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Hello, and thank you for listening to the math teacher educator journal podcast. The math 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 today was Jennifer Ruth, who is an assistant professor of mathematics education in the College of Education at the University of Oregon. We will be discussing the article visions of the possible using drawings to elicit and support visions of teaching mathematics, 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 she shared in the article, her successes and challenges and how these lessons relate to her other work. Jennifer, thank you so much for joining us. Well, thank

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you very much, Eva, and Happy birthday to you.

0:53  
Thank you. Alright, so let's get started into the questions, we always start with a very brief summary of the article, including the results.

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So this article actually came out of work that I was doing as a doctoral student. And we at the time were collecting drawings from pre service teachers that were designed to capture the title of the task at the time was to draw the quintessence of teaching. And this work evolved from project designed by Genie lift coat, who was the science teacher educator there at the time and previously at Columbia. So I owe a lot to genies ongoing work in this field. So to cut to the point, one of the pre service teachers that I was working with produce this before drawing and an after drawing, and they're the first few drawings you'll see in the article. And I saw this Stark change between what began as this idea of teaching as telling with students sitting in rows as passive recipients taking notes. And by the end, you see students clustered around a table full of manipulatives. And the teacher is now folded into the conversation. But when I asked the teacher what she's doing, I'll call her Sheree. She says that she's basically facilitating discussion and that the students are driving. It's very much student centered, inquiry driven learning. And this was really exciting to me, as someone who was learning to become a math teacher educator, after many years of being a math teacher, I was like, Wow, so if we have an instrument here that shows this much shift in what somebody thinks it means to teach and to learn mathematics, just across this time that they're in the program, I'd like to see what more we can get from it. I turned it into a study and I looked at the drawings from the next cohort. And in order to get more data, I coupled the drawings with an interview protocol. So I got a sense of what the artist meant for their drawings to represent

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this was a cross a one year master's program. Yes,

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correct. So the students came in, in mid June, and they graduated in early June, The following year,

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and you had them draw a picture at the beginning and at the end, and then you notice those big shifts, and you study those? Yes, that's it. All right. I will get to it later. But I just smiled. Now, because I was imagining in the article, you talk about how you laid out all those pictures in the living room, and looked at them. Alright, so who should read this article?

3:29  
If you're a math teacher, educator, and you're interested in capturing what your pre service teachers care about, and what they believe the roles of math teachers and math students to be? This is a useful instrument, it's a nice piece of triangulating data, because drawings sometimes reveal things that we don't get out of conversations or papers with our students.

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And can you talk a little bit I don't know if you mentioned that in your summary. What was the prompt for the drawing the

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drawings in this particular study were to draw the quintessence of teaching and like I said, this was Genie lift coats work and Jeannie had some very specific lead ups to how she got to what she considered to be Quintessence. So there's a lot of work around defining that particular word. In my experience, Quintessence seems to be a bit of a flashpoint for people, they're not quite sure what the word means it feels very fancy. So across the years that I've been using versions of this task, I've modified it and I've gravitated to Karen hammer anuses work and Karen was actually a part of the program as well as she did her postdoc there. And she worked with Jeannie lift coat, and Rachel tan. So all of these people were they were doing the foundational work that led to this article, parent talks about teacher vision, and you're going to see that show up in the conceptual framing of the piece alongside noticing. So if you think of vision, it is basically especially for new teachers. What is it that they envision their teaching practice being like? And then how are they going to get from where they are in current practice to what they think is going to best support their students.

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So your prompt for the paper was this Quintessence? Is that what you're saying? So

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for the data in the paper, it was for the because this was actually data that was collected as part of their teacher education program, they were still using genies prompt, which was to draw the contestants of teaching, I now talk about with my students about their vision for teaching. And I specifically asked them to think to envision an optimal day and you know, like a really good day in their math classroom, or someone's math classroom, what is the teacher? What is the teacher doing? What are the students doing? Can they please draw that and then write a brief description? So I understand what I'm really trying to get at is what's the role of the teacher? What are the roles of the students?

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So there's two aspects. One is this vision of teaching, but then it's also related to the math classroom.

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Correct. Okay. And to link it with existing threads in math teacher educator, I looked also at how mte has framed noticing across several issues. And so both the noticing and teacher vision have in common is this idea of a metaphorical lens. What is the way that math teachers look, both in reflection and in planning at their practice and their students practice?

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Thank you. So what is the important problem that you're addressing with this paper,

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when our students come to us to become mathematics, teachers, they are coming out of their own apprenticeships as students of mathematics. And so what we know is that, though things are changing, we have a long history in the US mathematics, education being primarily transition, or start transmission base where teachers tell people how to do mathematics and students absorb. They're very passive. They're meant to write notes, practice problems, and then regurgitate them on tests. And that this doesn't tend to lead toward deep, flexible, robust understanding of mathematics, it's not really getting at the conceptual pieces as much. So how do you take someone who is infused with that as their model of teaching and learning mathematics? And we often How are students as pre service teachers for one year? How do you shift that vision in the time that you have? Wouldn't it be nice if you knew kind of where they were at coming in and had an instrument for assessing where they're at on the way out? And this drawing task is one of several tools that a math teacher educator could use? So the problem of practice really was? How do I get a sense of what my pre service teachers beliefs are about teaching and learning mathematics? And how can I capture snapshots of that? And then how can I use that information to be a better math teacher, educator, and to evolve the courses that I'm using to help people learn how to teach mathematics?

8:11  
So you already mentioned whose work you were building on, but let's just make sure that we get out everything you want. I mentioned whatever particular you mentioned, noticing as a lens. What else is your article grounded in

8:26  
so there are several contributors to noticing frameworks, and one scholar in particular, whose work I'm drawn to is Lisa joke. She wasn't the first person to use noticing, but she does use noticing very effectively in her work with inservice teachers. Lisa's work is grounded in particular around how students are positioned and framed. And specifically, how can we look at students and frame them in terms of their strengths, their areas for growth, basic is asset framing as opposed to deficit framing. If you look in the article, you will see some of the more foundational work and authors for noticing as well.

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Alright, so let's chat a little bit more about this drawing activity. Just tell us a little bit more about you talked a little bit about that that was part of your dissertation work? And then what are you learning from it? How are you implementing it, those kinds of things, I

9:28  
use variations of drawing tasks with every class that I've taught, whether it's a methods class or a math content class, and it's in part because there are students who get to shine in this task in ways that they may not and other course activities. And along with that, I get this window into how people envision things. I will say that my initial excitement in that very first pair of drawings that Sheree produced and that was very Back in 2011, was tempered when I got the first full set from a whole cohort, because what I realized was, first of all, not everybody captured everything as cleanly and clearly as surely did. And I had to do a fair amount of digging to make sense of things. And as you mentioned, yes, several times those drawings were laid out across the floor in my living room. So I want to be clear that if you want to use this as a tool for research, you know, it's like any form of research, it does take some time, you have to build your conceptual frameworks and your coding schemes and do all that work. If you are using them as a math teacher educator, it doesn't take quite as much time because you're not doing your analysis to that level of scrutiny. But I would encourage you to really marinate with the tribes for a while, because things start to emerge from them as you look at them as a set. And also, if you pick up an individual drawing, and really resonate with it for a while, it's kind of like walking through an art gallery. Every time you pass through it, you're going to notice something new or different. So I know that well, the evolution of the drawings as well, we use two different variations and part of the 2017 service teaching and research cohort star for mte. And I met two other scholars while I was there, Shannon Sweeney was at Northern Arizona University, and Chris Willingham was at James Madison. And the three of us have been working on a suite of instruments to try and get a sense of how people envision mathematics and what they think it means to be good at mathematics. And so we've used two variations on that task. And I believe you've used one of those as well,

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I have.

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One of them is to draw who mathematics is to you. And the foundational work there that came from a paper that Dobbs asked us produced in 2015. So it's this personification of math, and it can be another person, some sort of, it basically has to have some sort of living entity quality to it. And then Shannon's work specifically around growth mindset pushed us toward drawing a variation on the draw scientist tool, which is where a lot of these drawing tasks come from. That's good enough work. And she started this in 1928, we asked our students to draw someone who is good at math. And then again, with all drawings, it's very helpful if you ask the artist to write at least a brief description of what they're drawing and why. Or point out salient features in the drawing, they want to make sure you notice all of these things. And this is in part because drawings don't come with dictionaries, we have to it's a fairly interpretive form. we're much more comfortable, I believe, as humans with words and language. And so if an artist tells you in words, what they want their drawing to mean, I think you can make stronger claims about that, especially if you're trying to use this as data.

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So I have two follow ups to this question. The first one is I have used various versions of this task, and I love using them in my class, I have not used them as much as an assessment, potentially in the way you have. But I what I have done, which has been kind of cool is after I had my students draw pictures, I then put them all collectively on a PowerPoint and then debrief with them. And that was really cool way of just getting out some of the thoughts that people had. And some people like in my classes have drawn themselves like a really nice portrait of themselves, which is kind of cool. The other follow up? So this was a comment, not a question. But the question that I have is, I want to be a little bit more clear on your second drawing. If I understand the article, right? Do they see their first drawing and then they create a second drawing and comment on that,

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I think you could go either way, I would really encourage anyone who wants to take this task up. There's an appendix to the article where I talk about variations and implementation, I would say make this work for you, as a math teacher, educator and make it work for your students. And along with that, it's important to mitigate the fact that not everybody likes to draw. And so there's some anxieties about it, encourage people to draw stick figures, whatever they need to do I tell my students, I'm not putting these on my refrigerator at home. Don't worry about that.

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Just under living on the bedroom floor. Right. Yeah, exactly.

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And then I will pick them up when I'm done. Your art is very precious to me. And I say that with, you know, it sounds joking, but I'm actually sincere about that. When I'm asking people to create drawings in some ways, they're sharing a little bit of their soul with me and I really do take this seriously. That said, you asked this question specifically about the after drawing do students look at the first drawing when they draw the after drawing? And the answer is it varies in the case of this study, they did They had access to the first drawing. And so what you'll see if you go to one of the dependencies has all of the drawings from this cohort, except for the one person who asked to be exempted, she really was uncomfortable sharing her drawing. So we just okay. Yep. And so you'll see is that there are themes that go from the first to the second drawing that make a lot more sense. When you understand somebody had access to their first drawing, it's useful to compare what you do at the beginning with what you're going to draw at the end, if you're using this as a reflection tool, which is one of its most important and powerful values. Take a look, this is what you were thinking when you started the program. Do you agree with yourself? How is your vision changed? And in the case of I think, what is called Eric, in the article, you will see, like the core idea of what brought him into teaching was that he wanted to open doors for kids to get into STEM fields. And he hung on to that and his after drawing. But by the end, he was like, well, it's not just stem. So my job isn't just to get kids into a field that I valued personally, because he was a stem person coming in, but the field that they value. So how do they envision a future career that includes mathematics? It's my job to help them get there.

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Yeah, as you're speaking, I remember this picture. There's like a pathway to a door that said stem on it in the pre picture. And then what wasn't the post? How it did a post picture change on that one? Well,

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yes, that's important, actually. Because that claim about his vision at the end of the program comes more from the interview than it does from the after picture. Okay, the after picture actually shows his evolution as in his beliefs about the roles of teachers and students. So in the after picture, you see three vignettes from go from the left to the middle to the right. And you see this evolution from teaching as telling and learning as passive recipiency. And Eric came into the program as with a stem background, he'd also been a tutor. So he was literally paid to sit and tell kids how to do math. And that was his understanding of what it meant to be good at teaching math. And then you see, by the end this, he's off to the side, it's much more student centered. So he's, he's shifting his role from telling the facilitating and the students are becoming much more agentic by the end of the picture.

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So this is actually a nice lead into our next question, which is, what were the research questions that you've studied to document the effectiveness of this innovation? Though, I'm

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actually gonna return to the article to answer that question. Good, because there were four, and I don't have them memorized, but I'll go ahead and read them. How did the pre service teachers depict or describe the roles of students and teachers, if there was a central metaphor in the vision that indicate a core belief about teaching and learning mathematics, which depictions were revised, and which remain static, and where they're problematic depictions in any of the after drawings? Alright,

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so just let's just tackle that one or two of those. So what do you find? And how do you know you found us? So

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let's start with the darker side of it in the article, I refer to it as stuck kinases, I'm not sure I would still use that term, but it's in print. And it's out there. You know, like, if somebody comes into a program, and they really think the job of a math teacher is to tell kids how to do math. And they still think that at the end of their teacher education program, that's a problem. It means it's, it's a great problem to have as a researcher, because it's data. It's a not great problem to have the math teacher educator, because it means that I haven't successfully done my job in preparing that person to go out and teach. So I did find one is especially striking case of that in this data set. And it made me wonder, my own reflection then as a math teacher educator, because part of the problem for this pre service teacher was the head, he had a cooperating teacher, who is reinforcing some problematic beliefs he had this pre service teacher had about I think I'm referring to him as he and in the article. So Ian believed that they had really had some strong deficit framing for kids who were put into low track math classes in his pre service teaching placement. And he had equally problematic beliefs about kids who were in high track placements. So the kids in the high track classes, were getting student centered inquiry based instruction, and in the low trek classes, they were getting worksheets and repeated lectures. So whatever the lecture was in their algebra class, they would come to this support class and the lecture would then be repeated For them, and this was being modeled by his cooperating teacher. So upon reflection, I thought, well, how might Ian's experience have been different if he'd had a different cooperating teacher? And this is most definitely a problem with practice that all math teacher education programs face? How do we find cooperating teachers whose visions of teaching and learning align with the vision that we're trying to help our students understand and step into as newly minted teachers. So that's one of the darker sides on an upside. And I'll just go back to how did the PST is depict or describe the roles of students and teachers, there's just so much lovely stuff in there. And there was so much creativity in terms of how people were drawing what it meant to learn mathematics. And one of the cool things that popped out of it was the use of the lightbulb for the aha moment. And I suspect that that's kind of a metaphor that you'll see popping up if you use this task, because you don't have to have a lot of drawing skills to depict the light bulb and it sort of usefully universally understand is the moment understanding takes root in a person.

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So this leads nicely to our wrap up question, which is, how do you see other people using the innovation, and you already talked a little bit about everybody should use it the way they see fit, but maybe you can give a little bit of guidance? Sure, I would direct

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back again to one of the dependencies does have some guidelines for ways that I've used it ways that Genie lift code, and it gives the backstory about why Genie was co created it and how she created it. One of her gifts to me was it large stack five binders of years worth of people's quintessence of teaching of drawing, so I have a large vault of data to go back to if I if I want to at some point. So I guess I would just say, if you find drives compelling, this is a great task for you, I guarantee you that some if not all of your students are going to find drawings and interesting departure from a lot of the formal academic work that they do. And this is a great place to elevate. And to show this is this is truly as a low floor high ceiling task. And so it's a way that we can model for our teacher educators as we do in so many ways, how you can be using these kinds of tasks with students. In fact, I had a conversation with one of my doctoral students yesterday, who is also a math teacher educator in our program, she got an email from one of our recent graduates who is asking her permission to modify one of her classroom tasks for his own students. And so the ways that we use these tasks in our classrooms, they can propagate out for our own graduates as well.

22:55  
Before we started recording, I mentioned to you that I'm currently teaching a class with a blind student. And I was wondering if you have any thoughts for how we could accommodate students who may not be able to see with a task like this, you also

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described to me what you had done, and the accommodation. So I'm going to repeat back to you what I understood. And you can validate for me if I've got it, right. One of the things I've been stressing throughout this interviewing undue stress as well in the article is that drawings are highly interpretive. And so it is very useful to have words, whether it's a verbal interview that you audio record or a written description, and I will tell you that a brief written description is worth an awful lot of an MTS time. So I highly recommend asking people to include a written description of what's going on in the drawing. And what I remember you telling me was that you had your students write a description, in your case, it was the personification of mathematics, I believe, you know, and written descriptions can easily be comfortable descriptions, and vice versa. So if we want to modify this task for especially for accommodation purposes, yeah, I mean, there are pictures that is worth 1000 words, but you can easily create 1000 words, or some other variation of it. I would love to see an interpretive dance.

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So as you were talking, I was thinking because I'm sure most people notice already, but I didn't know how to use the alt text to put a picture description into slides. And one of the things that I saw my daughter do yesterday, as she's submitting homework now online is that for math, she has to explain her reasoning, but she can put a voice explanation into her math worksheet. And so actually, that would be kind of cool. Rather than a written description, have them draw a picture and then record a voice description of the picture, because then you get a little bit of their own voice. With it rather than just a text. So thanks for the idea you just gave me by reflecting back to me what I thought I was doing. Before we close up, is there anything else that you would like? Like if people use this task? Or if they're interested in the work? How can they contact you? Or is there anything else that you want to say

25:21  
I would be delighted if people wanted to use this task or something related to it. And as I mentioned, I've been working with Shannon Sweeney and Chris Willingham. And we have other publications on other uses. of the drawing I am as I say this to you quickly flipping through the mte piece to see if my email addresses on it someplace, it should be on there. Okay. So I would say emailing me is probably the fastest way to be connected with us.

25:51  
And your email is also linked to the podcast so people can find,

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yeah, send me an email, I'll send you stuff or we can have a conversation about it.

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And one of the things that I believe mte is trying to do is figure out a way for people to be able to talk back to authors, if they have used their in interventions. I'm not sure exactly where that's at. But if anybody wants to use it, it would be nice to hear from that, right. Oh, absolutely. Yeah.

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I would very much like to hear how this goes out into the world.

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All right. Well, thank you so much for joining us. And for further information on this topic. You can find the article on a mathematics teacher educator website. This has been your host, Ava Sennheiser. Thank you for listening and goodbye.