Room to Grow Podcast

Season 6 Episode 7: Erin Waller Cleveland

00;00;02;02 - 00;00;35;24

Joanie

In today's episode of Room to Grow, we speak with Erin Waller Cleveland with an expertise in early numeracy and elementary mathematics. Aaron shares how one central challenge of teaching math is maintaining the awareness of the mathematical complexity that has become invisible to us as teachers through our expertise. With strong questions and a deep understanding of mathematical learning progressions.

Teachers can ensure that student understanding lasts more than a minute and give students the routes they need to soar in math understanding. So let's get growing.

00;00;35;27 - 00;01;11;24

Curtis

Well, hey, Joanie, I am super excited to be recording the Room to Grow podcast again with you. And today we get to record in the office together. We're sitting at the same table. We don't get to do this very often. And so I'm really excited about our guest. Today. We have an opportunity to

chat with a person who has some experience in elementary math, which is not a space where either of us have any really any experience other than just helping our own children.

So I'm very excited to learn and to grow today. So I'll let you introduce our guest today and then get us going. 

00;01;11;24 - 00;01;57;00

Joanie

I’m really excited for our guest. We have Erin Waller Cleveland with us today and Erin is a fellow Colorado educator, hey Colorado here in Dallas, sitting across the table from Curtis when I should be in Colorado sitting across the table from Aaron. Erin, we're really thrilled to have you. And like Curtis said, we don't get to talk about the elementary topics very often. So really excited for our listeners to dig into a little bit to the work that you're doing. And I know I have some privy and have known you for a couple of years now, so I know the great work that you're doing across Colorado and across the nation. So why don't you start out, we typically asked our guests to introduce themselves by describing some of your life and professional personal experiences that have brought you to where you are right now.

00;01;57;00 - 00:05:54:23

Erin

Absolutely. I'm super excited to be here with you guys today as well. So yeah, I'm Aaron Waller, Cleveland, and I was a teacher and a school leader in New Mexico and Colorado for 19 years. And for most of those years I taught upper elementary and middle school math in Colorado Springs, which is where I still live. And the math that I was teaching made perfect sense to me, but that was not the case, I would say, even for the majority of my students. And I also saw, you know, that my teaching methods were pretty much what I remembered.

And you know, what my upbringing was as a teacher was I do we do you do with, you know, good engagement strategies to keep kids focused and, and talking to each other a little bit. And what I saw was that led to mastery for like a minute. And it was really obvious hearing from teachers above me and just also from being in my own classroom through the rest of the year. That learning that seemed to have been there at one moment wasn't sticking at all over a longer term. So that kind of set me out on a learning journey to understand what it was and kids experiences that one. I mean, it led them to question if they were capable of learning mathematics. I saw that a lot, you know, like just not even thinking I can do this, especially at the middle school level. And then I also was trying to see, like, what needs to change in my teaching to lead to enduring learning, because I think I was several years in a space of this. There's got to be other ways, but I actually don't know what they are yet. Right. And so, you know, that involved me seeking out a lot of excellent math educators, a lot of podcasts, a lot of online resources, and also to the Masters in Teaching Mathematics program at Mount Holyoke College, which was definitely a life changing experience for me, like from the first moment, just was able to see how much more I had to learn about K through eight math, you know?

And at that time I was teaching middle school, but it was like, okay, this really goes back all the way. And at the same time I became a parent. My kids are four and six now, but, you know, it was sort of right at these the end of my years of teaching that they were toddlers and, and coming into preschool, and I could just see that really early side of the math journey from the front seat. And I was like tuning into their development from that, from that lens of like, what is what is mathematical reasoning? What is learning the verbal counting sequence have to do with struggles later on in math. So, you know, flash forward to today, I established resilience math in 2024. And with that I partner with schools to provide professional learning and coaching.

And I just say that that's to improve the experience and the outcomes of math learning. So both of those things, the experience and the outcomes. I'm also an adjunct instructor in adjunct instructor at Community College in Denver in the math department, and I teach the the high school diploma for adults, the math course for that program. And then I'm on the board for a nonprofit called Early Family Math, which is all about supporting families to understand the early side of math from birth through age eight, and has a ton of resources for free for everyone. And then I am the president elect of Colorado Math Leaders, which is our states and CSM affiliate. So I've got ongoing experience across a really wide range of settings, and I continue my own learning journey across all of them. I love the name of your guys's podcast, because that is definitely a mentality that I lean to, you know, in all spaces.

00:05:55:16 - 00:07:13:22

Joanie

I love it. That's very obvious as you tell us your story and so much I want to dig into with you. Some of that I knew, but a lot of that I didn't know. So even more excited, I would love to have you for a second. Just reflect. Because as our listeners know, as you both know, I have 14 month old twin grandbabies.

And so thinking about the timing of your your master's work at Mount Holyoke with the birth of your children and thinking about them as mathematical learners from infancy. And, you know, as Curtis mentioned, he and I, of course, both have kids, and we both had the experience of supporting them through learning the elementary level kinds of math. And yeah, I think with our teaching background, with our understanding of higher mathematics, that certainly I feel like that helped me contribute to supporting my kids.

But I think you bring an expertise in that early numeracy space that neither of us have. So I would love if you could just kind of elevate some of the things that you paid attention to because you were learning them in your master's program, or had learned them in your master's program. As you were thinking about your kids in their first 4 to 6 years of life. Tell me a little bit about that. What, what, what does that look like to support them mathematically, to set them up for later success.

00:07:14:24 - 00:10:15:12

Erin

It's actually not very complicated. And that's what I love about it is, you know, the message for, for parents and caregivers is you don't really need to be doing a whole. You don't need to be doing anything special outside of what you're already doing, other than noticing and commenting on certain things and having conversations about math when it pops up, or asking kids, your kids to count in situations that they're already in.

So for my for the program, one of the elements was the capstone project. And for my capstone project, I developed a session that was aimed at parents preschool age children about early math roots and how to foster them, and also why they're so important. Because this is something, you know, and this will come up later, too. It's like a fact that I have internalized so much that I kind of walk around the world under the impression everyone knows it, and I feel like actually, almost no one knows it. But kids math abilities, especially in advanced counting concepts when they enter kindergarten, is highly predictive of a whole range of outcomes, not only about how well they're going to be doing in math later in school, but actually it's better predictor of how well they're going to be reading than they're reading. You know, level or their reading abilities is at that age.

And then also just life outcomes, you know, things like mental health, income, likelihood of incarceration, a lot of things. And we could spend hours and hours talking about why that is. But it's correlated with all of those all of those things. But, you know, I think for your question, literally, it is as opportunities for counting things arise do that.

You know, I my favorite tip for people is just to use if you already have, as most families do, a bedtime or a storytime book reading routine that you just always look for an opportunity to count something, or to have conversations about math during that because you're already doing that. You know it's not. And there's always there's always something to count. And then, you know, just because I was paying such close attention with my own kids and learning about the early number concepts like 1 to 1 correspondence and cardinality and number conservation, you know, they I could just see how, for example, like my three when she was three years old, my younger daughter could count really well and accurately if things were in a straight line, but if they were scattered around the page. Less accurate. Or her 1 to 1 correspondence may break down. She might double count something. And so I could. And so the other benefit of that was just because we were doing it all the time. You could kind of see how things were developing in a really clear, in a really clear way. 

00:10:16:25 - 00:11:20:20

Curtis

Wow, wow. I just my brain is just I don't even know how to conceptualize the questions that are coming up in my head because I'm thinking about things like, you know, I'm envisioning my own kids every time I have these conversations. Anytime we have the podcast, I end up thinking mostly about Teagan and Truett, and I'm specifically thinking about true it and spatial reasoning. And when you described your daughter counting along the line versus being doing the spatial reasoning piece of of things are scattered around the page, and keeping track of where I've been on the page to count them feels like that. That just feels like something like, that's a that's a big deal. It is. Right. And just that, that the depth of that is, is huge. I'm I'm fascinated. I've always said this, but I'm fascinated to hear you talking about research that confirms kind of this necessity of numeracy and necessity of mastery of the, of the cardinal counting and of the concept of number being so foundational, so incredibly foundational.

00:11:20:22 - 00:12:28:10

Erin

Yeah. And I think I think that was like a key insight for me was just like, I think that some of the issues we see when we see it show up as problems with fractions or algebraic expressions. I actually think a lot of times it backtracks really far. And I just think that, you know, at the societal level, we put so little focus on early math because it feels simplistic, it feels basic, you know, it feels like something that is just going to happen.

And we don't, you know, I didn't learn the nuance of it even in my master's program. And still until I sort of like, chose to focus on that and build up my understanding by going in a direction around that. And it's and then we panic, you know, we panic in upper elementary and middle school when kids have gaps or deficits or I just think that, you know, that's literally why I named my my business Roots and Wings.

Math is like, we need to actually attend to the roots before we're asking kids to have wings. And I think that we are only looking for the wings without attempting to the roots, I love it.

End of Segment 1

00;12;28;00 - 00;12;34;03

Music break

Start of Segment 2

00:12:34:13 - 00:13:23:12

Curtis

You brought something up that it struck a chord for me. Anyway, this idea of we get, we feel like it's simple or we feel like it's easy and I don't know, maybe that's the reason why we sometimes rush through things, right? Like we see kids have some success. We. Hey, they counted along in a line, so they must be fine. They're doing they're doing okay. And and we miss the opportunity to deepen that, that understanding. We miss the opportunity to really make sure that that is secure before moving on to more complex kinds of things. Is that something that you kind of see? Is that something you've been able to to dig into in your research and in, in the things that you're talking about?

00:13:23:12 - 00:15:07:01

Erin

Well, yeah, I mean, I think when we talk about, you know, early childhood mathematics or elementary mathematics, many of us is that we're talking about something simple. But I think that simple and elementary or simple and easy are not actually the same thing. So is it okay if I kind of dive into a framework that I like to use to use to really kind of anchor that for us, because some of these words are subjective and some are objective.

So if you imagine a coordinate plane with the x and y axis, right. And you have on the x axis easy all the way to the left and hard all the way to the right. Those are subjective descriptions of difficulty of anything. We'll stick to math right now, but really, you could say for anything. And then if you put on the y axis, you've got simple down at the bottom and complex up at the top.

And those are objective descriptors. So the more simple something is, it just means that it has less steps, or it cannot be broken down into more basic parts than what it already is. If it's at it's the most simple end of that continuum, okay. And then the more complex it just means the more interconnected components or steps there are as we move up towards that, like upper side of the of the y axis.

And so then that creates these four quadrants where we have things that are easy and simple, easy yet complex, hard and complex and hard yet simple. 

00:15:07:01 - 00:15:52:07

Curtis / Joanie / Erin

So I have to trace that in my I'm moving my hands. I'm moving my hands around. So easy. Simple. Lower left can corner. Yep. Right. Easy. Simple and then easy. Complex. Upper left hand corner.

Yes. So hard. Complex. Upper right hand corner. You got it. And and then. Simple. Hard hard simple in the lower right hand corner. That's hard to even say. Yeah. Something simple if something is simple and hard. Yeah. And I know that maybe. I don't know if we prepared you for this question, but I'm I need examples, examples. I need things that are going to help me that doesn't even have to be math. Don't have to just kind of talk through just talk like samples.

00:15:52:08 - 00:17:18:20

Erin

So again, because we have easy and hard in there, it could land different places for for different people. Because, you know, whether something is easy or hard for you depends on your experience level, your interest with it, your natural abilities for sure to some extent. But examples that I like to use, you know, for easy and simple.

What I typically say is something like turning on a light switch. Okay. Yeah, yeah, like the switch. Hard yet simple for me is leading my garden. That's not that's not a lot of effort. But I don't want to do it. And it's hard for me to get myself to do it. Okay. Okay. Hard and complex. Learning a new language as an adult especially.

That's a good one. A lot of effort and it's complex. Okay. Yeah. And then easy at complex is where, you know, I would put anything related to early numeracy. And you know, for secondary math teachers, maybe most things related to elementary math as an adult teacher with lots of work in the number system behind me and internalized.

00:17:18:20 - 00:18:20:15

Curtis

So easy yet complex the early numeracy concepts and can I can you expand a little bit on why complex? Because I think I think I agree with you, I do, but I'd like to just have you elaborate a little bit on what are the complexities, maybe that you see that maybe things that seem and feel easy because we are experienced now, they're not necessarily easy for your four year old or for your, you know, yeah, kindergarten or whatever.

But for us, because we have grown up and we've got math degrees and we've done things, they may be on the easy end. And that was another thing that came up, came to mind like something could move along that. Yeah, horizontal hard and become easy. But they really topics don't move vertically, vertically. They move left and right, but they don't necessarily move vertically. And I think that's an important concept to have in your head.

00:18:20:15 - 00:21:11:02

Erin

Absolutely. And I think that the perception we have is that it's moving vertically sometimes. Right. We it beats things. Something big begins to seem simple to us as it becomes easier to us because we kind of lose sight. We forget how much there was to learn. We we may not even be aware of all what all of those subcomponents were.

And again, talking about early math here, you know, the the insight that we have as teachers into learning trajectories and all of the components required to, like, truly understand something at a conceptual level, we probably didn't have that understanding of all those subcomponents as a learner. But so the example that I like to use in my sessions is find the seventh number in the pattern starting with four, eight, 12.

And when I do it I ask people to place it. And so typically you will have a group of people that puts it in an easy and simple, and typically those tend to be the teachers of older grades. And you'll have people that put it in easy yet complex. And those will tend to be teachers of younger grades that are kind of have the front row seat to, you know, that is a multiplicative concept, right?

Or a you're extending a multiplicative pattern there, which you could do in a variety of ways. But if you're teaching, you know, at a an age that is before that is like really fully expected to be mastered, you're probably more likely to be aware of the complexity. So if you think about that, like you said, Curtis, for a four year old or for a kindergartner, and you begin to outline all of what is required to have mastered to do that, to finish that activity.

Yeah. And you go all the way back to, you know, verbal counting sequence and knowing that the number or number structure is going up by one plus one plus one plus one all the time. And, you know, we could probably make a list of over ten things that you would need to have understood in order to complete that pattern.

Yeah. If we really were going to the early childhood, early elementary age range. And I think that is something I did not have the opportunity or even the thought to do, you know, as a middle school teacher is to really dig into not just like, what's the one checklist skill that comes before adding fractions with unlike denominators, but like, what are the yeah, kind of 50 concepts that are all involved in this now more complex kind of number problem.

00:21:11:20 - 00:21:42:02

Curtis / Joanie / Erin / Curtis

Can I be really vulnerable right now? Yeah. Do it. Of course. When you ask that question, of course, immediately I'm thinking about how I'm going to solve this question. Of course you are. And I as you continue talking, I realized. A that I definitely would have put this in easy complex and be that I went to add it, I went I used an additive method. And it wasn't until you said something and then I kind of turned, I don't know if you guys saw it, but I turned red. I started chuckling to myself and I thought, oh my gosh, you said so. You would have put it. You would have put it in easy and complex. Well, so when you said it easy, complex first.

00:21:58:10 - 00:22:30:01

Curtis / Joanie

Absolutely. That was that was definitely my reaction. But it was my reaction, partly because I didn't immediately know exactly the response. I didn't have an answer. I actually had to think about it. And the way that I thought about it, frankly, is not the maybe mastery level version of it. Did you just think of counting by four? I counted by I thought, well, now I've got to iterate this silly sequence, and it didn't even dawn on me that I would land.

00:22:30:02 - 00:23:32:15

Curtis 

I had no preconceived idea as to where I was going to land. Is it in the hundreds? Is it on 50? Like, yeah, and maybe I haven't had enough coffee yet this morning. Maybe. Maybe I'm not firing on all cylinders, but I just, I want to kind of put that out there as, hey, look like even now when you just verbally presented that to me, a because you use the word sequence and B because it was a sequence with four, eight, 12, which weren't immediately, I didn't immediately recognize those as multiples of four.

I didn't, I just didn't I don't know, today that was the way it struck me. And so I just want that to be comfortable for folks and for people to realize and maybe even feel what that is. My thought is going back to our conversation with Pam Harris and the way that student, the circles in which we or she has like the thinking, yeah, that complexity of thinking or not reasoning.

00:23:32:16 - 00:23:55:01

Curtis / Erin / Joanie

Yeah, yeah. So anyway, I just wanted to put that out there that I did not multiply by seven. Well, it's interesting too, because, you know, I often do this in sessions with slides. So it's visual. And now we're doing this all just verbally. And that also makes a really big difference. That makes a huge difference. That changes the complexity for sure and how we're thinking about it.

00:23:55:03 - 00:24:39:08

Joanie

I would love for you to like make a give us a tight connection about how I mean, you're starting to go there. Excuse me? You're starting to go there around. Why having that early foundation be so solid contributes to success later on. So can you help us draw some direct line? Some connect the dots from and I'll let you pick the topic.

But something that feels maybe easy and simple, that has to be in place for something that is more complex, whether it's easy and complex or hard and complex, like how do you how do you see from your studies and your experiences? How do you see that mathematics dot connection happening? 

00:24:39:20 - 00:25:13:06

Erin

Yeah. So the example coming to mind is I have a series of quick look routines.

Are you guys familiar with that routine. So actually I am where you kind of just flash something and then cover it up again. Yeah. Yeah. Yes. Some people it's also called quick images. So it's what you just said Joanie. So you're flashing something. And typically the prompt is how many are there. Right. And so it's a a subtyping or a group routine.

00:25:13:08 - 00:25:43:15

Unknown

And so you can do that, you know, and I have a, a series of them that's all embedded in ten frames. So a ten frame just being the grid of two rows of five. Used a lot in early elementary math. Right. And the first one is just the ten frame with seven circles in it. So something that would be appropriate for preschool kindergarten to to show that.

00:25:43:15 - 00:26:07:13

Unknown

But even with adults, you know, you can have a good conversation because it's still in top row five two in the bottom, so people can say how they saw five and two to make seven, or how they saw four and three to make seven, or how they saw the absence of three spaces. They know it's ten frame. Yeah.

00:26:07:15 - 00:26:38:19

Erin / Joanie / Erin

But if you keep that ten frame structure and then you start to put in dot patterns. So like the four dots on a die. Yeah. And now there's four dots on a diet in, in seven of the ten spaces of the ten. Oh gotcha. Now all of a sudden people can instantaneously. So now they're thinking in groups of four instead of thinking in groups of one. Yes.

00:26:38:20 - 00:27:12:00

Erin / Joanie

And then we put circles that have that are representing three fourths in there. So three shaded pieces with a fourth missing. And now we have seven sets of three fourths. Oh cool. And then we put integer chips in there that progression. Yeah I can totally see what you're talking about now. 

00:27:12:02 - 00:28:34:20

Erin

So now I start putting some red in some yellow chips to to stand for positives and negatives and negatives.

And we can see ties instantaneously what the total amount would be by looking at zero pairs or, you know, being able to process that pretty instantly. And then as I have done that sequence, folks have come up with some other really cool ones where they put algebraic components of algebraic expressions in there. So they put some xs and some Ys where they put x plus two in several of the different spaces of the ten frame.

And now you have the connection to the distributive property. Right. And so I think that what that comes down to is the idea of number conservation, that we can break apart and put together numbers in different ways. And that is a thread that goes all the way from age four, you know, or you know, say even maybe age three for some kids all the way up into dealing with algebraic expressions, when we know how to flexibly decompose and compose numbers and expressions, that is a toolkit that serves us across classes of numbers and across, you know, and then on into algebraic expressions.

00:28:34:28 - 00:30:56:15

Joanie

Right? Okay. As a as a secondary level teacher, I love this because I think what we often don't do and in my experience, I worked for seven years as a district math coordinator, worked with teachers across all grade levels. And I just noticed that especially at the secondary level, teachers are not making connections to what kids learned at the elementary level. And, you know, we talk a lot about I remember maybe my fifth or sixth year teaching, I listened to somebody talk at a conference about how expecting kids to just memorize math, like we expect them to memorize their math facts, right, and expecting them to just memorize. Here's how you do an addition of fractions problems. Here's how you do the distributive property. Like if we just continue to teach, as each kind of math problem is a series of steps and you have to memorize the steps, you you hit a maximum and everybody's a little different, but you're going to hit a maximum where you just can't memorize anymore. Right. And what you're talking about like, Aaron, I'm drawing a little pictures of my notebook in front of me as you're talking.

So to see the structure underlying of that ten frame and helping kids understand that, you know, I'm going right to like an algebra one concept or eighth grade math concept, like seven times the quantity x plus two. Well, if I put x plus 2 in 7 of those ten frame boxes and compare that back to seven dots in the ten frame box, like that connection between, we're just talking about seven of a thing. And the thing can be an integer, or the thing can be the it can be the quantity four. It can be the it can be the number three fours. Like we don't teach fractions as numbers, we teach fractions as parts of a whole. And it's like, well, yeah, but it's also a number on the number line. And we can have we can have seven, three fourths. And what does that mean. It's not seven and three fourths. Not the mixed number but it's seven of this number three four. So like yes that thread I love how beautifully you took us from integer values counting that, you know, A34 year old would probably be fairly proficient at all the way up into algebraic concepts and how it's just the same idea. I love it, it's just the same idea level of complexity.

00:30:57:00 - 00:32:08:02

Erin

Yeah. I think, you know, in my recent learning, I think maybe one particular stumbling point is the idea of unity and being able to see something both as one whole amount and as composed of different amounts. And I think that is, again, it's something that seems rather obvious to us as secondary math teachers, but it is actually it involves switching your lens on something and seeing it in two different ways simultaneously.

Yes, that is really complex. Yeah, I have a hunch just, I just a hunch that that is a particular maybe early root that doesn't get grown and that can then become a stumbling block later on. Or, you know, if you think about fractions is probably a really a place where that explodes, you know, for some people, yeah, for sure.

It becomes a bigger stumbling block there if you have not if you have not really solidified that concept. And I don't know that that's, you know, I don't have deep familiarity with the early elementary standards, but I don't know how much that's lifted up. 

I wonder. Well, I was just thinking about this, you know, the idea that, you know, people call it and I'll put quotes up, skip counting or even that sequence that you brought up earlier, just unitized the four was a thing that I clearly I didn't do right away, or maybe I was doing and I, you know, because I was going to count by them. But it just I think you're right. I think there's this idea of being I love the visuals, by the way, Joni, I got to. I was drawing as you were talking. I got to watch Joni, this drawing, and I was I also and I don't want to get down in the weeds, but I do have a question about those ten frames and the way that they're, the way they're being filled out.

Is there is there a I don't know if the right I don't know if the right is the way, but is there a structure? I think there must be a structure because you purposefully started in the in the upper left and worked across the top right. 

00:33:03:08 - 00:34:01:08

Joanie / Curtis

I don't think it has to be though, because I've seen them. I've only drew them that way because that's how Aaron was describing the seven when she started.

So I wanted to like draw what you were describing. But I've also seen the ten frames going certainly for two columns of five rather than two rows of five going. I would have filled in the bottom row first. Yeah, I don't think it matters because yeah, I don't know. I don't want to I don't want to jump in.

But yeah, when you're working we start doing this concretely with kids, right. So we might even rotate the ten frame around on the table or on the page so that they get the sense that, like, there's not something magical about the orientation of the ten frame. It's about the ten frame just helps us with that subtyping of recognizing a five really easily. So I don't have to go to the single counting every single object one at a time, which is developmentally appropriate at a stage. But you want kids to be able to then go recognize that that's five and not have to count, not have to. Do they have a visual reference just like the dice.

00:34:01:24 - 00:34:48:04

Erin

 I think Curtis, my response on that would be, I think that the typical way of filling it in, you know, from the top, from left to right, and then some in the next row is like that matches how we read.

And, you know, young kids are learning to read. But if I were doing this ongoing in a classroom, I would mix it up. You know that once you know that it's a ten frame you you can work with that no matter what the you know, which boxes are filled in or, or left as, as negative space. So but for the purposes of showing that, that through line we do always I always do is keep five on the top and I do actually move around the bottom row a little bit, but always keeping seven filled in.

00:34:48:07 - 00:36:13:05

Joanie

Yeah, I think it's important to do like I just want to jump in here because I know in my experience to when we present the mathematics always in the same way we can create a misconception in kids that right, that there's a dependency there. So I'm even thinking about I'm tapping back into all my memories of supporting elementary when I was in my district role, and I remember somebody saying, oh, this is a huge research question.

When you give kids like five plus three equals blank plus one. Yes. Right. And if they fill in, if they fill in eight, it's because they think oh equal sign means do the operation. Right. So they're not seeing this as an equivalent statement. And they're looking for a number that when added to one, is equivalent to the number that five plus three is equivalent to their viewing it as do an operation.

So it's a it's like a kind of prompt you can give to identify that misconception and then you can address it. But if, if all they ever see is five plus three equals blank, then of course that's what they're going to think. And so even just as simple as when they're doing those early, you know, what is that second grade addition problems.

Be sure sometimes the solution is on the left hand side of the equation. You don't always put the you don't always show it the exact same way. So they understand this is a relationship. It's not a pattern. 

00:36:14:04 - 00:36:39:04

Erin

Yes. My my older daughter just finished first grade and I definitely saw that they mixed that up. Now from the yeah throughout first grade. The equal sign is not always after you know the five plus three. Sometimes it's eight equals blank plus five or right blank equals five plus three. So I have noted and appreciate that that shift that seems to have happened in at least the curriculum that she was working with.

End of Segment 2

00;36;39;05 - 00;36;47;15

Music break

Start of Segment 3

00:36:47:20 - 00:37:23:25

Joanie

I'm wondering if you can help us think a little bit about how, how can a teacher uncover how students are thinking about these things to know, like, I loved what you said at the beginning about talking about your experience where kids had mastery for a minute, right? Where the the, the learning wasn't sticking. So if I recognize that in my students, like, how do I get to how do I get to understanding what they might be thinking that isn't correct?

Like, what would that even look like if I, I don't know what you know, I don't I don't have awareness about this as a teacher. What is that? 

00:37:23:25 - 00:40:53:25

Erin

It really comes down to questions. And that is another piece that I feel like just as constantly under development for me, I, you know, I used when you all did your series on the effective math teaching practices.

I really honed in on the posing, purposeful questions. I shared that one with a lot of folks, and I just when we think about trying to teach, other than how we were taught, yeah, we have to become students of how kids understanding develops, and the only way to do that is through hearing them and asking them questions. And so if we go back to something like that quick look progression, we were just talking about the one that gets the most conversation and is the most fruitful for hearing, thinking.

And I'm talking about, you know, thinking of adult teacher learners in a in a session, but would be true in a classroom as well is the fraction one. Because there are a lot of different ways of seeing that. And so it is allowing a person to say how they see it and then prompting them to say more. And even the recording, the teacher recording of of what is being said to confirm that that is what the person means, or to get them to clarify further.

In my grad school program, we latched on to this term that we had seen written in a case that a teacher had written up. She used the term overhearing not as like overhearing, like eavesdropping on someone, but overhearing meaning over assuming understanding based on a correct answer without pushing and prompting for further explanation and thinking. And so that was something that we brought up over and over we would question over. And am I overhearing this student? Oh, I heard the right answer. I feel really excited. And, you know, we have a bias to asking questions when answers are wrong. And, you know, that's another piece is let's ask. Let's ask about the thinking whether right or wrong. Yeah, I find that in the the high school diploma program course that I teach.

When I do that, people get really freaked out because they assume it means they're not used to it. Yeah, they assume it means they're wrong. When I ask them to explain how they came up with something. And when I do my poker face, you know, when when a response is given. Right. And so, you know, I think that so it keeps us from overhearing if an answer is correct from over assuming some understanding from something that may have been gotten without without real deep reasoning or understanding, right or with a misconception.

Right, right. So so I think the the honing of the the craft of asking good questions is a huge part, because that is what is going to allow us to, to, to know what kids are noticing is going to be the thing that allows us insight into what relationships are they seeing or not seeing yet, and maybe ideas that are in progress that need a nudge but don't necessarily need right me to start class with 15 minutes of telling you something.

00:40:54:08 - 00:41:24:10

Joanie / Curtis

Right, I love that. I think the importance of exposing for students, even because I think it's a practice, and I see this in my own kids and in the conversations that I get to have, I think way more often, just culturally, we focus on folks response, people's response to name your favorite topic, whatever it is we're talking about, we don't often think about in anything. Mathematics aside, we don't often think about what we're thinking. Right. And so exposing students to the practice of I arrived here, this is the response that I got here. How how did I get here? What was my thinking process? And having students think through that, even if they're thinking is wonderful and we want it, but it exposes, especially when their student, when they're thinking is wonderful and they've thought through this in all kinds of really unique mathematical ways.

00:41:55:09 - 00:42:16:00

Curtis / Joanie

But I think the importance of being able to expose and get used to thinking about what I'm thinking, it's not just a math skill, but it's a life skill. Yeah, that we really need to be able to develop. Yeah, yeah. That's fantastic. So I want to kind of transition us toward the end because we're running out of time, unfortunately.

00:42:16:02 - 00:43:20:19

Joanie

And I can't wait to have you back on. I think we need to have another episode with you where we really dive into some content conversations and so much cool stuff to unpack here. Let me know. But I love learning elementary math. I really, really do. Me too. And and I love that. I can like, see the connection.

So this I'm maybe only starting to answer my own question. You talked about how that questioning is such an important skill for teachers that are really interested in trying to shift their teaching and ensure that, you know, students are getting these really well developed roots or at the secondary level, uncovering which roots weren't developed so that I can circle back and really help students have that foundation for learning.

So aside from developing out their ability to ask purposeful questions, what other shifts or ideas would you recommend for teachers who want to, you know, really think more carefully about what students know and creating positive foundations. What would you suggest for an elementary level teacher? What would you suggest for a secondary level teacher to get more in line with the kind of teaching you're talking about?

00:43:21:04 - 00:46:24:18

Erin

Well, I think, you know, a challenge for elementary teachers is seeing where the mathematics that they're responsible for teaching is going. And a challenge for secondary teachers is seeing what are those roots that I am supposed to be building upon. You know, my most of my years of teaching, I was operating without a deep understanding around that and just being confused about why kids thought some of the stuff we were doing was so hard. Why don't you think about it the way I think about it? Yeah. And so I really think, you know, you talked about diving into some content. I do think that's it. It's it's actually engaging in experiences where you can be in the seat of the learner, learning elementary math content or the content that you teach from the seat of the learner.

And those opportunities are not always easy to find. But when we talk about teaching through asking lots of good questions and not, you know, the the funneling questions that are just the and then what's the next step and then what's the next step. Right. The questions that are like, if you don't know what our the sub concepts, then it's hard to come up with good questions to reveal which ones are there and which ones aren't.

And so, you know, there's a real gap in how we are being asked to teach now, and how the majority of us experienced learning ourselves. And I just don't see a lot of effort or opportunity being put into giving educators those experiences, to have those experiences from the seat of the learner themselves. For me, that was transformational and it put me into.

So I always pinned this back to the concept of the. It's called the Dunning-Kruger curve. Yep. I know what you're talking about is Jonny. So you know my understanding of it. It's like a psychology concept where when you're a complete beginner with something, you know, you don't know anything about it, but then once you know, like a little bit about something, you way overestimate your confidence around your expertise with it.

And I think that a lot of us sit there, even though we're not beginners, even necessarily, but maybe we're beginners in understanding student thinking around conceptual understanding and how conceptual understanding is developing. And if you can have an experience that kind of like pushes you over the cliff down into the oh, wait, hold on. There's a whole bunch of stuff.

Actually, I don't know about this. Yeah, yeah, that that is where the magic is at, in my experience, for people to really, really have an understanding of like how much there is to learn about using kids thinking to to guide instruction. 

00:46:25:00 - 00:47:14:16

Joanie

Yeah, I love that. I love that especially because I saw I don't saw Aaron Curtis raised his hand when you were talking about that phenomena.

And I think it's one of the things I love just to have a little I love my podcast host moment here. It's like Curtis and I think about the math so differently that I feel like we have that experience together all the time. When we talk about math, it's like, oh, oh, I never thought about it that way.

Like we really think about it differently. So even just even as an educator, just putting yourself in the position to have conversations with other people about how you're making sense of the math and how they're making sense of the math, like, that's not even hard to do. Like, I don't have to go take do a master's degree. I can just have more conversation and ask more people how they're thinking, and ask more students how they're thinking and that’s how we learn.

00:47:14:20 - 00:48:13:15

Erin

Well, you know, I think that that again, in my experience, something like that fraction quick look, where maybe the concept is more in elementary math. That actually tends to bring it out more in secondary math. I have had the experience many times where I'll give a task and everyone does it the exact same way.

Everyone does it procedurally and nothing else organically comes out of, you know, because it's like we just were drilled on procedurally or we know because it has become easy for us. We know the procedure, whether or not we have made sense of the procedure, maybe we have, maybe, maybe we haven't. But we know it's the most efficient. So I don't know if it's in there.

It's so that's a great point. But maybe, maybe, you know, bringing something out that has a visual component that can be seen in different ways helps make that a little more true. But I'm glad you all do have different ways of thinking about it. So you can have that experience frequently. 

00:48:14:00 - 00:48:39:09

Joanie

Yeah, for sure. I think maybe, maybe the caveat on my suggestion is cross grade level.You know, that vertical articulation component. Yeah. It's that that's going to help on both ends. Right. Like asking an elementary teacher how they think about, you know, that that ten frames idea with the fractions in place is going to they're going to say things differently than a secondary teacher would say. So yeah, 

00:48:39:09 - 00:49:12:02

Curtis

I think and just to follow up one, one maybe step further is you brought up the visual.

But I think there's also just the the effort or the intention of making connections. I think if we go into these investigations cross grade level or even within the same grade level, but making connections and thinking about what are what mathematics is this connected to either in the future or how where did we what were the roots that built up to what we're experiencing?

00:49:12:02 - 00:49:41:03

Curtis / Erin / Joanie

Are the procedures or the concepts that we're being able to demonstrate here? I think that part of that should be part of that investigation. I think so too. Yeah, absolutely. I have enjoyed this so much. Yeah. Me too. This is just so, so good. I really feel like we've been able to grow a whole lot. And so really appreciate you taking the time to hang out with us and chat with us today.

Yeah, absolutely. You back for having me any time.

00:49:43:16 - 00:50:01:13

Joanie Outro

Well, that's it for this time. Be sure to check the show notes for the resources we mentioned and others you might want to explore. We would love to hear your feedback and your suggestions for future topics. And if you're enjoying learning with us, consider leaving a review to help others find us and share the podcast with a fellow math educator.

See you next time!