0:00  
Hello, and

0:00  
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 fan Heiser, and I'm talking today with Matt Campbell, from the Department of curriculum and instruction, and the College of Education Human Services at West Virginia University, and co authors Aaron baldinger, and foster graves from the Department of curriculum and instruction at College of Education and Human Development at the University of Minnesota. Do I just learned that faster graduated Yay, congratulations. We'll be discussing the article representing student voice in approximation of practice, using planted errors in coached rehearsals to support teacher candidate learning, which was published in the September 2020 issue of the mathematics teacher educator journal. We will begin by summarizing the main points of the article, and then discuss in more depth the lessons they shared in the article, their successes and challenges, and how these lessons relate to their other work. Thank you for joining us.

1:15  
Thanks for having us, Eva.

1:16  
It's great to be here.

1:17  
Absolutely.

1:18  
All right. So let's jump in. Can you give us a brief summary of the article in the results.

1:25  
So in this article, we focus on sharing some of our learning around our use of coach rehearsals in secondary mathematics methods courses, and our particular focus has been on supporting teacher candidates around their work and facilitating whole class discussions, and particularly around the moments of discussions where students say things that are what we call errors, but that could be anything where a student says something that's incomplete, imprecise or not yet correct. And to focus on that practice, we have incorporated into those rehearsals, what we've called planted errors. So a teacher candidate is leading a rehearsal and their classmates in the methods class are serving as the students for that rehearsal. And one of those teacher candidates that are playing a student, we actually give them a more defined role a contribution to provide as a student that is either imprecise or incomplete or incorrect. And so we've considered how we've designed those and the implications, those have an opportunities for learning for teacher candidates across a number of components of skill. And so we found that there are certain ways in which we design those planted errors that seem to better support teacher candidate learning. So we've contributed examples of our plants that errors and what some of those outcomes were, what we try to bring forward in this article as well is that really any time a teacher educator is engaging teacher candidates in what's called an approximation of practice. So a coach rehearsal would be one example of that. So an opportunity to engage in the work of teaching in a setting that's much less complex than an actual classroom, that teacher educators need to think about how student voice is being represented in that approximation. And so our use of what we call planted errors would be one example of that. And we try to put forward some considerations for others, not just around, planted errors in code rehearsals, but how student voice is represented in other kinds of approximations as well,

3:44  
that was a lot of stuff. So let's kind of deconstruct that a little bit across the next few questions, let's kind of start by digging into what is the problem of practice that you're addressing,

3:58  
there are a couple layers of the problem of practice. But the first layer is, if you want a teacher candidates to learn how to respond to errors, they need to respond to errors. So if you're, you know, taking a practice based approach, and you want your candidates to to approximate practice, you have to get the error in that space, in the context that you want it. And I think what we found, as we started doing rehearsals in our methods class is that sometimes errors would come up, but not always. And as the teacher educators, we wanted to be in the role of coach, which made it difficult. If we were the ones contributing the errors, then the teacher candidates would kind of know, right, they know that. That's whatever comes from the teacher educators kind of a plant and using planted errors was a way to get the errors into the student voice and to Make it like a bigger focus of what was going on in the rehearsal.

5:04  
I just think that students, there were these teacher candidates that are rehearsing their peers are acting as their students. I think we also just found through working with them that they needed, I think, a sense of guidance to play students authentically as well, because we want to create a space where they can engage in these practices, but we want to be as authentic as possible, and asking teacher candidates in a postback program to be playing, you know, middle or high school students, and to do it in a way where they maybe have ideas that are, you know, not yet complete or mathematically imprecise or incorrect, to have those be present authentically. I think the planted errors allowed us to keep them in there in a more authentic way.

5:45  
So let me try to get a vision of what's going on. So I think, if I'm understanding correctly, the problem of practice your dressing is you want to allow teacher candidates to lead a discussion that includes errors, and you want to bring in those errors into the discussion. But do you want them to be authentic? Am I close,

6:16  
we look at authenticity in a few different ways. And so like Foster was saying, we don't necessarily want to rely on the teacher candidates who are playing students to kind of come up with these possibilities on their own, from our own work and talking with others who have engaged in rehearsals, sometimes there's this inclination for teacher candidates to sort of throw those curveballs themselves, they think they're doing their peer a service by kind of providing this novel situation. But we don't really expect that an emerging teacher has the clearest sense of what a student might do, not only what they might say, initially, but how that student might carry through that idea. And so when we think about authenticity, we're thinking about that in a few ways, what a student might be likely to say, but also what some of the reasoning is behind that that would actually play out in the rest of the conversation.

7:15  
And just building on that we are also taking on this idea that responding to errors isn't just what you say immediately, the response to the error is extended as long as that idea has been surfaced. And so part of what we wanted to approximate is how that ongoing conversation sounds. And to get that to be something where the teacher who is rehearsing is accountable for those continuing ideas as they emerge in the course of the discussion.

7:51  
So if I want to do this, how do I imagine setting up the class? Like how do I give one of the students this error? Can you talk us through what that looks like a little bit?

8:04  
Sure. So what we do is we have these errors planned in advance. And I choose a student among my teacher candidates who will be the planted error, and I give them the slip of paper, or this year, I gave them the Google Doc link with the information about the planted error. So they have a little time to think through it. I asked them not to share the planted error with other folks, but that they're taking on that role for the whole of the rehearsal. So the planted error tells them when they're going to make their contribution, what will convince them what won't convince them to change their mind or change their reasoning? And yeah, then they take that role on and

8:52  
how are the other contribution to the discussion? Just perspective, teacher contributions are what are the other contributions?

9:02  
Yeah, we set up the expectations for the rehearsals in general that people are otherwise responding as themselves. They are engaging with the mathematics not in a sort of play role, but sharing their own thinking, which sometimes might actually have imprecision or something that's incorrect. But otherwise, but people are engaging with the ideas of the activity themselves. And then there's this one role that is the planted error, where around that one idea they are contributing what we said and like Aaron said, what we and what we highlight in the paper is part of the evolution of these planted errors was to incorporate more information on what might convince this student of different idea and also what might not convince this student have a different idea, this play student if you will, because we realize that the teacher candidate doesn't necessarily know that So early on, we found that they would kind of throw out this contribution, and then not really know what to do next, depending on how the teacher or other students responded. And so that was part of the evolution of these planted error roles that we put into place. Yeah,

10:17  
I think the idea was, we weren't just telling them what to say. I think the idea is we wanted to provide as clear window as possible into the reasoning behind that. And it wasn't just simply a matter of, yeah, this is what I want you to say when they call on you.

10:30  
So it sounds like that first person who gets the planted error, but then the whole class gets to really engage with that error, because of the way it is done presented. In the real setting,

10:47  
does it become clear pretty quickly that way? Like, who is the planted error, or is that not become clear when you work? Your students, one of the examples we provide in the paper, and there's also an accompanying video available online, it's sort of our first phase example that we show, it's pretty clear in a few ways that that the student is contributing some idea that is not there. In part, because the student didn't really know what to do once he said the initial thing. Other teacher candidates in the class were kind of like laughing at this premise that this idea came forward. And then it just kind of got left. I think, by and large, people catch on to the fact that someone has contributed something that they were kind of told to contribute. But in the ways we've improved upon setting up these planted errors and the information that that's given to the teacher candidate, it's become less of a farce, and more of an opportunity for the teacher candidate who's rehearsing and the other teacher candidates who are serving as the class of students to really engage with this idea, as if it was contributed just like any other idea, because really, our goal is that in responding to errors, what we call errors, we still want to position these student contributions as assets and as resources. And so we've seen over time that that has improved, even if they kind of know that that individual was given something specific to say.

12:28  
So two follow up questions to that. One is how do you select the errors that you introduce into the discussion? Yeah, let's start with that one,

12:39  
that has been part of our work as a research team to develop these errors. And they are based on the content that we're focused on, as well as types of errors that students typically make. So in the paper, we have three different sort of types of errors. So one error is where the student correctly identifies an equation as an example of a linear function, but does so for incorrect reasoning. Other kind of common errors would be taking a card that is, it's a line, but it's not a linear function, because it's not a function and calling that a linear function. So that's another kind of error that we have included. And so the errors are really targeted at the mathematics. We're also intentional about how we situate them within the in the context of the discussion, we kind of know what the routine of our sorting task rehearsal is. And so we say during this portion of the discussion, here's where you're going to share this thing. So that it's clear that this is an error that needs some attention. And it's also a space then for Yeah, for people to ask questions when they're not sure what to do.

14:00  
So are the errors based on your analysis of the mathematics or on what the research tells you is difficult for students or a combination of those or something else?

14:16  
I'd say a combination of those.

14:19  
Okay. And you introduce these errors? Why are the planted errors because they would not come up? They don't reliably come up in the conversation anyway. So correct.

14:37  
Yeah. Yeah, I

14:38  
would say at least not authentically.

14:40  
So these are errors that would not come up authentically in your methods course, but would come up authentically in a secondary classroom.

14:51  
That's our goal. Yeah. Sorry, man. Go ahead.

14:55  
No, I would I would say that's right. Even though we don't claim that Any of those would reliably come up in any classroom or that there wouldn't be other ideas. But along with all this, we want teacher candidates to not just know how to respond and action, but begin to see student contributions, even if they're not complete or correct, that these are still resources, they are still sensible, and that there are ways to respond to them, that can be part of the ongoing discussion of the classroom toward whatever mathematical goals are set out. So they become these examples of how to position student thinking in that way, based partly in research based partly in our own experience, knowing that that isn't necessarily the full representation of ideas that might come up in a discussion, but it's, we're more targeted with that.

15:49  
So the idea is not necessarily addressing the mathematical concept or the particular error. The idea is to learn how to incorporate errors in whole class discussions,

16:05  
I think that's accurate, I think that's a good way of putting it responding to errors is particular. And it's not particular, it's something that transcends content and activity. And something that research has shown is a place of struggle for teachers, because they have to balance so many competing demands in the classroom. And we want to make responding to errors, unintentional object of study for our methods, students, and we have found that the planted errors in the coach rehearsals really helped us to do that.

16:39  
Alright, let's jump into the question of why this is important. Why is this an important skill for your method students to learn?

16:48  
I think Aaron just highlighted that a little bit that we know that these moments come up in a classroom where a student says something that here we classify as an error, and that teachers often respond in ways that position those students or those ideas in negative ways, so as something that can be that should be avoided, or moved on from or quickly corrected, when instead, we want. And then in turn, there's a resulting impact that has on the student themselves, where they are not positioned as kind of a value to the mathematical discussion or the mathematical community. And so we want to change that and have teachers responding in ways that position students more productively. And so that's why we've focused on this practice, in particular, and again, you know, and then needing to create the moments where that can be a practice that is developed through these rehearsals.

17:48  
So I think we're wrapping our head around what you did. How do you know it works.

17:54  
So just to start what was important for us in figuring out if this intervention work was figuring out what working meant for us. And to that end, we adopted hammerless, and colleagues framework for teacher learning, which talks about several different components of what is encompassed by teacher learning. And it's something that other scholars have taken up in this space of practice based pedagogy as well. And so the components of that framework include, like tools and practices, right, the sort of doing of teaching, which was what you would expect. And it also includes vision of teaching what you imagine teaching ought to look like, and your dispositions towards teaching towards your own learning as a teacher, and it includes the framework includes understandings of content and students as a big component of what teachers learn. So once we kind of unpacked that framework for ourselves, we were able to use that framework to help us analyze the video from these rehearsals to really make sense of what was going on.

19:10  
In addition to videos of these rehearsals over our two sites over two years, we also had additional data such as teacher candidates video annotations of those same rehearsals as part of the class activities, and some written reflections from the teacher candidates in general, but also particular from the people who rehearsed. And so altogether, we were able to look across those data and particularly around the moments where these planted errors were being discussed, what we call planted error episodes of the videos and map opportunities that arose in those moments, to these different components of teacher learning, tools and practices, vision, dispositions, and understandings of students and students and content. And so what we saw was that as our design of these planted errors improved in a way that we defined as more authentic, that these opportunities to learn across these components were increased. And we see that either in things that happened in action during the rehearsal, or things that people were highlighting either through their video annotation, or through their reflections that map either to tools and practices, vision, disposition, or understandings.

20:36  
So I think I skipped over the question that typically comes earlier, but I want to give you a chance to dig into it now. How does what you do connect to prior work or literature in math education?

20:52  
This work builds on a couple different strands of the literature in math education. First and foremost has been the focus over the last decade or so on practice based pedagogies. And Coach rehearsals in particular, that work, especially in elementary math Ed has really shaped this project from the beginning, in thinking about what should coach rehearsals look like, in the context of secondary math, teacher preparation, how can they be opportunities for teacher candidate learning, in addition to that literature, shaping our design, just of the of the coach rehearsals themselves, it has also shaped our approach to how we're analyzing what comes up the types of decisions we're making as coaches and trying to contribute to that body of literature. The other big body of literature, literature that we're drawing on here is the robust work around responding to errors. It has helped us to conceptualize what we mean by error. And to be really intentional about naming errors as this kind of springboard for inquiry, a place that is a worthy and is an asset that we want to keep bringing up and bringing to the fore. And it's a place where there is struggle, there's a struggle in different parts of the framework for learning that teachers have. So they may struggle around what they want to do with what they have the practices or tools and feel equipped to do. And they're competing demands around what one student needs versus what another student needs. Those are the two main strands of literature that we're drawing on. And please met and foster add on to what I just said,

22:44  
we build on a long body of work around practice based pedagogies, and teacher education. And like Aaron said, specifically around coach rehearsals. And so while including out of this journal, so there's been previous articles that have focused on rehearsals as a pedagogy in secondary math, Teacher Education. So we certainly see this as adding to that and giving some specific examples that teacher educators could take up in their own practice, where what we see this adding to also is, again, this this acknowledgement that anytime you engage a teacher candidate, or a practicing teacher, in an approximation, a practice or something that's engaging them in the work of teaching in some simplified or less complex form, that you have to take into account, what the students in this approximation are doing, whether they be written into a script, or whether they be presented in an animation, or whether they be playing roles in a rehearsal. That interaction between the teacher and the students is really how you engage in the work of teaching. So it requires a lot of thought. And so we saw that as a way to contribute, both with the tools of these rehearsals and these planted errors, but also a broader discussion about what teacher educators need to be thinking about as they engage in this practice based work in general.

24:17  
And then in relation to when Aaron was talking about like the work on errors in previous literature and about how errors often are viewed as, like, problematic by by teachers. And oftentimes, they want to be corrected quickly. So they can kind of keep moving on and through other work with these teacher candidates. We've gotten a bit of a window, that those tensions are things that they feel as well, and those tensions that they feel of wanting to correct the error in that poll to do that. Those oftentimes can come from a good place of their they have good goals around you know, not wanting students to latch on to incorrect ideas or not wanting to you know, kind of get their lesson off the course that they expected it to be on, or, or things of that nature, but Through those these other discussions and trying to see them as opportunities to unpack student thinking, we've been trying to find ways to also realize that they can kind of resolve that tension a little bit as well. We're trying to, we're starting to understand our reason as to why teachers feel that need to kind of correct the error and move on. But trying to open a window into them being able to see that they can kind of serve two goals here with the types of moves they can make.

25:26  
All right, so let's wrap up by giving some advice. So I listened to this podcast, and I'm all intrigued. And I want to do this now. How do you see other people using your innovation? or What advice would you give? While we give some specific examples of planted errors that we've used in coach rehearsals and we have other types of coach rehearsals that we've done and in turn other planted errors that are associated with those? I think we would want. And certainly, we would want people to feel like they could pick those up and use them in their own practice. We'd also want people to consider, in general how they might design these planted student contributions in something like a coach rehearsal, and we found really important that it's not just what the error is, or what the answer is, maybe it isn't even an error. You could have novel student contributions that are correct. But regardless of what the answer is, or the solution strategy, we found it really important to provide that additional context of what is the reasoning behind that, especially if someone is playing that role? Because that's what informs them to know how to respond next, next, and that that's a way we've thought about what often authenticity really means is that it's not just does it sound like it came from a student? Or does it seem something reasonable that a student said, but does it seem to result in an interaction that is like what you would see or would want to see in a classroom? So for people to think about these types of planted contributions, using some of those broader design principles, I think would be a key takeaway,

27:21  
that kind of nicely Connect, I think, to your goal as well, which I believe is that your prospective teachers learn that there is reasoning behind errors. And that's really what we want to get out into the discussion. So that we can build on that reasoning.

27:47  
Yeah. And in the moment of rehearsal, not everyone is looking at that planted error roll that we give to one teacher candidate. So it's not like they're seeing the bulleted list of that reasoning. But where the reasoning comes out, is in the discussion itself. And we've almost we've talked about it in a way in our own work, where and in our own discussions like, how is the person who's leading the rehearsal, the teacher? How are they held accountable? How can you know, it's one thing if they use some talk moves, or they respond and seemingly positive ways, but then they kind of just move on, and maybe leave that idea and that student behind. But if there's more robust reasoning behind what this person has contributed as a student, it sort of holds the teacher accountable to confront that reasoning, and how the conversation would play out. And so while we might engage the group with those planted error cards, after the fact, for that additional opportunity to engage with the those versions of student reasoning, in the moment, this has created a much more robust way for teacher candidates to kind of see that reasoning play out in a discussion.

28:59  
I agree. And I think that one of the important ideas that we brought into this and also are taking away is the importance of thinking about the interaction over an extended set of talk turns, because that's what leading a discussion is. And so that's where the learning is going to happen. It's not a one and done move. There's no magic approach to responding to errors, but it's, you know, thinking through what it looks like in an extended way, and building that interactional nature into our planning and into our thinking about how to support teacher candidate learning.

29:41  
Well, thanks so much for joining us today.

29:44  
Thank you. It's great to talk about this.

29:46  
It's great to be here. Thank you.

29:48  
For further information on this topic. You can find the article on the math teacher educator website. This has been your host, Ava Sennheiser. Thanks for listening and goodbye.