliters and grams, and there’s no fractions.

Kyle:

Hey, that’s like us. That’s like us.

Andrea:

I know. I know. But so even very beautiful baking assessment that I came up with, I was like, “Okay, there are these things called cups. There’s a whole cup. There’s a half a cup. That’s half the size.” I had to give all this background. So, it wasn’t even it wasn’t-

Kyle:

Having a measurement. Not even to mention that.

Andrea:

It’s a decent task. But then I had to bring in all this other stuff. And it was not nearly as intuitive as it should have been. So, even baking is what I have, it wasn’t great. I guess it just doesn’t land here. It’s the wrong context.

Kyle:

So, can I ask you, do you think if you were able to do this, so if this was something like we could get that magic wand that we always talk about and wave the magic wand to make this problem go away? And this is more curiosity for me. Do you think that it could lead to would it just satisfy your want for the meaning for students? Or do you think it might actually help them in terms of learning even deeper about or actually understanding even at a deeper level, the mathematics? Is that actually a hindrance towards the success in learning the mathematics?

Kyle:

Or is it just that you want it to be something that is more meaningful, and therefore maybe lasts longer in their mind? What else? I think we all want things to be meaningful, the things that we do, but do you see that being something that would be beneficial to actually helping with the students’ success in your classes as well?

Andrea:

Yes. I feel like my students tend to remember. When I ask them what they knew about graphing and data analysis, they had a ton of background knowledge. When I ask these exact same students what they knew about fractions, there was just a lot less background knowledge. And it’s probably because fractions are presented often like this is how you add fractions.

Andrea:

Here’s a number, like even if there are visual models that go with it, it’s still like, here are these numbers called fractions and this is the magic we do to them. Here’s the number one. This is where the magic happens. But with data, if you want to find out how many students have a dog and really learning about people or your students or your friends or it’s grounded in more like something that students can connect with better, and I feel like my students would do a better job of understanding it when they’re graphing with it independently, and they have more recall of it in the future.

Jon:

When you think of those units that you’re feeling successful and there’s that unified theme, I want you to imagine the students faces. I want you to imagine your students reactions to getting to the end of say, a unit, and what they’re feeling when that happens. And I want you to think about what do you think is the real underlying cause of their feelings towards that end result? What do you think has made them go, “Ah, this is where I am?”

Jon:

Because it sounds like they’re seeing a lot of purpose here. Is that what it is? Is it purpose, or is there something else there that’s made them feel and made you feel that this was more worthwhile?

Andrea:

I came from the elementary world. We love a good craft project. We love something that looks pretty, looks nice in the end, but I want something that’s mathematically meaningful, even if it also does look pretty. I’m good with that, too. But when I see students really have pride in completing something, and when they ask me a clarification question, and I can give something and they’re like, “Oh, yeah,” and then they’ll dive into it. And you’re really thinking about mean, median, and mode right now. And that’s really fun.

Andrea:

Or to really see a student be like probability statement. I’m like, “Something like 10 out of 16 students like the color red,” they’re like, “Oh, yeah.” And then they’ll go into it. And the next time I look, they’ve got 10 probability sentences written down. So, it’s like they can feel successful and they’re applying it in a way that is interesting and meaningful to them. Because I mean, with my data analysis unit, they got to pick the question that they asked. So, I ask like, “We’re going to do this for several weeks. So, pick a question you care about.”

Andrea:

So, they do and they want to understand. It’s more frustrating to them when they don’t get it because they’re like, “But I want to find out what the median is. Which one is the median again? Is that the middle number? I don’t remember but I want to know, because it’s my data.” I feel like there’s more, instead of just like, all right, the test says I have to find the medians. And that’s the middle number, whatever. I feel like they care more because they’re more invested in the outcome. Because there’s more ownership with it.

Andrea:

I have had a lot of engagement with this data analysis unit. And without there being a lot of bells and whistles, our lessons have been pretty basic. I don’t mean boring or not mathematical, but just simple conversations. And they’re able to dig really deeply with it. And I feel like with fractions, I was like, okay, what magic am I going to play? What magic are we going to do today in order to keep them interested in it? And I just don’t need that with this current unit.

Kyle:

Yeah. And my wonder that I have is, and I went through my own experience as well, and I was just thinking about some of these topics that are these, got to teach this topic again, and I’m not really sure why. And what I wonder is about our own comfort with the actual concept from a conceptual standpoint.

Kyle:

And let me articulate what I mean by that. When we’re trying to essentially take these ideas that we’re teaching from a unit of study that are obviously related, so use fractions. So, we’ll use fractions as an example. And if I don’t necessarily have a really clear idea of how they connect, it makes it harder for me to understand a scenario where they would connect, if that makes sense.

Kyle:

And then it just makes me wonder about an idea of whether if you were to go through and obviously today, we’re not going to put you on the spot and have you do this, but makes me think maybe an activity that you might engage in is like jotting down just very briefly, just an overview of like, what are the topics that I am getting this cupcake, this connected product at the end?

Kyle:

And then I want you to do the same over here with fractions and any other concept that you’re struggling with. And then I want you to be honest with yourself. And I want you to look at those concepts, and I want you to look at them. And then I want you to almost like rate on a scale of one to three, one to 10 whatever scale you want, for each of the concepts of which concepts do you feel you are the most comfortable with that you’re able to model and represent in multiple ways? And the ones that you understand various students’ strategies.

Kyle:

Now, if you go and you find out that it’s all tens. First of all, I think you’re lying, because I don’t think there’s anyone out there who has that ultimate understanding of everything. But if you did, I wonder if some on the cupcake list side have maybe some just little gaps that are there. And I’m saying from a conceptual understanding. I’m not suggesting that you don’t know how to multiply and flip for division of … All of those things.

Kyle:

But what I found personally, and I’m wondering if this might be similar for you is that I really didn’t enjoy teaching the concepts that I actually didn’t really understand as deeply as I wish I did. And only after engaging in some of that content knowledge and that real like actually trying to get at the root of, what does it mean, when you blank? So, dividing fractions is perfect. What does it mean when I divide a number by a fraction? And you might start with a whole number by a fraction first and go like, what does that mean?

Kyle:

And what you might start to see is that oh, my gosh, where it led me was, wow, I can see that division could look like a certain way when I’m dividing fractions. But then you start to realize there’s actually two types of division, which makes it really hard when you’re dividing fractions if I wasn’t even aware that there were two types.

Kyle:

And the reason I’m using that example, we talk about division all the time on the podcast, one of our favorite things to talk about, or at least mine, is that I wonder if there’s some other pieces there between that whether it’s the fractions unit, or any of these other units, where you’re like, at the end, you’re struggling to find something meaningful to do with it. It makes me wonder, at least, and I feel pretty strongly that maybe this wonder could be accurate as when I can’t see how it applies to the world around me, maybe I don’t understand it as well as I thought it did.

Kyle:

And this happens to me all the time, when I’m like, I actually don’t know why that works. And this could be true with solving systems of equations. Algebraically, I can see it, but then does it always work or what happens when the context changes, or when I try to make something that connects multiple ideas together, we tend to segment it and throw these procedures into these little holes.

Kyle:

So, I’m going to flip it back to you just to get maybe your thought on what I’ve shared there. And I’m wondering if when you think about this, is there a possibility that maybe so for example, for me, I was unable to model what it looked like to divide fractions? I could only do it with the algorithm. And when I started modeling it, and I started to go, “Oh, my gosh, this is hard.” And then I was like, “Wait, this is easier. Wait, this is starting to make sense to me.” Then I was able to start actually crafting real world experiences like the salting the driveway unit that is on the website.

Kyle:

And actually, I shouldn’t say that context is actually event Lehman who came up with that context. But the idea that we can actually think about fractions in that way and division of fractions and go, “Holy smokes. Now, I understand what the heck’s going on here.” So, I want to flip it back to you. Is there any cupcake to my wonder here, or maybe I’m just way off base on that?

Andrea:

Yes. And that’s pretty humbling because fifth grade is the class that I’ve taught the most like, this is the grade that I’ve taught the most. So, I feel like I know, I should know them the best. And I have certainly grown over the years that I’ve taught fifth grade math I used to-

Kyle:

And you’re not alone. So, that’s the other thing.

Andrea:

… not know a ton of strategies, but I feel like I know more. But I’m actually maybe now that I’m thinking about it, not sure that I’ve made the connections between all the different strategies that I’ve learned. I’ve got a good Jo Boaler activity, I’ve got a good base or a fraction bar activity. I’ve got all these different good visual models, but I’m not actually sure I’ve ever like connected them.

Kyle:

Connected it.

Andrea:

And I’ve had students be successful with them. And I’ve worked with them enough to know, to feel pretty comfortable with them. And I had one student really get dividing fractions when we were using fraction tiles. He was like, “Oh, my gosh, how many times does three fourths fit into two and a half? I see it. It’s right there.” He got it. And that was amazing. We’ve cut paper and made cards, but sometimes I’ll be explaining something.

Andrea:

Some students will be nodding, cutting. They’re like, “Yeah, I got it. It’s two divided by a third, it’s six. I got six cards.” And then there’s another student that’s like, “But how was the number bigger?” “You’ve divided each paper,” and I’m explaining it. And then I feel like what I need to do at this point is like, I need to bring out a different strategy. And I don’t have it because I’m so used to working in like the context within that. Every context I have comes with a visual model. And I have connected the visual models to other contexts. So, I’ve got all these visual models, but they’re all siloed instead of like-

Kyle:

It’s like I know this one works for this one thing.

Andrea:

Or it’s like I could do this one, and this one, and this one instead of [inaudible 00:29:48].

Kyle:

Absolutely. But that’s so far ahead for so many people too, though.

Jon:

That’s where you have to recognize. And I think when you start to go down that path, what you’re going to realize is that the unified theme that you’re after is really about celebrating the ownership that we’re all taking in your students are going to be taking in building their own understanding of the skills in the sense of models and strategies in fluency.

Jon:

So, at the end of the unit, we took a test. And that’s where we’re working towards. Well, maybe that’s not what we’re working towards. We’re working towards how am I demonstrating this understanding of how I can represent different strategies with different models? And how can I do that? It’s all about now celebrating the ownership. And that’s really what your students are already doing when they have the data management unit, or the other unit. They’ve owned this math all the way through. And now here’s my record of ownership.

Jon:

When we do a test, a test is like, “Okay, I can own this, but it’s very abstract.” But if we start to build a way to capture our ownership throughout this process, and at the end, I’m like, “Here, this is my understanding, my collection of understanding of how to divide fractions with a visual model. How do I multiply mixed numbers so that we’re using these strategies? How can I demonstrate this? If I can start to build that, then that is my ownership. That’s my underlying theme that says, ‘Look at, we’ve done this unit, and we have a sense of pride about the work that we’ve done.'” It’s really the same as the other units too.

Kyle:

I know, as I’m picturing that. It’s almost like, and with data management, you can see why it would be so much more attainable or easier out of the box for students to take that ownership, because it makes sense. We’re using data in that case, they can even pick their own data with stuff from their real world. And that’s super accessible, and obviously, a great example. Whereas on the other hand, it’s like when is the last time I divided fractions in real life.

Kyle:

And it’s not as easy for a student to just look around them and go, “Oh, I’ll just grab that scenario and do that.” Not to mention that I’m going to guess that you’re like us. And that when you started trying to model and it sounds like you’re doing a fantastic job, by the way, in terms of having models and strategies, I would say probably where I hit this place with a lot of the things I was doing where I was like, I am modeling mathematics, but I was still only able to do it, maybe one way, I’ll say two, because I could do it algebraically. Or I could do it this one visual way.

Kyle:

And that’s still amazing. That’s like double the number of methods I had when I first started teaching. So, that’s a huge, that’s 100% gain. Amazing. But the reality is that fluency and flexibility that comes from exploring the content knowledge. And as you feel more fluid and flexible, it puts you in a position where you can then start to craft that journey for the students a little bit more clearly.

Kyle:

So, you can almost see it, and you go like, “Oh, that makes sense.” The reason we’re doing this one next is because this is going to help us and serve a purpose to do that. And until we make those connections, and I heard you say it already, until you make that connection for yourself, that’s going to be a little bit harder.

Kyle:

So, what I’m going to recommend for you is that you don’t overburden yourself and think like, by this year, I’m going to fix this for all of these different ideas. But maybe it’s starting with one. And you put that on your personal learning plan. When we talk about our math program and the tree, your own PD structure, your own plan, professional learning plan would be like the limbs of that tree, those big, thick, thick main branches.

Kyle:

And if you focus on that, the beauty is it’s first of all, you’ll be focused. You’ll have a goal that’s going to help you stay on track and it’s attainable. Because what’s not attainable is to try to like do that for all the content pieces. It’s just not realistic. The reality is, is that we didn’t learn about how to do this in our preservice programs. So, it’s not like it’s anyone’s fault.

Kyle:

So, if you pick something, and then you were to go down that rabbit hole and go, huh, when you start to see that path, and you start to see like those patio stones, like more stones come together along the way, and then all of a sudden, it’s now it’s like a paved path. It’s like, “Oh, my gosh, this is so clear to me now,” then it’s actually going to make it easier for you when you turn to another idea. Because you’re going to look to the next idea and go like, all right, clearly there’s some gaps between here and here. And it’s almost obvious to you now, whereas right now, it might be really fuzzy in all these areas.

Kyle:

So, if you make that commitment to yourself, and it doesn’t have to even be a one-year commitment. This is work that oftentimes it’s never over. But at least you can start making pushing those patio stones together and getting closer to a nice path that’s going to make it easier for you and then in turn easier for your students.

Kyle:

And what I’m thinking here, Jon, on the fly is I think what we can do for you is we’ll get you an Academy membership, so that you have access to our concept holding your students back course, which is a roadmap course, that starts in pre-K, all the way up into high school. But there’s a big section in the middle that’s focused on fractions and it’s focused on the two types of division, because they’re very closely linked. And then when you’re ready, it actually will help you link towards ratios and rates, which are all dependent on fractions and division.

Kyle:

So, you’ll have that as a guide or a tool. You won’t be rushed through. It will keep you on an annual, so that you can like, take some time, maybe over the summer time and dig in and go in and dip your toe in when you’re ready, and how you’re ready. But hopefully, that will be one thing here that can guide you along this journey. Because I think you’ve recognized something that most educators of math, I’m going to say, the majority don’t recognize ever in their career. This is how we do it and that’s what we’re going to do.

Kyle:

So, I just want to give you a big hat tip, and say the work that you’re doing is awesome. And you’re recognizing that, hey, maybe there’s some of that cupcake missing. And slowly, but surely, we can get it baked up nice, big and fluffy.

Andrea:

Sounds great. I think I’m going to focus with I think when we do multiplication and division with fractions, because I feel like that’s the newest material for the fifth graders. Most of them come in with some basic understanding of addition and subtraction, even if they haven’t quite mastered it yet. Most of them have some background knowledge with that.

Andrea:

But multiplication and division is it tends to be even less concrete, not that there aren’t concrete strategies for students to use. But in their minds, they can see half a cookie and half a cookie or like a pizza. It’s always pizzas. How you multiply pizza, it’s harder to think about it like that, instead of like, well, here’s one slice of pizza and another slice of pizza. Look, now I’ve got two-eighths of the pizza. I’ve got a quarter. That’s a little more intuitive. So, I think I’m going to try to find some ways to connect the strategies for those.

Kyle:

Excellent. I love it.

Andrea:

So that I can see if there’s a context that comes out with that and work my way back, the more basic not the simpler things, but the comparing fractions and adding, subtracting, because that still needs to be there for students. But if I can pick a good context that fits with multiplication and division, then I think I can put it the other stuff. The other stuff will be easier to fit back in.

Jon:

I was going to say is she got a great plan moving forward, for sure. And I would encourage you to also look at a couple of units that we have in the Academy, the resources, the tasks on salting the driveway and pizza party, which are all around what you’re leaning towards. Andrea, moving into the conclusion of our chat here, what would you say other than what you’ve already told us your pathway forward. But if you reflect back into the discussion, what would you say would be your biggest takeaway from this discussion?

Andrea:

If something doesn’t feel right, it’s probably something that I’m missing. That sounds like it’s all my fault. And that’s not true. But if there’s something in my teaching that just feels thin, it’s probably not because I’m not trying hard enough to find the right context. It’s that the contexts are there and I just haven’t pieced it together yet.

Andrea:

So, there’s something incomplete about my understanding of what I’m teaching. And that knowing that I have a lot of good strategies, not only I’ve got a lot of good stuff, it’s not that I have to scrap it all and start over. But as I tell my students all the time, it’s really important to make connections. How does this connect to something else we’ve done? It’s important to make those connections with the understanding and the learning that I have, so that the connections become more apparent, because it’s going to be more obvious once I see, oh, look, I can do this strategy for both this and this instead of just this context. And just this context gets this strategy.

Kyle:

That is so well said. And I’m telling you right now that it is so easy, not just in math, but in life. When something isn’t feeling right, obviously, there’s something missing. And it’s really hard because it’s missing, and you don’t know what it is because it’s not there. And so, I think that is such a huge, huge takeaway.

Kyle:

And it really makes me, I’ll give an example. There’s a topic that I know a lot of different representation. I’ve been working on it a lot, but I’ll say on and off, because I haven’t actually had to teach this concept in over 10 years. But it’s been on my mind because we are on this journey. Jon and I through Make Math Moments, and that’s trigonometry. And what I would argue and I bet you, James Hansen, if he’s listening to this, James Hansen would probably nod his head. It’s like the story of trigonometry for me is incomplete. I know how to teach this idea. And I know I could do a good job teaching this idea.

Kyle:

But when I tried to put the story together, I describe it as that path. But you think about the storyline of this idea that is obviously taught in a unit together, because it’s all interconnected, the story is incomplete to me. I got to walk into the movie, the movie already started. And then I walk out of the room, and then I come back in and then like it’s at a different part. Now, I know all about that scene. And I’ve actually seen that scene 900 times. So, I can tell you all about that scene.

Kyle:

But then I leave again, and I miss this other pivotal part of the storyline. And I feel like mathematics, that descriptor is so true for so many concepts and so many educators, because again, maybe no one has that clear start to finish understanding of the massive story that is mathematics. But it also makes me excited, because there’s so many educators like yourself and others that we’ve chatted with on this podcast, who want to keep pushing forward, want to keep learning, want to keep doing better.

Kyle:

And if we keep doing that, think about the students that we’re serving. And if we can help them to piece the story together, then they’re going to be better suited when they get into our position when they come in. They’re not going to be flailing around like we did for over a decade wondering what the heck’s going on here and why can’t anyone understand what’s happening.

Kyle:

So, good on you. And I want to say, thank you so much for taking some time to hang out with us here. We didn’t even ask you what time it is over there. Hopefully, it’s not too late.

Andrea:

It’s after 9:00 now. But I mean, the sun set 15 minutes ago.

Jon:

All right. It’s coming close.

Kyle:

That’s not too bad.

Jon:

It’s getting close.

Kyle:

Not like when we’re chatting with our friends in Australia and New Zealand.

Jon:

It’s getting close to summer.

Kyle:

So, I want to thank you, again Andrea.

Andrea:

Definitely not the middle of the night.

Kyle:

Thank you, again, for spending some time with us here today.

Andrea:

Thank you.

Kyle:

Thank you for your dedication to education and mathematics in general. And we’re jealous. So, we might show up at your door and come visit you in Germany sometime.

Andrea:

We’ve got a guest room. So, it’s ready for you.

Jon:

Thanks so much, Andrea.

Andrea:

Thank you.

Jon:

And hey, don’t be surprised if we reach out in about a year or say nine months just to check in with you and see how things are going.

Andrea:

Great. Thank you so much. I look forward to it.

Kyle:

Take care. Have a great one.

Andrea:

Bye.

Jon:

Take care.

Kyle:

As always, my friends, Jon and I learned so much in every single episode. And as of late, you probably know, we like to make connections to the six parts of an effective mathematics program. When I think about those sprinkles and I think about the cupcake or the missing cupcake at the end of the unit, I’m drawn to a couple of different parts of the tree.

Kyle:

One of which I would say is this idea of our leaves helping us to engage students lead them down a path, but maybe having a bit of a disconnect from the roots of the tree. This is that content knowledge piece. This is the mathematical proficiency that we want our students to actually develop over time.

Kyle:

With some topics if we’re just teaching them, we’re engaging them, but we’re not helping them to see the horizon or we’re not seeing the recently we were on a podcast with Bill McCallum coming up in a few episodes. And he described it as the story of the mathematics and how it all connects. Then at the end, I feel like all you have is just some sprinkles and no awesome, wonderful end result of that cupcake ready to go and eat.

Jon:

Yeah. And I think also the branches of the tree, the pedagogical moves the teacher is making helps bring all of those things together. This is why the tree is connected all the pieces, all the six components which are connected. We can’t discount one versus another. We can’t say one is more important than the other. But the branches, those moves, bring that leaves in line with the limbs and connect it all the way down to the roots. So, those teaching moves are so important.

Jon:

Folks, we want to thank you for joining us here on the Make Math Moments That Matter podcast and listening to this episode. If you found this episode on social media, or maybe a friend has shared it, or maybe someone passed it or told you about it at the lunch table, please share the episode in the same way, pass it around. Tell people about it.

Jon:

Our goal here is to help and serve eventually a million students. A million students, this is one of our main goals here. And we can only do that through folks like you who are taking action on what they hear here on the podcast and spreading the word so that more teachers can start changing their moves in their classrooms building their own content knowledge in their classrooms to share that with those same students.

Kyle:

Hey, Jon, folks know that you and I are all about numbers. We’re all about the data. And actually, we were just digging into the data on the podcast and something we noticed a trend we saw, a little data literacy over time is that right now as we record this, this is the end of May, which is where we see a drop-off in listens as people start wrapping up the school year. Standardized testing is happening. It’s a very busy time. Some US schools are even out by now. And the Ontario schools, Canadian schools and I guess the upper USA or Northern USA states, they’re slowly wrapping things up as well.

Kyle:

But what we see right after that is a massive spike over the summer months where educators are looking to refuel, energize and start thinking about their practice. So, now is the time you want to be sharing this podcast. And guess what? The data also told us, this shocked us but our podcast is in the top 3% of weekly downloads of all podcasts, like of all podcasts. That is incredible to us.

Kyle:

So, we want you to share that with your educators, with your friends, your colleagues, and you can do a huge solid by making sure you leave a rating review, hit that subscribe button. All of those things are helpful to share the message with more educators out there.

Jon:

Show notes and links to resources from this episode, the transcript is over there as well. Head to makemathmoments.com/episode235. Again, that’s makemathmoments.com/episode235.

Kyle:

Thanks for letting us help you grow your math classroom tree. Until next time, I’m Kyle Pierce.

Jon:

And I’m Jon Orr. High-fives for us and high five for you.

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