Today on the Heinemann Podcast… think back to when you were first taught math. Did you feel engaged? Did you feel you had a deep understanding of the material in front of you? Or, like many students, did you memorize and regurgitate facts that you struggled to explain the meaning of?
Authors Patsy Kanter and Steve Leinwand want to change that narrative.
In their new book, Developing Numerical Fluency, Patsy and Steve present what they call pivotal understandings, and put an emphasis on a school-wide culture that values and nurtures numerical fluency.
We begin today’s conversation by asking, what exactly is numerical fluency?
Below is a full transcript of the conversation.
Josh: So one of the things was when titled this book, both of you were really clear that you wanted numerical fluency to be the focus. So just talk a little bit about what numerical fluency is, and why it's such an important topic for elementary teachers right now.
Steve: We both have been working in lots and lots of schools before the Common Core and in the Common Core era. And we just see far too many kids being hurt by traditional instruction that puts memorization and speed ahead of strategies and understanding.
And so we were driving together somewhere talking about it, and Patsy says, "Well, we really ought to write a book. We ought to develop a roadmap on making this shift towards strategies and understanding, and away from memorization and speed." And at about the same time, I had been working on the NCTM Principles to Actions book that I had the honor of being the head of the writing committee for, and we wrote there.
And we also have on page ix in Developing Numerical Fluency this wonderful quote that says, "Fluency is not a simple idea. Being fluent means that students are able to choose flexibly among methods and strategies to solve contextual and mathematical problems. They understand and are able to explain their approaches, and they are able to produce accurate answers efficiently." We read that and we said, that's wonderful. We agree with every one of those words. But how do we help teachers do it?
Patsy: I think it's always been important to have numerical fluency in elementary school, because that's where students start. And I think that it keeps kids grounded and engaged in mathematics if from the beginning, they're involved in thinking numerically, and not just, as Steve said, memorizing.
Steve: So this notion that Patsy talks about of hooking the kids early, of setting the stage ... I've always been struck by the fact that number talks, done well, are safe places for teachers to experiment and focus on understanding.
I think that in our book and in our way of thinking, when you look at our chapter on process or the processes, and when you see how these processes play out throughout the book, it's the same thing. It's like the Trojan horse. It's like the lost leader. Those processes apply for all the mathematics we teach.
But it begins with the numerical fluency where we talk about contextualizing. You've got to put it in context. We call it storytelling. You've got to have kids physically constructing things. We call it building. And so, again, we know that this applies for all mathematics. But we think that in highlighting these processes early on, and using them in kindergarten and the first grade and second grade, so that all kids say, I can do math. I'm smart in math.
The process of representing, both graphically and symbolically, we call that drawing and using symbols. Our attempt here is to use that process, and then visualizing, verbalizing, and justifying are part and parcel of what, in so many great ways, Patsy has brought to this book. This idea of, you have to see it. Then you can describe it, and then you discuss why, is really the essence of moving simply from regurgitation and minimal understanding to a depth of understanding.
And so we've taken all those ideas and applied them to this notion of fluency over one-digit facts, addition, subtraction, multiplication, division, and then computation work. The power and importance of place value understandings. That's what has excited both of us about both the idea of developing numerical fluency and building this foundation, and doing it with processes that are as important here as everywhere else in mathematics.
Josh: One of the things that is interesting is you spend time in the beginning of the book where you talk about what fluency is. But you also take the step to talk about what fluency isn't. In the introduction, you say, "We often hear a range of definitions that neither we nor the research support."
Steve: I think that what we've learned, by looking at Principles to Actions, again, is the importance of examples and counterexamples. We know that applies to teaching, but it also applies to professional growth.
And so I think in Principles to Actions, some of the most powerful work is in the tables that say, teachers do, teachers do not do. Students do, students do not do. So that's what this is built on. And so it just helps be clear that you may not know what fluency is or what we believe fluency is if you still harbor some of these ... what we think are misguided notions.
Patsy: Also, in talking to teachers, these are the most frequent responses they give us when we ask what fluency is. So we know that we've taken them and turned them around of what it is not.
This instantaneous recall is such a misnomer, because many children can instantaneously recall them and do nothing with the facts. On the other hand, you have a child who takes a few seconds because they have a strategy, and they can go on from there.
Steve: So that means ... The work that exemplifies this book is what great teachers have been doing all along. We understand that when we simply expect kids to memorize eight plus nine, and that nine plus eight is a different thing to be memorized, that we are hurting kids. We are depriving them of the critical connections and linkages to mathematics, and we are telling them that mathematics is memorization, and it is about speed rather than the kind of instruction that we advocate, that we talk about with the activities in this book, and that we know are part and parcel of the best classrooms that we're in.
It is absurd to think that 100% of humanity, simply with practice, can remember that their brains automatically convert six times seven or seven times six to 42, when we know that for some kids, getting there is a journey. That is the kind of understanding that, to us, constitutes fluency. And when you have a classroom and have discussions about, how did you get it? How did you get it? What strategy did you use, what strategy did you use? Then you're building the kind of community of learners that we value so, so much.
Josh: One of the things I've heard both of you talk about is that incorrect answers can be just as valuable as correct ones. You write about that in the book. So why is that, and why should we value incorrect answers so much.
Patsy: Well, I think that incorrect answers ... First of all, they provide an avenue for children on either side to participate in. For example, if a child thinks they know the answer and they don't, but it's acceptable to say it out loud, then more kids are willing to do that.
Also, when you point out why six and seven is not 12, then you're practicing other skills in an effort to help the child get there. So I know that six and six is 12, not six and seven. That would be an odd and an even. So you're getting to point out some other mathematical relationships that you wouldn't get to point out.
Steve: This idea of mistakes is also what we stand on the shoulders of our colleagues ... In this particular case, Jo [Bowler 00:08:01], who wrote a beautiful, beautiful blurb about this book that's we're so appreciative of. But Jo and her work on mindsets and on mistakes is just so important and so powerful, and not enough teachers fully internalize the idea that mistakes grow your brain. That if all you do is regurgitate answers on a [inaudible 00:08:23], you are not processing. You are not making neural connections. Your synapses are not firing.
Great teachers understand that mistakes are, more than anything, a learning opportunity, just like Patsy says.
Patsy: My grandson came to me one day and said, I have to make 100 on my speed test today. And I said, why are you going to school? He said, what do you mean? I said, well, why do you need to go to school to make 100 on a speed test? You won't learn anything new. You'll just go and tell what you already know. So I think that teachers don't understand that children that never make mistakes aren't learning.
Steve: That's my coauthor, and subversive grandma.
Josh: It is an interesting thing, though. So how do you think a book like this, other people's work, conversations in schools ... How does that culture start to change, where maybe teachers, the culture and the school, is very much focused on fluency, like you just said, Patsy?
Patsy: I think the way that you do it is you get some teachers involved in meaningful problems, helping them see that if kids are deeply involved and really digging in, that the math just unfolds beautifully, and if they have some numerical thinking, they're even better off. And you continue to share rich problems and experiences with teachers so they feel that they can share them.
And as long as the activities of the classroom are meaningful and produce learning, they're okay. But when we get stuck on boring drab, we're not getting anywhere. The culture's greatly enriched in the classroom when teachers talk about problems themselves.
Steve: So Chapter 11 speaks exactly to this notion of culture. We know all about "it takes a village". Most of the important things in society cannot be done alone. And Chapter 11 is where we talk about, it takes a school to develop numerical fluency.
And so it certainly applies to more than just numerical fluency, but the key elements of better use of meetings. Grade level meetings, grade band meetings, faculty meetings. To use videos, to have models, that teachers model the behavior. To turn a faculty into a class, for a teacher to be able to show what it looks like and what they do. To really build the teacher collaboration. Those are the kinds of strategies that strengthen a school regardless of what we're talking about, but certainly when you have to change mindsets the way we are talking about, change practices the way we are talking about ... Without meetings, collaboration, sharing, and then the last page is administrative support.
Teachers are so busy. Teachers have so many demands upon them that somebody's got to intercede and say, wait a second. This is one of our priorities. It is no longer acceptable to have kids say, I'm dumb in math because I don't know my facts. I'm dumb in math because I cannot find a sum or a difference. And that's a school-wide culture that builds off of transferring some of our passion into what goes on in a school.
Patsy: I definitely agree with Steve. I think that school-wide passion for mathematics is really important, and sometimes it takes one person in a building heralding it. But if you don't, you will have pockets of numerical thinking, but you won't have classrooms that are fluent with it.
Steve: We're talking very generally about what we believe ... At the heart of this book are the six chapters, three on developing addition subtraction fluency, and three on developing multiplication division fluency, that really run in parallel. And the meat of this is the discussions about building a fact fluency in coherent ways. Recognizing that all mathematical understanding begins with this critical notion of part and whole. You cannot solve problems, you can't solve addition subtraction problems, if it's not clear to you that you have part, part whole. That is foundational for them, two years later, turning to factor, factor, product.
No way that students will be able to, with fluency, divide or solve a problem involving division, if they don't have that depth of understanding that they've got the total and they've got one of the factors. Now they have to go forth and find the other. We spend a lot of time on different strategies for developing addition subtraction facts, and then multiplication division facts, and spend a great deal of time on helping teachers see that a kid who says, I don't know their facts, a teacher who says the student doesn't know their facts, is often only talking about 10 out of 100 facts.
And then finally, in both of those sections of the book, we say, there is as much reason to use alternative algorithms as opposed to the traditional algorithm, or we make the case that this is one of the few places where we are in disagreement with the Common Core, and we do not believe that there is the standard algorithm. We think that there are several standard algorithms. We hone in on that and very, very strategically give teachers permission to ignore what is not testable ... That is, this notion of the standard algorithm ... and to help kids who need to see it and do it and process it differently.
Again, that theme that runs throughout, that started with Patsy's deep belief that we've got to differentiate. We've got to recognize that kids are all different. Their brains are different and our instruction needs to accommodate to that.
Josh: In the middle of this book, there's a lot of information about developing addition subtraction fluency, and developing multiplication and division fluency. But really, one of the big ideas about this book is, we have to rethink the way that fluency instruction is done in schools throughout the country.
So what is it about this rethinking that's so critical, and then where are some places that you've seen this done really, really well?
Patsy: I've seen fluency taught really well in places where mathematics is valued throughout the school. And every teacher, every day, wants to fit a piece into that puzzle by doing a number talk, playing a game, having kids make their own manipulatives. Doing something that contributes to their numerical growth.
And I think those are the places where there's been the most energy, where kids hear you talking about math and they run over to see what you're talking about. Where kids are always seeing counting objects, and there's just this feeling that math really matters, and you can see it by the materials in the classroom, by the attitude of the teachers, by their bulletin boards, by the discussions that are going on.
Steve: Patsy has beautifully summarized Chapter 3, where we talk about the structures that we see in schools that recognize that numerical fluency is part of ongoing instruction, that it's not just the fact chapter. It's not just the multiplication chapter. But all of the work with number gets developed through number talks, through games, through problem solving activities, and student made materials. Those are some of the tools, and they are always available and always used as an ongoing basis to develop and strengthen students' numerical fluency over the course of the year.
Josh: As school starts back up, what would say is something that a teacher could start thinking about? If they're saying, I want to take a fresh look at fluency instruction in my classroom, I want to take a fresh look at the culture of fluency that I'm building in my classroom, what's something that a teacher could start thinking about tomorrow?
Steve: Well, first of all, I try not to put any of those beginning of the year responsibilities on an individual teacher, because teachers have got just so many things on their plate. This is where we need a teacher leader. This is where we need a principal working with a teacher leader, or this is where we need the coach. Again, it goes back to a school. We know that when it happens randomly, incoherently, and in a couple of classrooms, it's not a school-wide focus so the kids don't get the same benefits.
Josh: So what would you say is one of the things that you are the most proud of in this book, one of the things that you're the most excited for people to read and think about?
Steve: I really love Chapter 1. I think Chapter 1, which are the nine pivotal understandings for numerical fluency ... This whole book began with a workshop that Patsy and I gave almost three and a half years ago at NCSM. But these nine pivotal understandings ... If teachers and if students don't eventually understand that everything begins with counting, if they don't understand that all quantities are composed of parts and wholes, that the language of operations, these ideas of joining and separating and equal groups and groups of equal size ... It's language and pictures that become so key.
And I'm not going to go through all of them, but I think that if people were just to copy, discuss, those nine pivotal understandings, which ones do they accept, which ones do they already do, which ones do they need to focus on ... and that really became the way the school looks at this large domain of developing numerical fluency, we'd move things giant steps forward.
And then the last one, again, is place value. Place value is essential. I am forever amazed how much expanded notation is done in September, never to be revisited. And yet most of estimation and all of an understanding of the algorithms is based upon place value. The separation is deadly. So that's where I go, and that's perhaps what I'm proudest of.
Patsy: I, too, like Steve, am very proud of this list. One of the ones I have found myself so excited about is understanding the role of properties in elementary math, because I always thought that properties were what you learned when you got to high school. And yet they're so innate to all this numerical understanding that I think for teachers to become comfortable with properties and see how they're used and not used will be very insightful for them. And if they don't shy away from them and don't make them a list of words to memorize, it will be very helpful for the kids.
Josh: So you've got these nine pivotal understandings for numerical fluency. The chapter that follows that talks about six processes that can be used to help develop numerical fluency. But one of the things that you both talk about in the introduction is that there is a challenge here, that a lot of teachers ... This is a shift for them. What's hard about that for teachers, and how would you coach them or encourage them, or leaders, to help move beyond that? This is different than maybe the way you learned some of these things.
Steve: It goes back to things I have written since 2000 in Sensible Math, that people cannot do what they can't envision. They don't do what they don't understand and don't believe. They can't do well what they don't practice and get feedback on. And it is unlikely to be sustainable if it's not done collaboratively.
And so I think that the message here has got to be that simply mandating this is not going to work. People need to be given the opportunity to envision this kind of instruction in action. I am so heartened by the increased use of video. That every day, there are more and better videos available for professional development with discussion guides. People can't do this unless they practice it. It's why I spend so much of my life advocating for coaches and for coaching, because this is hard and it's strange and we start to get squirmy when a kid doesn't immediately say, seven times three is 21, and has to think about 14, and therefore, 21. I just did it in two seconds, what's wrong with that? That's fluency.
And then I ask, how did you get it? What did you think? That's a matter of being exposed to these things, and then within a grade level and within a school, recognizing that that's just the way we do things around here.
Patsy: I have a lot of people ask me how I feel about "new math", and I keep telling them over and over, there is no new math. We're just trying to help children understand what we never understood. That math makes sense, that it's reasoning, that children can all do it. They can do much more than they think they can and much more than we can do if they're given the tools to guide themselves through it.
And I think that this notion that math is new or different is what we have to wipe out. I remember when I was in Italy and I was in a church and I looked down on the floor, the floor was made of all the pattern blocks that we use in our classrooms. But the church dated back to BC. So I took a picture of the floor and said, math's been around for a long time.
And this is not new. It's just making really good sense of it.
Patsy Kanter is an author, teacher, and international math consultant. She worked as the Lower School Math Coordinator and Assistant Principal at Isidore Newman School in New Orleans, Louisiana, for 13 years. Patsy is the co-author of Every Day Counts: Calendar Math and a consulting author for Math in Focus.
Follow Patsy on Twitter @patsykanter
Steve Leinwand is the author of the bestselling Heinemann title Accessible Mathematics: Ten Instructional Shifts That Raise Student Achievement.He is Principal Research Analyst at the American Institutes for Research in Washington, D.C., where he supports a range of mathematics education initiatives and research. Steve served as Mathematics Supervisor in the Connecticut Department of Education for twenty-two years and is a former president of the National Council of Supervisors of Mathematics.
Follow Steve on Twitter @steve_leinwand