Marilyn: I’ve been thinking about that question for a long time. In some ways there are similarities in that in both of the areas, in reading and math, we’re trying to help children learn to bring meaning to black marks on paper. And so I’m astonished that when you read things, that literally black marks on paper will evoke huge emotional responses in me and hopefully in kids too. I’d like to think the same in math without it being just negative responses.
But I think there are differences as well. Read is a verb. We want kids to read, but we don’t want kids to math. Reading requires the ability to decode, after which you have access to all the books and the library in the world. I could even read Spanish. I don’t always understand, so the comprehension part is so important. So in reading, it’s accuracy and fluency and comprehension. And in math it’s similar. We want kids to be accurate. We want them to be fluent, and we want them to understand. And in math world we don’t use the word comprehension. We use the word understanding. But basically I think of them as the same thing.
But I think there’s a difference, and I’m not sure about this. In reading, the ability to decode depends on learning the system. Learning what the sounds are, having some skills to attack words, having ability with sight words. And in a way that’s kind of social convention learning. You have to be taught that from an outside source. We’re not born with the ability to decode words on pages.
But in math it’s all based on logical structures. Kids come to school knowing that one and one is two. They learn the counting part. Those are words, but young kids count in kind of weird ways, but they have an understanding of some of the mathematical relationships that they’ve encountered. They sometimes they have wrong ideas like my half is bigger than your half. But the ideas in math are rooted in logic. So the logic and the learning about the symbols and the equations, the black marks, are kind of meshed together.
And I think in reading, I think kids who can read even accurately and fluently have no idea what they’re reading. So they’re similar and they’re different. I think the people who are devoted to thinking hard about math and people who are devoted to thinking hard about reading, we can learn from each other. I’m not sure we always do.
Brett: Oh absolutely. Well with comprehension in mind, if a student in math has memorized math facts, how can that be misleading to a student’s proficiency?
Marilyn: We all memorized our times tables, and you really need to know them from memory if you want access to them. It’s really laborious to say, okay, 7×6, let’s figure out, 6×6, whatever you have to go. You need to know 7×6=42 and it helps you reason mentally with numbers. It helps you make estimates. It’s a part of what you need to know. But without understanding, that kind of memorizing isn’t going to serve a child very well.
So I suppose the same is true in reading. When you memorize the sight words and you’ll have had skills to get them, and you want that to be as fast as possible so you have more fluency, but the difference in math, it’s different somehow.
It’s kind of hard to make sense of it, but I think that the understanding is always … The logical underpinnings are always there in all aspects of math. So for kids learning how to write numbers and count without reasoning, it’s not going to work. It doesn’t work for them.
Brett: You had a couple of tweets go viral recently, and I’m going to paraphrase the two of them together here, but essentially one of them said, “Correct answers can hide confusion, while incorrect answers can mask understanding.” What is happening there?
Marilyn: What I have seen on paper and pencil assignments and quizzes are correct answers that when I talk to a student about what they’re doing, it doesn’t seem to be any understanding. It’s kind of like the dumb luck approach. I Interviewed a boy about comparing fractions and he said, “One-fourth and five-sixths,” he said, “Without question, five-sixths is greater.” That was my question, which is greater. Now if that was on a worksheet where it would have said circle the greater fraction, he would have circled five-sixths. I wouldn’t have learned anything. Then I said to him, “Well, how’d you decide?” He says, “Well, one is far away from four, but five is real close to six.” And I’m thinking like, “Whoa. That was interesting.” What I didn’t ask him as a followup, which I kind of wish I did, what if I had asked him about five-sixths and six-sevenths? They’re each one away. How would he figure out which was the greater one there? There’s always a way to keep probing and pushing, but I would have assumed from the worksheet that Michael understood comparing fractions, and I would have been woefully misinformed.
Brett: And that really gets to the power of interviewing. Why is interviewing so important?
Marilyn: Interviewing is important because I really get a window into how a child is thinking. I sometimes say to kids, when I’m interviewing them, “I’m going to ask you to explain your reasoning. If I could, I would like to open up your head and look inside and say, ‘Oh, that’s how you’re thinking,’ but I can’t. So you’ll have to tell me. So I’m really interested because how you explain your thinking will help me understand you better and that’ll make me a better teacher. So let’s go.”
So the interviewing is always about how did you figure that out? Even when a kid is right or especially when the student is right. I think in classroom, my early teaching you would ask a question and a child would give an answer and then you’d say, “Are you sure about that?” Or, “Would you say that again?” It’s the hint like, “You’ve got it wrong kid. Go back to work.” So I’ve shifted that in interviewing where we ask the kids how they think, whether they’re right or wrong. We don’t ever give any feedback about whether they’re right and wrong. We could talk about why that’s important.
But it’s a question of hearing how the student is making sense. Math is all about making sense. And I think that too often students think that math is about following the rules.
Brett: Well and you take in your interviewing process, you take the emphasis off of getting the right answer by focusing on thinking.
Marilyn: Well, yes, but I’m careful about that because the right answers are important. So sometimes people say, “Well, the answers aren’t important. It’s more important how they think.” No, the answer is important. Well, what the answers are, are important because sometimes we ask kids for an estimate or sometimes you have to be accurate. I think knowing when you have to be accurate and when an estimate would suffice is part of learning. But whatever the question I’m asking, the answer that I get is my first indication into what a student knows, but I don’t stop there. I’ve got to find out what the student really knows. So the answers are important, but they’re not the Holy Grail. I’m not looking just for the answer. I’m looking for answers and explanations. Like in reading, I’m looking at can a child read a passage accurately and fluently and can they understand what they read?
Brett: When we think about interviewing, someone might compare it to a running record, they might compare it to conferring. Interviewing is different.
Marilyn: I think it’s similar and different in both of those situations. Now I don’t have a lot of experience with running records, but I did go and spend some time with teachers to learn more about running records when we were working on the math interviews. So in running records you are checking for accuracy. So are we in our interviews. We are checking for fluency. So are we in our interviews. And you’re checking for comprehension, but the comprehension is related to the specific passage that the student has read. So you have the questions you’re going to ask.
In the interviews with math, I think it’s a little more open. When I say like I did to Michael, “Which is greater, one-fourth or five-sixths,” then I have to say, “How did you figure that out?” Or, “How did you decide,” or, “How did you know?” And I don’t know what the answer is. With the running records, you kind of know what you’re listening for. I want to not pay attention to what I hope he would say and listen to what he might say. So it’s similar but I think it may be subtle, but I think it’s a profound difference.
Brett: One of the things that’s very special about your interview style, you stay in the silence. Why is that so important?
Marilyn: I learned about wait time years ago, and it affected my teaching amazingly. Ask a question and then shut up. And I think as teachers we talk, we talk, we talk. And so if I stop and give kids time to collect their thoughts, then they have time to collect their thoughts. Otherwise, I’m the one who’s doing all the learning because I’m doing all the talking. Whoever is doing the talking is probably doing more learning than the person who’s listening because you’re working it out in your head.
So I’ve learned in my classroom teaching about wait time. I’ve learned about ways in the classroom to make that wait time productive time when kids are thinking. And so it seems natural in an interview situation, especially when you watch the kid … You’re sitting face to face. You can watch them thinking. Their faces are scrunched up or they’re kind of subvocalizing or they’re tapping. Whatever they’re doing, as long as the kid’s involved, why would I interrupt the child?
It’s fascinating. In the moment it’s not hard to do because you’re sort of curious about what’s this kid going to come up with. But sometimes I rewatched interviews on video tape and then we have plenty of opportunities for teachers to see them. When a child sits for 20, 22 seconds, that is a long time. That’s the amount of time we’re supposed to be washing our hands that we’re worried about the coronavirus. That’s the time that I learned yesterday is how long it takes to sing the Happy Birthday song twice. Let me tell you, that’s a long time to sit because I was interested in showing teachers some of these examples of wait time and I’m thinking, “Well that would be the most boring professional learning session. It’s like watching paint dry.” But you’re not. You’re watching wheels turn. I’ve become comfortable with silence.
Brett: And through Listening to Learn, you’re showing us just how vastly different every student processes, how every student thinks. Talk a little bit about how different each student can be.
Marilyn: So here’s a question that’s our first question in our interview three which focuses on adding and subtracting. And interview three is numbers within 20. Interview two is numbers within 10, and then four and five, and the numbers get greater. So the first question is how much is 4+9? Well, the reason we selected that question is because when we started asking that question, we were amazed at how many different ways kids came up to solving that problem. Ways that I had never even thought were possible. From the kid who starts with the four and has to count on, five, six and gets all the way up maybe to 13 and that’s a risky strategy. So the kid who says, “Well look, I’m starting with a bigger number nine,” or a kid who says, “Well, you know, 4+9, 9+1=10.” Every time I think I’ve heard it all, a kid comes up with it. There’s this girl says, “Okay, I take 4+4=8.” I’m thinking, “Well, all right, kids know their doubles.
Where’s she going with that?” So, “Eight and then I took that four out of the nine so there’s five left over. So eight and two more makes 10, and then I had three leftover and it was 13,” and it’s like, “Go girl.” So I showed that clip to a lot of people and they say, “Well what do you do, because it’s not particularly efficient.” I said, “She’s eight years old. Every child has a right to be eight for a whole year before they have to be nine.” That’s her thinking. She made sense of mathematics. I’m ecstatic.
Brett: So what is our takeaway from that in terms of what it means for classroom instruction?
Marilyn: One the things that I think many teachers are comfortable and excited about doing on number talks kind of mini-lesson discussions, 15 minutes at beginning of the class. One possible way for number talks is you put out a problem on 4+9 and you’d say, “What are all the ways we can think about figuring out that answer in our heads?” And in our heads, it wouldn’t really, there’s nothing to do on paper and pencil. You’re just going to have to count. So one of the things on number talk, put up the problem, 4+9, you might say, “How many different ways can we think about this?” And then I want kids to hear and learn from one another. I want a community of math learners. So when the child does the counting on, I want to honor that. I want to put it up there in the board and say that’s where Abel did it. But when the little girl Hanasis took out the 4+4, I want to say, “Oh, this is how Hanasis did it.”
Something happens in those number talks. I had a kid do something. It’s one in those areas of like, “Huh, I never thought about this.” He took the nine, he took one off the nine and he put it on the four and he changed the problem from 4+9 to 5+8 and said, “That’s 13.” I’m thinking like, “All right. Yes, that’s right,” but I still to this day, don’t know. Did he know … Was 5+8 his friend and 4+9 wasn’t? It’s sort of like the fact that he was compensating in addition, seeing if I take one off one end and and put it on the other, the sum stays the same. He didn’t verbalize that generalization, but that’s pretty sophisticated mathematically.
Brett: Why do we want students to develop number sense?
Marilyn: It’s part of life. I mean, I think about all the decisions that we make where we use arithmetic in our lives. What we have to add, subtract, multiply or divide to solve some problem that we need. It could be keeping score when you’re playing a game. It could be buying wallpaper when I actually suggest that you overestimate, not underestimate because as soon as you run short, that dye lot’s going to be gone. Understanding our finances, figuring out the tip in a restaurant.
There’s so many times when we’re in the supermarket, “I got $20, I don’t want to go over, I don’t have my credit card with me.” Keep track of where things are. We think about numbers all the time. It’s just part of our world. So if we’ve done this with teachers where we ask them to list all the ways and we get long, long lists about it, about what they do and then you say, okay, if we sort those and whether you figure it out with paper and pencil or you use a calculator or you do it in your head, how does that come out? And you know, most of the things in our daily life where we use arithmetic, cooking, gardening, playing games, we do it in our head. You figure out, “Okay, what time do I have to leave to get to the movies? The Showtime is 7:15. Will the commute traffic been done? What time do I leave?” Look at all the math you’re doing.
The biggest math problem of the year? What time do you put the turkey in the oven to have it ready for Thanksgiving? Because now they say so much a pound. So much a pound with the stuffing. Turkey seems to cook more quickly these days, but what are you going to do? That gives people anxiety, but basically we think about math all the time. So people say, well, “Oh yeah, that’s why we should learn fractions because of cooking.” I say, “Well if I’m doubling a recipe and it calls for two-thirds of a cup, then I just take two-thirds and two-thirds. If I’m halving a recipe and it calls for three-fourths of a cup.” But then again it’s not really important. You kind of can measure it.
So I want kids to be playful with numbers. Think about numbers, find numbers useful to them. Not be afraid of numbers, be curious about numbers. There’s so much that I want math to be playful and exploring and interesting and enjoyable.
Brett: And along with that, to you it’s very important that students see the meaning in all the math. You redefined your teaching to specifically get there. Why?
Marilyn: Yeah, well without meaning, what is it? I mean, math is such a weird thing. I majored in math for an embarrassing reason. I didn’t like writing. I figured you’d just had to do the problems. I was kind of goody two-shoes in school, and I was good in math. I memorized my way through a lot of college math courses.
It didn’t work. Made me a better teacher. I know what it feels like to feel really dumb as the professor’s filling up the board and all of a sudden, “Uh-oh, I missed a step.” And then when you miss a step, you’re out. And then I had to keep writing everything down that was going up on the board so I could try and figure it out later. So making sense is something that the kid does. That the source of the understanding is in the kid’s head. My job is to stimulate the child with ways that they can create that understanding so they can bring meaning to math. What else is it about?
Brett: I mean selfishly as a student I would be that student that would say to the teacher, “Why are we doing this?”
Marilyn: Yes, and for many teachers it’s hard because they learn math by “Yours is not to question why, just invert and multiply,” and get threatened or annoyed at the kid who says that whereas you get thrown. I mean, I get thrown by questions.
A kid, third grader was convinced that 90 was an odd number because it had a nine. And so I said, “Brayden,” I can still see his face, I said, “I’ll get back to you tomorrow. I got to think about this.” I really wanted to think about why did he think that? Well, nine is an odd number and the 90 starts with nine, must be odd. So I could have said, “No, no, no. If the nine comes first, it doesn’t count. The zero is an even number.” Then I realized many teachers I’ve asked, “Is zero even or odd?” And there are people who aren’t sure or maybe it’s neither because it’s kind of that pivot number. So we all know what that flutter is. “Uh-oh. I don’t get it.”
Brett: You can see us thinking there, even in the question. I love that.
Marilyn: Kids are my laboratory. They never, never fail to amaze me and surprise me.
Brett: How did the role of mental math come in?
Marilyn: Well, that was part of the common sense thing when you realize that everything you do in life is mental math. Okay. Reading, writing, arithmetic. That bothered me so much as a child because arithmetic didn’t start with R. Reading. Writing didn’t start with R, but it sounded like it. But it bothers me now for a different reason. It should be reading, writing and reasoning because arithmetic is … I don’t want it to think of it as what we do with paper and pencil with following algorithmic procedures. Algorithms are important. They have an important place in mathematics, but I want the reasoning to be the leader of it. So it’s the reasoning and that requires thinking mentally. But I think that as teachers, we’re under a lot of pressure to send home work.
And I know there are many teachers who give their kids their work packet at the beginning of the week, and they have all week to work on. And there’s a lot of practice, and it’s a good way for kids to practice their skills. But what do you send home if you’re doing mental math all day? The parents are going to think you didn’t do any math. Especially if they say, “What’d you do in math today?” And the kid says, “Nothing,” because all they did was talk and reason and exchange ideas. So I think that workbooks and textbook pages have really not encouraged teachers to give mental math the due it deserves and requires.
Brett: How do students react to your interviewing? To the interviews themselves?
Marilyn: When we developed these, I often go and borrow a class and ask the teacher if I can interview some kids. I’ll go and introduce myself to the kids and explain my usual thing. “I can’t open up your head. I can’t look inside,” the kids are a little nervous. And I’ve always said to the teacher, “Don’t give me your top kid. Don’t give me your kid who everybody knows is really having a difficult time with it. Just give me a kid, a regular kid.” I take the kid out and they come back and everybody’s looking, and the kid comes back and we have fun. When do kids get all this attention? This is a person they don’t even know and I’m friendly, and we’re going to talk about this and I’m interested and they come back. Sometimes I have kids come back, they say, “That was good.” Then they’re all raising their hand, “Take me, take me.” I have never met a kid that didn’t want to be interviewed.
I remember one time a kid did not want to be videotaped and he whispered the whole thing, which I thought was a very creative solution to this avoiding my videotaping him. But I don’t know. We’ve been teachers for a long time. You walk into a classroom and we know how to relate to children. We know how to deal with mobs of children, and we know how to deal with individual kids. So, “Come, we’re going to have a conversation. Help me out here.”
Brett: Marilyn, you’re developing a new project called Listening to Learn. You call it a new digital interview tool. What can you share with us about it?
Marilyn: Well, I’ve been interviewing kids for years, and I want teachers to have the experience of really finding out what their students understand and encouraging them to do interviews. And the reason that it’s a digital tool is important because what I’ve figured out is it’s hard to listen and learn. You have to learn to listen in order to listen to learn.
And that means really hearing what a student says and capturing that. But what am I going to do with that information? As I’ve thought about it, and it’s kind of embarrassing to think that it was later in my career that I thought about, there are certain reasoning strategies that became evident to us as we interviewed kids that we want kids to be able to have access to. So if I think about addition and subtraction as a content focus, which is separate from multiplication and division or then fractions and decimals, or even before that, some foundation. So if I focus on addition and subtraction, I want kids to be able to decompose numbers. I want them to understand the inverse relationship between addition and subtraction. I want them to be able to break numbers apart and to place value parts. I want them to use benchmark numbers.
I’m rattling these off, but we have the 10 reasoning strategies that we want all kids to have access to in order to be able to add and subtract mentally with numbers within 10, 20, 100 and 1000. The same ten. This is not a lot. And once I figured those out I said, “Wow, this is pretty slick.” But when I ask a kid, “How much is 4+9,” the child’s answer that I want to help teachers is listen to what the child said. And we have captured all the different ways, over 1000 kids that we’ve interviewed, of how they answered them and we give them, so this one is the most close to that. Then we take, in the digital tool, which I couldn’t do otherwise, is the heavy lifting of mapping the kid’s explanation onto one of those 10 strategies I just said.
So I want all teachers to know the 10 strategies. I don’t want you to teach the strategies as if they’re separate isolated skills because something will always emerge as connected to more than one strategy. If a kid says, “Well, 4+9 okay, I’m going to start with the nine and then I’m going to take one off the four, make it 10 so it’s 10+3,” the kid is reversing the two addends. So it’s demonstrating applying the commutative property. The kid is using the benchmark number of 10. The kid is decomposing numbers within five. They’re doing so many things. We capture all that. So what the digital tool can do, which has got me so excited, is it can tell you which of those strategies the student has demonstrated. And I can look across my class and say which of the strategies that kids seem to have access to and which do I need to spend more time with? That is truly information that could inform my instructional decisions.
Brett: Well that was what I wanted to ask you next. What can a teacher do with this tool in their teaching?
Marilyn: So we were interviewing a third grade class at interviewing and we found out that none of the kids use the inverse operation. For example, 11-9. That’s within 20. A kid may count back 11, 10, nine, eight. It’s a risky countback strategy, but 11-9 like, “9+2=11.” That’s the inverse of, “I can use addition to solve a subtraction problem.” I have had kids, older kids, you ask 1000-998, it should be slam dunk two. For some kids it isn’t. So imagine if you had that information about your entire class of who had chosen to use the inverse relationship to solve problems because it was the most efficient.
Now, if a kid says to me, “1000-998. Okay, 1000-900=100. 100-90=10. 10-8=2.” A little long winded, but I’m impressed. But I want to know if they could have access, so we have several questions in each interview, which should give kids the chance to show that they could use that. But suppose I have no kids demonstrating that or suppose in my class I’ve got all but a third of the kids aren’t. So that tells me, “Okay, in my number talks, let’s talk about problems where it’s more efficient to subtract, to solve it or more efficient to add, to subtract the problem.” It tells me what I need to have so that kids who are either developing understanding of that important strategy or cementing it or really can extend it to even more complex numbers, greater numbers.
I have to have that information, but I want a community of learners so that I can engage it. It helps me make intentional decisions about my instruction, and that has been the most stunning thing to me. My instructional choices have become so much more intentional once I have the information from interviews.
Brett: And you’ve seen it strengthen the classroom community as well.
Marilyn: Oh gosh. Yes. I understand the need for differentiation and it’s hard. Yes, I understand the need for classroom community. I know how hard teaching is. Some years you get a class where everybody’s sort of within a range. Some years you get a class where you have kids who are so disparate in two different groups. I’ve seen it all. They’re all tough, but at all times we are a community of learners. And that’s when I need all the tools I have pedagogically.
I need to know how to do number talks. I need to know how to engage the kids in problem solving investigations that we can all learn from. I need to know how to give them independent work and truly differentiate. I need time to confer with kids and help them. I need to do it all. They do that in reading and writing instruction. I need to do it in math, but it needs to be a community even when kids are coming to it with their own way.
Brett: I’ve heard you say, “Math can be a way in for a teacher to really see their students.”
Marilyn: Yeah. I remember a principal in New York City told me that math was easier because you can make it concrete and it’s most astonishing thing when you’re sitting with a young child and we have 10 counters in front of us. I agree, the child agrees we have 10. And I say, “Watch what I’m going to do,” and I take one of the counters away and I say, “How many are there now?”
Some kids have to count them. Other kids will just say, “Well nine.” The difference between those two kids, I would not know that one kid understands that nine comes before ten, nine is nested in 10, 10 … There’s so much that happens. We think of counting as being simple, but it’s really pretty complex and without the ability to see it, I could actually see it there. That gives me understanding about what that child does and doesn’t know.
So the child who isn’t sure probably needs some work with smaller quantities. I tweeted out recently a question that came to me one morning and it was like, if you look at the pips on a di e, if you look at the arrangement of dots, you don’t have to count one, two, three, four, five. Well you don’t, but some kids do. It’s a brain function. You can see five so it’s subitizing. Is that the same as sight words? And I tweeted that out. A bunch of people sent me some really interesting information to think about with this idea of subitizing. What do I see? What do I know? And I want teachers to be curious about that with their kids.
So one of the things we suggest in Listening to Learn, we help teachers with the protocol of giving interviews. We interpret the students’ explanations and map them onto the strategies. And we also provide some “what next” instructional suggestions. Teachers have their programs and they’re basically, those are their default roadmaps for instruction, but we give some suggestions. It’s common for many teachers to do dot patterns where you show kids random dot patterns. Sometimes few enough to see if they will … You just sort of show-hide, take a quick look. But if you have a bunch of them arranged in such a way, a quick look where you can’t subitize that many, but you could see perceptually a group of four and a group of two and you’d say, “Oh, that’s six.”
So it’s kind of like a conceptual subitizing rather than just perceptual subitizing. I mean, I’m amazed how complicated counting is, and I taught secondary math. I encourage secondary math teachers to go down the grades and talk to the kids about what they understand. Things that we took so much for granted. When the professor in the math class said, “It is now obvious that,” I was the kid to whom it was not obvious.