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Hello and thank you for listening to the teaching math teaching podcast. The teaching math teaching podcast is sponsored by the Association of mathematics teacher educators. The hosts are Eva, Sennheiser dusty Jones and Joel Amidon. Today we're talking with Amanda Jensen from the University of Delaware about using rough draft math in online classes. Amanda Jansen just published a book called rough draft math, and we're excited to hear her discuss this in general, and specifically for the use in online classes. If you are interested in integrating some of this into your classes, Amanda and I just this week recorded a seven minute video to be used with students to explain what rough draft math is and why you would use it in math classes. To find this video, you can go to my website, Eva sennheiser.wordpress.com. Welcome, Andy, we'll be tackling four questions. So let's get started. Hi, I'm happy to be here. As an extrovert at a time of social and physical distancing. It's nice to be able to talk to some friends Today, we are super glad you're here because this rough draft math is such an important topic. So we're going to talk about four questions. The questions are what classes are you currently teaching? How are you moving them online? How are you using rough draft math in your classes? And how is it going? So let's get started with the first question, what classes are you currently teaching? So this semester, I am teaching one class, I have a course released through grants, so I only have one class this semester. And it is a math content course for elementary teachers. It's the second of third of three math content courses in our elementary teacher education program at the University of Delaware. This course focuses on rational number concepts and operations as well as proportional reasoning as compared to the podcasts that we recorded. The last two weeks, you are moving your class online synchronously. Right, synchronously. Sorry. That's right. So I decided to have my class as a synchronous online course. And this was a really difficult decision that I pretty much agonized over I spent a lot of time thinking about it talking to folks reflecting but also serving my students multiple times, I heard a lot of conversation about asynchronous is more humane right now. Because we don't know about what people's lives are going to be like, and adding more flexibility is better. And I believe that to be true. But my students when I surveyed them, even though it's hard to know what you're going to want and need, the students said almost pretty much unanimously, please can we meet synchronously live. And so I thought, Okay, this is what they seem to want to try. So we'll try it for a week and see how it goes. And we're three days in at this point, the University of Delaware, canceled classes starting March 12, because we had our first case of COVID-19 on campus on March 11, or the first case in the whole state. And then we started classes again on March 30. My students are mostly freshmen and mostly 19. And they thought it would help them stay on track if we were having face to face meetings. So my university has zoom. And my university has Canvas as its learning management system. And so we could use breakout groups and zoom just like we would do in face to face. And I really had to make the decision by considering what my students preferred. And I surveyed them multiple times. They already had a course pack for the class that had the problems that we were working on. The students all reported having strong Wi Fi at home, so I thought synchronous was worth a try. And all of my students live in the same timezone. They live in Delaware, Maryland, Pennsylvania, New Jersey and New York. And they preferred texting reminders, rather than email reminders. So use remind calm for that. And you'll notice that I said New Jersey and New York, so there's a portion of my students are in a region of the country that's really affected by COVID-19 right now. Oh, yeah. Yeah. So one of my students tested negative she had she was sick, she had to be tested. One of my students, two of her relatives, tested positive and one of my students, her mom's in the healthcare industry. So I'm going to have to constantly re evaluate and to see if this situation with synchronous is going to work. But right now the students are appreciating it. Maybe with your synchronous class. Does it meet twice a week, three times a week, five times a week? How often do you guys get together so we have a tuesday thursday class twice a week, and we are meeting on our regular course time, so it's two o'clock to 315

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Then I have office hours before and after class for students on zoom in that way, I just hope that the people that are listening here, and I know from interacting with you on social media, or just seeing your social media presence, that you're not just not just out there posting that, I mean, you do post some nice pictures on your box, I appreciate that. But then also, you're also asking questions, probing questions to professionals, like, Hey, I'm dealing with this, what are some asynchronous versus synchronous decision? like looking at that and think, like, what are the different things I need to consider and, you know, thinking about it from abroad as a professional like to other professionals, and also to the amount of time that you spent serving your own students? It's like the to all the input that you took in, I just love hearing that from a care perspective of thinking, like, how can I best serve my students? I'm gonna look at it from what are other professionals doing? And what are the things I need to be thinking about? But then also, hey, from the students themselves, like, Oh, you want me to text you rather than email you? Okay? That's great, are synchronous versus asynchronous, or, you know, considering all the different inputs and even their, their own situations with where they're located, geographically does matter, even from an online client. And so I just love hearing all that and all the different input that you had into your decision making. Thank you for sharing that. Oh, thanks. Another relevant aspect of that is that I learned that a third of the students don't have printers. And so that was stressful for them. They're like, well, how can I take a quiz or a test? And I said, that's totally fine. You can just do it on regular paper. And they submit assignments by we did some practice with this, where they use their phone, they take a picture, they turn it into a PDF, and then upload it to Canvas. And then I can write on it with my Apple Pencil on the Canvas app. And yeah, so thanks for saying that. Joel. I think that it's time for compassion. And it's time to dissenter and understand where students are coming from, but then also be really flexible and know that this is an evolving situation. Okay, so let's get to the meat of today's talk. How do you use rough draft math? And I would like you to tackle both in general. And it's not an emergency situation. But also, how can we leverage this in the current online situation? Oh, thanks for asking. So rough draft math, is when we encourage students to think about any of their ideas or their work as a draft. That means that you can share your thinking when you're not sure, you can share your thinking while it's still in progress, you can share your thinking when you're not done, you could just share how you got started. And in everyone's draft, we can learn something from it. So doing some things during the year to set that up just talking with the students about it. And then when people share their drafts everyone orienting ourselves to finding strengths in the work and pointing out what we can learn from it. And then building in revision, revising is a big part of rough drafts in math class. So revision can be anywhere from when someone presented their problem with the document camera, you then can go back and think about how you want to revise your own problem, too. We have midterms and quizzes in this class. And so they can revise their quizzes as many times as they need to up to full credit if they want. So, building in revision times, that's been a nice part of thinking about rough drafts and math classes. So those are two examples, in general of what I've done with rough drafts, briefly, I've been using this idea of rough draft as well in my classes, and I've been pushing it a little further, in the video announcement I gave to my students this morning. That's using it beyond just when you're not finished, but also considering finished work as a draft. And I was comparing it to peer reviewed papers. So that like even if you finish something, feedback can still make it better. Mm hmm. I was just kind of wanting them to be able to say, Hey, this is a draft, but they also this is finished, but I'm still open to feedback. So let's move into the online version. How would rough draft work in online and let's kind of pull both cases apart. So in your case, you knew your students and you had relationships with them, and you moved online in the middle of the term. But in my case, I have a brand new group of students and they're, I'm going to need them to become comfortable with rough draft without having the classroom community to do so. So I'm wondering if you could tackle both of those. Yeah, so in my case, I did know my students a bit but we had only been in class about a month at Delaware this semester started around, I think, February 10. So I knew them a bit though, and I think that makes a difference for having some of those relationships pre established. But when we came into the zoom environment, I didn't

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Another launch with them about rough drafts are still welcome in this space. And I talked about it a little bit, and then I had them reflect about what rough drafting means to them on a Padlet. Just kind of reorienting, rough drafts are still a thing, and how this plays out in our student work. In the class, I'd like to give a little story from last class, if that's okay, that'd be great. We love stories, yay. So I would really depend on a document camera when I teach a lot having students work as an object of discussion and reflection, but I don't have one anymore. So what we do have are Google Slides, and we have cameras on our phones. So when I move them into breakout groups, I would have them put their group straps on Google Slides. So in the last class, we had been working on understanding multiplication of fractions for a while. And so having a group where the task, in this case, it was the objective of the class was to think through the standard algorithm for multiplication of fractions and what it means. So for example, if you have two thirds, multiplied by four fifths, you would multiply the numerators and get eight and you multiply the denominators and get 15. And the answer is eat 15. But that's kind of a mystery. So I wanted them as a group to use their knowledge that we've developed so far to think about why. So we had been using an area model to solve story problems. And I wanted them to think about how they construct the area model, and how that helps them think about the algorithm. So they were in groups, and they wrote their understandings. And as a group, they collected their thinking and put it on a Google Doc, they even had some times like this picture from one person about what they think and so on. So you could have like a Frankenstein explanation with pictures from different people. All right, so you can figure things out, like you could say, Well, the first thing I need to do is draw that quantity. So if you think about the example of two thirds times four fifths, the first thing you should do is model that quantity of four fifths, if you think about the problem is two thirds of forfeits. So drawing four copies of a measuring unit of size, one fifth that you're going to work with. So then the next step is you would partition each measuring unit of size one fifth into thirds. And that's like multiplying the denominator, because you have one third of four fifths, that's four copies of one third of one fifth. So that's for 15th. So that's where the 15 comes from one third of one fifth. And then next, next, because it was two thirds of forfeits. Now, you have to find two copies of one third, or four fifths, or two sets of 415, which is 815. So that's why we multiply the numerators. Right? So I figured that they had enough experiences with area models that they could come up with some explanation. So each group I gave them about 20 minutes. And then they posted some explanations on Google Slides that were a draft. Then we came back from the breakout groups. And they wrote, they reviewed other groups explanations, and they wrote notes to each other in the notes section of Google Slides. And while they were doing that, I was doing some selecting and sequencing. So I selected one group to talk about how they drew their quantity, one group to talk about how they multiplied the denominators and how they thought about that, and their diagrams and explanations, and one group to talk about how they thought about Multiplying the numerators. Then I sent them back into their groups, and I had another set of slides that were like their revision slides. And they could then think about how they wanted to revise their entire explanation. And they could crop and take and steal parts of explanations from other people's groups and put them on their revision side. And they could type out on their slide with how they would have revised their first explanation and why. And then for the next class, what they're doing right now is each individual's taking a different multiplication problem. And then explaining and uploading to Google Slides, how they would explain the, the algorithm themselves and then I'll choose to to have a whole class discussion about in the beginning of class, next class, that's really cool. I like how you're having the students kind of take advantage of the technology, and be able to, you know, cut and paste or crop from somebody else's explanation, that something they might not be able to do if you were using a document camera, in face to face setting. That's really cool. I think, yeah, an advantage the online is that students get to see even more of their colleagues work, not just the ones I've selected to talk about. And students can use their sentence starters or questions for giving into their feedback in the same way from the from what we did face to face. I would always ask them to point out what they notice what they wonder and what they appreciate about each other's work. All these ideas about the Google Slides and how to use them including stealing parts of others explanations. I have to give a shout out to Teresa will

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She is at George Mason. And she has been doing the most amazing online PD for K 12 teachers and professors about asynchronous instruction. So I just give her so much credit. She's just been the hero of this whole thing for me. She's taught me so much, Mandy, I would like to follow up with some questions. So can you tell us how many students are in your class? Yeah, so I have 22 students this semester. And when you group them, how many students are in one of those groups with zoom breakouts, you can, I have always done random grouping or not always, but for quite a while I've done randomized grouping, taking from the work of Peter Lilly at all and his work on building thinking classrooms. So the idea of if you randomly select groups all the time, students will feel more comfortable working with more people, the knowledge will spread across the classroom, and the unit of the community becomes the class and not their group. So I can use that and zoom and set up the breakout groups randomly. And so I'll ask for five groups, and then it'll be about four or five. Okay, so you have five groups. So when you send them to give each other feedback, how many groups are students expected to give feedback on? I haven't been very systematic about that yet. Yesterday was just the third day I've been doing all this stuff. So at some point, the fact that they're trying it at all, like giving anyone one person even longer feedbacks, good, but I'm noticing that the feedbacks being distributed differently, like one group might get one comment and other group would get like six comments. And I don't know why. So that's a good thing to think about. Do you have thoughts about that? Well, I was asking, because I have been thinking about this, because, as opposed to your approach, mine is completely asynchronous. And my students are creating rough draft slides as well. And I have them comment on a certain number of slides. So I have 31 students. And they each create a slide and then I have them comment on five other people's slides. And I pulled out the five randomly from my head. I don't know why I said that. But sometimes I'm thinking I would like them to read all the solutions. Sometimes I think that's overwhelming. So I was just wondering how, especially with revising, right, so once you get feedback, so one of the things that I've done in terms of making sure everybody gets feedback is, and this is asynchronous, right? So I'm restricting them to they have to give feedback to five people. And they should give feedback to only people who have received less than five feedback comments. So in the end, everybody should have five feedback comments, right? But I don't know, I was just thinking about it. That's why I was asking you and you're in real time. So that's a different imposes different restrictions on how much feedback they can give. But I do feel like it sends a message of whose solution is valued. So if someone if one group is getting more interaction than others, I would be concerned. And because you don't want that to sound like the other groups ideas weren't worth thinking about. So that's a helpful push for me. When I talk face to face, I used to have a clipboard with me and I would mark down information about who did I specifically select to share who was volunteering to comment who was asking questions. And then when I selected in the future, I would try to hear from more people that weren't marked off on the clipboard that day. But I haven't gotten that organized for the online class yet. And that's something I want to think about. So we can hear from more people in the whole group space. But in some ways, all these Google slide interactions are getting more interactions from more people. Because in real life, we can't speak at the same time but not real life, but you know, in local space what I have a question then. So have you vocalize that intention that you would like to see like the every group's work get comments, like having equitable amount of comments so that if someone goes oh, this group's already got a number of comments to move on to another group, it's a really good idea of I've just been happy that I've been getting a functioning class. So So now I'm at a point now where I'm I think, three class periods in that I am ready to start thinking and talking about that. So that sounds like a great idea for Thursday. Well wonder to also if like, I was just looking thinking about your notice wondering appreciate like having shorthand ways to do some things like noticing is a certain symbol that you put on a slide, you know, like a like an arrow and then wonder is a question mark appreciate is a star, you know, sort of like social shorthand them and they can start using some of that I love that emoji sort of language that they've perfected on their phones.

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You know, to get them rockin that way in order to shorthand it. So it's not as you don't have to think like you're writing a paper or anything like that, but then they can really get, I'm just curious, like, after you do this for a semester, you're gonna have all sorts of ideas. Like you're, you're giving us your rough draft thinking about your teaching right now. Okay? And so like to see like, what's gonna and this is why I'm glad that we have you on because I know Mandy, your mind is working. You've probably written down like a few ideas already, just like Ava probably has, as well, thinking about different ways to tweak what you're already doing. But yeah, that was I was thinking like shorthand ways to do some sort of feedback, but then also, just even being vocal with the intention, right? Like, hey, we want to make sure everyone has these, the thinking and because that's communicating to their future classrooms, like, hey, everyone's thinking they'll have some value within our class. And how do we make sure to honor that thinking? I think that's exactly right. Because it's one thing to tell people that their drafts are okay to share and valued. But where people get the real message is how the interactions play out. And so if we interact with students thinking in a way that elevates their strengths, if we help students see how they can learn from each other's drafts, and if we have it as a community value, I think that you're exactly right. This is helpful push, and also awesome ideas about how to signify an index, the noticing the wondering, they're appreciating. And I'm also thinking about this steal and idea thing where they profit from each others. That is another way to show that you valued somebody's thinking in addition to the comment. So also, I need to figure out how they, I guess I could trace the handwriting back, but like, whose idea was elevated by their peers? I'm curious. Okay, so let's slowly wrap this up, I wanted to give you space to kind of talk about something else that might have popped up throughout answering these questions that you would like to talk about. So is there anything else you want to share that this made you think of? Yeah, I one thing that was really important in this class was, how was I going to handle midterms. So we have two midterms and a final in this class. And we used to have a two hour session in the evenings. And everyone from every section of this course would come on a Thursday evening, from five to seven and a lecture hall in a timed environment. But that doesn't make sense in this space. And so we decided to open up flexibility for it to be a multi day assessment, open resource, they could collaborate with their colleagues, but they had to write down their own work. And I survey my students at the end of every class to ask them things about their experience, like in the last class, I asked. So in this class, I tried not doing screen share with the Google Slides. And in the last class, I did share my screen, what do you prefer? You know, I'm always asking them a question about something I tried and to get their take on it. And then I always ask them open question about some feedback about how their courses going. And they told me that in most of their classes that professors are still doing, like time constraints around assessments. And that's really stressful for them. And they appreciated that this class is more open with the assessments. And then I guess the other thing I want to say is I want to give a shout out to Jen wolf at the University of Arizona, she's been an amazing thought partner for me, because she's also teaching a math content course for teachers synchronously. And I've been learning a lot from her about how to be really explicit about how to use the teaching students how to use zoom, they didn't have a lot of experience with it. And so she's been awesome helping me think through how to do that. She's been great at helping me think about different ways to do self care check ins at the beginning of class and building. She's just such an amazing person with building classroom community, focusing on building collective understanding using complex instruction. So I think having a collaborator at another units institution has been really useful for me, in addition to the collaborators I have locally, yeah, Jen's really great about I mean, even before this, of using like, Instagram, she used Instagram to post some pictures and to tell her class that things are updated. I just I love with some, you know, interesting looking math pictures and stuff. Like, yeah, she's, she's already she was, it's like she was building that community. And so it's not like, oh, all of a sudden now you need to build this online sort of presence, but like she already had it to lean on and like, yeah, she's, she's someone I'd like to think about things with as well. Definitely. Okay, Mandy, if somebody wants to find out more about rough draft thinking, where would they go? Well, for one thing, rough draft thinking and pre service teacher content courses. You wrote an article with me You're the lead author and an article in the mathematics teacher educator about rough drafts in math content courses for teachers that's in that I believe them March 2016 issue i

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I have a book that just came out like he said called rough draft math, published by Stenhouse, and you can actually read the first chapter for free on the Stenhouse website. So if you want to have an opportunity to read something for free online, that's a space. I have an article with three, Delaware middle school teachers and math teaching in the middle school called rough draft talk. So those are some options. Okay. Thank you. Any other questions? Joel or dusty before we close up? I'm good. Yeah. Thanks so much, Mandy for helping me. I've got so many notes here that things I want to try. So thanks. Awesome. Thanks for letting me come on. Thank you, Mandy. All right. Well, thank you, Mandy. And thanks again to everyone for listening today teaching math teaching podcast, be sure to subscribe to the podcast. We do hope that you're able to implement something that you just heard about and take an opportunity to interact with other math teacher educators.