Eva  0:00  
Hello and thank you for listening to the mathematics teacher educator journal podcast. The mathematics teacher educator journal is co sponsored by the Association of mathematics teacher educators and the National Council of Teachers of Mathematics. My name is Eva fan Heiser, and I'm talking with Megan Wickstrom from Montana State University. Today, we will be discussing the article developing pre service teachers understanding of area through a unit intervention published in February 2022 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. Megan, can you briefly introduce yourself?

Megan Wickstrom  0:47  
Thank you for having me on this podcast. I'm Meghan Wickstrom. I'm an associate professor of Mathematics Education and the Department of Mathematical Sciences at Montana State University in Bozeman, Montana. And I've been there for eight years.

Eva  1:00  
Yay. And you're in sabbatical right now. All right, so let's jump in. Can you give us a brief summary of the article including the results?

Megan Wickstrom  1:12  
Yes. So in this article, my goal was to document an intervention, mathematics teacher educators could use to bolster pre service, elementary teachers understanding of area and from Pastor search my own and others, we know that pre service teachers often enter content courses with limited understandings of area. Yeah,

Eva  1:33  
I agree with that. I feel like that's one of the things I've been really struggling in the content courses, areas, always hard for them.

Megan Wickstrom  1:43  
Yes, and most of the research out there are many of the studies have focused on uncovering formulas and thinking about pre service teachers understanding of formulas and unpacking where they come from. But we know from chill research on children's thinking that before we even get to formulas, we have to understand area, and unit structuring. And so that's where my intervention takes place is in that space of thinking about units thinking about area units, and where to those ideas about multiplication and area overlap. And to document whether or not this intervention was effective in the article I draw on the different strategies that they use, and the correctness across strategies. And I zoom in on specific case studies, to really help the reader see some of the things that pre service teachers are grappling with.

Eva  2:31  
Yeah. So as I'm listening to you, and as I was reading through the article, I was thinking about a task I used to pose my students. And I feel like if I asked them, What is 15 times 32, they can figure it out. But if I give them an area model, and say, This is the area of I should have calculated, what 15 times 32 is, whatever that is, what are the side lengths a day cannot do it. And it's just really interesting. And I feel like in your article, you touch on a lot of issues that deal with that. So I was super excited to read this. So let's jump into who is the article for.

Megan Wickstrom  3:16  
So I would say the articles for two different audiences. The first audiences, I would say is math teacher educators teaching content courses, so anyone who's interested in incorporating hands on activities into their content courses to really unpack and explore area. The second person I think this might interest is someone who might be teaching your graduate course on learning theories, or thinking about learning trajectories, because I think the TAs does a really nice job at eliciting different types of strategies, and helping pre service teachers or whomever you're working with. Think about the strategies and how they're connected to each other and how they could build off of one another.

Eva  3:55  
Cool. So why other than I clearly care because I struggled with this. But why is this an important problem for our community that you're addressing?

Megan Wickstrom  4:08  
So what I found super fascinating about this problem is that this has been something that we as a field have wrestled with for a long time. And and by longtime, I mean like 30 years or 40 years, which feels like a long time. But it's something that someone people in the field have talked about, and then it kind of goes under the surface for a little bit, and then it revives itself. So when I first got interested in this, I had found an article that was really that I drew a lot from which was an article, it was a material and nascent article from 1996 and Educational Studies in mathematics. And I believe both of them are Australian researchers, and they had done this in depth look at all of the different strategies that pre service teachers use when solving area tasks. And they showed that pre service teachers really they really struggled with some of the things that you were talking about. Like, if they multiplied two side lengths together specifically, like if they multiplied, you know, two decimals together, like two and a half times two and a half, they struggled with figuring out, how can they see the area inside of that square. And then we there were further articles. So there was an article by Simon and Blum, I believe, in 1994, and Jeremy, and then what was super interesting is when I wrote this piece, one of the reviewers actually identified this conference proceeding from the early I think, 1990, where another researcher had talked about these issues. And so as an early career teacher, when I saw this in my classroom, I started wondering, how can I help pre service teachers? How can I help them make sense of area? When what activities and insights can I draw on from the field. And when I went to consult different resources, what I found was that those calls from the early 90s to develop task and to think about pre service teachers understanding of area they really hadn't gotten answered. And so that was what kind of motivated me to dive into this field and think about how can we support them in understanding concepts of area.

Eva  6:09  
And as you were talking, I was also thinking, especially with the two and a half times two and a half example. They will often do four and a quarter, right? Where they multiply the whole number and a fraction, it like, a plus b squared is a squared plus b squared kind of idea, right? Yeah. And so that's one of the fundamental reasons why we need to really understand how and why multiplication works. Because it's, it's everywhere, right? Yeah, like, it doesn't matter where you go down the road.

Megan Wickstrom  6:44  
And for them, in particular, I think one of the things that I stress to them is that when we look at things like the common core, the area model is interwoven throughout all of the Common Core. And so as teachers, they need to be able to look at something like that and use it flexibly as this tool for their students to be able to understand and visualize multiplication.

Eva  7:06  
And the Common Core actually does a really nice way or the they have these progressions through the Common Core. And there's like a lot of nice color examples that use color to make some of those connections. All right, so the next question is What how do you build on prior work, which you just answered, I think, did you want to add anything else?

Megan Wickstrom  7:32  
I think the other thing that I wanted to add was Where did my task stem from? So my work, the theoretical underpinnings of the study, were really based on this idea of learning trajectories, or mapping how students come to know a concept over time and what the instructor can do to support them in that understanding. And so as a graduate student, I was fortunate enough to work with Dr. Jeff Barrett at Illinois State University, and he does work with children's learning trajectories. And as a graduate student, I worked a lot with teachers and helping teachers integrate their understanding of learning trajectories into practice. So listening to students and figuring out how can we respond to students. And one of the things that I wondered about as I worked with the teachers, was I wondered about how they reasoned about these concepts. And this became even clearer. When I was working with pre service teachers in the classroom. I wondered, I wondered, like, how are they thinking about these topics? And how do they understand these different concepts of measurement. And one of the things that I find really interesting about pre service teachers is that they reason in a lot of, in a lot of ways similar to students, but in a lot of ways, they have this extra conceptual baggage that they bring along for the ride. And for measurement topics, a lot of it is these formulas, they cling on to these formulas. And they haven't really received that in depth understanding of where the formulas come from. And, and kind of the conceptual underpinning. Yeah, I have

Eva  9:01  
found that to that is kind of a drawback for pre service teachers. Not only do they know the formulas, they also trust the formulas more than they trust their own instincts. So it's an interesting dilemma that you face when you get to work with them. So let's get to your intervention. Can you tell us a little bit about the intervention, what it is and how it addresses this problem of pre service teachers? Not yet understanding?

Megan Wickstrom  9:32  
Yeah, so my intervention is a three day activity. And here at Montana State, the course that I teach, the each day is about a 50 minute class period. So if you're thinking about it in terms of timing, it's usually about you know, maybe 100 to 150 minutes of classroom time. And so, in the first day of the activity, the pre service teachers are presented with this 32 by 32 Stud Lego mat. And so when they're presented with a mat, they're given several different types of Legos, so different rectangular and square Legos. And what they're asked to do is to determine how many Legos it would take to cover the mat completely, was built off of a study in 2017 that I did with my colleagues where we looked at different strategies that pre service teachers use when they're trying to tile a space. And so the purpose of that first day is for the pre service teachers to really uncover some of these strategies. And if it's helpful, I can kind of unpack those strategies to that. Yes.

Eva  10:35  
So I'm envisioning, I think your image shows like one of those green Lego

Megan Wickstrom  10:41  
foundation.

Eva  10:42  
Yeah, the flat thing. And so you say it has 32 by 32? And then I think on the image, there's like different sized Lego pieces. Yeah. So are you asking the students are to just fill the whole thing,

Megan Wickstrom  10:56  
this is where it gets really interesting and fun. What I have on the table for them is these bags of Legos. So let's say like maybe I have 102 by two Legos sitting in a bag. And so they have to decide what they want to do, they can take a little bit of each one, they can take a whole bag. And I purposely structured it so that there aren't enough to cover the whole space. And so as they're working through that task, usually there's five different strategies that kind of pop up. The first strategy is just a counting strategy, like what you mentioned. So where they try to just kind of fill the whole space, or what does it face?

Eva  11:33  
Does it matter which size they're using?

Megan Wickstrom  11:37  
Yes, yeah. So they're asked to think about one Lego in its entirety. So each Lego, how many of of that particular Lego would it takes to cover?

Eva  11:46  
And so there is square Legos and rectangular Legos, right, but you it doesn't matter, they can pick whichever shape they want, they just need to cover it with that shape.

Megan Wickstrom  11:56  
Yes, yeah. So, um, what they want to start with. So some of them, they might use counting, some of them go back to this idea of length times width, and use multiplication strategy, where maybe they outline the length and outline the width. Some of them, especially with the rectangular units, they'll do something that we call addition of parts, where they separate the map into different regions. And maybe they have a multiplication statement for a region. And then they're counting another part of the region. What the reason that I developed this was actually to get at the comparing unit strategy, which is based on Susan laymans idea of unitizing, where another strategy that they can use is to compare units. So if they know that it takes maybe one, I don't know, if it took 10, two by two blocks, and you have a two by one block, you know that that's half as big, so it should take twice as many. And then the last strategy is this idea of dividing. So on the Legos, there's studs, is what they're called. And there's a certain number of studs on the mat. And so sometimes the pre service teacher will take the number of studs on the mat, and then divided by the number of studs on the Lego piece to determine the quantity needed.

Eva  13:04  
Okay, so let me see if I can re state so the first one, I could imagine filling that whole green Lego piece with like, one by one. Sorry, there's no one by one two by twos? Or is there one by one or one by one? Okay. So I could fill it with one by ones? And that answer would give me what we would think usually of area, right? And then I could fill it with two by twos or two by fours, and but always with the same. Yeah. And so then there's different strategies to get it like if I already covered it with one piece, then I could just compare that piece to another piece and then use that information. Yes. The other thing you said is, I could take the whole green thing, which is 32. And divided by the size of what I'm calling it with, which is really kind of having one dimension and finding the other one, though, it's not really the dimension of the square, you would find a different area dimension. Yeah. Wow, that is really intriguing. So that's the first part. Yeah, that's

Megan Wickstrom  14:12  
the first day. And I think a really interesting about the first day that I didn't expect was that we spend a lot of time talking about the Lego pieces and how we should call them. So should we talk about them in terms of their area, like the number of studs on the piece? Should we talk about them by their length and their width? And so it gives us an opportunity to talk about these different attributes and the way that we can talk about the pieces. And then we spend

Unknown Speaker  14:38  
somebody suggest color. Yeah, they're also all a different color.

Megan Wickstrom  14:42  
Right? Yeah. So yeah, so sometimes people will suggest color. But yeah, so that first day we we really just talked about the dimensions, the attributes, and then we spend a lot of time just trying to nail down like, usually I have a whiteboard, and I have them articulate their strategies. And we think about like, is this the same as the strategy or is this different, and getting them to really voice what they're doing?

Eva  15:03  
So I want to come back to the two naming conventions you just mentioned, which could be like either we refer to Lego by its area. So I could say it's a two or I could do the dimensions, which could be a two by one or one by two. Yes. And that is super related and super difficult for like, even that connection. I think that's the crux right of them understanding area. So the fact that that comes naturally out of that task is pretty amazing, because that's usually so artificial to talk about.

Megan Wickstrom  15:39  
Yeah, it was really exciting. And then like, the other thing I thought that I had not anticipated would come out of this task is on day two. On day two, we usually talk about where where do these strategies stem from? And does the strategy relate to length? Does the strategy strategy relate to area? And would you want to use the strategy on all the Legos because some of the strategies naturally lend themselves better to different Lego pieces. So multiplication, for example, the square Legos, they know right away how to multiply and find the area. But the rectangle, the rectangular Legos are much more challenging, because most of them don't tile the space completely. And what I had not anticipated was the same thing that the Simon and Blum article showed in 1994, was that they want to take the rectangular Legos and lay out the longest attribute along the length and the longest attribute along the width. And so every single time I teach this, we get into this interesting conundrum where, for example, I had a group of pre service teachers, where two of them said, Well, it's going to take the same amount of two by one Legos as to buy two Legos. Because look, I have, you know, let's say at 16, spanning the length and 16, spanning the width for each, and when I multiply those two together, I get the same. And then I have the, the other two and the group saying, Well, no, that doesn't make sense, because this two by one is half the size of the two by two. So it should take more, it should take a lot more. And so we have this every single time I teach us, we have this conversation, and I bring it to the front and have the whole class talk through this, like what do you think? How can we interpret this through these two different strategies, and the rectangular Legos really allow myself as the instructor and the other instructors who have taught this, to get at that idea that multiplication and length times width, how that connects to groups and number in a group?

Eva  17:36  
Yeah. And I guess the other thing that is really difficult is to understand that area is measured by units of area and not by units of length, right, which is where all of this confusion comes from. Yeah, because that's I think it's the first time we switch on our students. Right, what because when we do addition and subtraction, we add NV The answer is the same unit as each of the parts. But in multiplication and division, we like think about length and width. And then we get area. And that's like, confusing, right?

Megan Wickstrom  18:17  
Yeah, it's super confusing.

Eva  18:19  
I love this idea that you just mentioned that, like, if we use square pieces no matter what size, it's easier for them to see it because their size matches, at least in this case, the square matches the square, green. So when you talk about area here, it gives you the opportunity to talk about area with respect to different units to which is also something we don't usually do, right? Because usually areas are one by one. And so area for a two by two versus two by one. So that it gives you this, this idea of also clarifying that we're actually measuring with area pieces, because look at answers. That's cool. So I explained a to a little bit more in

Megan Wickstrom  19:07  
depth. So in day two, most of them, I mean, most of them after day one, I would say if for anyone else thinking about trying this task, they get drawn to those rectangular units, they get kind of frustrated with them, then I usually tell them, you know, pick a different unit, try something else develop some different strategies. And So day two is really kind of going back and clarifying those strategies and working with Legos that they found challenging to begin with. So we spent a lot of time and day to just solidifying those strategies, being able to articulate what we did, and just giving them time to like clean up their thinking and be able to revise their ideas to be able to put them out there.

Eva  19:47  
And when you talk about strategies, is that two main different strategies that one is you get the length times width, and one is you separate it down into regions.

Megan Wickstrom  20:00  
I would say there's four. Besides counting, there's four different strategies. So one is the multiplication, getting the number of units that you need for the length and the width. One is what you said the addition of part strategy where they're separating it into different regions. The third strategy is comparing units. So like, if you've already tiled with this, buy two unit, can you infer how many four by four units you need? And then the fourth strategy is thinking about division, and just focusing in on areas and comparing the areas of the space, when I think about the four strategies to really boil down to ideas about multiplication, the addition of parts and multiplication, and to really boil down to thinking about and comparing areas, which is the computer. And then, yeah,

Eva  20:44  
cool. Yeah. And so day three.

Megan Wickstrom  20:47  
So on day three, this one I added on in the last couple implementations of the task, and I thought it was a really nice opportunity to come back to multiplication, and thinking about the area model in terms of fraction multiplication, mixed number, multiplication, and revisiting some of those ideas now that they have the idea that different size units can exist, and that we can work with these units in different ways. So for example, on day three, we do tasks, like, draw me to different area models of three, something like three fourths times seven eighths, and think about different units that you can use, and what does that show about the multiplication?

Eva  21:26  
So that seems like a huge jump from days one and two. So talk about how day three goes.

Megan Wickstrom  21:33  
What's interesting is that most of the students have had the Number and Operations course prior to this course, they've learned this before. And so it's almost like revisiting past material. And most of them feel pretty comfortable multiplying three fourths times seven eighths and generating something that they've seen before, like from the previous course. So maybe draw. So

Eva  21:55  
I'm trying to think about, like, if I had this experience with Legos, how would I think about three fourths and seven eighths now?

Megan Wickstrom  22:02  
Well, so one way that they could think about it is that you have this denominator that could be partitioned into same size units. So instead of thinking about three fourths as three fourths, I could think about three fourths as six eighths. And now instead of thinking about three fourths, times seven eighths, I could think about it as six eighths times seven eighths. And now I have this, I have the same size units. But what's really interesting is then they start trying to grapple with what is the like, they understand that these different units and different sizes of units can exist. But they begin to grapple with, how big is that unit? So I have 42 of these things. But what are these things? Yeah, and how do I understand what the denominator is?

Eva  22:48  
So there's a few things I'm thinking about, because in our system whole numbers isn't a different course than fractions. In a sense, it seems like I have to almost do the fractions after the whole numbers here to really understand the whole numbers as well. So I like how you're putting them together as a task. Yeah, it's cool. It's fun.

Megan Wickstrom  23:14  
I mean, it's really fun. And I think that third day, for me, at least, it's not too much of a stretch, because they've seen this before. So it's almost like revisiting something, again, through a new lens.

Eva  23:25  
All right, so my next question is, how did you research or What question did you look at to document that this is actually effective this innovation?

Megan Wickstrom  23:35  
So I actually spent a lot of time I really wanted to write this piece up for for MTE. And I spent a lot of time thinking about how could I structure my data collection in a way that, that it would tell the story of this task? So the two things that I looked at the two research questions were one about the correctness. So the first thing that I I looked at was, I had my pre service teachers solve these area tiling task once before the intervention began and once after the intervention. And I think it's important to kind of talk about what a tiling task is. So a tiling task is asking them to use a particular kind of tile to tile a space and to explain their thinking. So for example, one of the tasks that I would give them might be something like how many, three by three inch tiles would it take to cover a square foot, draw a picture and tell me how you know, and so I always would try to choose one tile where that tile would cover the space completely, and one tile where there would be partial tiles. So for example, I might choose something like a three by three tile and a nine by nine tile, or a four by four tile and an eight by eight tile. So that way, I could get a sense of how are they grappling with the problem when you know there's a hole number three the side lengths versus like a partial number you that. So I asked that before the intervention and after the intervention and the things that I attended to, were the strategies that they use. So for each one, I asked them, can you try to do this in two different ways? So I looked to see what are the strategies that they're using? And then I looked at the correctness across those strategies. So, you know, are they able to use a wide array of strategies? Are these are they able to use more than one strategy? And how correct are they in these strategies? So what did you find prior to doing the intervention, it's really important, I think, to think about after like prior to the intervention was still after they had received their typical instruction. And our typical instruction is still pretty robust. So they're still getting multiple weeks on studying area formulas, thinking about different principles of area like moving and additivity. And what I found was that prior to the intervention, that pre service teachers, there are a couple of things that I found pretty surprising. So first off was that a lot of them for like a three by three tile that could cover completely, there were still three of them using what I call linear reasoning, where instead of even thinking about area, they're applying formulas for length to area. So three of them are still doing that, a lot of them are doing counting, so they were trying to draw the picture and count up. And then a couple were using multiplication, but not very many, like maybe one was using these other strategies. And for the tile that doesn't cover in a whole number of tiles. Most of the pre service teachers, they had no idea what to do, they would try to draw a picture and maybe draw like a nine by nine tile over the 12 by 12. tile, and then maybe write me a note, like, I don't know where to go next. I don't know what to do next. And so they were really struggling with trying to coordinate number and space. And the thing that I found super interesting if we had focused so much prior to the intervention on multiplication and area formulas, and when faced with that nine by nine tile, only one pre service teacher tried to use multiplication, to actually make sense of how many nine by nine tiles would go into a square foot,

Eva  27:15  
though, can I ask you a quick question you distinguish counting from multiplication? Can you share quickly how you think about multiplication?

Megan Wickstrom  27:24  
So I think about multiplication as them being able to talk about, well, for the most part, how it surfaces is then being able to talk about the side lengths in relation to the area and thinking about the side lengths being indicative of like the groups and number in a group. And so a lot of times when I think about coding for counting, it's more that they are, you can see evidence of them just like building units on and kind of counting them and guessing and checking. And there isn't evidence of this idea of a systematic approach, where they're comparing maybe side lengths of the tile to the area. So if

Eva  28:00  
they did a thing, like they tiled a space and counted how many are in one row, and then counted how many rows there are that would that still fall into counting? Or would that fall into multiplication?

Megan Wickstrom  28:14  
That would fall into multiplication? Okay, so

Eva  28:16  
counting is more counting one by one idea.

Megan Wickstrom  28:20  
Yes, yeah. Or in an unsystematic? Way, where there's not evidence of them thinking about structure? Okay. Yeah. And I would say, I don't see that as often. Like, for adult learners, what if they're covering a space with tiles, like the three by three tile into a square foot, most of the time, they'll start by drawing a picture and start counting, and then multiplication will kind of dawn on them, like, Oh, I could use multiplication to describe this, where I see a lot of counting going on is where they're in a space, that they can't make sense of it. So like the nine by nine, or eight by eight into a 12. By 12, they're not sure how to quantify those pieces that are remaining. So then they start just trying to count the pieces and bring them together.

Eva  29:04  
So this leads into two questions. For me. The first one is, even if they did like a structured counting of like, how many are there in a row? And then how many rows are there? How do you know that they're thinking about area with respect to area, you know, like, with the larger area with respect to the smaller tile,

Megan Wickstrom  29:26  
that's something that still kind of remains unknown, like until then one on one, and then I kind of know a little bit more about what they're thinking. For me, I think that's kind of a stepping stone into them being able to think in a more complex way about how the attributes relate to each other.

Eva  29:42  
So if I'm stepping back and thinking about what is the goal of your intervention, it is really, and I'm gonna say something you'll correct me okay. It is about seeing the structure of multiplication within the tiling is that One of the aspects you want to get at, yes,

Megan Wickstrom  30:03  
yes, and understanding how those how the attributes, those two attributes, the length attributes, and the area attributes can inform each other. So giving them ways to play with space to understand how those, what are things that we can infer what are things that we can understand just by looking at those different attributes.

Eva  30:23  
So you told us that they struggled before your intervention, what happened after.

Megan Wickstrom  30:29  
So after they were one of the things that I saw was that they improved significantly in their ability to solve the tiling task. And in the number of strategies that they implemented. So more of the pre service teachers were able to solve the task in two different ways correctly. And most of them shifted from primarily using counting or multiplication to kind of dispersing across the different strategies. And one thing I was really excited about was that the linear reasoning which I wanted to kind of eradicate was gone. So none of them were thinking about area measurement in terms of length conversions. And most of them, were also able to really draw a picture to leverage and explain their calculations. So you can see kind of changes in the article, where they're able to draw in the particular units, look at the side lengths, and they're able to really leverage those pictures as ways to justify their reasoning. In the pre assessment prior to the intervention, it was like they would try a particular strategy, and they weren't sure of it. But they didn't have another way to kind of leverage their understanding to double check that what they were thinking was correct. So you know, one of the case studies, he did something correct, but he wasn't sure about it. And so he was trying to do calculations to make sure that what he had done was correct. And I didn't see that in that second post intervention assessment, where now they're able to kind of draw on these different strategies to help support their thinking.

Eva  32:05  
All right, that's exciting. Yeah. So how would you summarize the contribution that you're making to our field?

Megan Wickstrom  32:13  
So I think there's two contributions, the major contribution, I think, is that call. So in the 1994, Simon Blum article, they said, they basically made a call to the field that we need more tasks to support pre service, elementary teachers, and reasoning about area. And we need tasks that promote in promote them playing, promote them thinking, promote them talking with each other about these ideas. So I would say that's the first contribution is just getting this out there and giving mathematics teacher educators the opportunity to try this and see what they find and see how their pre service teachers understand this task. And then the second is to think through to add to the, to what we understand about pre service teachers, and their understanding of measurement. So like I mentioned before, a lot of studies on area measurement have been from more of like a dynamic perspective of thinking about formulas, and where do they come from, but not really focused on those building blocks and understanding the units. And so I think from this article, we can really see that pre service teachers aren't that different from elementary students, they need time to play they need time to explore, they need time to really visualize what they're working with. And then by giving them those times that time to play in a space and really reason about a task. With tangible tools. They're able to make great leaps and bounds and their understanding about area.

Eva  33:41  
Awesome. Usually, we close with two questions. How do you see other people using this intervention? I think that's pretty clear. Since you have a lesson plan attached there. Yes. And then how does your the work you describe fit with your work in general? So

Megan Wickstrom  33:59  
Well, one thing I wanted to put out there, as I thought a lot about the lesson plan, Dr. Karen Harlow brands encouraged me to create that as a appendix. And so anyone wanting to try this task, I would encourage you to look at the appendix because that was one thing that I poured a lot of thought into is like how would somebody else jump in and try this task. So it gives much more detailed and the article of different strategies to expect and things that you can see from your students? My work in general. So my work in general falls in two domains. So thinking about area measurement, and working with teachers and measurement ideas in general and then mathematical modeling. And I would say that the way that this task fits in with my research is that my goal is to create spaces where students can bring their, their their lived experiences, their authentic self to the space and they can share those ideas and grow. And so I think this task really fits in well with this idea of how can we promote pre service teachers as knowers and Two years of mathematics while also kind of exploring and building on what they know, and documenting that.

Eva  35:05  
So let me push a little bit here. How does this task, help teachers bring their authentic selves?

Megan Wickstrom  35:14  
I think that the idea of play, so my pre service teachers when they come to the task, I think a lot of times in specifically in measurement task, we're often given a formula and told, you know, here's the area of a triangle formula. This is how you calculate the area of a triangle. But no one really asked students like, here's a triangle, how would you find the area? What would you do? And so I think it puts students and pre service teachers in the driver's seat where they're actually making the choices, and they're able to think about what strategies that they use, and how can they contribute to class discussion. And it's always interesting every semester to because they're always trying to find different strategies are like, can we find another one outside of these five that I could use? Let me

Eva  36:02  
try to rephrase what I think I understand. What you're saying is, because there's materials they get to play with, that they come in, they enjoy it more, they have more ways of coming up with different ways of thinking, that kind of how you're seeing the connection.

Megan Wickstrom  36:19  
One thing I think is surprising is that a lot of times they see it as play at first, and then when they realize that it's cognitively challenging, I think they have a different response. But through the tasks, they they gain this insight. And what I see is that they gain more confidence in their abilities to really think through these problems. And so it's in a space where they feel comfortable trying things. So they realize that we're in the space where they're able to try out different things. And you know, I think most of them as kids were used to seeing Legos as a creative tool that they can build things and try things out. And so I think in many ways, they bring that same stance to the task where they don't see it as right or wrong. They see it as kind of a play space of we're going to try this. We're going to see what works and then we're going to report out.

Eva  37:11  
Okay, that cool. Well, thank you so much for joining us today.

Megan Wickstrom  37:16  
You're welcome. I really appreciated being asked to be on the podcast.

Eva  37:19  
So for further information on this topic. You can find the article on the mathematics teacher educator website. This has been your host Eva Anheuser. Thank you for listening and goodbye.

Transcribed by https://otter.ai