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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 fan Heiser, and today I'm talking to Alison McCullough, we will be discussing the article, creating a third space for learning to design technology based mathematics, which she co authored with a graduate student, Nina Bailey at the University of North Carolina, Charlotte, and Kristin phi and Gail Scott, who are teachers who participated in writing the paper. This paper came out in the September 2020 issue of the math teacher educator journal. We'll 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 work. Allison, thank you so much for joining us. Thanks for having me. So let's jump right in. Can you give us a brief summary of the article including the results? Sure, I will try to summarize that quickly. So this articles, right, I wrote about a project I shouldn't say I wrote about we wrote about you mentioned my co authors, we wrote about a project that we and this is the first time we had tried this out in a course that is focused on learning to teach with technology. And I worked with two practicing teachers, Gail and Kristin, who are outstanding partners of mine, to come up with a way of getting my students involved in designing technology tasks for real classrooms. So for years, I had been trying to figure out how to my students that take this course, are often relatively early in their program, the course doesn't have a field experience, and they would have them Design Technology tasks, but they were so separate from any real students that just weren't very good. I don't know how else to say that there was they were so disconnected from reality. And it was hard for them to imagine how their work could support student learning without real students in mind. So trying to find a way to bring my students work closer to a real students in real classrooms with feedback from teachers who would use such tasks in their classrooms. That's where this project was kind of built from. We talked about the partnership that we created. And we adopted the theory of a third space, trying to really think hard about how we can bring the practitioner and academic knowledge together, and ways where everything is put on even ground and really support our students learning about the language of both spaces, the discourses of both spaces, and how they work together to support student learning and the things that they're designing for students. Essentially what we had them do, I went to work with the teachers to the teachers identified mathematical idea that they thought that technology would be really helpful for teaching this idea, but they just hadn't had time to design a task themselves. Nor had they found one. My students kind of worked as almost their partners in they shared what their learning goals were how they envision technology could really support the students and my students work to actually designed those tasks. And we went through a series of revisions. For this process, they went through revisions with me, they went through revisions with the teachers. And eventually, the teachers taught us their tasks in the real classrooms. And my students got to be in there and see their task with students. And actually were involved in the teaching of that as well. So about halfway through the semester, we realized that there were some really neat things that were happening. So our decision to study ourselves actually came about halfway through the semester, we realized how much we were all learning from this process. So we quickly ran the IRB and got permission to do some retroactive things after the semester was over and keep all the documents that were part of this. And really pay attention to we had two big research questions. One was we were wondering about and what ways to pre service teachers that were working in this space that we provided that we came up with, they were coordinating the what we're calling the practitioner and academic knowledge in their designs process, and what they were learning from that and then to what were the aspects of our partnership that made it successful or presented challenges for us, because we saw very quickly, this is something we wanted to continue to do. But we also wanted to refine as it went. So we kind of have two goals. Their first findings go, I'd say on for the first research question related to what the pre service teachers learned. We only focused here on two big things in the paper. One was the way that they coordinated practitioner and academic knowledge related to the test design itself. So things they were learning from things they're reading or learning about, you know, very technologically specific things, balancing that with feedback and to

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questions they were getting from the teachers who were bringing the real life situations that they could imagine this happening their students and how my students were then balancing the two and making decisions about the revisions going forward. So there were things very specific to the design of the technology that they were balancing there. And then also very specific to the posing of questions, which was my I think one of my favorite things through all of this while it was the course was focused on designing technology tasks, they came, my students said this over and over again, what I learned more than anything was it's not about the technology, it's about the questions that I pose, given the space that we're in, the emphasis on that came from that working back and forth between things that were happening in our classrooms and things that they were getting into in the classroom with secondary students in those teachers. So those were kind of some of the big things that really improve their designs as they were coordinating. Let me take a step back and understand the course better that this was situated in. Sure, sure. So this particular course is a course on teaching mathematics with technology, specifically for future high school teachers. Most of the students in the course, were sophomores. So second year, undergraduate students, there were one or two that might have been during the third year, juniors, actually, that semester, the room, I apologize, this was done this four times since then. So it gets muddled in my head. This was actually written the various first semester we did it. And that year, I had two students, and only two students, which was a great year to try this out. They were both sophomores. So I'm asking because I'm thinking about the results, and who those are going to be useful for. Not necessarily for people who just teach courses like this. But for anybody who incorporates technology in any course stuck cry. Yes, I would definitely agree with that. And it doesn't have to be about technology. I think that type of partnership, that the way we're trying to work together in this partnership, for us, it was in the context of this course. But I think it would work for any course, where you were trying to have an informal field experience if you didn't have that opportunity to have students out in real classrooms. But working closely with teacher partners in this way, would be very valuable. So that kind of leads us nicely into the next question. Because there's a lot in this paper, right? And you could read it from the technology angle, or from the collaboration angle, or probably a bunch of other angles. Let's pick one, what was the important problem of practice that you were trying to solve? Or that you're addressing in this article? For me the important problem of practice? Was that informal? How do we find a way for pre service teachers to work within the informal field experiences when those opportunities aren't there for a formal field experience? What other options do we have, in ways and I really wanted to work closely with teachers who wanted to be partners in this work. So I think the biggest thing is that that for me, I think the biggest takeaway from this paper for others, that'd be some that would be interested is more about that partnership. How do we design that partnership? Kind of what makes it work? Well, so the problem you stated at the beginning was that when you had originally worked with students in this technology task design, that it was difficult for them to envision good task, because they didn't have any feedback from how that would work in the classroom. Is that correct? Am I not necessarily the good envisioning a task a lesson? I think it could be almost any part of the pedagogical skills we might be teaching in a methods class, where there is value in seeing in an authentic setting. So connecting the university classroom to the K 12. Setting? Yes, the idea? Yes. Okay. So the idea of making something authentic with this notion of, it's not just something I'm doing for my class to pass, but it's something I could actually use. Yes, exactly. Okay. So I think we can wrap our heads around this and most of us who teach at university in a not field placement setting have struggled with this idea.

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Okay, so what did you do and how could other people do this? Thank you for pulling that back together. So the I think the starting point with this particular partnership, and these are Kristin and Gail are incredible partners I still work with today. And we've expanded other teachers in their school, my class was getting larger other teachers wanted to work with some of my students. So we there are more of us working together now. But it started with actually meeting with the person

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Have that school and just introduce him I was new to the city at the time, I'm new to the institution went and introduced myself here the things I do, and expertise I have, and is there anything that I can do to help you, you just kind of getting to know each other sort of situations. These particular schools were very close to our campus, when he and I met, it was knowing that he is looking to he wants to get to know, future teachers sooner because he wanted to make good hires. And I was wanting to meet teachers and principles and so forth, because I was new to the area. So we just had an informal lunch, about the things that we were both excited about and goals we have. From there, he introduced me to us from his favorite math teachers. And we just sat and talked about ideas that we had together of ways we might be able to help each other. That's how we started this, it was more of a research practice partnership, when I did some of the work I do in other spaces is more of a you know, I knew in the back of my mind, that was one of the problems I was hoping to figure out. But it started as a conversation of what do all of us need, support them what we're doing, and just developing that relationship before anything else that we did. So I think one of the things that's often difficult to imagine for people at university is, how would I find schools that would want to work with me, and I have done a lot of work with family math nights, and my experience is that you can actually call any school principal, and they will meet with you. And if you want to work with them, and it does not cost a school anything. So far, my experience has been really positive, right? I mean, the budget is a different issue. But if you're setting up a collaboration that is mutually beneficial, I would just encourage people who have been thinking about it to just send out an email to a principal at a school and do what you did in today say, hey, I want to introduce myself, I do this, would you have some time to meet? And my guess is most of the time, maybe not during COVID right now, but most of the time, you would get some callbacks. And I think a key point that you make there is the mutually beneficial part. One of the findings that came out of our study we were looking at why did we feel like this turned out to be successful. Um, you mentioned my co author, Nina Bailey earlier, she was instrumental to this part of the study, because we were studying our partnership. I didn't want it to be me doing that part. So she came in and did the interviews and analysis really, really, really helped with that part. So we had an impartial person from the outside, asking the questions and trying to help us make sense of this beyond anecdotally, the successes and challenges we face. But that feeling valued piece was really, really important, especially to Kristen and Gail, they are at a school that is near University, they get hit up all the time for research opportunities, but it's usually somebody wanting to come and do something with their students or in their classroom. There's no give and take that there was never anything in it for them. They didn't get that often, they never even found out the results of those studies. For them, they talk about how they felt valued, that they had something to offer, that they were also going to really benefit from this themselves. And they felt like they were they were being helpful to future teachers as well. So they felt like there isn't it wasn't just their space that they were offering up with their expertise as well. So there was a big part of that was that mutually beneficial piece and being really careful, all of us talking about it very openly. And honestly to make sure that we all felt that we were valued felt like the work share was being done, distributed fairly, making sure that everybody felt like they were being heard. And not just assuming those sorts of things. That was really critical. So let's jump into learning a little bit more about the innovation. So you call it the third space. And when I was reading the title, I was like, What the heck is this? So can you tell us a little bit more about that? Yeah, building on what you already said. So the third, the idea of the third space comes from hybridity theory, which is the idea that indiv individuals make sense of the world by attending to different discourses, and zeichner started talking about he may not been first but that's certainly where I've read a lot about third space in education, but pulling this in and thinking about how do we bring together that practitioner and academic knowledge in ways that they're uneven ground, that one isn't being valued more than the other. And that doesn't mean going out and holding up methods class in a school, it actually means bringing the people together and working together towards this, whatever that common goal happens to be. But with everybody on even grounds, that's where that works.

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Trying to create that space, where you've got the practitioner knowledge, and the academic knowledge coming into one space where both is valued. And our pre service teachers have ways of interacting with all of it together, and not completely separate with one class being over the university, and other experiences being out of the school with no connection between that to between the two. So that's what he means by third space. So give us a little bit of a sense of I'm assuming this was a semester long class, how often did you meet? What did different people you said the teachers requested some assistance with a certain so how can you just give us a little bit of a visual of how that happened? Sure. So we start. So this was a regular course that we had our regular course meeting, which is twice a week, about an hour and 20 minutes, the project was posed to the students, it was probably close, I mean, that they knew it was coming. But we didn't start talking about specifically until about a month in. So because it was a course on teaching with technology where we're working specifically with math action technologies, I waited till they had a little bit of introduction to the big technologies that we would be using throughout the semester. So they at least had that in mind, even though they weren't experts on them yet, when they first met with the teachers. And that first semester, we did it, we went through two cycles that students work together and a pair so they were co designing. And the first one that I did with, I don't remember which order we did, but we did one with Miss Scott and went through that whole process. And then we did one with Miss phi and went through that whole process. So they first met let's say it was with Miss Scott on the first one. She and I had spoken ahead of how did they meet did the teacher come to your class and the students go out to the teacher like the first meeting, the teachers came, the teacher came to my class introduced herself. He was about a 10 minute introduction just to develop a base. And from there, I set the students out to set up a meeting with her in her classroom, they came up with an interview protocol of the questions they wanted to ask her they knew the topic that she was hoping that they would design something for but they didn't know the exact learning goals. They didn't know the experiences that her students had had using technology in the past, they didn't know what technology she tended to use. So their first interview with her was a pretty wide open one just trying to get a sense of what her goals were for the the lesson and or the activity what how her she, how her typical lessons ran, how she hoped this one would go, what technology she felt comfortable with, or what technology she was hoping to learn more about. So it was a pretty open interview like that. They then did, we worked through initial drafts where they were first working on just small little bits at a time Can we use, I use the task planning guide from the taking action books or focusing on just the task itself. So here are my learning goals. Here's how I envision the task, I've chosen a technology, they came up with a rough draft of it. They talked about how they would launch it that talked about they had anticipated student responses. And so it was first around just the time, just that task. That first came to me for feedback. And after. And this is where it got a little we had to be, I made some decisions that I wasn't always comfortable with. I didn't want it to be just me giving feedback. But given the teachers time, I didn't want them to look at it when it wasn't worthy of their time yet. And it's the first time they've done anything like this. So they did around a feedback with me first revised, and then it went to the teachers and the teachers had a document where we worked in Google Docs, we had records of everything. So kept track of all the different things that they hoped that they'd like to see addressed and why. And then they also had a sit down meeting with the students and I wasn't there for any of that they would go to the classroom have a meeting, they would come back and reflect on that with with me in the classroom. They would make revisions and in the revisions, we do something similar like we would do for a journal article and make a table. Here's the suggestions or comments. And here's what I did in response and why, which was a wonderful process because they didn't necessarily make changes just to make them. And they were really careful. If something was suggested they think really critically about why that might have been suggested. Do they agree with the suggestion because sometimes it was something related to the technology was that it may be not possible to do sometimes it was related to students and it's a classroom of students that they don't know yet. So we went through this revision cycle. I want to say we did it twice, two full times with the teachers. And then the final time when the task was pretty solid that the students enrolled in the full lesson plans. They could imagine the timing of a full lesson. And from there, they finally got to go in and see something that they had designed

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being taught by this did de modified after desaad implemented the debrief talked about how they would modify it. We did not I didn't require it that semester, but they all presented these at our State Conference in the fall. So they did with the teachers, they did do all the revisions before presenting, oh, if I'm calculating, while you were talking, this would require the work of the teacher required is to come in, whether that's zoom or in person and introduce themselves to class, and then two or three meetings with the students to give them feedback. But the teachers requested the task. So they would be bought in because it's something they would want, right? So yeah, just some random thing. Yep. Yep. So let me ask you a practical question. Because I've done family math night, and I've almost like fallen over from feedback. So what is the largest group you've had? And done this with? The largest group is 15 students. And you did initial rounds of feedback for all of them before I went to the teacher, yes. But they were working in teams. So that semester, I had them work in teams of three, which I will not do again. But yeah, I did. They are working in teams, it's still it's a lot of time on everybody. Yeah, cuz I'm thinking if if you had a class of like 30, which is more typical some places, then maybe this would just have to be revised a little bit where what I ended up doing in my work was having them give peer feedback before I was giving feedback or anyway, yeah, I've tried that too. And then also, you would probably need that many teachers, right? Because each teacher can only work with one pair of students is my guess. Ideally, Kristen and Gail have both taken on two pairs. In the past. By request, they they asked to take on two pairs, one for each class, they were teaching into two different classes. But typically, one parents do teacher asked the students are creating these activities, are you collecting them somewhere for the teachers to have access to? Yeah, we have a shared goal folder with every teacher's folder within the overarching one. So yeah, lots of very carefully shared Google folders and documents in between. I also learned that first year that if we're going to be working, designing these tasks in technologies like GeoGebra, code up Desmos, etc. It's in one person's login, only one person can work on it, that became problematic. So we have a general Dr. McCulloh account that we all work within that everybody has login information for. So teachers and teams and everybody else can get in and work on these particular things. So it doesn't fall on one person to do the, the technology side of it sort of thing it ended up was a little kind of things we learned early on. Okay, so I think that like we're starting to understand the process. So the students work with a teacher partner, and the teacher partner requests, or a can request a kind of tasks that they would like to have. And then the students develop that with your support with their peer support based on research that they've read. And then the teacher gives practitioner feedback. And so they get to integrates both what they learned academically in your class, and what they're hearing from the practitioners of what would actually work in a classroom. And then finally, they get to be present for the students to engage with the task, and then reflect on that. summarize that very well. So sounds like a great way of doing that. And I know in the article, you have a visual that kind of goes through this part. So this third space, is the idea that the student doesn't just live in the academic space or just in the classroom but negotiates space in between, or like drawing on both the practitioner and the academic. Yeah. So the the idea is exactly that, that this student, is it not seeing these as completely separate from each other. Many of us have heard that what I learned in my classes doesn't apply when I'm out teaching that we're trying to write play against that. There is not we're trying to erase that divide between the two. And by having them tied in together here. That was really helpful in doing that. I have to say on kind of the theoretical side of that this idea of rejecting hierarchy and we've got these decades commies of the academic space versus a practice

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innerspace that's kind of an artificial dichotomy in some sense, especially amongst the incredible people that I work with, they, the teachers, I work with all our academics themselves, you know, they've, they've all gone to school, many of them have higher degrees, a couple are now doctoral students. And the same time, I taught high school for 12 years. So I've got some practitioner experience as well. So we all have experience from all these places. One of the teachers I work with, she's a Desmos. Fellow, she knows more about Desmos technology than I do. So this idea of the separate knowledge basis, belonging to particular people, was something that we also wanted to break down those barriers, quite honestly, for our students. So while we're saying we've got that practitioner space in the academic space that we're wanting to get rid of that dichotomy between it's also helping them understand that the people in all of these spaces have a lot of that same knowledge base. They're just using it in different places at different times. But they all share that. So that's one of the things that I've really read. So you are pushing on, you also mentioned that you all learned a lot. So it's very clear to me how the student learned a lot. Can you talk a little bit about how you and the teachers learned a lot? Sure, the teachers both mentioned how much they learned about the technologies themselves. So one particular teacher Gail hadn't been using technology much in the last year or so. They are in a one to one school, she uses their LMS a lot. She had been using some of the simpler quizzing systems and so forth, like they did Cahoots, and they did shared Google Docs, but she had kind of gotten away from some of the math action technologies. And she's like, mentioned how much one this is reminding her why she used to love to use them, but also how far the technologies have come. So she needs to go back and revisit them. So that was one thing that Gail mentioned, Kristin learned a lot. She was a Desmos expert, because she learned a lot about GeoGebra. And she'd forgotten about some of the things didn't realize some of the things she could have been doing in that, that she had been using a paid program to do and had lost access to that paid program. And now with this, we reintroduced her to something she could use for other things. So I'd say the two of them, it was more the things I talked about were on the side of the technology. For me, a big part of it was kind of the context of the infrastructure of doing these sort of things in a school on Chromebooks with paywalls. With only certain number of outlets, those kind of things where we I had not been in a one to one classroom with Chromebooks. Prior to this, the one to one classrooms I've been in we're head, iPads, or some a few shared computers here and there, but PCs, not Chromebook, but Dells or something like that. So understanding kind of that the move towards Chromebooks, which is something that our local district is using, I don't know how widespread that is, and some negotiating some of the things around that infrastructure. All of us, I would say, nothing right about the paper, we've talked about a lot how we it took a little while to develop a common language. First to realize all the things that we were actually talking about the same thing, sometimes just using different vocabulary, which just came from where we learned about that thing, or the name that we had assigned to whatever that thing was give an example of that my students came into their classroom just talking about assessing and advancing questions. And you know, that was language that they weren't familiar with. So they came to me was I want to make sure I'm using similar language. Well, what's this advancing and assessing question thinking? So we said, Oh, well, when I was in school, we talked about these kind of questions. We talked about how those, which the different language we've used in different literature to talk about similar sorts of things, but what the big ideas are, what's that we are posing good questions that we are trying to understand our students thinking and also posing things that help push your thinking forward a little bit more to. So that's one of the examples that comes to mind. What was the research question that you try to answer and how would you answer it? And more interestingly, since we already talked about that, what were some of the major challenges? The major we kind of talked about as a major challenge, for me personally, was in the times that I was sitting with the students and they were working on revisions based on feedback from Gail and Kristin. And Gail and Kristen weren't in the room with us. And they had questions about it and they had ideas for how to move forward and I was having to help support them in that in making assumptions about maybe what Kristin and Gail were looking for there and why they made it

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Looking for that, rather than having them there to to address that themselves. So part of that is necessary because everybody only has so much time to give. But in trying so hard to make sure that everyone's perspective was being taken seriously. I felt sometimes like I was overstepping boundaries, because I wanted that to come from them. So for me, that was a challenge sometimes. So did you bother Gail and Kristen at all afterwards? You're like, Yeah, no, you know, don't worry about it. So since you have that tension, I'm wondering, is there something you did about the way that the students elicit feedback that alleviates that a little bit? Oh, is this just a perpetual problem? We put in more cycles of feedback. So the first year we did so when we wrote about this, they went through one cycle with one of the teachers and then they did it again, we no longer do two in a semester, that's just not possible. So yeah, that was way too much. We start early for something that's going to be taught very, very late, very, very, very late, which allows a lot more time for smaller cycles of revision over smaller bits of things, and doing that more often. So adding a second cycle with the teacher seems like a nice intervention so that they could revise based on how they understood the teacher, but then they can check in. Yes. And we had that originally, I think we now do that at least three times. So it smaller chunks before the big one at the end, we found often that they didn't under they students sometimes aren't interpreting the learning goals, even in the same ways that the teachers intended that so we do a real early check in of, here's how I'm thinking that we're going to we chose this technology, here's why. Here's what we're thinking we're gonna do. And we think it will dress your learning goals in this way, before they even start building the technology piece. And that was something that wasn't there before. So that's where the third one comes in, to try to address that. So let me ask you a bit off script question. Yeah. So you're doing this in your classes. And you you wrote this paper about something you did a few years ago, and you've been doing it? So presumably, you've gotten better at it? How do you know, it works? Or what makes it work? Or, like, why would I as I'm listening to you on the podcast? what gets me excited about trying to do this? It's a really great question. One, I know it works, because I've worked with this in the cycle of courses that we take, I often teach the methods course that comes after this. And the difference that I saw between students who had taken that technology course and those who had not in their attention to students thinking in their test selection in alignment to learning goals, it was night and day. So it focuses them on student thinking that's one of the outcomes definitely did, yes, because a lot of the feedback from the teachers was about, well, my students, my students, my students, so it's no longer this hypothetical student that might be out there. It's an actual group of students that it's being designed for. Okay, that was definitely one of the big things, the quality of the tasks they've designed, have been of such high quality that the teachers have continued to use them year after year after year. So it did, it hasn't turned into something that all right. student teacher came in, and they did this. That's great. It was designed for me, for my students. And we're going to continue to use this even if we continue to tweak and revise it. So these tasks they've designed to have stayed past them out in the schools being used. So they must be good quality tasks, which is great. And see me line. Sorry, go ahead. No, say at the same time, we see the difference in those students in the courses that follow as well. I was just gonna say that you might have some task collections that you could publish at some point out of that. Probably could they've been sharing them every semester every year at our State Conference, that's part of what they've been doing is a conferencing, which is great. Okay. So let's close out a little bit by asking you so I've heard this, this sounds fantastic. I want to use it. What's your advice to me? How do I get started? What do I do? The first thing is finding some partners, finding partners that would like to work with you reach out to principals, like I mentioned, is a great starting point. And sit down and meet some teachers it whether it's for a technology class, like mine, or some other type, of course, but have a conversation about here's what I would like my students to learn from you and how we I think we could work together but how could this help you as well, and I think that that

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It is really, really important so that it becomes something that's mutually beneficial that everybody really wants to put the time into because they see the benefit for themselves of the K 12 students for the pre service teachers for every for everybody involved. All right. Well, thank you so much for joining me today. Thanks for having me. And for further information on this topic. You can find the article on their math teacher educator website, this has been your host, Ava Sennheiser. Thank you for listening and goodbye.