0:00  
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 Thanh Iser, dusty Jones, and I am Joel Amidon. Today we are talking with Mandy Jansen from the University of Delaware because she is the author of our teaching math teaching summer book club choice for June, we have been reading and discussing rough draft math revising to learn every week on Instagram Live. And it's been a pleasure. And we've been doing that throughout June and are wrapping up the month with this culminating podcast episode. Welcome, Mandy.

0:38  
Hi, Joel. Thanks for having me. And I know you've been on a previous episode of the teaching man teaching podcast, but anything else you want to share in a like a brief introduction.

0:49  
So I am a teacher educator in the School of Education at the University of Delaware and I have a joint appointment in the math department at my university. And I work with secondary teachers and professional development in service teachers. And then I teach undergrads studying to be elementary and middle school teachers. Fantastic. So we are talking about your book, rough draft math, and we'd have been talking about it, like I said before in the intro, and you know, this idea of rough draft math, it's been something that I've heard you talk about at conferences, and I mean, well, we all have ever heard you talk about it at conferences and in other papers and things like that. And now we're excited that you have the book. But can you share the history of this work and the evolution of the idea of rough draft thinking? I'd be happy to. So I originally started to think about the idea of rough draft talk, probably back in graduate school, actually, when I was taking a class about analyzing discourse. And the framing of talk can be in a rough draft or a final draft space from researchers such as Neil Mercer, Douglas Barnes, that idea kind of remain in the back of my mind for a long time. And historically, I've always been really interested in understanding how teachers can promote productive discourse. I conducted a study group, I would say maybe five years ago, with secondary teachers in Delaware, and we were all really invested in trying to improve classroom discussions. So they felt more exploratory. So it wasn't just show Intel, it was a discussion where people would continue to learn during the discussion. So we read chapters from a book called exploring talk in school.

2:45  
And the teachers in the study group said it would be more useful to call the kind of talk instead of exploratory talk, rough draft talk, because rough draft is a term that has meaning for kids. So we went with that. And in the study group, teachers generated ideas about how they could use the concept of rough drafts in their math classroom in different ways. So everything in the book is inspired by or informed by ideas that teachers had when they were playing with a notion of rough drafts and math class. So that's where the book evolved from was from working with teachers. Also, Ava and I conducted a project in one of her classes, a mass content course for teachers where she developed an intervene intervention that we call the label where students would label their posters that they might create in small groups, as in terms of how complete the work was on the poster, and how confident they were in their work to really normalize the idea that we can share our thinking when it's not finished and when we're not sure. So I Eva and I have an article in the mathematics teacher educator about that. The teachers, in my study group, a few of us wrote an article in math teaching in the middle school about rough draft talk. At that point, I thought I was kind of done. I had written a couple pieces with even with the teachers. But then the book came into being because I had been giving some talks about this, and people wanted to know more. So then, I had the opportunity to write this whole book. And I didn't know I had enough ideas for a whole book. So it's pretty exciting. I it's so exciting to hear about it. And I really wanted you to answer this question because, like the idea like the kernel of an idea or whatever, taking an idea and thinking about all the different ways you had to work with the idea and people you got to collaborate with and then leading to the book and like the idea is not done, you know, and I think sometimes out there with this idea

5:00  
of academia and things like, for everything is a new idea, right? There's not this one to one correspondence between idea and product, right, that you can, the evolution of ideas is going to lead to several different things along the way, and then it could actually be the trajectory of a career too. And I just love that part of it. And I don't know, did you get any? Like, I mean, the, I guess the the mile markers or the encouragement along the way of like, what were those key pieces of encouragement that you got?

5:32  
That's really interesting. I, for me, the key pieces of encouragement were a few different

5:39  
moments, mostly around a teacher, showing me how the concept of a rough draft would generate inspiration for them. And so my thinking about what rough drafts look like in math class kept expanding, because people kept telling me, this inspires me to do this in my classroom. And then somebody else would say, this inspires me to do that in the classroom. And so I felt like it was a snowball, I just kept getting to learn and learning, learning, learning, learning, learn more. And so then it felt like I needed to preserve and share what I was learning. So I would say it's a series of moments. And it can happen, say, after a conference presentation, a teacher will come up to me and say, Well, why have this idea? Do you think it would make any sense? And I'm like, of course, wow, I never thought about it that way that's really useful. So I think those interactions really buoy me and make me feel like I'm a part of a learning community with anyone who's interested in the idea of rough drafts. Yeah. And just to pivot a little bit, I mean, I'm gonna put aliens. But Eva, how did that? How did the collaboration between the two of you come to be?

6:49  
Hmm.

6:51  
I think we were just chatting at one of our get togethers about ideas. Mandy, is that right? Was that one of our writing retreats? I think so I think that you were, you were you had heard me speak about rough drafts at the teachers Development Group leadership seminar, and you wanted to come up with an idea that we could try in our classroom and maybe study it together. So we brainstormed together, what could we actually try?

7:20  
That's awesome. The power of community coming together and support some some new thinking, that's awesome.

7:28  
Um, so we've had a chance to do a lot of talking you and I, through the summer book club chats on Instagram, live and had some great conversations and you're able to really dive deep into this content. So what are some highlights that you took away from those weekly book chats? Well, Joel, first, I want to say, I'm really thankful that you spent so much time thinking with me about the ideas in the book, because it helps my own thinking again, continue to grow. So one of the things that you were asking me about was, what sort of broader orientations or perspectives or conditions need to be in place for rough draft, thinking to be a flourishing experience for students in the classroom. And together, we brought up this idea of just being really open in our orientation to other people's thinking, assuming that what other people have to say has value, that it makes sense that there's something that can be built upon, and you brought up this idea of the concept of open hands, receiving other people's thinking with open hands. And that metaphor has really stuck with me. So I'm thankful for that. Another

8:43  
was, you pointed out to me that the process that I put into place for doing this work, was an example of rough drafting and revising

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the example early on about, we used to call it exploratory talk, picking up on the work of Douglas Barnes and then we revised it to the label of rough draft talk. And you You laughed, you're like, well, that's that's the same, that's revising, you're doing the rough draft work through the work with the teachers. And it's really funny, because I hadn't made that connection so directly before. So thank you for that.

9:21  
And I guess I want to say one more thing that stood out to me from something that you shared with me, chapter five, is about my reflections on the concept of humanizing the math classroom or re humanizing the math classroom. And I go through a series of revisions to how I might define that concept. And the revisions are inspired by work of classroom teachers and the readings I've done. And so you gave a really nice example, and work that you've done with teachers about asking teachers to elicit their current philosophy of teaching and then over a period of time they would revisit and revise that philosophy of teaching and I applied

10:00  
appreciated how you were able to point out how the way I wrote the chapter could be related to a way that we engage with teachers and their ongoing revision of their thinking about their practice. So those are some a few connections, I really appreciated that you brought up. Yeah, I really like the, you know,

10:20  
I'm just gonna, you know, pile some praise on you for a second, just like, there's certain things about the book that really make it approachable to you know, like, you're not presenting some like, you know, super clean version of examples. These are real examples from real teachers doing this work. And it's like, not saying like, here's the answer, but here are ways that you can do this work, which I really appreciate. The second thing was that, were that last chapter when you do talk about, or the second the last chapter, chapter five, when you talk about your definition of humanizing pedagogy, and, and the way you like, treated it as like a draft that you're keep going through, as you become more and more informed and thinking about all the conversations and interactions with folks that kind of led to, or even with the literature. I mean, it's just it's kind of a nice inside peek. I did a lot of comparisons between that and Megan Lampert's work in teaching with problems or problems, a teacher where you kind of get inside her head to her thinking about things as they're kind of moving along with the classroom that's in that book. This book, I think, does some of that same stuff, where you kind of you get inside Mandy's head a little bit, which is great, it's a great head to be in, and to think about, like how you're, you're, you know, sorting out some of these ideas, which I really enjoy. Thank you. Magdalene. Lanford writes about intellectual courage and intellectual humility, right. And

11:42  
I think that if we ask people to share their rough draft thinking, it requires both courage and humility. And so if we're going to invite people to be in this rough draft space with us, I think it's essential that we model that kind of humility and vulnerability about our own thinking. So people feel safe themselves to engage in that kind of thinking. So that was, I don't know, something I hoped for. Yeah, thanks. Well, so you, you obviously heard my thoughts on the book, and then throughout our Instagram live chats, and but I'm curious to hear what dusty and Eva might have to say about some of their highlights from the book and thinking about some of the ideas that you put forth.

12:31  
Mandy, I just finished this book a couple of days ago. And I just want to say I really like it.

12:40  
That's, that's, that's my starting statement.

12:43  
Thank you dusty, moving on.

12:46  
As one of the things that I was thinking about just now, as we've been, as we've been talking is, somewhere I've heard, and I wish I could find the source that

12:59  
people tend to learn things. But if we don't, Mark, like you had mentioned in the book, you know, where our understanding is at a certain point in time, and then reflect back on that we might not see how we've been progressing in our learning through that through through whatever it might be. And so I thought that was a really nice thing that as I read through the book, I was like, well, I agree with this. And this is a good idea. And yeah, I like this. So I thought, actually, I'm probably learning. I know, when I was first, a new teacher, I didn't, I wasn't doing all these things. And I hadn't even thought about doing a lot of these things. But now as I've been going through it, I feel like I've really been learning through your book about my own teaching. And specifically,

13:42  
as I read the last chapter about how

13:46  
the title, putting it into practice, or we are all works in progress. I really like that. We're all works in progress.

13:55  
idea. And as you reflect on

14:00  
the semester that you had hoped to do all of these different

14:05  
things, and then life happened, whatever life might be, and you were kind of honest with that. But then the fact that you wanted to look back and see what you have done, and you were thankfully, I think pleasantly surprised that your students really did pick up a lot of that, that just reminded me of how every time I get course evaluations, I tuck them away, I don't open them, I don't open the email until I've you know, got my favorite beverage in my hand. And I'm ready to you know, take a look at it. And it usually is not bad, but there's just a part of me that doesn't want that evaluation. I could just avoid that. So I thought that was really brave of you. Not only to do that, but then to I really liked how you wrote about that process that you went through. Thanks, I

14:57  
I got a question from someone

15:00  
That's why it says it like putting it into practice somebody, math coach said to me, how often do we have to do it? How often do we have to do this rough drafting? And I was like, oh, what's what does that mean? It. And I realized that the it could be asking students to revise explicitly how often you have to do that. But the other piece, all the aspects of building the classroom culture, so students can communicate to learn, and how their draft ideas are welcome. That's more like a spirit of rough drafting that I'd like to I'd like that to be in place all the time to permeate the experience. And so some of it, you would only need to do, sometimes the revising experiences aren't going to happen all the time. But other parts, it's more like our orientation toward each other, and how we set up the classroom environment and how we interact, that that's more of an all the time thing. I put a lot of pressure on myself. I thought, Okay, this book is an act of curation, right? I've learned so much so many different teachers. Now, I'm going to have the responsibility to show that it can all come together.

16:15  
So look at me, the perfect math teacher greatest of all time, no. And so then it was like, Oh, no. And so I said, My reaction was if I can't even do all of this stuff, who can? I mean, not to not to say that I should be able to do all the things, but I had a lot more information right about how to do this. And but the students took away more than I thought I asked them some really open ended questions about what they took away from the course. And I didn't prompt them to talk about certain things. And they've brought up many aspects that I hoped that they would take away. So then I trace back, okay, what was I able to do, and trying certain practices sometimes ended up having an impact. So I learned that you don't necessarily the book isn't saying you need to do every single one of these ideas to have a successful experience. It's more a menu of options. And if you have a spirit of rough drafting, and you choose some other options, it'll make a difference. And I just I needed to be real about that. Because I remember being I, when I was a teacher, and I went to professional development, and I would get very overwhelmed. Like I can't overhaul all my teaching to be like this. And so I feel like it's more liberating, if you can go to a professional development experience. And you could say, Oh, this complements what I already do, I can pick up these aspects and gradually extend my practice that feels more realistic to me.

17:48  
Yeah.

17:50  
Eva, did you have any, any thoughts to add, I know you had a chance to play with these ideas with the study. And I know you've done some, some work with it and your own thing, anything that you wanted to add with regards to this book and some of the ideas that stuck out to you.

18:05  
Yeah, so I incorporate this in a lot of my teaching, and it is something that my students really enjoy, and especially this last term where we had to go all online. And when you share your thinking, everybody can see it, and you don't see reactions. I think it was really important for my students, and a lot of them mentioned it later, that they really enjoy joy, the idea of

18:39  
being able to provide rough draft thinking, one of the thoughts that I had was, so as we go through the revision process, so one of the powers of rough draft thinking and sharing rough draft thinking is that you can say, Hey, this is where I'm at.

18:58  
And this is what I know. And I know, it might not be all the way there yet, but here's where I'm at. And then we start into the revisions process. And I've been starting to think about like, what,

19:13  
what do my students base their revisions off? of? So where do they look? And this question comes from like an authority perspective, like, where do they look for how to move forward? Do they look to their classmates? Do they look to the teacher do they look?

19:31  
And then as they provide revisions, how do they?

19:38  
How do they decide whether the revisions work for them or not? And I have in my classes, so far gone through several rounds of revision. So and then I have them at some point, create a final draft that is like so here's the best work that you can do, versus here's my reference.

20:00  
Thinking. And so I've just been thinking through like, Where? How does it work that they work together to create more sophisticated solutions? Or bring in more ideas? And how do they decide which ideas to bring in? So I'd be curious what Mandy thinks about that.

20:21  
I really am intrigued by this notion in terms of an authority perspective, right, and, and whose ideas get taken up and whose ideas get positioned as valuable in the process of revising? One related idea that isn't exactly what you're talking about, is something that concerns me about revising, is sometimes when people are exposed to an idea that's different from theirs, they might reject their draft completely. And think, Oh, this is the better way? What does it take to revise our thinking in a way that honors your ideas, the strengths that were in your ideas, don't discard that, right, like, see strengths in your thinking, but then also be open to learning from other people's thinking, I wonder if that's related to something about this orientation toward everyone's ideas have value mine and other people's.

21:22  
That's one reason why. So one of the revision routines in the book is from the math language routines out of Stanford, stronger and clearer each time. And so it would be, maybe I would write a draft, and dusty would write a draft, we would trade and read each other's and take our papers back, and then write a new draft each of us would, but I added this piece where you would reflect

21:49  
what you changed. And what inspired that change. That way, you're thinking about, you're assuming your ideas of value and your partner's ideas have value. So what inspired the change?

22:04  
It doesn't get at this notion of how do you know if your change is an improvement? So that's another level to analyze, which I really love.

22:14  
This got me thinking

22:16  
a little bit or Eva, did you have a follow up to that? No, I'm good. Okay. We're just gonna be thinking like the you didn't think about this idea of, you know, we kind of talked a little bit in the book chats about rejection versus revision, right? And like, or the abandonment of an idea. And it's like, there's so much, from my perspective, it feels like there's so much on the teacher of like, when an idea comes out, it's like, No, you can't see me, but like, having two hands held out, you know, kind of like all state like, where you're, you're grabbing that idea. And you're making sure like, I've got this idea, and I'm going to treat it as valued. And, you know, even if even if you don't think too highly, I'm going to treat that idea as value and show it to other people, like look at this idea. And so that there is this, there is this like culture of we're going to value ideas. And I think that's where the beginning of the book does a really good job of thinking about how do you set up your classroom to, to value ideas, and so that even if it's your own idea, you will value that idea. And that's what we do here. And so I just think that that's that's a big thing that came through with regards to the book, but also something you need to be aware of right? If you're seeing that people are rejecting their ideas, well, how do we how do we make sure that they see the intrinsic value of them?

23:34  
Definitely, I want to go back to a was wunderman, about whose ideas get taken up with revision. And I think that as a as an instructor, one way I tried to think about that was I would have a clipboard with different students, like everybody in the classes names. And during class periodically, I would make notes like who I was calling on, and for what reason. And then so I could study that at the end of class, and then the next class, kind of make notes to myself, well, I didn't position this person is having strengths yet, you know, and kind of prioritize myself to make sure certain students ideas were being taken up, playing a role in that, to make sure that authority could be distributed in different ways. Imani Gaffney has helpfully pushed me she's at the University of Maryland, to say it's important to notice whose ideas are being positioned as drafts, and whose ideas are being positioned as further along. And are there differences along those dimensions by race? So I need to keep track of whose ideas I'm positioning in what ways and then I'm hoping the students will internalize that and think about like, am I listening to other people or am I always listening to the same

25:00  
People, I could probably be more explicit about that with my students that I'm trying to really

25:07  
observe and notice everyone's strengths in different ways, but and then they could maybe start to try to observe and notice strengths in different people's perspectives. But

25:17  
anyway, that's something I think about in relationship to Ava's wonderment. Nice. Well, I think this is, this is the point where we want to pivot a little bit to thinking about using the book to teach math teachers, right. So we have maybe some folks out there that think, okay, I'm all in, I want to start using some rough draft thinking in my math class, or what I'm doing some things within service teachers and whatnot. How do you, how do you? Or how would you use the book? And I'm thinking, how would you for myself? And for anyone else out there that's not doing? How would you? Or how do you use this book to teach math teachers in either methods, content courses, or even in service professional development? How would you, how would you use this book?

26:00  
So the two types of courses I teach at the undergraduate level are math content courses for teachers and math methods, or pedagogy courses for teachers, and so I would use the book or ideas differently for each one. So in the math content course, what's great about that is it it's an opportunity for students to experience rough draft math. So I would be wanting to try to model some of the ideas in the book, like, carefully setting up a culture of rough drafting that I talked about in chapter two, maybe I could assign the students to read chapter one, it's free online@stenhouse.com. So they could read about the ideas in chapter one about what is rough drafts, I could talk with them really explicitly about my orientation toward how people learn. So dusty was saying what we did earlier in our career versus what we do now in our career. And one of the ways that I've changed a lot is I've been more upfront and explicit about what I think it means to learn. And

27:05  
so they would then understand why I'm asking them to do certain things in the classroom. So I'm also thinking about how I could create opportunities in the math content course, for students to revise. So in my class, they can revise their quizzes, for instance, how does my assessment structure match up with revising? How do my lessons incorporate opportunities to revise so giving the pre service teachers the experience of drafting really, in the math content course.

27:36  
In the methods class, I would like their planning process to incorporate some of these aspects. So planning what you're going to do to try to explicitly build culture with your students, such as incorporating the idea of generating maybe a classroom set of rights of the learner, from the work of crystal clinic, Craig and all the tourists. I talk about their work in the book, like the right to make a mistake, the right to say what makes sense to you. Maybe that's something you want to plan for, or what other strategies do you want to plan for with your class, we can talk about that in the methods class and develop strategies for that. In the methods class, when we talk about task selection, math, task selection, and implementation. In what ways

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does the task provides opportunities for students to communicate their thinking to keep learning so it can be one of the criteria for choosing tasks, if your goal is developing conceptual understanding.

28:37  
I'm also explicitly planning some revision experiences for students. So they could read chapter four and think about, oh, how can I build in revising at some point in this lesson, and they don't have to necessarily try out all the ideas in the classroom, but they could plan some or they could do some micro teaching. Yeah. But I would say those are three goals that I would have in a methods class that I think would be useful. And then chapter five, being about humanizing the math classroom, I'd hope that that would be a theme that would permeate my methods class or other folks methods class because it's, it's in our ways of how we orient ourselves to our students in lots of practices.

29:23  
Well, and I know we got a couple other math teacher educators on the line, so even dusty, you have any ideas on on using the book or the ideas in the book, and I know, Eva explicitly, you have done this.

29:35  
So I

29:38  
worked with Mandy and recorded a video that is available for everybody to kind of use. It's a very brief description of what rough draft thinking is. So if you want to use that instead of the other ones, you can but there's other ones available. And then just making clear, like I think it's really helpful for

30:00  
to students to understand that it's okay to just share your thinking. And I think the reason why I'm so interested in the authority question that I mentioned earlier is because often when the elementary content,

30:15  
teachers come into our classes, they

30:20  
don't yet know often that mathematics can make sense. And that it's not just a bunch of rules you had to memorize. And so this whole notion of, I can make sense of an idea, and I should not accept an idea, if it doesn't make sense to me, is something they have to learn in the class. Because so far, I mean, if we all remember, or we work with any children, usually, when you ask them a math problem, like what is two plus two? The answer is not four, the answer is four with a question at the end, right? Because there's so used to like, for what I think it's for, is it right, and they always wait for somebody else to decide whether it's right or wrong. And so this notion of deciding on your own whether something makes sense is something that we have to really dig into. And I think rough draft thinking is one very useful way to kind of say, Well, what does make sense to me and try to figure that out? And where am I stuck. But you also have to, like, understand that something can make sense, which is like a whole different idea.

31:36  
One of the things that I have learned from this book, and the practice I'm going to change is allowing my students to revise on some assignments. In the past, I would have students asked me if this isn't in a college classroom, you know, can do we do test revisions? Or can I revise my homework? Or they didn't maybe use those words, specifically, but they were looking for? Can I get some extra points if I go back and redo some of the stuff that I got wrong? And in the past, my idea was, no, this was the assignment, you had this certain amount of time to do it. That's what it is.

32:14  
and examining my own motives on that, I think I was mostly thinking it's easiest to grade, if I just have to grade it once.

32:23  
And then what will happen, you know, the world will collapse if people get a chance to work together and try to figure out, you know, what's the right answer and what goes in there. So I really liked the description that you gave. And it made me think, yes, and my class I'm teaching next month, I'm going to allow for that. Not on every assignment, but on some assignments, just to, to see how it goes. The other thing that a problem of practice that I've been working with is in a class that I teach for, is for mathematics majors. And it's in Euclidean geometry. And most of the students there want to become teachers, I started a an assignment where they would present a proof or a couple of proofs throughout the year, and students worked really hard to make sure that their proof was just perfect, or at least that's how I wanted it to be, I was hoping that they would work really hard.

33:22  
Recently, I think I made a part of the rubric that they needed to come with me and talk about their proof before that, and that has been really helpful in trying to see what their ideas were, a lot of people were coming like, you know, five hours, four hours before class to say, here's my ideas, what do you think, and they were really nowhere near as thought out. But I think what I want to do next time I get a chance to teach this class is not have them present their final proofs, but present a rough draft of a proof. So maybe they don't have, you know, a week or two weeks to prepare for this. But I just want to see I just want to value the, the rough draft of that proof because I have most of my students come in and they say to me, I passed the proofs class. That was the prerequisite but I still don't know how to prove geometry stuff.

34:16  
And so I think rough draft thinking would really benefit students there, especially if they're going to be high school teachers.

34:27  
I love that. I think it's so important to make this shift of learning experiences as a performance I need to perform as well as I can and shifted more toward learning is this ongoing iterative process. So I may have performed well in a previous proofs class. But knowing that proving is a challenging activity and we can keep growing. Even if we got an A and approves class, we can keep growing in our capabilities of proving our reasoning with each other.

35:00  
So that orientation, I think, is really shining through as you talk right now. And it's inspirational. So thanks for sharing. Yeah, and I'll be sure and use some of those sentence stems that, that you provided for students too, especially with my secondary majors. They're not used to talking with each other

35:20  
about this sort of thing. And maybe that's their prior experiences in the mathematic classroom. But even if something is great, you know, what else can we do? What could How could we make this more clear? Right? So I found that people are more used to talking about each other's thinking when there's something to pick apart and critique, maybe point out a mistake. When a solution looks pretty good. People are like, well, what's there to say, and so putting up some sentence starters helps us learn how to talk with each other about ideas that can help them continue to grow. So if we could have other goals for revision, besides fixing a problem, we could say, maybe the explanation could become more elegant, or more concise, or we could layer in different kinds of terminology that's more aligned with the discipline, you know, maybe it could be more elaborated, more, more connected different kinds of connections, maybe it could be more illuminating with a different kind of visual representation. So thinking about a range of ways that we could revise work. And then sentence starters, like, I don't know, that makes sense to me, because that makes me think of, I appreciate how you, so getting a number of different ways to get started talking about something that hopefully people will eventually internalize, and then you won't have to prompt them as much. But we all need support to learn how to interact differently. So I view the sentence starters as kind of an inspiration. And then people can kind of riff on them and go from there and not necessarily feel controlled by them, but maybe inspired by them.

37:02  
Yeah, the sentence starters, have a pretty good role within the book, and might be a way to help folks do this any other like, tips or maybe cautions when using the ideas in this book? So one thing that periodically comes up for me when I talk with folks about this is, is this just for honors students as if this is a type of kid, right? It's not everybody has mathematical brilliance. But can I just use this only with the students who are in this track? Or is it for everybody, and then I get the other flip is this for students, maybe I would need it for my enrichment class, or remediation or something like that. And that perspective doesn't really align with how I think about rough drafts. I think about it like, this is how people learn things by making their ideas and objects that they can reflect on and then seeing other people's ideas. And then iteratively improving our understanding, I think that's how a lot of us learn most things. So I wouldn't say that I would save these practices for certain certain types of classes, certain. I don't believe in types of kids. But you know what I mean, I would say that this is how people learn generally. So that's one thing that's come up for me in the past that I've had to think through how to talk about.

38:33  
Nice. Um,

38:36  
if I know again, I know you use some of these ideas, like any any tips that you have, if people are thinking about using this idea of revising their thinking when working with math teachers?

38:49  
I don't know that I have tips. But I think reflecting on revisions has been really useful. Like how was I thinking, what, what did I change? How did I change it? or How did I build on it?

39:06  
Yeah,

39:07  
I've been thinking about how rough draft thinking has created a different kind of lens for me when I select and sequence student work to talk about in a conversation. So I'm looking at their work, whether it's it was in the Google Slides of my students in the online class or walking around the room and looking for student work. And I look for examples that have ideas that can be built upon have strengths to them, and then where there's room to grow. So I use the possibility for amplifying strengths and opportunities for revision when I'm selecting and sequencing students work now, and I wouldn't have done that a few years ago. Nice.

39:52  
I mean, I just overall I just I like the him kind of talked about this during our chats is the idea of marking something so then you can see the

40:00  
growth, you know, in, in wanting teachers to, to see, to see the growth and not only just for the student to see their own growth, but also for teachers to see it, and to see like the how the ideas evolve. And I just think that's pretty valuable in order to be able to market and see, see what happens after that. And so that we're not just because I think you're there's just sometimes we just,

40:26  
we freeze each other in time a little bit, even our own teaching. And so to see even like the evolution of our own teaching, or the evolution of our, of our students and their development as doers of mathematics, it's like, this helps you see that evolution. Yep. And so some of the professional development I've tried with teachers, one thing that's been useful is after we do a first session about what are rough drafts and how we can incorporate it in the classroom, the next session would be, they would bring in student work. And so they would bring in student works, where you where the students would have multiple drafts of solutions to a problem. And they would look at the students drafts and talk about how the thinking evolved. And so we have a different kind of artifact of practice. Now, if you can see multiple drafts of students thinking. So that's been really interesting to see how the multiple drafts provide us insight into developing our own math knowledge for teaching and what it means to come to know and understand an idea.

41:27  
Well, and we have come to a very good understanding of this idea, but we always have more to grow. And so if people want to learn more about rough draft math, how might they be able to get this book? What's the best way? Mandy? Thanks for asking. So rough draft math is a book published by Stenhouse. And the cheapest price for this book is on the publishers website. Set Stenhouse calm and the version that's on Amazon is a little more expensive. So I think you can get free shipping. It's done house too. If you want to learn about the ideas and not read the whole book, the first chapter is available on the Stenhouse website, it says preview this book. And the first chapter is there along with a foreword by Dr. Robert Berry, which I'm very thankful that he was generous enough to write that and very honored by that. So that's a place where people can think about the ideas. They could also read the article in mass teaching in the middle school. It was at the end of 2016, the beginning of 2017 called rough draft talk.

42:33  
They could go to the march 2016 issue of the mathematics teacher educator for an article first author Eva Sennheiser. Second author, me. So these are some places where people could read more if they would like. Yeah, and we'll we'll also post a link to a video along with the podcast notes here. So yeah, um, and so thank you. Yeah, again for being the the first the first in the summer book clubs for the teaching math teaching podcast. Yes, is great. And so thankful. I'm humbled and honored that you selected my book for this process. I learned so much from talking with you. So thank you. And this conversation was really enlightening today. So I appreciate that people shared how they think about the ideas was an easy decision. It was great. So and a shameless plug for the podcast is that we are continuing the summer book club in July we are going to be reading high school mathematics lessons to explore, understand and respond to social injustice. So follow the teaching math teaching podcast on Instagram at teaching math teaching and twitter at teach math Teach For more information about the book club, the reading schedule, and where to find our weekly chats because we're going to adjust our we're probably going to Facebook Live, but just want to see if we can because there's a lot of authors in this book. So we want to see if we can get more than one because Instagram Live is good for one but not good for Yeah, you and I were on Instagram Live. Right when a way to talk with multiple people. Yep. Great. So thank you, Mandy. I appreciate it. And thanks again for listening to the teaching math teaching podcast. Be sure to subscribe to the podcast, we hope that you're able to implement something that you just heard and take an opportunity to interact with other math teacher educators.