In this newest installment of the Teacher's Corner Podcast, Cathy Humphreys and Ruth Parker, authors of Making Number Talks Matter and Digging Deeper: Making Number Talks Matter Even More talk about their journey writing these two important math books and what they've learned along the way.
Listen to it here.
To preview or buy Making Number Talks Matter and Digging Deeper: Making Number Talks Matter Even More go to Stenhouse.com.
Nate Butler: This is Teacher's Corner from Stenhouse Publishers.
Ruth Parker: We want number talks to be all about sense-making, and for most kids, the standard algorithm has been recipes that they've memorized without having any idea why.
Cathy Humphreys: There's no perfect number talk and that what number talks are about thinking, listening to kids, and always keeping an eye on getting students' thinking to the forefront rather than my thinking.
Nate Butler: That's Ruth Parker and Cathy Humphreys, authors of Making Number Talks Matter and Digging Deeper: Making Number Talks Matter Even More. Using number talks in your math instruction will not only create number sense, but also transform how a teacher teaches and a student learns. In today's podcast we'll hear from Ruth and Cathy about their journey writing these two books and how what they learned along the way can help you in your math instruction.
Cathy Humphreys: One of the things that we did after we wrote Making Number Talks Matter, is we started watching lots of number talks. And we watched our own number talks and we watched number talks that are online, and we started to see some patterns of things that made us wonder what was going on. We were thinking about equitable participation, for example. How hard it is for teachers to do that. We especially noticed teachers and students both being uncomfortable around making mistakes, and teachers feeling the need to validate students' answers and not feeling comfortable not doing that. And so we wanted to make a new book with some video to address some of the issues that we were noticing.
Ruth Parker: We also saw kids who were excited about sharing their ideas, but rarely were they interacting with other ideas that came from other students. So there was kind of a flatness that we were seeing and feeling. I have to say both in my own work that I was doing and in what we were watching in classrooms and watching on videos. Was hard book to write because we were having to deal with, so what's troubling us here, and what can we do about it? And it was like, with Making Number Talks Matter, we were writing about things we'd already understood and with this book we were trying to figure things out. So we'd have ideas and we'd think, "Well that's pretty good idea." And then we'd have to go try it out with kids or with teachers to see if it was a good idea or not. So we really learned a lot in the process of writing the book.
Cathy Humphreys: We have a chapter on mistakes, and what we wanted to do is help teachers see how to deal with some of those mistakes. And it's hard to just talk about that. So for example, we have a middle school teacher who has a student explaining his reasoning, and he makes a mistake. She doesn't bat an eye, she just waits and keeps recording and then he self-corrects. He figures out, oh, he did it wrong. He does it all by himself. She doesn't say anything about it. She just keeps right on going. So the focus on the teacher as a listener and someone to record what kids are doing, but to downplay every little mistake that might happen. So that's one example that's another one.
Ruth Parker: Well, one of the things people who watch the videos is going to see that was hard for both Cathy and I to say, "Okay, let's put it in the book," is our own mistakes that we did in number talks. But we thought if we're asking teachers to acknowledge that mistakes are going to happen and they're natural and unavoidable, we should probably be using ourselves as subjects. We wanted to be pretty careful that we weren't using video tapes of teachers that were critical of teachers and we found plenty to be critical about in our own work.
Cathy Humphreys: That's true.
Ruth Parker: Video tapes were our own work.
Cathy Humphreys: That's true. Yeah. The whole first chapter, as a matter of fact, is around one number talk that in high school where this student named Frosty advocates for his own mathematical error, wanting to know what's wrong with it or why it's wrong or if it's wrong, and how I had to navigate that really tricky territory of just what to do. And so I was unsure all the way through that number talk about what step I should take. And that uncomfortableness I write about, because every step I take is based on some beliefs I have, but I never know what's going to happen next. So that was also an example where I feel like I made some moves that I wish I wouldn't have, which I talk about in the chapter as well. Some other things I'm happy that I did, but it just shows that there's no perfect number talk and that what number talks are about thinking, listening to kids, and always keeping an eye on getting students' thinking to the forefront rather than my thinking.
Ruth Parker: We realized in writing this book that the teaching practices that are part of number talks, kind of turn on its head what a lot of teachers have come to believe is good teaching. So we're putting them in a place where their struggles are really understandable because we're asking them to change in some pretty deep ways, in terms of how they're interacting with students. Leaving wrong answers unaddressed or on the board at the end of a number talk is really hard for some teachers. So they want to do things like have the class agree on what the right answer is before they close down a number talk, which in our way of thinking, that just pulls the responsibility for the thinking right back to the teacher when we're really trying through number talks, to turn it over to students more. But that's one of the things that teachers told us they really struggled with.
Cathy Humphreys: We as teachers have learned that our job is to show kids how to do things correctly, to validate right and wrong answers, and to be the one that sort of arbitrates what's right and wrong. And so of course teachers have trouble when we're asking them not to do that. We now are not validating right answers. We're accepting their ideas, asking them to explain how they're thinking without any judgment. Hopefully without any judgment. Without any evaluative comments whatsoever. Not even things like, "Great, good job," things like that. Because what we're wanting to do is keep the thinking right on the student. And not to validate answers is hard. It's hard, it's hard, it's hard. Because that's what we, as teachers, many of us, learned in school that the teacher's job is. And that's what we've been trained in teacher school to do. And so what we're doing, it's a major shift in our conception of what it means to teach. And it's really more like teaching as listening than teaching as showing.
Ruth Parker: We focus a lot on wait time in the book. We have a chapter on it. And we watched a lot of videos trying to figure out what's what, before we realized that we weren't really using something we've known as really important for a long time and that's wait time. So we actually added ... we were all pretty good and most the teachers that we saw were pretty good at doing the first wait time, when we ask kids to think about a problem and we give them time to think about it. But there were some other wait times that we were missing that we really had to make a lot more explicit. For example, asking if anyone's willing to share their thinking. Well if we don't have a pretty nice wait time there, the same few students are the ones who are quickly ready to share their thinking.
Ruth Parker: So into the new, I want to say protocol, but routine that we wrote. We wrote things like, "You might want to say to kid something like, 'You're probably wondering why I might've even made eye contact with you or seen your hand up and haven't called on you.'" And the reason why is we're trying to give more kids the time to really decide, do they have something they want to share, and to get the courage to share it. So we built in several new wait times and tried to say it's important to tell your students while you're doing this, what you're doing and why.
Cathy Humphreys: Education words are so problematic and nudging has come into the common language now in all kinds of different ways, as so many words in education do. And what we saw as nudging was, as the chapter's called Nudging for Strategic Thinking, nudging in this book and in our thinking is not asking questions. It's not trying to push or gently guide students to any particular place, but rather our nudging strategies, I guess you'd call them, are particular problem choices that can help kids open up the world of their own thinking and so that they have ways of figuring out problems without being shown. So that's our nudging. It has nothing to do with leading questions or guiding kids to a particular place. What it has to do with, is opening up kids' thinking based on their own ideas.
Ruth Parker: Getting them to begin to think flexibly.
Cathy Humphreys: Yeah.
Ruth Parker: So the strategies are really purposeful around that.
Cathy Humphreys: Yeah.
Ruth Parker: One of them is designed basically to take the standard algorithm off the table to begin with, and say to kids, "That's really not what number talks are about." They're about you having your own mathematical ideas and posing problems in a way where it's accessible for them to find different ways to pull those apart and put them back together.
Cathy Humphreys: Our nudging chapter includes strategies for opening up kids' thinking and letting them rely on their own ideas. One of them is to pose a really simple problem like seven plus eight and say, "Pretend like you don't know how to do this. If you didn't know what to do, if you didn't know what the answer was, what might you do to figure it out?" And interestingly, lots of different strategies come up whether we've tried this with kids and with adults, and a variety of strategies can come out with a problem like that.
Ruth Parker: And then the second one is really, so if you hadn't been taught how to solve this, how could you solve it? We didn't before and it seemed like a lot of teachers, a lot of kids, would use the standard algorithm. Sort of like, what do we do with it now? We don't really want to honor it because we want to move away from it. But it felt like a, "Gotcha," to kids to say, "Well that's not really what we're interested in." So we just thought, if we start with the nudging and say here's the way we were taught. But that's really not what number talks are about, we can get rid of that problem.
Cathy Humphreys: One of the problems with the traditional algorithm is most kids have been taught how to do it, but don't really understand why it makes sense. And so if you were to use a standard algorithm for multiplication, and I said, "So why does that make sense?" "Because that's what I was taught. That's what I'm supposed to do. They told me to do this," or "My teacher said to do this and that's what I've done." Sometimes kids do understand the algorithm, but in general, what we're wanting is kids to make sense, break the numbers apart, make sense of the quantities involved, rather than follow a procedure that they remember. So it's the idea between thinking and figuring out based on number of properties and operations and remembering.
Ruth Parker: There a couple of other problems with number talks. You're working mentally all the time and once the problem gets at all big, the standard algorithm is usually the most cumbersome way to mentally work through a problem. And kids have a hard time giving up that algorithm. That's what they've learned sometimes for years before they start with number talks. That's what they've learned real math is about. So they tend to cling to it and not be able to get answers. Where kids who aren't clinging to those algorithms are happily taking the numbers apart and putting them back together. We want number talks to be all about sense-making. And for most kids, the standard algorithm has been recipes that they've memorized without having any idea why. The arithmetic properties undergird so much of mathematics, but they're obscured by the standard algorithm. Almost always they're obscured by the paper and pencil algorithms. And the processes that that kids invent for adding, subtracting, multiplying, dividing often reveal those arithmetic properties.
Ruth Parker: So it feels like number talks not only get us farther in terms of kids coming to believe that they have mathematical ideas worth considering and that they can learn from ideas that other kids come up with if they're intrigued by those ideas. But they also get us farther ahead in terms of building the mathematical foundation that's so important for any kids, who, as they work their way through mathematics. And I feel like math very quickly by middle school on, if not before, becomes a big bugaboo for a lot of kids because they just can't hold on to all those recipes anymore. And kids that are inventing something that makes sense for them and using it repeatedly in number talks, there's not a forgetting process because they understand the relationships involved.
Cathy Humphreys: Seventh grade teachers will say, "Why are we doing number talks with subtraction of whole numbers? Why don't we do it to match what we're teaching?" And our thinking about that is, I want to build on what Ruth said, that we're building foundations for their learning. The properties that are involved in subtraction of whole numbers are the same properties that are involved in the subtraction of fractions, of integers, of polynomials. And the better a student can understand and build on those foundations, the more fluidly they'll be able to move through the higher levels of math. So the idea is building a strong foundation of understanding, which many of our kids lack.
Ruth Parker: There's the added problem of, if we don't do number talks with whole numbers and start with mentally working with fractions and decimals, that puts the mathematics out of reach of a whole lot more of our students, when it comes to making sense of the problems. I think it's really empowering when kids realize that they can make sense of mathematics in their own ways and that it comes down to a sense-making. They have to relearn, and trust that we really want them to make sense because they, again, for years sometimes trust that we want them to do it just the way we taught them to do the problem.
Cathy Humphreys: Yeah, actually that's right. They need to come to believe that they can make sense, that they don't need to have someone tell them how to solve every problem, that they can figure things out. That's a huge, powerful idea for every student to have and every student can have it, we think.
Ruth Parker: We're doing a statewide project in Washington that's kind of relevant and really interesting. And we have elementary through university teachers of math together in the project. And one of the things we wanted to figure out is, would taking a deep dive into number talks help transform all of mathematics time? And we're halfway through it, but we're not sure. Because the puzzle for me is a lot of teachers have programs that they're supposed to be learning to teach that are very much recipe-bound. And if a teacher is trying to do something with number talks in 15 minutes, but the rest of math class is really about, "We fix mistakes, we want you to do it this way now." We don't know. Can teachers or students, what does it take to get ... can they live in those two worlds? We don't know yet.
Ruth Parker: And we certainly know that there are cultures that exist in the classroom that allow teachers to take to number talks right away because it just feels really supportive of what they've been trying to create in their classrooms anyway. And there are other cultures in the classroom and it's not just classroom. In the schools or in the districts where teachers feel like they're being asked to do one thing and they don't feel like they have permission to do what we're asking them to do with number talks.
Ruth Parker: So some teachers are liberated by it. Even some teachers who are fragile and working in programs that aren't aligned are liberated by it because they're understanding some of the math for the first time themselves, but it's pretty tough to buck a culture if you don't have support of colleagues or administrators or parents. And this is a deep change we're talking about. Deep changes.
Cathy Humphreys: I think it can work both ways. I think that we start number talks in a culture that is not necessarily friendly, but it can transform how a teacher teaches and how students learn. Teachers are learning a new way to teach and kids are learning a new way to learn and new responsibilities.
Ruth Parker: Empowering kids to believe that math makes sense and not being content when it doesn't. But I've become so convinced that it's a deep journey and a long time journey to try to figure out how to optimize number talks and kids, and kids talking to kids. That part of my motivation is, I don't want myself or other teachers to rest content doing number talks at kind of a surface level and thinking, "Oh, wasn't that fun?" Because they're really powerful and I think they're powerful for all kids. I think they're powerful for language learners. I think they're powerful for groups of disenfranchised kids. I just want folks to understand that there is a lot to try to get to here.
Ruth Parker: I think number talks are wonderfully forgiving of both teachers and of students as they're trying to negotiate this new culture in the classroom. So there have been many, many times that I've made mistakes and thought, "Phew, I'm glad that didn't matter all that much," because we're in here trying to figure out something new. So there are challenges and there are also many delightful things that happen along the way too. And that's-
Cathy Humphreys: I think one thing that gets lost along the way is that no number talk is a lesson in and of itself. That because it's such a short amount of time and because just a few students talk during that time, the power of number talks is something that emerges when they're done regularly over time. The power of number talks emerges through their use over time as the teacher starts to really watch and listen to the students, what they're understanding, what they're not, helping students interact with each other.
Ruth Parker: I still think that doing them on consecutive days is really important. If you have a limited number of days that you can devote to it. Three days, let's say three days, and then doing a Monday, Tuesday, and Wednesday or Wednesday, Thursday, and Friday is a lot more powerful than Monday, Wednesday, and Friday. Because of that idea of kids having a new idea and wanting to test out that idea and see if it's going to work or wanting to try something they heard from somebody else. It's really hard for kids to remember something that happened two days ago or three days ago. So having number talks frequently enough that they really get to test out and build on ideas becomes really important.
Cathy Humphreys: So we do want all students to participate during number talks, but because number talks are so short, only a few students are going to be able to participate at any given number talk. This is why doing number talks regularly over time is so important. But the second important idea is that we don't believe that you have to be talking to be participating. A lot of students really do think and learn from listening to others. So we're not worried about having everybody participate every time.
Ruth Parker: Yeah, we stay pretty true to the notion that kids are in charge always during number talks. I'd like to say always during math, but we'll keep this to number talks, on deciding if and when and how to bring their own voice to the public space. And I think for some kids who have been injured by math or who are shy or for a lot of different reasons, it takes a long time to, first of all, come to believe that you can have math ideas. And then get up the courage to share it publicly. Building that trust, it's a long process and it's a really long process for kids. Kind of what happens often is that students who don't talk in class will come up to the teacher some other time and say, "Can I tell you what I did with that number talk?" So they test out their thinking in a really one-on-one situation or a small situation before they're ready to share in the large group.
Ruth Parker: But I think the beauty of number talks though is whenever a student does feel ready, they're there and they're welcoming and it's a nonthreatening environment. And mistakes have become so naturalized that it's not something to flee from or to feel bad about.
Cathy Humphreys: Interestingly, one of the hardest kinds of number talks to do is those dot talks. So that's the one of the places where video was so powerful because Ruth did a dot talk with fourth-graders, I believe. And she was able to isolate the different recording moves that she uses and talk then in the book about why using that kind of recording is so important. Because everything that we've suggested in Making Number Talks Matter, there's a reason for, but in this book we were able to dig more deeply into, why are we really doing it that way? And how could we be more helpful to teachers? Not to provide an algorithm for them to do number talks, but to help them see the reasoning behind some of the moves that we have. Because number talks are widely called math talks. They're being used in a whole bunch of different, I guess, iterations. And what we're trying to do is talk about why the particular kinds of moves that we make as teachers and the choices that we make, why we make them. And that's what this book is about.
Cathy Humphreys: I think you need a buddy when you're doing a new practice like this. Ideally you have somebody that's doing the same number talks together and then you can go talk to each other in the hall. There's not a lot of time for teachers, teachers don't have extra time for planning, but if you've a number talk study buddy and you can share it, it'll help you through the times where you get discouraged, you don't know what to do, you need ideas. And so I think the first book can help you with ideas of what number talks to do. The second book helps maybe a little bit more with the particulars of why we do certain things and how to keep number talks going for example, or how to help kids have new strategies for doing problems.
Ruth Parker: In the best of all worlds, teachers would be doing number talks at back to school night so that their parents were on board also. Because I think it's really hard for a typical parent to support something they don't understand. And yet I've worked literally with tens of thousands of parents around the country from very privileged communities to very impoverished communities and given an opportunity to learn, almost universally, parents say, "Gosh, I wish I'd learned this way," and they want it happening in their kids' classrooms. So teachers, don't be afraid of parents. There'll be your strong advocates if you give them opportunities to learn.
Nate Butler: Making Number Talks Matter and Digging Deeper: Making Number Talks Matter Even More are available from stenhouse.com. We hope you enjoyed this episode of Teacher's Corner. Send your questions and comments to the Stenhouse team at membersatstenhouse.com. Thanks for listening.