Secondary Keynote: Rough Drafts and Revising in Mathematics

a joint appointment. Uh she got her PhD from Michigan State University. So she did kind of her rounds here uh in the Michigan area, but she's originally from Arizona. her undergraduate was at the University of Arizona where she was a math and English secondary major. Um, and then she, you know, she did her, uh, she taught middle school math for a little bit before going back to her PhD program. Um, and then since that time has an interest in the area of motivation and engagement as well as teacher education. Um, just kind of a fun fact that she shared with me a little bit early earlier is she loves pinball and she owns her own pinball machine, right? And in particular, she said it's based off of the game Roller Coaster Tycoon. It's a sim video game. Have you played this one from early 2000? Yeah, there are some folks like you're the respect pinball machine for that one. Yeah, that much more. So, anyway, so super fun person uh an amazing researcher and teacher. So, uh, it's just going to be a great session. Without further ado, Mandy. Thanks, Paul. Yep. Hi, everyone. Thanks for coming. This version of the talk will have examples more from, uh, middle school, like grades seven to nine, but there are always ideas that you can take into older grades as well. So, I have been working on the ideas of rough drafts and revising in collaboration with teachers in Delaware, and then it's expanded since then. If you want a copy of my slides, I think they'll be available online in some way, but you can also um go to this tiny URL, rough draft copy, too. And if you're on Twitter, please follow me. I'll follow back and then we can stay in touch beyond today. So, my Twitter handle is Mandymath Ed. So, yeah, it's fun to be back in Michigan. I appreciate this. Just a little bit of snow, not too much, so I can feel like I'm back in the Midwest. So when I tell people I was your name is Mike, right? And you were saying, "Oh, I didn't really know where Delaware was." Lots of people don't. I say, "Oh, I live in Delaware." And people say, "Oh, I really like New England." And I say, "That's cool. I don't live in New England." Um, Delaware is like the east coast of Maryland. This peninsula peninsula here is called the Delm Marva Peninsula, and it's got three states on it. The bottom's Virginia, and this is Maryland. This is the Chesapeake Bay. And this part of the peninsula is Delaware. Oo, almost all coastline. And you guys relate with the Great Lakes and your coastlines. Uh, where I work at the University of Delaware, New York, Delaware, is positioned here, south of Philadelphia, north of Baltimore. Those are my two airports. Delaware doesn't have a commercial airport. So, it's south of New York City and north of Washington DC. So, this is like the one hour radius of driving. And then this circle is like a two-hour radius so you can get a sense of where I live. And the president of the United States is from Delaware too. His wife got her Jill Biden got her doctorate in my department in the school of education. Okay. So that's a little bit about Delaware. I've been working on the idea of rough drafting and revising a math class for six years or so in collaboration with teachers. And some people may have heard of it, but lots of people may never have heard of it. So thinking about where you're at and you can show with a fist. If you've never heard of it, it's like a zero. If it kind of rings a bell, it's like a one. If you feel like you can explain it, it's a two. If you've even ever tried it before, it's a three. And if you could maybe teach someone else about it, it's a four. So could you could just show each other where you're at with this. Awesome. Love that. So, there will be something for you. If you've heard of it before, and if you've never heard it before, then it's all going to be new to you. So, that's great. Thank you. Dot talks. People ask students to think about dots. Why in secondary would you ever do that? So, I want you to think about it this way. You might ask the students, how can you tell the total number of dots without counting one by one? How are you chunking the image? Mathematicians look for structure. There are different ways to chunk the image to tell the total number of dots. And one of the reasons why we ask people to revise their thinking is not to fix something that's correct or incorrect, but so we can see differently. So when we compare and contrast how we chunk the image to see to think about the total number of dots, someone else will chunk the image differently and you can see structure differently and you both have correct thinking. It's new ways of seeing. So part of doing math together is to develop new ways of seeing relationships. So think to yourself, how might you chunk this image to think of the total without counting one by one? Now please turn and talk to a partner. How did you chunk up the image to determine the total? Or you can talk in threes, twos or threes. I thought I was trying to make it. I was like, "But if I try to make it the top. So to bring us back together, I'll raise my hand and when you see my hand, you can also put your hand up so we all know that we're together. Thank you. So you might ask students, how did you chunk it? How did you see it? And I like to to collaborate with people on Twitter about teaching. And so this uh math coach was working with kids and had the kids record their thinking maybe up at a document camera of how they chunked the image. And then you can if you want you can take it into symbolic representations of different ways of expressing developing equations or expressions to expand the relationship. Maybe thinking about distributive property. But the point is not about is it 14, right? How did you see it? How did you chunk it? Can you get a new way of seeing? Because then you can imagine if you're working with students to generalize patterns and you see the next step in the pattern, the next step in the pattern. If they're practicing chunking in different ways, when they have a pattern, they might be able to generalize from there once they've started to chunk the image differently. So seeing differently, that's one important idea about revising our thinking. So what I'm going to talk about today has three parts. The first part, how do we create a culture in our math classrooms so rough drafts are welcomed and where students feel safe to share their thinking when they're not sure? The second step in the talk would be how do we explicitly and purposefully and intentionally incorporate revising into math class. And then the last portion would be how do we honor strengths in students early drafts and in their revisions. One of the powers of this term rough draft is that people have heard it in other contexts, right? Like even without me talking about this, you already have some ideas of what this could be. And so you can ask students, what would it be like if we brought in rough drafts into math class? Why would we do that? What would it mean? So we can also think about it. So I have a Padlet that we started this morning that you can add to. So, if you have a device like a computer, like a phone, like a tablet, I'm going to put up both a URL and a QR code, and you can look at other people's draft ideas about why and how we would welcome rough drafts in math class, and please add your own. Why would math teachers incorporate rough drafts and revising into math class? There's already some draft ideas. Please also add your own. You can comment on the ones that are currently there, but you can also add your own ideas. Please collaborate on the document. It encourages students to make an attempt. It's just a rough draft. You're expected to revise and update it later. Success is the destination of failures. That's beautiful. Math is learned through trial and exploration. Somebody wrote, "Give students the permission to be bold with their answers. It's not one and done. The most important thing to do is get started. It's safer to share. It may feel safer to share a rough draft than a higher stakes final answer. We don't just have one person run numbers one time and then send a rocket full of people into space. Multiple people, multiple drafts and rechecks. Thank goodness. Wow. So you can see that if you ask people why would we use drafts in math class, people already have thoughts. So you can build on people's prior experiences with that concept and ask students what this means to them. So to introduce drafts and revising to their students, a seventh grade teacher in Newcastle, Delaware just asked her students to reflect. She made a prompt in Desmos and asked the students to share in our math class this year. You will do a lot of rough draft thinking and rough draft talk. So, what do you think this means? Before I even tell you, just like what do you think? Her student said things like they mean we're going to try it, possibly mess up, won't be a big deal. It's going to be a rough draft. I think they mean just like the first writing I will make. That one doesn't have to be perfect. And then you make your second draft and then until your final. Your final should be perfect just like that rocket. and then you keep on improving through different drafts. I think rough draft thinking in a math class means your first thought on something when you just discover it. And rough draft talk means discussing your opinion with other people's opinion to gather new information. I think rough draft thinking is unfinished thinking. Like you're not completely done about what you're thinking. You don't think you're thinking is right. And rough draft talk is what you think but it isn't complete and you're not correct. But you pretty much estimate what you think. But it so the students have ideas and you can help they can help you build the ideas about what rough draft could be. So I would say that rough draft thinking happens when students share unfinished and in progress ideas and also when students remain open to revising their ideas. Even when they think their answer is correct, their work can still be improved. They can make a new connection, a new representation. They can refine their arguments. I spent a chunk of my career as a researcher trying to understand what it feels like to be a student in a math class that's having a discussion and a conversation. And students seem to make a distinction between it's my job to perform what I know. So people might think I'm smart or I don't want them to know that I don't feel smart in this moment versus talking in order to keep learning together. And we want conversations to be an exploratory space where we all learn together. So it really makes this contrast between final draft talking to perform versus rough draft talking to keep learning together and we want conversations to be that learning space. I spent a semester working with teachers in Delaware and we were reading this book, Exploring Talk in School and we were thinking together about how to create classroom discussions that were more exploratory in space. And the teachers in the study group said it could be more useful to use the phrase rough draft talk rather than exploratory talk because that label made more sense to students. So rough draft talk sounds like students will say, "Oh, I don't really know. I'm not really sure yet or my brain hurts or I can't find the words." And the idea here is you're communicating to work on developing your understanding. Anytime you try to put something into words or to represent it, that process helps you develop your ideas. And students pick up on the term right away and they start to use it themselves. Oh, can I just share my rough draft ideas? Can I share my rough draft thinking? So, they find the term useful for them to clarify to other people where their thinking is at, what stage they're in. One way to build a culture where people feel safe is to talk with students about their rights and even develop a set of rights with your students. So I draw from the work of Crystal Craig who lives in San Antonio and she learned from a math teacher in Tucson, Arizona, Olga Torres. And Olga develops a set of rights with her students. And these are some rights that you can communicate to your students. You have the right to be confused in here. You have the right to make a mistake. You have the right to communicate in ways that make sense to you. You have the right to make mistakes. You have the right to revise your thinking. You have the right to share unfinished thinking and not be judged for it. and asking students to say today what right do you want to work on in acting at the end of the lesson reflect what right do you feel like you enacted today and how and when or think about your classmates what right did you observe someone enacting and how even more what rights do we want to build in this classroom what other rights do we want to have and building the set with your students. So, thinking of these rights, are there certain rights that you value or any rights that you would like to add to this list? We're going to do some math together in a little bit and think if you want some rights in this room. What rights would you like to have? Please talk with each other about these questions. I like So sometimes my job is the role of interrupter, right? Like you're having these good conversations. But I'd like us to hear a little bit from each other about your thinking about the rights. And when I pass this around, it doesn't amplify in the room, but there are folks who are in the hybrid situation and it allows people to hear you. So, would anyone like to share something that you talked about in your small spaces with partners or in groups of three? A right that you appreciated or right that you wanted to add and why? Hi. With apologies, I tend to go outside the box. And no, it's encouraged. You have the right to say what you want to say. And the way I was going outside the box here was I I was thinking, are there a collection of rights that the teacher has to complement these? And how do they fit together? Because these are all meant to give students as much liberty and such as possible. Are there other ones that the teacher would have? Maybe the right to make you feel uncomfortable, to challenge it, various things like that. Now teacher's in a power position. So you got to be careful. But but anyway, that's what I was wondering. That's beautiful. More thoughts about this. Yay. Um I was just saying I think that a lot of these things hit on like an environment that I try to create, but I don't know if I've ever explicitly stated those things as like succinctly and easily, right? Like I think that all goes towards having like a good classroom environment, but stating it would be another level of I don't know usefulness in that I guess and helping students know right you do have this right in here and I want you to experience it and enact it. I I can share some online ideas while here. So some online ideas was the right to take risks and make mistake. Um, someone said, "I value the right to respond with I'm not sure yet." Um, the right to understand even though I'm not a math educator. Um, and the right to learn are some ideas that uh were shared online. That's beautiful. Thoughts from this side of the room. Any thoughts from over here? Yes. That's okay. Um, so I was kind of thinking, my district's been talking about like math smarts and like different ways to show how you're smart in math versus just being quick. Um, because a lot of times students think, "Oh, I don't have the answer right away. I'm not as smart as so and so." So, I think kind of tying some of these into like how can I show I'm smart by like completing the rough draft and working and revising, asking a question that helps the whole class learn. having a revision to offer. Yes, there are so many ways to be mathematically smart, not just getting the answer quickly. So, right, I think I'm hoping a lot of this today will help you make things more explicit to your students in different ways. And some of these ideas you'll feel like, oh, this resonates with some of the things I already value, but there's different words to communicate to folks about that. So, one of the ways I advocate explicitly building that culture is talking with students about what it means to learn. It takes a lot of time to revise your thinking, to get your thinking out there. Everyone learns from trial and error and gradually evolving their ideas, but we have to put something out there to keep learning together. So, normalizing the drafting or revising is something we all do. You can invite them to tag their talk as rough drafts. This idea that you have, if you call it a rough draft, totally fine. And then we can all learn from your draft. And then inviting students to exercise those rights as learners to think about when they're exercising those rights. Encourage each other like it's fine. You're enacting your right to be confused right now. I'm confused, too. allowing those rights to be a normal thing. Another layer to rough drafting in the math class is explicitly and intentionally incorporating revision into the math classroom. So, I want to talk about that a little bit. For me, this is one of the newer things about rough drafting is when and how can we invite students to revise in math. I like this image. It shows that uh you can continually revise and improve but every time you know that this is a butterfly, right? So the butterfly itself keeps evolving but it's not like one is wrong. They all represent a butterfly and you just keep improving. So in math you can have an answer that people agree is correct but your response and your thinking can continually grow and be revised. So when we think about revision, what might we be trying to do? What are some of our goals? One could be fixing a mistake to become more correct. But there's all kinds of other goals that we could have for revising. Maybe we could be seeking new insights for a new solution strategy. Maybe we could be developing an explanation to become more precise or more detailed or more concise. So you could improve the quality of the argument or the explanation even if it's already correct. You could become more illustrative. Can you add a drawing, an illustration that helps the ideas make more sense and communicate to others? Mathematicians may be striving for a more elegant proof. The proof might work, but how could it be more elegant, more illuminating? When I'm writing my first draft of something, it often makes sense to me, but then I have to keep working on it so it'll communicate to other people. What about becoming more convincing? Convince yourself, convince a friend, convince a skeptic. So you can keep revising in lots of ways other than fixing errors. Um working with teachers around the country around these ideas. A teacher in Tacoma, Washington said she teaches sixth grade. She said in her class they try to reject the language of correct and incorrect and just talk about whether or not they want to revise. And she said that this gets her students to try more because it's not about this like who gets the right answer first, but everybody keeps revising even if it's correct. And so more students feel safer to keep working. And then when we're working on building the classroom culture, we're going to do some math in a minute. And so I would say this is a nice set of norms to build with your students. An eighth grade teacher in Boulder, Colorado, uses this phrase with his students all the time, and it's his one set of classroom norms. We're just going to be brave and be kind. Be brave enough to share when we don't really know. Be kind enough to take everyone else's thinking seriously and assume that it makes sense to them. And our job is to try to understand each other. People want to be understood more than they want to be changed. So, we want to try to think with each other and understand why their thinking makes sense to them. Okay, cool. We're going to do a routine that I just call a multiple drafts routine. It doesn't take that much time to imple implement. It doesn't require you to have any special curriculum materials, but it really helps people's thinking grow. So, you're going to have a piece of paper. Do you have any paper? Math teachers having paper. You can take a piece of paper and you're gonna just draw a line down the middle and the left side of your page is gonna say first draft and then the right side of your page can say second draft. So if you have any sort of scratched paper, I'm not collecting it or anything. It's just a space for you to write in a minute or two. And you're not going to write quite yet, but you will in a minute. So the idea with multiple drafts, you're going to work once, think about it, and then instead of erasing, you're going to leave your draft there. And then your new thinking will be the second draft, right? So you don't want to erase. You want to leave your original thinking. And you're always going to be able to adjust your thinking no matter what. So even if you love your first draft, you'll still be able to adjust it for your next draft. And so if I had students come up and share their work, I give them sentence starters. It's interesting, even us as adults, we don't always have a way to talk about each other's thinking. If the goal is to understand each other rather than evaluate. So if you want people to work on understanding each other, I like to have things like this makes sense to me because I also say share what you appreciate about someone's thinking. Share what you're wondering or maybe you would like to add on. So these are things I like to encourage people to say when they talk about each other's thinking. Okay. So I'm going to show you a video, a graphing story. So, it's about functional relationships and you'll be thinking about the height of an amusement park ride over time. You don't need any writing utensils yet. So, go ahead and put your writing utensils down. And I'm just going to show you a video and you're not going to write anything. And you're going to just think, what do you notice or what do you wonder about what's happening in the video? and we'll just collect a lot of noticings and wonderings and you're not going to worry about anything to write yet. This is a ride that's called the Slingshot. What do you notice? What do you wonder about this ride? Nothing yet. But can you imagine? All right. I got this video from a high school teacher in Ohio. She uses it to introduce functions to her students. And so she asked them this. I met this teacher on Twitter. She tweeted out, "Um, I used this slingshot ride to engage students in rough draft math." I was like, "What? How did you do it? What did you do?" She said, "Well, the first thing I did was I just showed them the video and asked them what they noticed and wondered." So, when you watch this, what did you notice? What did you wonder about the video? I would like to take some notes. Who would like to share? Yes. How many people are in the ride? How many people in the ride? Yeah. Like how big is that ride car? Right. And how many people fit in there? What else did you notice and wonder? Yes. What is the maximum height that the ride gets up to? What else did you Yes. Say it again. How did they get back to the ground? What was the geforce? These are very cool noticings. More noticings or also very cool wonderings. Any also noticings are okay to say too. Yes. moves and bounces a little less. Noticings, wonderings, more thoughts. Was wondering what was the highest and someone else had commented um saying, "At first, I couldn't really tell if they were tethered to the scaffolds on each side. So, one pass through when you're just thinking about what you're noticing or what you're wondering just kind of orients you to the situation without feeling pressured to come up with whatever the solution might be. So, you saw one look at the video just like what's even happening here, right? Then so she told me that was the first thing that she did. Leah Simon and you just talked about what you noticed and wondered. Then she had them draw multiple drafts. So we're going to watch it again. This time please draw a draft of height over time. Could have had you do speed over time, but we're just going to do height over time. And we're going to watch the video again and just draw your draft. Like it's okay. They're all going to be a little bit different, but you're thinking about that relationship. I'm gonna go back to the [Music] beginning. In big. All right. Okay, cool. Please draw a draft. You're going to watch the video again, so you don't need to feel too committed to your draft. If you feel like you have a draft, I want you to see at least one other draft. So, can you turn around or look through some people near you and and share and just realize that everyone's draft you're going to have something to learn from. So, please share with a colleague. You have a nice beginning there. A little bit above the ground. They're not exactly on Yeah. Yeah. Yeah. Yeah. They're in that rock. And then what's happening up and then kind of go down a lot slower, but I ran out of space. Yeah. And kids talk about if this is like a direct point or they're lingering at all. I don't actually know. And they get like closer and closer. So you some kids start it up here and then they have to talk about if it's starting down there. Some people start at the origin, then they have to talk. By comparing each other's drafts about how they're similar and different, they think a lot about the features. So, that's cool. Can they see yours, too? Can she join you? Now that you've seen one um at least one other draft, you probably have ideas for how to rethink this. And we're going to do it one more time. Okay, your second draft or for some people it might be your third draft. That's cool. I got to make it big. Okay, I want to ask you to think about this as you're making your second draft. Oops. I want you to think about similar and different. I'm going to collect your thoughts about similar and different and it can be similar and different from your first to second draft or similar and different between yours and your colleagues. So what was the same or what was different in your in the graphs your first and second or yours and colleagues? What would you like to share? Yes. and the different representations allowed you to notice and represent different things and see differently. More thoughts of similar similarities and differences. Yes. I don't even know How many were there? Five. Five ups and downs somewhere in there. So, what counts as one? Like what's countable? Cool. Thoughts? The platform is above the ground. So then on my second draft I put that height above the platform so that it kind of like starting at different spots. Yeah. Start relative to what is it at the origin is above is it above what? Yeah. How do we interpret that initial height? Cool. more noticings, similarities, differences across your two drafts or your drafts and a colleague? Yes. happens over a different stretch of time. Like it might be stretched out. So that was one way your thinking kind of evolved. Very cool. So Leah, Leah Simon, who was doing this with her students in the Cleveland area, she would strategically, if I had the document camera hooked up, right, I could have strategically chosen some people's drafts to share. This is one set of three drafts from one student. She had multiple students come up and share their initial drafts. You can see right with this student their initial height is different in the two in the three graphs. The end point is different. The way they think about the oscillations might be different like the number of them. And then you would ask the students to reflect. How did your thinking about this relationship change over time and why? So it's a complex situation with a lot to notice and think about, but then you can help them think about functional relationships. So there's a website called graphing stories. If you Google graphing stories, you can see a bunch of videos like this and the students can think about them. Whether or not they're discreet or continuous is something to talk about, too. Some of the graphs in the graphing story videos are step functions. So, it's a it's a nice experience. Another thing to think about related to revising in math class, how does our assessment culture in our class, our assessment practices relate to revising? This is from an eighth grade teacher in California. He has the students revise their tests and then has them reflect. When they're revising their tests, if you He gave them feedback, no points. He just took their test. He did screencasts for every kid where he commented and audio recorded. Just feedback. No points, just feedback, verbal feedback. They would like listen to it in earphones and then correct. And they could decide. and he wouldn't give them feedback on every problem, just a couple. And they could revise and then he asked them to think about reflecting on their revision. For each problem you revised, use this sentence frame. I used to think and now I think so you can record how your thinking grew and why. And then some general reflections like one way I would like to improve as a learner and in order to improve I need to. And then he said, "For those who learn the most, reflect the most." So be thoughtful. So thinking about when you revise, why were those changes improvements? Being a little metacognitive about it. And do we allow test revisions in our classroom? Why or why not? And what feedback do students get to decide if they need to revise their thinking? It's just something to think about. The last section of this session will be about students strengths in mathematics. We want to honor the strengths people have in their ideas at any stage because we can all learn from each other whenever we share. So how do we let students know that we value those insights? So there are all kinds of mathematical strengths that we could have. What strengths might students have? How do we see them? How do we help them see strengths in each other? And how can we amplify the strengths that people are bringing into our classrooms? Uh in Illinois, there are teachers who are reading a book I wrote called Rough Draft Math, and they are having a book study, and they're tweeting about it. So, this was just earlier this month. Heather is a math coach and she wrote, "By highlighting strengths in a student's draft, students might even feel proud of their drafts. The author of the rough draft is no longer positioned as having an incorrect answer or process, but as making significant contributions to the classroom learning. So if someone's sharing their ideas at any stage, we should assume there's something that we can learn from those insights and then everyone can grow because someone was brave enough to share their ideas. So thinking about if students are coming to us and we know that they have brilliance to share, what strengths might students have to support doing math? I'm going to revise this question right now. So, I want you to think about the function carnival slingshot ride. What strengths might students have to support their doing of mathematics? Thinking about the slingshot ride, what strengths would students need and bring to help them make sense of that task? Go ahead and discuss the kinds of strengths you might think students could bring. 20 seconds. Let's name some. What strengths would you think that students could bring to help them work on that task? Who can name some? Yes. It was your idea. Oh, yes. Oh, great. Awesome. Um, you were talking about personal experience on the ride, like some student sitting there and you can have a discussion about uh I don't know how like the first 20 seconds you're just waiting for it to go, but how long does that feel when you're sitting on the pad? And you' said like the first 20 seconds is an eternity just to watch. Imagine sitting there like waiting to go. So that experience kind of gives you a different insight into it, right? And so some students will bring that. They will have been to a theme park and they've ridden that kind of ride and it helps them kind of have that suspense. Awesome. More thoughts. Yeah. An attention to detail the strength like he was looking at the time specifically to kind of pace his graph. So having that strong or that strength of paying attention to the little details so you can make your graph more accurate for for greater precision if you're uh noticing detail more ideas different different strengths students could bring. I heard some good conversations over here. What were you talking about? and visualizing I think I heard. Yeah. Yeah. And so when you think about strengths, some of us might default towards certain kinds of strengths and miss others while other people will notice other kinds of strength and miss others. And so what kind of framework can we use to think about students strengths? I read this book and it's great. Even though it's geared toward elementary grades, it helped me teaching adults and it could help you teaching older kids, too. Because there are these different categories of strengths. Dispositional like I persevere and I'm curious or processes and practices like I want to make connections all the time or I want to I'm able to organize my information well. Or maybe they're content strengths like I'm really good at number sense. So like attention to detail and visualizing might be over here, but are you noticing other kinds of strengths too? Right? Are there certain strengths you're more inclined to see and others that we need to stretch to see and encourage students to see in themselves and each other? And so we want to do everything we can to let students know that we see those strengths that they're bringing. uh specifically highlighting the strengths of students that other students might not know have those mathematical smartnesses. So assigning competence to students that might have lower status in the class for their math knowledge and helping everyone see like oh gosh we're all math people. If we recognize the strengths in each other they can all see each other as math people because we are all math people. So you can disrupt who is seen as a smart person. Everyone in the room can be seen as smart people. This is like what you were talking about recognizing more ways of being smart in math. So talking about the process of doing math. So there's lots of mathematical smartnesses. Asking a great question, representing something differently, lots of ways to be smart. Ammani Goffnne is a friend of mine who works at the University of Maryland. And Dr. Gooff has helped me think about being careful about who we're positioning as being in a rough draft stage. We don't want to always position the same kids as being in the rough draft and other kids as being far along. Specifically thinking of making sure we are not acting out of implicit bias. Students of color of course should not be positioned as only students in a rough draft stage of thinking. We want to make sure who are we positioning in what ways. Also the nature of mathematical thinking. We don't need to position traditional ways of doing math as being the final draft. There are all kinds of ways of doing math that are valuable. So integrating and valuing strategies that may be outside of our default assumptions of the ways of doing math is important. So what do we do next? It's never too late to revisit our classroom culture. Okay, it's February, but we can still revisit and rebuild and reassess the kind of culture we want to have in our classrooms. So, thinking about how you're going to let students know that their drafts are welcome. Emphasizing that this is about learning and growing our thinking over time, not just about performing. and how are we going to invite students to revise their thinking and their work in our classrooms. So these are all things that we can start doing Monday and also I learn more if I get more access to your thinking. So this is a Google form and it has the prompt I used to think and now I think. So, you could either use the QR code and go on your phone or you can type in on your computer or tablet. And if you'd do me the honor of sharing some of your thoughts with me so my thinking can continue to be revised, I would love to hear from you. So, please share some thoughts. And thank you so much for spending part of your saf Saturday afternoon with me. If you want to stay in touch, this is my Twitter handle and my work email. This website has a link to lots of other resources about rough drafts in math, um, articles, podcasts, other videos if you're interested. So, thanks so much for being here and thank you for sharing your thoughts in the Google form. [Applause] Awesome. Great. Thank you. Thank you so much, Mandy. Sure. Um, and again, keep typing if you're still one, uh, wondering. Uh, as I just say, just a couple concluding comments. Uh, the next session, uh, Mandy also has a kind of a Q&A discussion session, and it's in this room right afterwards. Um, and the other thing that I want to say is, um, you know, if you need a pickme up for coffee or something, there's plenty of it there, self-s serve. Grab a cookie to go. And then if you haven't tried it yet, there's a lavender lemonade that is also self- served. help yourself to that as well. Um either between before the next session or on the way out. Okay, awesome. But um but yeah, uh we got about 10 minutes until the next session starts, whether it's here or in the other building. Thank you so much for joining us. I'm going to get coffee and I'll be right back. Does that sound good?