Eva 0:00 Hello and thank you for listening to the mathematics teacher educator journal podcast. The mathematics teacher educator journal is co sponsored by the Association of mathematics teacher educators, and the National Council of Teachers of Mathematics. My name is Eva Anheuser and today I'm talking with Esther Billings from the Department of Mathematics at Grand Valley State University, and Barbara Swartz from the Department of Mathematics at West Chester University. We will be discussing the article supporting pre service teachers growth in eliciting and using evidence of student thinking. To show me narrative published into September 2021 issue of the mathematics teacher educator journal, we will begin by summarizing the main points of the article and discuss in more depth the lessons they shared in the article their successes and challenges, and how these lessons relate to their other work. Esther and Barbara, welcome. And can you briefly introduce yourselves? Esther M. H. Billings 0:57 Thank you, Eva. This is Esther billings at Grand Valley. And I've been a teacher educator for 23 years. Barbara A. Swartz 1:04 Thank you, Eva. This is Barbara Swartz. I'm at West Chester University. This is only my second year there. I started during the pandemic, which was a whole exciting adventure, shall we say. And before that I was at McDaniel College. So some of the data actually came from my work there. Eva 1:22 Well, thank you so much. And let's get started by just with a brief summary of the article, including the results. Barbara A. Swartz 1:30 So this started with both of us kind of wanting to help pre service teachers understand the complexities and nuances of analyzing and thinking about students mathematical thinking, a lot of times novice teachers would go into the classroom have interactions with children, and we, you know, you'd come out and say, How did it go? And they'd be like, Great or fine, or I don't know, right? So really getting them to understand and pay attention to and be able to talk about a name, the details of Well, what did the child think? Why were they thinking that? And then what did you do as a teacher to respond? Did you build on it? How could we move it forward? Versus Oh, they were they said the right answer. They said the wrong answer and moving on. So that's kind of like the really big, where this kind of came from. And so in our I was teaching a content course at the time, and Esther had an integrated content pedagogy course. And we were thinking about, okay, so how do we get them to focus on the content, the mathematical content, more authentic ways, through the lens of student thinking and analyzing student thinking than just, you know, kind of very traditional textbook quizzes, that type of thing, even videos and stuff. So we had these mediated field experiences where we would take our entire class and work with the same classroom of I was in fourth grade, and Esther think you were in third grade. And so in that, we had them working regularly. Esther was there every week for 10 or 12 weeks, and I was there for seven weeks, kind of like every other week of the semester. And so through this experiences, we're, you know, again, having these conversations with our pre service teachers, and they're still having trouble kind of articulating identifying, discussing this mathematical thinking. And so Esther came up with, through collaborating with actually an English teacher, educator, literature teacher, and literacy teacher educator, this idea of this structured reflection tool, and that's what we, she named this show me narrative. And when she shared it with me, I was like, This is great, I need to do this. And so being able to use this in our classes, we were able to actually collect the evidence of our pre service teachers growth in their ability to not only elicit student thinking, but also attend to interpret and respond to so by what they wrote in their these reflections using this tool, we were able to actually identify like, wow, first, they were just saying these kinds of vague things. And now look at the way they're able to identify, you know, either given strategy, talk about how, what the student said, what they said, how they built on the strategy, and we talked about responding to in both verbally as you know, I'm going to respond to you via question or nice job or whatever, but also visually, so what did they write down? What did they record? How did that highlight the mathematical structure of what the child was saying? And both of us were just blown away by how great of the shift that we saw in our pre service teachers written reflections using the Show Me narrative tool, so we were like, we have to like we both felt it like It felt awesome reading and, you know, grading their papers, which you never really hear. But then we so we went in with a fine tooth comb after the semester to really analyze Okay, can we document what is the edit? Is there actually real evidence here? Is it just like that kind of good feeling that we had? And so we actually found that by the end, I think it was 41 out of 42. Every single one really moved significantly. 41 out of 42 pre service teachers move significantly in these kinds of these shifts. that we identified in the paper. Eva 5:01 That sounds excellent. So I can't wait to dig a little bit more deeply into what this all looks like. But let's talk first about who do you think should read this article? Esther M. H. Billings 5:12 Well, I think definitely teacher educators should read this article, even if and I don't think you necessarily have to be in mathematics to be reading this article, you might want to skim over a lot of deep mathematical details. But the basic premise of is named show me narrative, we want students to be able to show us what they've experienced not just tell us about it, right. So don't just tell me students were making connections actually give me evidence, show me through description, what was said How was interacted. So I think this is a really valuable tool for teacher educators to be able to provide a structure for their pre service teachers and teacher candidates or even practicing teachers, although I'm sure they would want to put this much detail into it, but maybe part of a more intensive professional development opportunity where you're really focusing in very particularly to attend to children's thinking. And I think that's what this does, it slows it down. We don't have to think about everything that happened. We're just concentrating on a particular confined moment in time and really unpacking that, and then using that to inform other interactions. So it's like we're taking a small sample, it makes it manageable. Yeah, that Eva 6:26 makes sense. So usually, when you write a paper for MTE, right, you identify an important problem that your work addresses, what is the problem that you guys were addressing? Barbara A. Swartz 6:38 I think it was really this idea of having pre service teachers being able to talk about right about an animal, you know, and in the AMT standards really analyze this student thinking about mathematics, they, again, were able to kind of talk generally or speak generally, oh, they were making connections, or oh, you know, we were using tennis frames, they're like, Okay, but what were they doing with them? And why is that important? And what are we building on? And where are they all using it? And how are they using? Are they using it in different ways. And so this tool helps to take all of the complexities of an entire lesson. And we really just focus, as Esther said, on one little instance. And you can define that however you want. That's why, again, I think she said, anybody can really use this or read this article and use this tool, you can define that instance, for this data, we were defined it as an instance of student thinking, and whether that's two different students comparing their strategies, or one student using two different strategies. So you can kind of define that in the moment or, you know, for your paper for your assignment. So that was the problem is that they can only kind of speak more vaguely generally about what's happening, or kind of how they felt about it, Oh, it went well, or it didn't go well, right. That's the language that they had, because that was kind of the feedback that they were getting on their teaching from weather field supervisors, mentor teachers, and with this tool, it really helped to scaffold, okay, this is what we want you to think about this is how we want you to think about it. How are you going to describe now this kind of interaction versus just telling about it? Eva 8:09 So would you say it's a fair representation to say, the tool helps them focus on student mathematical thinking? Esther M. H. Billings 8:19 Absolutely. Yes. And I think too, for both of us, right? Like we often have many goals in teaching these teacher education classes, but for both of us, right mathematical thinking was central to our content content, pedagogy classes, as it is in methods courses. And we were really wanting to hone our pre service teachers ability to elicit attend to respond to an interpret students mathematical thinking. So connecting it to that core practice of using an eliciting student thinking was really key. But as the research has shown around a core practice, you can't listen and attend to thinking if you don't understand the thinking or can't even articulate what the thinking is, right. So if I can't tell you anything specific about the mathematical interchange is going to be difficult for me to elicit beyond tell me what you did. Right? I'm not going to be able to respond in a way that challenges or furthers your thinking if I don't understand what you've done. And so, especially in these really early classes, and so for mine, it's their first experience really working with for many of them children, and then mathematics. For most of them, they don't know how to haven't developed the skills and it's been well documented in the research too, but they don't have the skills to really think about the thinking it goes too fast. They're so focused on what they're going to say next, right, or things like that. So how do we develop this skill to really notice and pay attention to the children's thinking? So that's that was, why this was created in the first place. How can we slow it down? And then the more you can practice that the more you can develop that skill. Eva 9:59 So just leads nicely into the next question that asks about what existing work in the field did you build on? And I think you just mentioned some of the noticing work and the core practices, right? Esther M. H. Billings 10:12 Yeah. So I think part of the challenge we had in this writing this paper is we were drawing from so many different types of research that how do we put it together in a coherent way? So in terms of the teacher noticing work, we were drawing heavily on Jacob seminal work about teachers noticing Vanasse his work as well, you know, what does that mean? But then we were also coming to it with a practice based perspective on what teacher preparation should look like, and in particular, a core practice approach. So we heavily relied on that McDonald's study about the learning cycle in order to think about what kind of experiences do students need in order to enact these practices and really be able to internalize them in a way that they can then utilize, right and develop their own mathematical understanding as well. And that knowledge needed for teaching and that core practice work? There's a lot of studies you know, we've got Kazumi has been a pioneer there. GROSSMAN Lampert. Deborah Ball, there's just a lot of studies. And so it was and then yeah, so the core practice work, the work on teacher noticing. And then because our lens was looking at that core practice of using and eliciting student thinking that was sort of the piece that held it together. Eva 11:31 So the idea is that your pre service teachers have a way, I think you use the term slow down to slow down and really learn how to notice student mathematical thinking. So can you tell us and remember that we're talking to people who might not have yet seen the paper, what the innovation is, and what it looks like? Well, it doesn't Esther M. H. Billings 11:59 feel very innovative. It's so simple. It's essentially a two column table. I mean, it's very simple. I think that's what I love about this tool. So on, if you imagine making a table, starting out with two columns, the left column, you're going to write a very descriptive narrative about what's happening. And it's going to be enough. So it's like you can visualize like a camera was rolling. It's not there. But what was what was being said, what was being done. How did the children respond? How did you respond? And you're just taking a snapshot in time, one problem, one solution of one problem is the way Barbara and I did things, we were working in groups of children, right? So you're just writing a description, and you're just trying to get down the details of what actually happened. You'll have notes of things that you recorded as the teacher, if if it's a discussion, and you're recording children's thinking on a whiteboard, or whatever you're using, or if you're using manipulatives, you might have a photo of what you're doing with them with the children. And then you go back later, and then depending on what it is that you're focusing on, for us, it was this idea of this core practice, right, then we have the students go back, that was later I guess, the core practice first with the mathematical noticing just to even name, what was the child doing mathematically? What types of strategies were they using? What where's the evidence, then? That was the initial piece of noticing the mathematical thinking, then we added a third column. So you would line it up, right? When you would have this description. If you get one across the like a row you would have the analysis mathematically is that was the focus of the course. And then the core practice? How were you doing whatever it was we were focusing on? And unpacking that practice? What does it mean? And where did you fail to do something? So you in looking back, you think, Oh, I wish I had asked this question. Oh, instead of asking a question, I told the child what to do. I took the manipulatives out of the hand and put it where it should be. And I realized now that's not really eliciting the child's thinking. And so if I could redo this is what I would do. So it's a chance for them to reflect back on the situation in a very specific way about whatever it is you're focusing on. Eva 14:11 So is the narrative something that you provide them with? Or is that something they create while they work with children? Barbara A. Swartz 14:21 They write it after? So So yeah, so the, the tool is basically as Esther said, just kind of a two column table, you know, you open up your Word document on the left, we want them to write this narrative, kind of like either a dialogue or a screenplay or however, you know, whatever analogy they need, and and we give them examples there. And in the paper, we have lots of appendices that have examples of what this is examples of the assignment. And so that way they can, it's difficult, it's completely different than anything they've been asked to do before it's usually like, you know, right a reflection on your interaction. You know, your lesson you're this You're that and so they're just using To the kind of free forming paragraphs, but no, this is more succinct in kind of like they do a name, and then what they said, then maybe some stage directions or what happened, and then a name and what that person said. So it's really just the narrative is just this dialogue, back and forth. And then in that second column next to it, it again, it's not this free flowing paragraphs form, it's now Okay, pick out the instances from the left column of what happened, and then now analyze them. So it's almost like kind of explain why that thing had been explained why the child said this, or did that or talk about what was kind of where the child is on the developmental continuum, they, they know this, but they don't know this, or we're working towards that. So that's kind of that second column to get them analyzing. And it's not. I know, narrative kind of makes it sound like this big written paper. But it's actually not a lot of text, there's more whitespace and kind of pictures of the evidence of the whiteboard, what was recorded, or the manipulatives that were being used. So that way, the again, the teacher candidate, or the pre service teacher can show what's happening so that we, as the reader, you feel like, Oh, I get this picture in my head, I can see this unfold these events unfolding. Eva 16:09 So I have done similar things in my classes, and I have found it difficult to get my pre service teachers to actually even just create the narratives. Esther M. H. Billings 16:20 Yet, that's what's been so amazing about this tool is that they do it and I think part of it is we condense it down, right? So it's, it might be a so for me, they're teaching a lesson with two groups of students. So that's 50 minutes of interaction with children. And maybe they're just telling you about five minutes worth, right. So they're not allowed to record, it has to be on memory. So it helps them pay attention in the moment to what happened. And it helps them take better notes I we always placed our students in teams, so one can take notes or and they take photos of their record. So it helps to jog their memory. But I think for us, we've spent a lot of time at the beginning, co writing a narrative together or looking at examples providing feedback. So they're trained to do it. And a lot of them have commented, oh, I have to ask better questions, because I know I have to write this narrative. And there's nothing to write if I don't know what the child's thinking. So even on a very simple literal level, it's helped to motivate more careful question asking, right, so they have something to write about, at least for one piece of the interaction. So I've been surprised how it's not just the really motivated students, everyone has written descriptions where you can imagine what's happened, but it requires giving feedback and developing the expectation. And I think they see the value in it, they see how they're growing. So it doesn't feel like busy work. Barbara A. Swartz 17:51 Yeah. And so as I said, we don't, this isn't a one and done. We do this every time every week, with Esther going 10 or 12 times to the school, she has them, right, at least five of them. And it's this feedback cycle of like, what do you want me to focus on, you know, or in a lot of it, we found giving written feedback hasn't been super helpful, but kind of just Okay, have like that 10 minute conversation with the teacher candidate. Okay, come to my office, let's talk about, you know, this is what I see. So tell me about this piece, because you can, you know, give feedback all day long, and it doesn't really help them. And our goal is not necessarily to have them be better at writing these reflections, right? We want them to be better, or, you know, more effective in their teaching. So this is just kind of that intermediary to help them to really, as we said, slow down, unpack and magnify, okay, we're just looking at this one instance, doesn't matter what, how the lesson started, how the lesson ended, we're going to look at this one instance. And now let's talk about it. And then what they've said, my pre service teacher said, Oh, that thing we did here, I used it in my literacy practicum I use that in my science practicum so it really once they understand what they're doing and why they're doing it, it does. They've anecdotally shared that they that it has transferred. So that was really cool to hear. Eva 19:07 Let's get into the research question, what was your research question? And then what evidence did you use to answer it? Barbara A. Swartz 19:15 Yeah, I think that will also help explain help to see this this tool. What we did we said that this show me narrative happened over multiple cycles. So they did I think for each of us, at least five of them maybe six and then they the final one that we analyzed for the show the evidence, we had them go back and redo one of those five or six and we call it the polished show me narrative. So that's where we really saw this growth these shifts because they were able to go back remember this instance and now really name oh, I, I said this, and I should have done this. And that that was really where we saw the growth in that final one, the polished one. So what a research question was is we're looking for what ways did the pre service teachers have ability to elicit, attend, interpret and respond to students my thinking shift over that, that it was just one semester over the one semester as they completed the Show Me narrative. So through these written reflections through with the Show Me narrative tool, then that's when we were able to use that as evidence. So we analyzed not their first one, but their second one, because that was still early in the semester. And the first one is like they didn't, yeah, as you said, they don't really know what they're doing yet. So we didn't, so we didn't start with the first one. We start with the second one, and then we did we compared it against the polished one. So now, it wasn't necessarily apples to apples, the same exact lesson, the same exact reflection, or, you know, lesson that they were reflecting on, but we were able to identify the instances where they were able to elicit the instances, they were able to interpret the instances they were able to attend and respond appropriately. And even in our analysis, we also identified missed opportunities. Were there ones that they named, oh, I should have done that. Or as we're reading is like, oh, you should have done that that does. So we coded for that. So that way, we could see how many opportunities they named as missed opportunities, and how many we would name as well to kind of identify those shifts? And great, so you want to add? Esther M. H. Billings 21:14 No, I think that that summarizes it. And because we were so focused on the idea of we need to really pay attention to how the children are thinking what their ideas are, right, by the second show me narrative. Most of them were getting much better at being able to take notes, right, rely on their partners, right, so that they could write something descriptive. So what we're really focusing on was a lot more to do with, okay, so then what do they do with this information? Right? How do they? How do we build up this capacity to respond in appropriate ways? How do we build up this ability to ask different kinds of questions in the moment, so that they can really build on children's thinking, not just report what the children said? But how do we extend that right to really delve into what this core practice is about? Eva 22:03 So I remember from reading the article, that there's a lot of numbers and a lot of changes. So I wondering if you want to pick like a few that stood out where you saw some changes happen? Barbara A. Swartz 22:19 I think one of the ones we use the the NCTM in the principles to actions, you know, they didn't eliminate all a teaching practices, we went into the our core practice of we're focusing was eliciting and using evidence of student thinking. But one way to elicit is to post purposeful questions. So we dug into that chapter a little bit more, and they had this great framework for the types of questions used. And so what we noticed is we were able, we coded them the types of questions that the the pre service teachers reported that they said, Oh, I asked this, the student this, I asked the student that you write, to demonstrate their ability to elicit student thinking. And so what we were able to do is see the shifts in the types of questions that they asked from one of them, we had to even so in the principles to actions, they have four levels, starting with gathering information all the way through encouraging reflection and justification. Well, in the beginning, we had so many pre service teachers saying, don't you think you'd like to use multiplication there? So and in that chapter by Fogle, and I forget her co author, they named this as a funneling type of question. Eva 23:27 I was verbalizing men, right? Yes, yes. Barbara A. Swartz 23:31 And so what we did was we lit we labeled those funneling types of questions as a type zero. So we know that they're not productive. They didn't help build on the students thinking, but they were asked by the pre service teachers early on in the semester. And so that shift from asking kind of those funneling questions and only a handful at most, I don't think we had any level fours in the beginning, towards by the end, we had a lot almost happened. It's not like, this was one semester. So it wasn't like, you know, zero to all, but it was at least half of them in the threes and fours. So if I could find that that table, I thought that was a really significant, really nice way to kind of provide that evidence of what shifted and how it shifted and why it was different by the end of the semester than at the beginning. Esther M. H. Billings 24:16 And I think what's also notable is, we provided structured lesson plans for the students, it wasn't like they created their own activities, and then just implemented them. So it's noteworthy to note that in the activities that they were doing with the children, we had given examples and discussed as part of our preparation. These are great connecting questions. These are great analysis questions. But our pre service teachers wouldn't ask them in the moment, right. I mean, now. So even though they had been provided and they knew what they should, maybe this was something that their instructor was saying you should do this is what effective mathematics instruction looks like. It really wasn't happening. They would just rely back on telling I think what was interesting was that the open sharing types of questions, those seem pretty intuitive, I think students, when they realized that was our perspective, this is something we think was really important to get students to share. Even very early on, they were able to do that, and then even more consistently by the end, but I think what they realized through this analysis is that is not enough, right? Just having the students tell me what you know, you have to have knowledge to be able to scaffold their thinking and to be able to probe more deeply to ask connecting questions, so they can make connections. So the fact that they really grew in their ability, or at least recognize the types of things they should have asked or ways to maybe represent thinking, so students could have made connections, especially when we did a lot of number talks, at least in the data we gathered for these semesters, it was really interesting to me that they, they actually grew in that area, like it was a hunch, but to actually see it in the data was really exciting. Like, wow, this is really helped them make these connections, while acknowledging we were doing a lot of other things in the class. We know, it's not just the tool, but the tool was a great way for them to reflect. And for us to document and see some of these shifts. Eva 26:12 This actually leads really nicely into my next question, which is kind of just wrapping up, like, how would you say, what would you say is the new contribution that your article makes to our field, Barbara A. Swartz 26:26 what I love about it is that it's easy and flexible. So we came at it with we want you to think about this one instance, you know, in your lesson that and highlight the student thinking because we were looking at it through that mathematical analysis lens. For this data. This past semester, I was working with some other research colleagues, and we framed it as we just wanted one moment in the lesson, that would be kind of a noteworthy moment. So you share a moment. And so while we had these two columns of the narrative, and then the analysis of students mathematical thinking, your second column could be whatever, right whatever frame, you want to connect it to have them identify those. And then we talk about, well, we have this third column, then we introduce kind of about two thirds of the way through the semester. And so it's what happened, kind of explaining the students child's thinking about it. And then the third column is okay, so what are the teacher moves that you did, to get them to say those things, right to make those connections, and that's where a lot of those missed opportunities came in. And so again, you can think about that teaching. That third column, as well explained in terms of equity practices, explained it in terms of anti racist teachings, but you know, you could use it to do you know, to focus pre service teachers, reflections in any way you want. And it really helped them to not only to develop the skills of posing, you know, purposeful questions, if we're going to, you know, quote, NCTM, effective teaching practices, to elicit that thinking, but also to, to really think about their role as the teacher and how that impacts the students thinking, right? The students don't just learn the math in a vacuum, it's only you. And it's not even just this kind of show and tell I just, oh, I just tell them what to do. It's really this interaction between the questions, I posed the examples, I use, the tools I use to support them. And this is how those children are now able to construct that actual rich understanding of mathematics. And for them, sorry for the pre service teachers that provided so much motivation to learn the math, because they knew they couldn't ask good questions or respond to students in ways that made sense without really understanding it in the first place. So that for me in the content course, that was like, the Holy Grail, because, you know, we had been doing number talks for how many weeks and it wasn't until they were with the student, the, you know, the fourth graders, and it was like, Ah, this is why this matters. This is why I want to be able to name the commutative property of the associative property, right, this is what it matters. And so it just gave them so much motivation to, to care and learn, because they got to work with real students in real classrooms and do what they want, you know, they love doing which is teaching so and they found another thing was that they got so much more confident in working with students, as Esther said, they never had my students same thing for the data in this article never had field placements in math before. And and because they work in pairs, right, we do everything. They get the lessons, they experienced the lessons, you know, through this whole learning cycle. They know what they're doing, they're prepared. It's not just like, Alright, go ahead in and figure it out. We'll see when you get back. So this tool was able to capture all of those complexities and nuances of that growth, which I thought was really, really exciting. Esther M. H. Billings 29:54 Yeah, and just to build on a little bit of what Barb is saying, like we said earlier, This McDonald's learning cycle was very influential. And in the analysis piece, right, they have these pedagogies of enactment, we felt like this is a way to really get it's a reflection tool really is what it is right. And the research clearly shows that the pre service teachers and teachers alike, right, if they're not taught specifically, they're going to think about things more generally, right, especially with the mathematical thinking. So it's a way to really hone in on the mathematical thinking, but like Barb was saying, you can expand out and look at that through other lenses. So it really is, I think, a simple flexible tool that can be adapted. It's not, it's not difficult, right? And when you see it, it seems so obvious when it's when I was collaborating with his English College English educator years ago that created this assignment, I was just frustrated that I had students at the very end who were unable to pay attention to the mathematical thinking, and I'm thinking we have failed as their instructors that they have like five courses in and still are having difficulties, what are we doing what, What haven't we done to support their ability to notice, and it was like this tool provided a structure and then from that structure, it created these habits. So then you don't necessarily need the tool so much anymore. But it's a it's a simple way to enter in and provide Barbara A. Swartz 31:18 opportunities for feedback and scaffolding. So and that's the whole point of the learning cycle, they get these scaffolded opportunity, or they get scaffolding to learn these teaching practices. But they learn from the scaffolding to understand what is the teaching practice, then scaffolding to plan and prepare? And then scaffolding in the moment of actual teaching? And then scaffolding in the reflection process in at no point are they just left to their own devices. And so the scaffolding and the feedback at each point of teaching, right teaching is not just the time in front of the class, teaching is all the time in front are all the time ahead of time that and then the actual teaching, and then all the time reflecting like, well, what worked, what didn't, what am I going to do tomorrow. And so getting them to understand that teaching is more than just the time also in front of the students is was also really important. Okay, so that Eva 32:12 finishes up the questions I had, is there anything else that you guys would like to add that we didn't get to? Barbara A. Swartz 32:18 I think if for any math teacher, educator, or any teacher educator that's interested in using a tool like this, it seems hard to do. And it is hard to do in the beginning, our advice that we've done, and what Esther did is just great it for completion and provide feedback. And those kinds of feedback conferences are key. Remember that getting better at writing this tool isn't necessarily your learning goal, right? So so give them a little grace, have a little grace for yourself. It can be hard, I think we're the learning is and then we grade like, like each one of his 10 points. And then the final polish one is like 50 points. So like that's where now they spend the time to really polish up one or maybe it's only 20 points, you know, whatever it fits in your course. But I think that that was really key because like I said, I use it in my methods course, this semester with a different frame, slightly different frame, we still use the three columns and had the students mathematical thinking in the second column. But I was working with two other colleagues that had never used it before. And we're like, the I don't know how to keep on top of this, they've handed it in. And I didn't do a good job at all, the first first couple of times I did it in, I didn't give them any prep. In this article. They're appendices, there's tons of resources for prep, and how to scaffold and how to introduce, I didn't do any of that. And this was still some of the data that came out of I think this is literally the first time. So it's like, even if you don't do anything, you just kind of have this expectation. Okay, I want you to write this. Okay, you got a 10 out of 10. But these are all the things you still need to work on. They by that polished one, you really saw evidence of the like, like, Aha, this is clicking this is what I'm supposed to do. This is why I'm supposed to do it. Eva 34:01 Yeah, I have to say reading the article. And the appendices, you do provide quite a lot of resources that it does feel comfortable wanting to try this out. And I just want to note that I have worked with collaborators on a similarly, I'm gonna use simple though, that's not what I want to use to all that's just a paper like three table three column table. We call it the student discourse observation tool. And people write down what they heard students say, and then they classified whether it's procedural justification or generalization. And just having that in your hand helps you pay attention right to what is it that I'm listening for, and how do I categorize it? And so I can see this tool being very similar in that sense, and also really powerful. So I really appreciate you all sharing and Thank you so much for coming on. Esther M. H. Billings 35:02 Thank you for giving us the opportunity. And yeah, I think a lot of times we forget that what can be very impactful is very simple, right? And so sometimes it's just returning to those fundamental principles of what is it that we're really trying to convey? And just focus on that over and over again and give students some agency and in thinking about how they want to grow, and then wonderful things happen. Barbara A. Swartz 35:26 Yes, thank you very much. Eva 35:28 For further information on this topic. You can find the article on the mathematics teacher educator website. This has been your host, Eva Sennheiser. Thank you, and good bye Transcribed by https://otter.ai