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Steve [Intro]: 00:25 Hello and welcome to the Steve Barkley Ponders Out Loud podcast. For over three decades, I’ve had the opportunity to learn with educators at all levels, both nationally and internationally. I invite you to listen as I explore my thoughts and learning on a variety of topics connected to teaching, learning, and leading with some of the best and brightest educators from around the globe. Thanks for listening in.

Steve: 00:52 Revision thinking in math classrooms. I’m delighted today to be joined by Dr. Amanda Jansen, a professor of mathematics education at the university of Delaware and the author of a book entitled “Rough Draft Math.” She is also currently a researcher through the National Science Foundation study in secondary mathematics “in-the-moment longitudinal engagement study.” Just excited that she is joining us here today. So welcome Dr. Jansen.

Amanda: 01:23 Thank you so much for having me.

Steve: 01:25 Would you start by describing the concept of rough draft math and it’s importance in student math success?

Amanda: 01:32 Sure. So the idea behind rough draft math is that every time we learn something new, talking through things or writing about something helps us crystallize and form our ideas. We’re used to rough drafts and other contexts like language arts or any time we’re writing, we get out our initial thinking, then we make it an object and reflect on it and then revise it and try to improve it. But we can do this in math as well. So we can ask students to explain why something’s true or why something makes sense. And then they can reflect upon that initial idea and then keep revising it. Because again, this authentically reflects the learning process and it reflects the work of what mathematicians do when they are solving a problem or trying to write a proof and they don’t know quite where they want it to go yet. You know, you put your initial thinking out and then you talk with people about it or you find, you know, what’s working, what’s not working and you iteratively keep improving it. So the idea is to make math classrooms a space that really aligns with more authentic learning processes. We’re sort of used to math classes where maybe students calculate correctly or incorrectly and then you fix the mistake and you move on. But there’s more to math than whether or not you’re calculating correctly.

Steve: 02:49 I remember in high school my math teacher telling me that I may never need to take a derivative of an equation, but the thinking process I was learning here was going to be valuable. So it turns out that’s true, correct?

Amanda: 03:04 Yes. Right. So there’s a lot in math about arguing, why something is true and coming up with an argument that has a clear claim and evidence for that claim and the warrants. There’s the logical thinking process of trying to solve a problem, be a good problem solver. We need to be a good problem solver and all kinds of areas of life. So math can help us with that, for sure.

Steve: 03:27 So we’re looking at a double payoff here of students working on this draft concept. And one of them is actually their, their math performance, but also a set of critical thinking skills that carry across to other areas.

Amanda: 03:44 I would say. So, so I would say because we’re using talking and writing to continue to make sense of why things are true, you have with the math performance, definitely it will help people calculate well, but it will help them retain that for longer because they’ll be understanding why it makes sense. So you’ll remember it. Because we’ve seen, you know, folks learn it for the test and then years later they don’t remember. But if conceptually the work makes sense to you, you’re going to retain it. And then also this notion of you’re developing a more positive math identity, because it’s safer for you to share your thinking when you’re not sure if you’re allowed to revise it, if your drafts are accepted and people see the value in your, in progress thinking. So students start to feel like they are more mathematically capable and then they’re going to put in more effort in the future because they are realizing that their ideas have strengths even when they’re not finished or they’re in progress. So I would say that’s an additional payoff with the critical thinking opportunities.

Steve: 04:47 So what would you describe as the most common changes that teachers need to make in their math classrooms when you look at introducing the concept of a rough draft?

Amanda: 05:03 I would say first thing that you would do would be, have a conversation with your students about the value of sharing your thinking while it’s in progress and the value of revising. Talk with students, like what do you know about rough drafting in general and other classes and why would you write a draft and then revise it? How does that help you learn? Have them think about when they learn how to do something else in their life. Like when you learn how to ride a bike, you had to try it and you would try again and you would go a little bit and then you might fall and then you think about what did I learn from that and try again. Thinking about other learning spaces where you have to make initial attempts, learn from those and keep revising. So help them understand learning in general and how a math class can be that way.

Amanda: 05:45 So that’s one. And then another is around culture building too. Setting up a culture where students want to hear from each other and understand each other and not jumping to evaluate. Not jumping to say right, that’s wrong, but spending time in this non-evaluative space where we want to understand each other. So one useful strategy might be teaching students to use these sentence starters where they’re trying to think about each other’s thinking like, “oh, it makes sense to me when you said that, because, or that makes me think of.” Helping students learn how to talk with each other. Orienting ourselves to make students thinking public so we can see each other’s ideas, think about each other’s thinking. Face-to-face – I encourage the use of document cameras a lot. And so what do teachers want to do to make the thinking public and online? There’s a lot of, you know, Google slides, jam boards, Desmos, things that students can do to see each other’s thinking so they can have those non-Evaluative try to understand each other conversations.

Amanda: 06:46 So as teachers, we’ve gone on a journey of trying to orient ourselves – the teachers I’ve worked with in Delaware, to orient ourselves, to see strengths in students work that’s unfinished, not completely correct, incomplete. We’re really oriented I think, initially, to see the deficits in student’s work that’s not right. And so orient ourselves to see what are the gems here? What are the strengths here that can be built upon? So if you’re going to revise, build from here. And pointing out those strengths to students privately and publicly to really honor each other’s thinking, so there’s a journey to go on, I think around strengths based orientations to kids work.

Steve: 07:29 As I’m listening there, it’s going through my mind as to watching a great language arts teacher give students feedback on writing that they can identify the strengths in what the student has done to this point and how they can build off of those strengths. I’m kind of hearing the same thing. Finding the strength in the math thinking the student’s done?

Amanda: 07:54 Right. Like what’s the potential here. Like, this is the direction that I think is worth building on. And so I think as we work with teachers and professional development, looking at students’ work together and brainstorming and thinking about where those strengths are. And then changing our mindset. Thinking about revising is more than fixing mistakes. Revising being, helping students communicate more clearly more precise or more concise or using different terminology or alternative diagrams or representation and some more elaborating. So as teachers, we want to shift our mindset of what it means to revise our thinking in math. That’s a big thing. And then building in revision, what do revision experiences look like in math, working together to develop strategies for that. And then thinking about assessment. If we really believe that math class is a place where students can keep revising our thinking during a lesson, so then what are the implications assessment so we can send a consistent message there. Do we let kids revise their tests? Do we have a portfolio based assessment system where students show their growth over time and say, “This is what I used to think. Look at how my thinking has changed. Here’s evidence for that.” So I think that that’s another space where teachers think about their work with rough drafts. They think about their assessments differently.

Steve: 09:17 I have to tell you what just happened. You’ve just given me an awesome aha. Well, so most of my work for the past 30 years has been in the area of coaching. Working with administrators and instructional coaches for how they work with teachers. And it just sunk deep into me as I listened to you that as a coach sitting down with a teacher, the lesson you taught today was a rough draft. I mean, I was so excited when I heard the title of your book and when I got the chance to visit your website and looking forward to talk to you, but it just hit me now that what we’re really talking is a metaphor of the whole coaching concept.

Amanda: 10:05 I love that. I so when I work with teachers on this, I also have us think about ourselves as we are works in progress. So if you think about our work as iteratively continuously improving, right? So there’s going to be strengths in your lesson. Like what went well? What were students seeming to understand? What about your lessons seem to enable those successes and what do you wish would go differently next time? And how do you want to document that so the next time you teach this lesson, whether it’s the next class period or the next year, what can we learn and revise?

Steve: 10:42 When I’m working with teachers in a PLC, I talk about looking at student work and asking the question, what do the kids need us to learn?

Amanda: 10:51 Oh, that’s beautiful.

Steve: 10:52 So before we can explore that learning for the students, we need to figure out what it is we need to learn.

Amanda: 10:58 Yes. I really love that.

Steve: 11:01 What do we change that now changes something for the student that creates that opportunity for the student to step in and do that learning on their part. I knew there was a reason I got so excited when I heard about your book and it just really concretized for me now that it really is a coaching concept.

Amanda: 11:20 And I think if we live out of this space as professionals ourselves, then it’s going to be really authentic when we create that space for our students, right? Like if we are constantly trying to improve ourselves, what can we learn professionally? What do we need to learn to support our students’ learning? Then the students will experience that as really true. They’ll believe you, because part of it is you can tell the students that their rough drafts are welcome, but then you have to actually welcome them in how you interact. They believe you more from your actions than your words, right? So if you truly believe that learning is a process of constantly revising, and you’re truly excited about the strengths in their thinking, and then you’re truly oriented to let them keep revising, they’re going to tell from your actions. And so if we’re practicing it in our own learning, it will be easier to interact with students about it for sure.

Steve: 12:15 I have frequently used that one of my favorite examples is looking at teachers in a higher math or a science classroom where they’ll post a problem and they’ll step back and say, I wonder where we should begin. And they actually mean it.

Amanda: 12:34 They actually mean it. Yes.

Steve: 12:34 The teacher doesn’t know where to start.

Amanda: 12:36 The students feel like they’re colleagues with you, right?

Steve: 12:38 Yea.

Amanda: 12:38 The authority for what it means to know and understand ends up being shared with the students. And it’s more fun to learn with your students. And so that’s why exposing students thinking, asking them to make their thinking public, it’s really fun for me. I wouldn’t have thought to write that representation and it actually works. That’s great. So I’m learning so much when I center my students’ thinking and instruction too. It’s really enjoyable.

Steve: 13:05 When I visited your site, I read a little bit about the National Science Foundation study. Give us a little descriptor of of what you’re looking at there.

Amanda: 13:14 Sure. So I’m working with a colleague at Arizona state university named Jim Middleton, and we’ve collaborated for years. We’ve co-written a book called “Motivation Matters and Interest Counts” published by the National Council of Teachers of Mathematics. So as part of that work, we’re trying to continue to understand what teachers can do to motivate and engage students. High school tends to be a time where a students’ motivation engagement drops off a bit for whatever reason. And one reason might be how math can be taught. So we’re trying to understand the successes that high school teachers have had in engaging their students with mathematics. And so, this is considered an in the moment study. We asked the teachers that we work with to conjecture what part of your lesson do you think is going to be the most motivating and engaging for kids and why? So they predict – because the lesson will have a series of activities and they’ll predict, okay, this activity is probably going to engage them for a particular reason.

Amanda: 14:15 Maybe it’s like a really interesting problee that’s going to challenge their misconceptions, or maybe it’s going to involve group work and students like to talk about their ideas, something like this, they’ll make a conjecture. And then when we observe them and we video record them, immediately after that activity, at whatever point in the lesson, we give the students a survey – it’s a short survey, we’re in schools that are one-to-one so they take it on Chromebooks, but you can do it on paper. And students would then share more about what they did and what they thought about that activity. And then we can analyze the video for instructional practices that researchers noticed that we think have the potential to be engaging, but then we can let the students inform us from their surveys about that activity and how it impacted them. So motivation is not a static thing.

Amanda: 15:03 I’m motivated in different ways for different reasons, depending on what I’m asked to do. So that’s why we wanted to look at exactly certain moments and to support teachers’ learning, we’re asking the teachers to make conjectures and then we can help them see if their conjectures play out and in what ways. So it’s been really exciting because we can learn both from teachers about what they think engagement is, what could engage kids, and then we’re learning from students about those experiences during the lesson. And then we have the video recordings to look closely at the instruction of lessons that really were motivating for the kids.

Steve: 15:41 Interesting. Because it matches a piece that I frequently work into my coaching, where I’m asking the teacher to make some predictions as to expectation of what they’re going to find during the instructional time. And then a post-conference, talk about what they did find. And so what was reinforced or what was questioned by that? So I love the idea of that as a study.

Amanda: 16:11 And so then the teachers end up having multiple data sources to them, think about their conjecture that goes beyond their impressions, right? So they get the student feedback that can then affect what they thought about how it went. And we actually also share video clips from that lesson with the teachers and then they reflect on what they noticed from the video. And so, yes, I think that it’s an interesting study because you can contribute to research and theory about what’s motivating and engaging but then the people who are participating in this study also find it educational.

Steve: 16:45 I was going to say, it’s also PD for the folks who are doing that during the study with you.

Amanda: 16:51 That’s what the teachers have told me. And so that makes me feel great because I definitely want the participants in studies to feel
like they’re also getting things out of it.

Steve: 17:01 I’m wondering if you have some wording for a question or two that might be good for coaches or administrators, or even teachers themselves to ask students that give you some picture of the motivation piece?

Amanda: 17:20 So, thinking about everything that I’m invested in intellectually, whether it’s the rough draft math or this study that I was describing, is about making students thinking being more at the center of the lesson. And so if I was to think about what coaches might brainstorm with teachers, well, first of all, I would ask the teachers for their best ideas because they already have good ideas. But if they wanted to know my opinion, I say things like, well, often we ask students, what did you get or how did you solve that? And you could shift and say, how are you thinking about this problem, right? And so this idea came – I’m active on Twitter and a lot of my ideas again, come from other people, right? They come from teachers and they come from math coaches. So this shift to ask, how are you thinking about this comes from a woman named Annie Federer. You know, asking teachers to think about what can we do to make students thinking public to one another. Like document cameras, but in the remote setting, jam board and Google slides, students taking pictures of their work and uploading it.

Amanda: 18:28 But what can we do to put student work available for their colleagues and peers to see? Getting people’s thoughts about that and then how they would debrief that. And what can we do to get students to talk with one another about their thinking? So what are teacher’s ideas about that? I tend to rely on, what do you notice about this person’s thinking, what do you appreciate about this person’s thinking and what do you wonder about it? And the appreciation question is newer for me. I used to go with notice and wonder, but the appreciation one is a nice addition because while you’re pointing out something you appreciate about someone’s thinking, you’re also making sense of it, but yet you’re helping people see your ideas matter to me. And so it adds the socioemotional dimension of interpersonal support. But the way I navigate working with teachers is I assume that teachers have a lot of great ideas already, and I spend a lot of time helping us all be in community and share. So your prompt, I would first start, like let’s just brainstorm together. And then I would be another person brainstorming in the group. So those would be the things I would add to the brainstorm.

Steve: 19:42 Great. We’ve got time for one last piece here. When I was on your website, I saw that you were making a connection between the rough draft concept, humanizing a classroom and connecting that thought with equity. And I know that’s a giant issue for many schools right now and I’m wondering if you could talk a little bit about the connection you’re making there.

Amanda: 20:10 Yes. Thanks for asking. So I really have a passion about creating math classrooms where students recognize that they have capabilities. Mathematical thinking is something that everyone can do. But sometimes we inadvertently set up math classrooms where students feel dehumanized, they feel criticized, they feel frustrated that they’re always wrong and it makes you not want to try. So the notion of rough drafts or making students thinking being at the center, it’s important for us to then orient ourselves, to see the strengths in each other’s thinking. And so we can raise status of students by sharing how their draft ideas have potential because all drafts have potential to be built upon. And then if an idea in draft form is worth learning from, then more students can be recognized for their mathematical capabilities. So one thing that’s important with that is recognizing whose ideas that we’re pointing out as a draft and whose ideas that were pointing out as further along.

Amanda: 21:17 I started in my own teaching – so I teach future teachers and sometimes I teach math classes for future teachers. I started making a note. I had I got a clipboard and I put my students’ names on it. And I started making notes of who I was positioning as having a rough idea and who I was positioning as having a further along idea. We want to make sure that we’re not constantly putting our students of color in positions as being shown as more in progress and our white students is further along. Or our boys in a particular way in our girls in a particular way. Or, you know, making sure that we are amplifying the strengths in all of our students. That’s really important to question our assumptions about who we’re elevating and who we’re not because ultimately all the students have brilliant ideas. And so we want to be able to point those out to one another.

Amanda: 22:07 So the students actually don’t always know that their ideas are so great. And so, they feel honored when you actually point out – this is what I learned from what you shared today, or asking the students to point out to one another what they learned from what their peers shared today. And so then it shifts this notion that to be smart in math, you have to get a correct answer quickly. Instead, to be smart in math, you can have an idea that helps someone else make sense out of something. You can ask a question, it helps students make sense, you can make a connection, your diagram might be more illuminating to someone else. And so then more students can be seen as smart in math and – because they are so we might as well point that out. So I have hope that orienting ourselves to see each other’s strengths is something that can be throughout the classroom community, students start pointing it out to each other, and then maybe that changes how we treat each other in the world too. Trying to understand each other before we critique each other and I’m assuming we have things to learn from one another.

Steve: 23:14 What a powerful message for teachers to be communicating to their students by the way we design the classroom and equally for administrators and instructional coaches to be modeling and how they work and think with teachers.

Amanda: 23:29 Thanks. And if I can add one last piece about that – this notion of revising our thinking in the face of new information, I think is another really powerful notion to be citizens in a democratic society, right? As the coronavirus, a novel virus, as we learn more and more about it, we’re changing how we practice social distancing in different ways. And revising our thinking in the face of new information is not flip-flopping or being weak, it’s being scientifically literate. And so if we can promote the notion of smart folks continue to revise their thinking in the face of new information, I think that would be good for society as a whole too. So let’s practice it in math classes, practice it everywhere.

Steve: 24:14 You just aligned with a blog I posted in the last two weeks on intellectual humility if it’s enriching teaching, but it has that impact across the whole world. Actually when I discovered I was wrong, it should be celebrated because it means I learned something new and I should create myself to always be at that point. And I’m hearing you describe a classroom where that message is being communicated.

Amanda: 24:48 Yes. I very much believe that without intellectual humility, we won’t keep learning. And also the intellectual courage to keep trying. So Magdalene Lampert is someone that writes about intellectual courage and intellectual humility, and also wise restraint, right? And deciding when it’s time for you to put your ideas out there and when it’s time to listen. So all of that is paired together for me. So it sounds like we’re kindred spirits in that way. So that’s great.

Steve: 25:19 Terrific, terrific. Well, thank you so, so much for joining us. We’ll put your your website into the podcast here so folks can find you. You want to just tell them what your Twitter handle is because I’m thinking some folks might want to jump right on to find you right away.

Amanda: 25:35 Oh, that’d be great. I always love connecting with colleagues on Twitter. So my handle is @mandymathed. So Mandy is M A N D Y – math
ed. So you can Google Amanda Jansen on Twitter as well, and you’ll find it that way. So thanks so much. Yes. I hope to have dialogue with folks who are also sharing some of these interests in making students’ ideas be the center and elevating students’ strengths and revising our thinking together.

Steve: 26:01 Well, I loved thinking with you today and getting my own insights out of it. So thanks for joining us.

Amanda: 26:07 Thank you for sharing your connections too. It’s nice to talk to you.

Steve: 26:11 Take care.

Amanda: 26:12 Take care.

Steve [Outro]: 26:14 Thanks again for listening, you can subscribe to Steve Barkley, ponders out loud on iTunes and Podbean. And please remember
to rate and review us on iTunes. I also want to hear what you’re pondering. You can find me on twitter@stevebarkleyorsendmeyourquestionsandfindmyvideosandblogsatmarkleypd.com.