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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 Sennheiser, and I'm talking with Teresa grant and Mariana living. Terry is a professor, and Mario is an assistant professor of mathematics education in the Department of Mathematics at Western Michigan University. We will discuss the article diverged and converge, a strategy for deepening understanding through analyzing and reconciling contrasting patterns of reasoning, published in the march 2020, 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 successes and challenges, and how these lessons relate to data work. Terry and Mari thank you for joining us.
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Thank you for having us.
1:03
Let's just jump right in. Can you describe your innovation, the context
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for our work is a number and operations course for prospective elementary teachers. So I had been involved in a curriculum development project for many years. And then Mari joined the faculty and began teaching the course that I had been working on. And we would have wonderful conversations about the course and then began a more formal collaboration, and the diverged than converge strategy emerged from the conversations that we had about our practice. In particular, it was a response to the following problem of practice. So when you do a task over multiple semesters, you start to see patterns in the way students react. And in one particular case involving whole number division, a task always resulted in these two answers, one correct one incorrect, the number of students who did each varied semester to semester, but the two answers always emerged. And in this case, the incorrect answer had to do with keeping track of the meaning and in particular, understanding the idea of remainder. So we began to always include those two responses, and have students try to analyze the thinking but but conversations though different from semester to semester, we're always a little unsatisfying, because once a prospective teacher kind of figured it out and explained it to the class, the rest of the class lost any reason any incentive to dig deeper. So the problem for practice for us was how to engage all of the prospective teachers into making sense of both the correct and incorrect reasoning chains of reasoning. So we designed what we call the diverged, then converge orchestration strategy, and it has four basic phases. So the first phase was to let students work on the task and observe how they did. And then at some point, we move to phase two, where we would interrupt them, we would tell them that we had observed these two different answers, or these two different basic approaches, and we would refocus them on analyzing those different ideas, and understanding and recreating the chain of thinking that led to them. Then in the third phase, we would ask either individual students or groups of students to go to the board, and present those chains of reasoning. And specifically, we asked them to do it without bias, that is without letting on whether they thought it was correct or incorrect, but just to get the chain of thinking out there. And after the chains of thinking we're out there, then we would allow them to go forward to the discussion of validity, whether one was correct or incorrect, or what was going on and why. So that was the strategy. And that's the innovation that the article is based on. And
4:11
from reading your article, it sounded like at least for the strategy that you discuss in the article that the students who solved the problem in a way that was not correct, really solidly believed that they were correct.
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Oh, absolutely, completely believed. And part of that had to do with the structure of the task itself. So as I said, we've been working on this curriculum with various people for years, and the numbers mattered. So this was a division task in which the problem was 189 divided by 11 assuming a sharing meaning of division, so if I have 189 things to share among 11 groups, how many can go in each group and We use in a lot of cases in this course, a first step as a way to prompt them to think differently. So in this case, it was very important that the first step, the estimate was knowing that if you had 220, things shared fairly among 11 groups, there would be 20 things for each group. And because they had to work down, they would lose track of what the hundred and 89 meant, and what it meant to have a remainder. And so they would honestly and truly believe we would have three camps, at least you'd have a camp that absolutely believed 18, remainder nine was the correct answer. You could not convince them otherwise, a group that might be teetering between that answer and the correct answer, and a group that understood the correct answer. But in each of those groups, they didn't necessarily understand the reasoning of the other. And so part of the strategy was making sure that they really understood the reasoning.
5:58
So would you say for the converge, sorry, diverged, converge strategy, that is a key component that the students strongly believe that they are reasoning correctly?
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I believe so. And I believe that one of the evidences for that is that every semester, these two answers come up every semester. And the problem of practice for us that Laurie and I discussed was over the semesters, it really depended who shared the reasoning and whether people were willing to actively engage in that, or whether they would just give up. And so we really wanted to find a way to as much as possible force them to engage in the reasoning,
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but yeah, they issues of status. So for example, as Terry said, you know, who's presenting and in what shorter the presentation happens, matters a bit, too. And so that's starting to diverge and converge strategy in general, not only in this particular case,
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yeah, I was asking this follow up question, because I think that all of us who teach know these problems, right, when we know when we teach this, this is gonna come up. And so I think, then the diverged converge would be a strategy, we could apply to any of those kinds of situations. And we do apply it in multiple situations. I think it's something I wouldn't necessarily do every day. Yeah. And I think you'll have an appendix where you have to go in depth into one example in the article, but then you have an appendix where you have additional examples. But I mean, walking away from the article, it's almost like, I know, I have situations like this in my class, right? Where I always know what happens. And so this could be a strategy that I apply for that. So I think you gave us a good sense for the strategy. Give us a brief summary, what were the results of your article,
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the heart of the article is first describing the classroom instance that Terry mentioned, showing the diverge and converge how it worked in a particular classroom instance with that task, and then figuring out a way to make a connection between our conjectures about how diverged and converge would prompt continued engagement and sort of allows students to get deeper into the conceptual issues and to connect, you know, what we thought should be happening once we do these phases of the orchestration strategy was students actual experiences of each of the phases. And so that required us to innovate a little bit the first time we tried studying our practice, we just videoed our classrooms, and we're trying to look for features of participation and reaction, but it was very hard to kind of make that connection. And so eventually, it led us to adapt this approach of using students confidence graphs, we adapted this from a study that jack Smith and colleagues had been using in studying the transition from secondary to collegiate mathematics. And then in work that I had been doing with jack, we were using it to study student experience at the level of a semester, as they were making the transition to proof sort of looking at students a little bit later. And I thought, Oh, well, maybe we can use this approach of having students report on their experience to give us feedback about how the orchestration strategy is working at the level of a classroom discussion. And so we used a structured version of that tool, this confidence graph tool, we ask students at the end of the class to reflect on each of the phases of the discussion and to give, you know, a reflection about how they were feeling at each of the stages in terms of their confidence and their reasoning. Can you remind us of what the phases are? Yes, so the phases there's the four main phases, they first engaged in their work on the task as presented in the as launched. So in this case, it was 189 divided by 11, with a sharing meaning, and that's
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individual individual difference,
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get some time to you know, get their wheels turning and get into the tasks themselves. It's at this point that the timing is somewhat important here as well, we don't want everybody to come to a complete resolution at that point, we just want them to get started. And then we interrupt when we see some people are already starting to have some more closure, but other people are still working. And we redirect them to thinking about the two answers that Terry mentioned, in this case, 17 millimeter, two and 18 millimeter nine, and the new task is to go back and to consider both of those answers and the chains of reasoning that would lead to them in the confidence graph,
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we separate kind of a phase two way and to be there thinking about the two different answers.
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And then there's the presentation phase. And again, we separate that out into how they reacted when they saw 18 remainder nine. So we usually we structure in actually having them talk about the incorrect pattern first, and then 17, remainder two, and then the discussion of the you know, validity wrapping it up. And so we did that for a couple of reasons. First, we wanted to just see, Was there some evidence of perturbation once they were confronted with a strategy that was different than their own. And as you mentioned, or we talked about before, you know, students did at least it was our intuition that students seemed to be pretty tied to the initial ways of thinking about the task. And it was something that being presented with a different pattern did appear to cause perturbation. So we wanted to use the confidence graphs as a way to actually see if we could see that pattern for students. So we looked for dips and variation in the shape of the graphs to substantiate that intuition. So
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let me interrupt you for a second for the listeners. So this graph had like different timestamps. And you asked students to reflect back how confident were you when you were working on your own at the beginning of the discussion as a different time points?
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That was an adaptation from the way it was used before where the research question was, what is salient to the students about the discussion or about their experience in these transitions that we were studying in these other contexts? For this purpose, we were very interested in mapping to the phases of our strategy. So we specified those points along the x axis, I'll sort of time the entire time.
12:08
Right. And then they rated themselves from low to high on a five point scale confidence. Yeah,
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yeah. And then you averaged their ratings across all the time points. And then you present that in the paper.
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For us. We're interested in the depths. And sort of, we're not comparing them to each other. This is more of a qualitative picture, a one step beyond looking at at our class and trying to get an impression of the experience. And so this was a tool for for hearing from students who are quiet and who might not respond and show us their thinking very audibly. So this was a way that we heard from absolutely everybody and had some feeling for what the experience was like. So
12:50
yeah, okay. And so how did you use that in your data? analysis?
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We look for evidence of variation. And we wanted to know, when they see 17, remainder to the correct answer, do they feel less confident in immediately, but then are they able to kind of grapple with and resolve so that was the very important part for us was moving through that dip to resolution. And so that was the evidence for us that this was something that enabled them to keep reasoning and also resolved their reasoning was, yes, there will be these dips and recoveries, but at the end, how are they feeling about their understanding of the task, and that was every student in the class came away rating themselves highly, but then we thought, okay, but that's their impression of how they feel, what other evidence can we use to substantiate how they actually understood at the end of the at the end of the lesson, so that was where the written explanations, and the analysis of those came in to give us a better indication of their ability to understand what the conceptual issue of the discussion was, and to articulate that
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and some might argue that that dip that you were talking about is essential to learning, right that we need some productive struggle that we need to go through. I think Joe bowler calls it like the valley or whatever, I just watched whenever, but this notion of productive struggle, right to like, have that dip. Okay, so to me, I have to say I love your innovation of divergent converge, but this confidence graph also gave me some ideas of what I could do, because I often ask my students to report in addition to do they think they are correct also, how confident are they in their solution? And as you were talking, I was thinking because what you did is you did it reflecting back great, but one could almost use an app where you could like every now and then say okay, now click how confident you feel which would be like more like timely, right? Anyway. Um, yeah, we diverging to a different paper. Okay, we,
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when we when we designed this, we debated having them do it in the moment versus at the end. But eventually Yeah, I did that it was too distracting.
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I don't know.
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Oh, possibly. I mean, I think it's something to look into more. Yeah.
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It'd be interesting to compare, right? Because often you'll remember things differently then from what they were actually like. That being said, they're gonna remember what they're remembering. Right. So that's their reality walking out. Yeah. All right. My next question says, What's the important problem or issue that you're addressing? I think we said some about that already. But let's summarize it briefly.
15:41
So for us, it's finding ways to make sure that all of your prospective teachers or all of your students, whatever the class is, all of them engage in making sense of chains of thinking, whether they're correct or incorrect, whether they're efficient or inefficient, but that they engage in all of them, rather than just focusing on either their own way of thinking, or the one that somebody presented that seems best. So to us, that's the important issue. And we
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like often, often students will go like, Oh, that's not how I did it and not pay attention. Yes. Um, so actually, this leads me to the question, Who should read this article? Because one could argue that this is especially important for teachers, right to be able to follow logical steps for various kinds of reasoning. But would you say it's beyond teachers, or
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I think it is, I mean, I happen to be married to a mathematician who teaches classes that are not for teachers. And these ideas are ideas that we discuss all the time of ways, because at the heart of it, it's helping you understand a particular mathematical idea. In this case, it's understanding the idea of division, and what a remainder means. And whether it's certainly very important if you're going to be a teacher. But one could argue that any student in any mathematics class, that this kind of approach allows them to get a heart of some conceptual issue.
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I mean, in fact, it's very connected with, you know, math, practice three. So, you know, construct viable arguments and critique the reasoning of others that connects right to something we already identified is really important for all students k 12, to be engaging in. But I think you're right, that there's additional value for teachers. And so you know, we in doing this work, recognize that there was a lot of direction from the K 12 literature to informing you know, orchestration patterns that we use them in teacher education as kind of ways of modeling for our pre service teachers, how they might then orchestrate discussions with their students, and this one we feel has additional value, because that is allowing them also this unpacking contrasting patterns of reasoning, in this case, too, but we also think of it maybe more than two, but the contrast and patterns of reasoning as being a really important skill for teachers to develop.
18:07
Let's jump into the research questions. What were your research question?
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The research questions stemmed from the hypotheses that we had for how that strategy actually functions for our students. So we wanted to know, you know, does juxtaposing the two answers encouraged perturbation in the pre service teachers thinking and support them and continuing to engage? Is this supported by the data is basically the question and then the second hypotheses? Will the pre service teachers as a result of engaging in this strategy be able to resolve the conflict that that they get into? And so will it be productive for them? In the end? Will it deepen their understanding? And does the data support that
18:49
I wonder if you could talk a little bit about how the resolution happened? And then you have that last step, where you have done follow logically through the incorrect solution as I
19:02
correct Oh, in the third stage, there is presentations of each of the chain of thinking that led to each final answer. And when we discuss those, the purpose of that discussion is very specifically on Do you understand the chain of reasoning, not whether you agree with or not. So there, we try to keep that conversation separate. And when everybody understands the two chains of reasoning, then we ask them to take a little time to talk about validity. Now, the way this unfolds happens differently in each class, sometimes we send them back to small groups first, before a whole group discussion. But I would say more often than not, and in the case of the instantiation, we talked about in this article. The conversation about validity is not trivial because They're trying to convince each other and find the words to convince each other. And this can wax and wane. And there are times when we've had to send our kids back to small group to have a little time to talk some more about the ideas that have been presented before they're ready to talk hold group again. But it mostly comes down to them being able to connect to the meaning of division, this idea that and in many cases, they feel more comfortable if it's in a story. So in this particular case, I think they created a story about balloons, you have 189 balloons, and you're handing them out to 11 kids, how many can each kid get, and then they'll start talking about? Well, if you start out with 220, that means you've started with some fake balloons, and then keeping track of the real balloons versus the fake balloons is what allows them to figure out that the remainder can only be real balloons. And although the wording is different, it really comes down to that in most classes, understanding this real versus fake, and that's usually when they're able to come to some closure. But even having done that whole group, and even having spent the time still then having them go back individually and write about it, you're still going to see variation in the way they are able to articulate that idea. And that's what we saw in their, in their written explanations. And while nobody was at what we consider the bottom level, which we've seen in previous activities,
21:41
how but not why. Right? So
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they were able to explicitly deal with this idea of extra items. But in our mind, and in the paper, we talk about whether they're able to come up with a convincing argument that we think would really convince somebody who held these beliefs. And so they vary in that. So I wouldn't say on the roll 100% there.
22:04
Yeah, a common thread between this task and others that we've you know, been experimenting with the diverge and converge strategy with has been the role of context in the discussion of validity. So the one that, Terry, the discussion that Terry was just alluding to, they start out without context, and they have to kind of add that in. And that's part of you know, what helps them establish the validity in this case. And so in some cases, and some of the ones that we have in the appendix, you know, they start out with a context. And it's partly just sort of once having an explanation crafted and discussed and understood, then bringing it back and kind of mapping it back to the contest in a careful way. But that's that's been a common thread for us in that last part of the discussion has been either introducing or redirecting to context, because
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context is what allows them, their explanations have to emanate from meanings of mathematical meanings. And we focus in the course on meanings that make sense for elementary students. So not algebraic definitions, but meanings that would make sense to a third grader. And they have a tendency to leave those, which is when they get in trouble. And it's bringing them back to those meanings.
23:16
Let's talk a little bit about evidence. We already talked about the confidence graphs that you had. Create, I don't know, if you had them create
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the graphs, you know, we gave them to them.
23:27
Right? You just had them re rate their confidence at different times. And what else did you collect? And how did it help you answer the questions?
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So first, we actually had some open ended questions about the confidence graph to make sure that they were interpreting it in the way that we hoped they would. So open ended questions about a high point a low point for them, and whether they were still struggling. So that gave us a little more confidence ourselves in that they were interpreting the graphs the way we intended. But then the second part that we've alluded to is where they we gave them a piece of paper with essentially the argument for the incorrect answer 18, remainder nine, we basically summarized what always happens and how that is arrived at. And then their task was individually on paper to fix the strategy and explain their fix in a way that would help somebody understand why their thinking wasn't valid and make it clear how and why they altered the strategy. So that was individual work that then we collected and analyzed in the same way that we analyze their work on quizzes and tests. So the first level is sort of, can they fix it? Can they fix the strategy in a meaningful way? And then can they explain why the fixes necessary and do it in a way that would help somebody who had this error? And in in sort of grading those papers, we had what we call the level one, which is they're able to describe how to fix the incorrect strategy but can't explain it. None of our students were there a level to where they described how to fix the incorrect strategy. They attempted an explanation, but it was either incomplete or unconvincing. So there were good elements there, but they were not there yet. And then the third level where they could clearly describe it, and offer a coherent explanation that hit the key points. And so in the paper, we offer some examples of student work in those different levels to get a sense of what we considered a convincing argument. And interestingly, convincing arguments did not always use context, we included examples in the article both of one that kind of did it in a more generic way, versus one that really used a story, his own story to make sense of it.
26:03
So would you say there was a difference between students that initially gave the incorrect chain of logical reasoning and how they performed versus students who initially gave the correct chain of arguments? Did you look at that at all?
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We did not look at that. But based on many semesters of experience, interestingly, those who start with the 18, remainder nine, some of them have a stronger sense of why it doesn't work as a result. But it winds up both ways you have those who are convinced of the correct answer, and likewise, may not have a strongest sense. So I get both kinds. But we didn't do that specific analysis. In this case, we did wind up with at least two thirds of them at the level two, and about a third of them had reached the level three.
26:59
And this was just one instantiation. Right. So I assume if you do this over and over that, that might be interesting to see what would happen. Looking back at your article, as I mentioned earlier, I feel there's like a ton of contributions in there. So there's the main one that you talked about the diverged and converge, but then there's also that confidence interval and other pieces. So I'm imagining, or I'm asking you to imagine people reading your paper, and what do you imagine that they can walk away with in terms of, Oh, I can use this now?
27:34
Yeah, I mean, I think as you pointed out earlier, you know, there anybody who is teaching is, you know, generating this knowledge of the patterns that arise in their class. And so anytime that that arises, where you have contrasting patterns of thinking that you want to make the focus of the discussion, you could use divergent and convergent adapted to orchestrate a discussion using those broad,
27:57
so to do so I would have to, I'm just trying to think I have like different and different example in mind. But I would have to have a good knowledge of the logical chain of reasoning my students usually do. So I would have to really analyze the incorrect strategy. So I could develop a way to share it that would really get at the why,
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yeah, this isn't something Terry and I have been talking about To what degree would you ever do this on the fly? To what degree is this something you know, that it really is drawing upon that anticipation work in that store of knowledge that you're developing over time and knowing you know, already have an idea for kind of the, you know, giving some more structure or ideas for how to structure a discussion that really emphasizes that thinking in a way that you may already be intending to do this with contrasting strategies, but there's the layer of not getting personal in it as well sort of evaluating Is this a time in the course where I want to, you know, remove the status issues of that and how they might play out in the discussion and really, really focus students on this skill of analyzing the thinking of others and in doing so, really deeply making that like the heart of the discussion. And then you know, another as you pointed out, a separate contribution is the confidence graphs those we are using in many different ways within teacher education. We used it here to analyze the the way diverged and converged functioned in our class, but I think any question you have about how your students are experiencing something that you're trying out, it could be adapted to fit, answering those questions or giving you information about those issues.
29:41
So to wrap up, let's just kind of look at how does this particular work fit into your large body of work?
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There's a lot of research orchestrating discussions at the K 12 level and we built on some of that, that work and for teacher educators for Future k 12. Teachers, it's important to be prepared for that work, not only by understanding chains of student thinking, but by experiencing those kinds of orchestration strategies before beginning to prepare to do them on their own. And so that's I think something that introducing to or emphasizing in teacher Ed and thinking about ways to leverage these ideas to help focus on important mathematical ideas and pedagogical ideas is an important thing. And we know from research that although these ideas have been out there for a while, they are difficult to enact. And so we need more examples of them or ideas for ways to make them feel like you can do them in your classroom.
30:48
Okay. Well, thank you so much for joining us.
30:51
Thank you for having us.
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And 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 for listening and goodbye.