Dr. Amanda Jansen, the author of Rough Draft Math, shares her insights and specific strategies for creating increased student dialogue, thinking, and engagement in math classrooms. What are the benefits of students beginning with the concept of producing a rough draft rather than a finished math answer?
Find Dr. Jansen on Twitter: @MandyMathEd
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Announcer : 00:01 Hello and welcome to the Parent Wellbeing and Student Learning During School Closures Edition of the Steve Barkley Ponders Out Loud podcast. With the extended closing of schools, we as parents have entered into a new territory regarding what we have known as “school learning.” With this and future podcasts, I’ll look to share my experiences as a teacher, educator, parent, grandparent, and continuous learner.
Steve: 00:34 Rough draft math. While recording an earlier podcast, I was introduced to a book titled, “Rough Draft Math,” and I was intrigued. So a quick search led me to the author, Dr. Amanda Jansen, professor of mathematics education at the University of Delaware. And I’m just delighted because she very quickly responded to my request and she’s joining us here today on the podcast. So welcome Dr. Jansen.
Amanda: 01:00 Thanks so much for having me, Steve.
Steve: 01:03 So jump in with a start off and tell us what is rough draft math and what led you to the exploration of it?
Amanda: 01:13 So I’ve always been interested in creating math classrooms where students ideas are valued as a part of the lesson. I used to be a junior high math teacher outside of Phoenix, Arizona. And when I was a teacher, I was always trying to create better classroom discussions where students were really involved and felt safe to share their thinking. And so, years later now, I am a professor and I work with teachers and I was leading a study group of high school and junior high teachers in Delaware about creating better classroom discussions. It’s an ongoing curiosity of mine. And we were reading a book called “Exploring Talk in School,” and it features the work of Douglas Barnes. And we were thinking about how to make math classroom discussions, more exploratory. And the notion of rough draft is familiar to kids from language arts and they’re used to experiencing math as you’re either right, or you’re wrong.
Amanda: 02:17 But really, when you’re understanding anything new, you’re thinking it’s continually evolving. So what would it be like to treat mathematical thinking as a process of constantly evolving your ideas, constantly revising, because that’s how we learn anyway. So why don’t we create a classroom environment where we’re doing that? So the teachers decided that rough draft talk was a better label for this process than exploratory talk. And then once they decided that that was the concept they wanted to work with, in the study group together, we sort of developed what would rough draft experiences be like in a math class? So this idea was co-constructed with teachers throughout the state and all the ideas in the book are from teachers. So the book is a set of curated ideas of what teachers have done to create a rough draft space in the classroom.
Steve: 03:10 Well, I have to say the the title really nails it because as soon as I heard the title, I knew that it had to mean you aren’t finished. And people, you know – math is finished, concept where you pretty much start that writing task knowing it’s going to take a couple of drafts and a couple of rewrites and you might even change your mind halfway through it and come from a different direction. So I think the title speaks great.
Amanda: 03:42 So then it works well with kids too, right? So if you say to a student, I just want to hear your rough draft about this, it does a lot of things. It sort of reduces – takes the air out of the pressure balloon, right? Like, oh, I just have to say whatever’s on my mind? My rough draft is going to be okay here? Yeah. And you can talk with students about how do we learn anything? We make an attempt, we make sense out of what we tried, then we try again. This is how we learn a lot of things in life so why can’t we learn math that way? And so asking students to share their initial drafts, talking with them about why that’s a useful thing to do to share your drafts, it just creates this safer space where whatever you have to say, we’re going to try to learn from it and it’s going to have value and it’s going to have merit and you’re going to have a chance to then revise that idea. You’re not going to be frozen in time by whatever you say right now, you’re not going to be judged for being right or wrong. We’re just trying to figure stuff out together.
Steve: 04:40 What impact are you looking at that having on student math performance?
Amanda: 04:46 Yeah. So it does a few things. For one thing, it helps students have a bigger view of the discipline of math. What is mathematics anyway? What does it mean to know and do math? Whenever I talk about rough draft math with mathematicians, they say, well, this is what we do when we are trying to solve a problem that we’re not sure about. We try it, we think about what we’ve tried, we try again, because we don’t know what the answer is going to be or what the solution’s going to be, or what is the appropriate proof or argument. And so students rethink what math even is. So that’s one really positive outcome. Another is that if students are asked to talk or write about why math makes sense, like explain why something is true or define a concept, it changes the learning goals from your you’re doing more than trying to get an answer quickly, you’re trying to understand a concept.
Amanda: 05:36 So students have opportunities to develop more of sense-making and conceptual understanding if they’re being pushed to draft and revise their thinking. So the nature of knowledge opens up to be more than procedures. You definitely want students to calculate correctly, but you want them to understand why it’s working. And so rough drafting allows to get into that space of why something is true. And then ultimately, students are developing more positive identities because they start to see the merit and value in their emerging ideas that their thinking has potential and that people can learn from their emerging, imperfect, unfinished ideas. And so they feel more valued as a thinker and learner, and then they’re more persistent and they’re more willing to put in effort because they realize that their thinking has merit and has strengths in it. And people are not used to that in math. People are used to feeling like they’re either a math person or they’re not, and that’s not true. Everyone is a mathematical thinker. So we want to create environments where they can recognize that they can think mathematically.
Steve: 06:42 So how does this lay out, looking at moving from the primary age students working with this through upper elementary, middle, and then the higher levels of mathematics in high school?
Amanda: 06:56 So this idea was initially targeted more for adolescents because it’s an age where students start to get very concerned about what other people think about them. And so it’s creating a more emotionally safe environment where everyone’s drafts are valued, but I’ve seen kindergarten teachers work on this with students. And so the main principles would be eliciting someone’s initial thinking, treating it as an idea that’s worth understanding, trying to understand it before you evaluate it, recognizing that you can learn from each other’s drafts and then going back to revise. So you can do that at any age, but it looks different at particular grades. So the elementary teachers have talked with would do things like maybe there’s an idea that’s an anchoring idea in that unit, like what is a fraction? And they might have a class definition that they post and as they continue to work throughout the unit, they would revise and refine that class definition for instance.
Amanda: 07:52 You might have a situation where at the beginning of the lesson, I’ve seen an elementary teacher have, like a problem that they use to launch the lesson. Then they work on the lesson for awhile. Then they go back, okay, take that paper of the problem that we were doing at the beginning, take a different writing utensil, like a pen or a marker and now put your new thinking on that page. So there’s different things that I’ve worked with elementary, but as you move into middle school and high school, you can get more mathematically sophisticated in different ways. I’ve seen middle school teachers have students make conjectures about what do you think is true about this situation? Like, a seventh grade students saying, well, how do you find a similar figure? Maybe you add the same number to every side. And in what cases is that correct?
Amanda: 08:40 And in what cases is it not correct? And so the class would take that claim and try to prove it or disprove it. So that’s more of an argumentation idea. As students get older, I’ve seen teachers do what they call a round of rough draft sharing. Give the students a challenging problem, they work on it, stop the groups before they’re done, bring your rough drafts up to share your thinking in progress. Like where are you all at? And then go back and now that you’ve heard from each other and just tried to understand each other, keep working and revise. So that’s some things that I’ve seen middle school teachers do, but an important thing with the middle school, well, with every grade, but with middle school, you’re trying to create that environment where they’re not judging each other, but they’re trying to understand each other. So setting up norms. An eighth grade teacher that I’ve worked with really likes this phrase, “be brave, be kind” as the anchoring norms in the class. So be brave enough to share your thinking when it’s not finished or you’re not sure. And be kind enough to try to orient yourself to learn from each other. So the norms that you set up are important at different age levels.
Steve: 09:51 That norm would fit all content areas.
Amanda: 09:55 I think so, right? It’s about taking intellectual risks.
Steve: 09:58 Yeah. Actually for the teachers meeting at a PLC too.
Amanda: 10:04 Yes. So, you know, as adults, we came up with these strategies for the students, but they actually very helpful for us. So it’s treating our own work as teachers as works in progress and giving ourselves credit for incrementally revising and improving, especially now with remote learning, people are teaching in ways that they hadn’t planned or predicted. So treating our own learning as a draft where we’re giving ourselves permission to constantly revise. So a lot of honestly, this rough draft work has helped me as a person, as well as in my teaching.
Steve: 10:39 I found a a webinar on your website where you talked about four elements for teachers and planning. And one of them was the norm setting that you just described. You then talked about test selection and explicit opportunities. Wondering if you could give us a little more detail about those two?
Amanda: 11:03 Sure. And one other thing about norm setting is that some teachers have found it really useful to work on this concept of the learners’ rights, the rights of the learner, and they could brainstorm things like I have the right to make a mistake. I have the right to use words that make sense to me. And so the students would generate rights and talk about what it would be like to enact those rights. So norm setting is super important and there’s lots of ways to do that. Task selection – you want to pick problems that are worth spending time on, problems that are worth discussing, problems where students can delve into understanding of concepts, problems that could have more than one possible solution or answer. So there’s things to compare and contrast and learn from. So the math problems that you choose, it’s really not going to be a problem where you just are calculating an answer.
Amanda: 11:52 There’s gotta be more to it. So there’s more to talk about. So that’s huge. Another aspect with the explicit opportunities around explicit opportunities to revise, taking time either during a lesson or during the week or during a unit where you pause and you look at work that you’ve already done or try and revisit it, rethink it. How has your thinking changed? And an important aspect of revision is that it’s more than fixing a mistake. You can have a correct answer, but your solution could be more concise or it could be more elaborated or you could use different terminology or an alternative diagram or representation. So pushing this notion of revising and and extending your thinking, you can have correct work and keep learning.
Steve: 12:39 Yeah. I like that extending. I’m seeing where I could have a right answer, mathematically right, but not have a depth of understanding as to as to why it was right or how it fits into a bigger picture. So I can see – I guess it’s probing on the part of the teacher that gets students there?
Amanda: 13:05 And comparing and contrasting. So say you solved a problem one way and I solved a problem another way and they’re both working. So what about the mathematical structure underlying those two strategies helps us get the bigger picture of what’s going on with the mathematics?
Steve: 13:21 I’m just really interested – I’m going to pass this on. I’m working with a couple of schools that are really focused on increasing student critical thinking across the board. And I can definitely see where this is a direct application to that element that teachers can be making.
Amanda: 13:41 I agree. I think that this aspect of pushing ourselves to keep trying to understand more, recognizing that I have something to learn from everyone else in the room we’ll then expand our horizons as well. So it’s partially this desire to keep understanding, but also how we orient ourselves to one another and what our colleagues have to say.
Steve: 14:04 The last element I found there on your website, you talked about humanizing the math classroom.
Amanda: 14:12 Yes. I think a lot of us have experienced mathematics in a way that’s dehumanizing. People have bad memories of being incorrect in public and not feeling good about it or having to accomplish a certain number of problems in a short period of time. So alternatively, orienting ourselves to one another thinking that everyone’s idea has potential that can be built upon. And so any draft idea that anyone shares is going to have something to teach us. So then if a draft or an unfinished idea is worth learning from, then more people in the room can be recognized as having an idea worth learning from. So more students can then see that they have capability, that they have competence. And acknowledging that learning is this ongoing iterative process of drafting and revising really normalizes that learning is a long process. Learning is ongoing, and this is true for everyone.
Amanda: 15:08 So many people have the idea that to be smart in math, you have to get a correct answer quickly, but that’s not true. And there’s lots of ways to be smart in math. Like you can make these connections between different people’s solutions rather than just having your own solution, right? That’s a great insight. You can ask a question that helps everyone understand more. So if we’re elevating students’ strengths, helping students see strengths that they didn’t realize they have, recognizing those strengths publicly so students recognize that their colleagues have strengths that they didn’t know about. And there’s lots of ways to be smart in math. I mean, this is directly informed by ideas and complex instruction, which is a set of concepts of teaching that is broader than math. And it’s by Elizabeth Cohen and Rachel Lotan. But this rough draft space really creates that environment where people start to realize more of their potential in mathematics.
Steve: 16:02 Well, thank you. I’m glad that I was intrigued and do you you you’ve pushed me to look further. I appreciate the time you spent here with us and I’d like to add your website into the lead-in to this podcast. So folks can go back find information about your book there and also correspond with you directly.
Amanda: 16:27 That would be great. Thank you so much. So again, all the ideas about rough draft math have come from classroom teachers. And so my learning is ongoing. As folks have their own ideas for how to make rough drafts work in the math class, there’s no one way to do this. So it’s been so exciting because people then come back and share with me other ideas that they’ve tried. So I hope to hear from people.
Steve: 16:49 I was gonna say – I hope some of the listeners will will send their thoughts and ideas on to extend yours. So thanks again.
Amanda: 16:57 Thank you.
Steve: 16:57 Appreciate it. Have a great day.
Amanda: 16:59 You too.
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