Amanda Jansen, PhD, Rough Draft Math: Revising to Learn

Mike Wallus, Vice President for Educator Support


What would happen if teachers consistently invited students to think of their ideas in math class as a rough draft? What impact might this have on students' participation, their learning experience, and their math identity? Those are the questions we'll explore today with Dr. Mandy Jansen, the author of “Rough Draft Math,” on this episode of Rounding Up.

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Dr. Amanda Jansen is a professor in the School of Education at the University of Delaware. She is a mathematics educator who conducts research on students’ engagement in mathematics classrooms and teachers’ learning from their reflections on their own practice. She is committed to honoring students’ voices through her research on students’ motivation and engagement. Her most recent book, “Rough Draft Math: Revising to Learn,” was published by Stenhouse Publishers in March 2020.

Before working in academia, Dr. Jansen was a junior high mathematics teacher. At UD, she is invested in continually improving UD’s elementary mathematics teacher education courses through research and development work.


Book - Rough Draft Math: Revising to Learn

MTMS article - Rough-Draft Talk in Mathematics Classrooms

MTLT article - Rough-Draft Thinking and Revising in Mathematics

NCTM article - Mathematics Teaching in the Middle School

Website with free resources


Mike Wallus: What would happen if teachers consistently invited students to think of their ideas in math class as a rough draft? What impact might this have on students' participation, their learning experience, and their math identity? Those are the questions we'll explore today with Dr. Mandy Jansen, the author of “Rough Draft Math,” on this episode of Rounding Up. 

Mike: Well, welcome to the podcast, Mandy. We are excited to be talking with you. 

Mandy Jansen: Thanks, Mike. I'm happy to be here. 

Mike: So, I'd like to start by asking you where the ideas involved in “Rough Draft Math” originated. What drove you and your collaborators to explore these ideas in the first place? 

Mandy: So, I work in the state of Delaware. And there's an organization called the Delaware Math Coalition, and I was working in a teacher study group where we were all puzzling together—secondary math teachers—thinking about how we could create more productive classroom discussions. And so, by productive, one of the ways we thought about that was creating classrooms where students felt safe to take intellectual risks, to share their thinking when they weren't sure, just to elicit more student participation in the discussions. One way we went about that was, we were reading chapters from a book called “Exploring Talk in School” that was dedicated to the work of Doug Barnes. And one of the ideas in that book was, we could think about fostering classroom talk in a way that was more exploratory. Exploratory talk, where you learn through interaction. Students often experience classroom discussions as an opportunity to perform. "I want to show you what I know.” And that can kind of feel more like a final draft. And the teachers thought, “Well, we want students to share their thinking in ways that they're more open to continue to grow their thinking.” So, in contrast to final draft talk, maybe we want to call this rough draft talk because the idea of exploratory talk felt like, maybe kind of vague, maybe hard for students to understand. And so, the term “rough draft talk” emerged from the teachers trying to think of a way to frame this for students. 

Mike: You're making me think about the different ways that people perceive a rough draft. So, for example, I can imagine that someone might think about a rough draft as something that needs to be corrected. But based on what you just said, I don't think that's how you and your collaborators thought about it, nor do I think that probably is the way that you framed it for kids. So how did you invite kids to think about a rough draft as you were introducing this idea? 

Mandy: Yeah, so we thought that the term “rough draft” would be useful for students if they have ever thought about rough drafts in maybe language arts. And so, we thought, “Oh, let's introduce this to kids by asking, ‘Well, what do you know about rough drafts already? Let's think about what a rough draft is.’” And then we could ask them, “Why do you think this might be useful for math?” So, students will brainstorm, “Oh yeah, rough draft, that's like my first version” or “That's something I get the chance to correct and fix.” But also, sometimes kids would say, “Oh, rough drafts … like the bad version. It's the one that needs to be fixed.” And we wanted students to think about rough drafts more like, just your initial thinking, your first ideas; thinking that we think of as in progress that can be adjusted and improved. And we want to share that idea with students because sometimes people have the perception that math is, like, you're either right or you're wrong, as opposed to something that there's gradients of different levels of understanding associated with mathematical thinking. And we want math to be more than correct answers, but about what makes sense to you and why this makes sense. So, we wanted to shift that thinking from rough drafts being the bad version that you have to fix to be more like it's OK just to share your in-progress ideas, your initial thinking. And then you're going to have a chance to keep improving those ideas. 

Mike: I'm really curious, when you shared that with kids, how did they react? Maybe at first, and then over time?

Mandy: So, one thing that teachers have shared that's helpful is that during a class discussion where you might put out an idea for students to think about, and it's kind of silent, you get crickets. If teachers would say, “Well, remember it's OK to just share your rough drafts.” It's kind of like letting the pressure out. And they don't feel like, “Oh wait, I can't share unless I totally know I'm correct. Oh, I can just share my rough drafts?” And then the ideas sort of start popping out onto the floor like popcorn, and it really kind of opens up and frees people up. “I can just share whatever's on my mind.” So that's one thing that starts happening right away, and it's kind of magical that you could just say a few words and students would be like, “Oh, right, it's fine. I can just share whatever I'm thinking about.” 

Mike: So, when we were preparing for this interview, you said something that has really stuck with me and that I've found myself thinking about ever since. And I'm going to paraphrase a little bit, but I think what you had said at that point in time was that a rough draft is something that you revise. And that leads into a second set of practices that we could take up for the benefit of our students. Can you talk a little bit about the ideas for revising rough drafts in a math classroom? 

Mandy: Yes. I think when we think about rough drafts in math, it's important to interact with people thinking by first, assuming those initial ideas are going to have some merit, some strength. There's going to be value in those initial ideas. And then once those ideas are elicited, we have that initial thinking out on the floor. And so, then we want to think about, “How can we not only honor the strengths in those ideas, but we want to keep refining and improving?” So inviting revision or structuring revision opportunities is one way that we then can respond to students’ thinking when they share their drafts. So, we want to workshop those drafts. We want to work to revise them. Maybe it's peer-to-peer workshops. Maybe it's whole-class situation where you may get out maybe an anonymous solution. Or a solution that you strategically selected. And then work to workshop that idea first on their strengths, what's making sense, what's working about this draft, and then how can we extend it? How can we correct it, sure. But grow it, improve it.

And promoting this idea that everyone's thinking can be revised. It's not just about your work needs to be corrected, and your work is fine. But if we're always trying to grow in our mathematical thinking, you could even drop the idea of correct and incorrect. But everyone can keep revising. You can develop a new strategy. You can think about connections between representations or connections between strategies. You can develop a new visual representation to represent what makes sense to you. And so, just really promoting this idea that our thinking can always keep growing. That's sort of how we feel when we teach something, right? Maybe we have a task that we've taught multiple times in a row, and every year that we teach it we may be surprised by a new strategy. We know how to solve the problem—but we don't have to necessarily just think about revising our work but revising our thinking about the ideas underlying that problem. So really promoting that sense of wonder, that sense of curiosity, and this idea that we can keep growing our thinking all the time. 

Mike: Yeah, there's a few things that popped out when you were talking that I want to explore just a little bit. I think when we were initially planning this conversation, what intrigued me was the idea that this is a way to help loosen up that fear that kids sometimes feel when it does feel like there's a right or a wrong answer, and this is a performance. And so, I think I was attracted to the idea of a rough draft as a vehicle to build student participation. I wonder if you could talk a little bit about the impact on their mathematical thinking, not only the way that you've seen participation grow, but also the impact on the depth of kids' mathematical thinking as well. 

Mandy: Yes, and also I think there's impact on students' identities and sense of self, too. So, if we first start with the mathematical thinking. If we're trying to work on revising—and one of the lenses we bring to revising, some people talk about lenses of revising as accuracy and precision. I think, “Sure.” But I also think about connectedness and building a larger network or web of how ideas relate to one another. So, I think it can change our view of what it means to know and do math, but also extending that thinking over time and seeing relationships. Like relationships between all the different aspects of rational number, right? Fractions, decimals, percents, and how these are all part of one larger set of ideas. So, I think that you can look at revision in a number of different grain sizes. 

You can revise your thinking about a specific problem. You can revise your thinking about a specific concept. You can revise your thinking across a network of concepts. So, there's lots of different dimensions that you could go down with revising. But then this idea that we can see all these relationships with math … then students start to wonder about what other relationships exist that they hadn't thought of and seen before. And I think it can also change the idea of, “What does it mean to be smart in math?” Because I think math is often treated as this right or wrong idea, and the smart people are the ones that get the right idea correct, quickly. But we could reframe smartness to be somebody who is willing to take risk and put their initial thinking out there. Or someone who's really good at seeing connections between people's thinking. Or someone who persists in continuing to try to revise. And just knowing math and being smart in math is so much more than this speed idea, and it can give lots of different ways to show people's competencies and to honor different strengths that students have. 

Mike: Yeah, there are a few words that you said that keep resonating for me. One is this idea of connections. And the other word that I think popped into my head was “insights.” The idea that what's powerful is that these relationships, connections, patterns, that those are things that can be become clearer or that one could build insights around. And then, I'm really interested in this idea of shifting kids' understanding of what mathematics is away from answer-getting and speed into, “Do I really understand this interconnected bundle of relationships about how numbers work or how patterns play out?” It's really interesting to think about all of the ramifications of a process like rough draft work and how that could have an impact on multiple levels. 

Mandy: I also think that it changes what the classroom space is in the first place. So, if the classroom space is now always looking for new connections, people are going to be spending more time thinking about, “Well, what do these symbols even mean?” As opposed to pushing the symbols around to get the answer that the book is looking for. 

Mike: Amen.

Mandy: And I think it's more fun. There are all kinds of possible ways to understand things. And then I also think it can improve the social dimension of the classroom, too. So, if there's lots of possible connections to notice or lots of different ways to relationships, then I can try to learn about someone else's thinking. And then I learn more about them. And they might try to learn about my thinking and learn more about me. And then we feel, like, this greater connection to one another by trying to see the world through their eyes. And so, if the classroom environment is a space where we're trying to constantly see through other people's eyes, but also let them try to see through our eyes, we're this community of people that is just constantly in awe of one another. Like, “Oh, I never thought to see things that way.” And so, people feel more appreciated and valued. 

Mike: So, I'm wondering if we could spend a little bit of time trying to bring these ideas to life for folks who are listening. You already started to unpack what it might look like to initially introduce this idea, and you've led me to see the ways that a teacher might introduce or remind kids about the fact that we're thinking about this in terms of a rough draft. But I'm wondering if you can talk a little bit about, how have you seen educators bring these ideas to life? How have you seen them introduce rough draft thinking or sustain rough draft thinking? Are there any examples that you think might highlight some of the practices teachers could take up? 

Mandy: Yeah, definitely. So, I think along the lines of, “How do we create that culture where drafting and revising is welcome in addition to asking students about rough drafts and why they might make sense of math?” Another approach that people have found valuable is talking with students about … instead of rules in the classroom, more like their rights. What are your rights as a learner in this space? And drawing from the work of an elementary teacher in Tucson, Arizona, Olga Torres, thinking about students having rights in the classroom, it's a democratic space. You have these rights to be confused, the right to say what makes sense to you, and represent your thinking in ways that make sense to you right now. If you honor these rights and name these rights, it really just changes students' roles in that space. And drafting and revising is just a part of that. 

Mandy: So different culture-building experiences. And so, with the rights of a learner brainstorming new rights that students want to have, reflecting on how they saw those rights in action today, and setting goals for yourself about what rights you want to claim in that space. So then, in addition to culture building and sustaining that culture, it has to do—right, like Math Learning Center thinks about this all the time—like, rich tasks that students would work on. Where students have the opportunity to express their reasoning and maybe multiple strategies because that richness gives us so much to think about. 

And drafts would a part of that. But also, there's something to revise if you're working on your reasoning or multiple strategies or multiple representations. So, the tasks that you work on make a difference in that space. And then of course, in that space, often we're inviting peer collaboration. 

So, those are kinds of things that a lot of teachers are trying to do already with productive practices. But I think the piece with rough draft math then, is “How are you going to integrate revising into that space?” So eliciting students' reasoning and strategies—but honoring that as a draft. But then, maybe if you're having a classroom discussion anyway, with the five practices where you're selecting and sequencing student strategies to build up to larger connections, at the end of that conversation, you can add in this moment where, “OK, we've had this discussion. Now write down individually or turn and talk. How did your thinking get revised after this discussion? What's a new idea you didn't have before? Or what is a strategy you want to try to remember?” So, adding in that revision moment after the class discussion you may have already wanted to have, helps students get more out of the discussion, helps them remember and honor how their thinking grew and changed, and giving them that opportunity to reflect on those conversations that maybe you're trying to already have anyway, gives you a little more value added to that discussion. 

It doesn't take that much time, but making sure you take a moment to journal about it or talk to a peer about it, to kind of integrate that more into your thought process. And we see revising happening with routines that teachers often use, like, math language routines such as stronger and clearer each time where you have the opportunity to share your draft with someone and try to understand their draft, and then make that draft stronger or clearer. Or people have talked about routines, like, there's this one called “My Favorite No,” where you get out of student strategy and talk about what's working and then why maybe a mistake is a productive thing to think about, try to make sense out of. But teachers have changed that to be “My Favorite Rough Draft.” So, then you're workshopping reasoning or a strategy, something like that. And so, I think sometimes teachers are doing things already that are in the spirit of this drafting, revising idea. But having the lens of rough drafts and revising can add a degree of intentionality to what you already value. And then making that explicit to students helps them engage in the process and hopefully get more out of it. 

Mike: It strikes me that that piece that you were talking about where you're already likely doing things like sequencing student work to help tell a story, to help expose a connection. The power of that add-on where you ask the question, “How has your thinking shifted? How have you revised your thinking?” And doing the turn and talk or the reflection. It's kind of like a marking event, right? You're marking that one, it's normal, that your ideas are likely going to be refined or revised. And two, it sets a point in time for kids to say, “Oh yes, they have changed.” And you're helping them capture that moment and notice the changes that have already occurred even if they happened in their head. 

Mandy: I think it can help you internalize those changes. I think it can also, like you said, kind of normalize and honor the fact that the thinking is continually growing and changing. I think we can also celebrate, “Oh my gosh, I hadn't thought about that before, and I want to kind of celebrate that moment.” And I think in terms of the social dimension of the classroom, you can honor and get excited about, “If I hadn't had the opportunity to hear from my friend in the room, I wouldn't have learned this.” And so, it helps us see how much we need one another, and they need us. We wouldn't understand as much as we're understanding if we weren't all together in this space on this day and this time working on this task. And so, I love experiences that help us both develop our mathematical understandings and also bond us to one another interpersonally. 

Mike: So, one of the joys for me of doing this podcast is getting to talk about big ideas that I think can really impact students' learning experiences. One of the limitations is, we usually spend about 20 minutes or so talking about it, and we could talk about this for a long time, Mandy. I'm wondering, if I'm a person who's listening, and I'm really interested in continuing to learn about rough draft math, is there a particular resource or a set of resources that you might recommend for someone who wants to keep learning? 

Mandy: Thank you for asking. So, like you said, we can think about this for a long time, and I've been thinking about it for seven or eight years already, and I still keep growing in my thinking. I have a book called “Rough Draft Math: Revising to Learn” that came out in March 2020, which is not the best time for a book to come out, but that's when it came out. And it's been really enjoyable to connect with people about the ideas. And what I'm trying to do in that book is show that rough draft math is a set of ideas that people have applied in a lot of different ways. And I think of myself kind of as a curator, curating all the brilliant ideas that teachers have had if they think about rough drafts and revising a math class. And the book collects a set of those ideas together. 

But a lot of times, I don't know if you're like me, I end up buying a bunch of books and not necessarily reading them all. So, there are shorter pieces. There's an article in Mathematics Teaching in the Middle School that I co-wrote with three of the teachers in the Delaware Teacher Study Group, and that is at the end of the 2016 volume, and it's called “Rough-Draft Talk.” And that's only 1,800 words. That's a short read that you could read with a PLC or with a friend. And there's an even shorter piece in the NCTM Journal, MTLT, in the “Ear to the Ground” section. And I have a professional website that has a collection of free articles because I know those NCTM articles are behind a paywall. And so, I can share that. Maybe there's show notes where we can put a link and there's some pieces there. 

Mike: Yes, absolutely. Well, I think that's probably a good place to stop. Thank you again for joining us, Mandy. It really has been a pleasure talking with you. 

Mandy: Thank you so much, Mike. 

Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability.