Kim Morrow Leong, PhD, Responsive interpretations of student thinking

Mike Wallus, Vice President for Educator Support


What are the habits of mind that educators can adopt to be more responsive to our students' thinking? And how can we turn these habits of mind into practical steps that we can take on a regular basis? Dr. Kim Morrow-Leong has some thoughts on this topic. Today, Kim joins the podcast, and we'll talk with her about three mental shifts that can profoundly impact educators’ instructional and assessment practices.

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Kim Morrow-Leong is an experienced mathematics education specialist and consultant with a demonstrated history of working productively with teachers in the area of mathematics education leadership. Skilled in lesson planning, educational technology, instructional design, leadership, and curriculum development, Kim holds a PhD and MEd in mathematics education leadership from George Mason University.


Children's Mathematics: Cognitively Guided Instruction

Warning Signs!

Never Say Anything a Kid Can Say!


Mike Wallus: What are the habits of mind that educators can adopt to be more responsive to our students' thinking? And how can we turn these habits of mind into practical steps that we can take on a regular basis? Dr. Kim Morrow-Leong has some thoughts on this topic. Today, Kim joins the podcast, and we'll talk with her about three mental shifts that can profoundly impact educators’ instructional and assessment practices. 

Kim Morrow-Leong, welcome to the podcast. We're excited to have you. 

Kim Morrow-Leong: Thank you, Mike. It's nice to be here. 

Mike: I'm really excited to talk about the shifts educators can make to foster responsive interpretations of student thinking. This is an idea that for me has been near and dear for a long time, and it's fun to be able to have this conversation with you because I think there are some things we're going to get into that are shifts in how people think, but they're also practical. You introduced the shift that you proposed with a series of questions that you suggested that teachers might ask themselves or ask their colleagues. And the first question that you posed was, “What is right?” And I'm wondering, what do you mean when you suggest that teachers might ask themselves or their colleagues this question when they're interpreting student thinking?

Kim: So, I'm going to rephrase your question a little bit and change the emphasis to say, “What is right?” And the reason I want to change the emphasis of that is because we often talk about what is wrong. And so rather than talking about what is wrong, let's talk about what's right. When we look at student work, it's a picture. It's a snapshot of where they are at that particular moment. And the greater honesty that we can bring to that situation to understand what their thinking is, the better off we're going to be.

So, there's a lot of talk lately about asset-based instruction, asset-based assessment. And I think it's a great initiative, and it really gets us thinking about how we can think about what students are good at and what they bring to the table or what they bring to the classroom culture. But we don't often talk a lot about how we do that, how we break the mold. Because many of our metaphors and our language about learning are linear, and they indicate that students are moving from somewhere to achieve a goal somewhere down the path, somewhere down the line. 

How do you switch that around? Well, rather than looking at what they're missing and what part of the path they haven't achieved yet, we can look at where they are at the moment because that reflects everything they've learned up to that moment. So, one of the ways we can do this is to unpack our standards a little more carefully. And I think a lot of people are very good at looking at what the skills are and what our students need to be able to do by the end of the year. But a lot of what's behind a standard are concepts. What are some big ideas that must be in place for students to be successful with the skills? 

So, I'm going to give a very specific example. This one happens to be about a fourth-grade question that we've asked before in a district I used to work at. The task is to sketch as many rectangles as you can that are 48 square units. There's some skills behind this, but understanding what the concepts are is going to give us a little more insight into student thinking. So, one of the skills is to understand that there are many ways to make 48: to take two factors and multiply them together, and only two factors, and to make a product of 48 or to get the area. But a concept behind that is that 48 is the product of two numbers. It's what happens when you multiply one dimension by the other dimension. It's not the measure of one of the dimensions. That's a huge conceptual idea for students to sort out what area is and what perimeter is, and we want to look for evidence of what they understand about the differences between what the answer to an area problem is and what the answer to, for example, a perimeter problem is. Another concept is that area indicates that a space is covered by squares. The other big concept here is that this particular question is going to have more than one answer. You're going to have 48 as a product, but you could have 6 times 8 and 4 times 12 and many others. So that's a lot of things going into this one, admittedly very rich, task for students to take in. 

One of the things I've been thinking a lot about lately is this idea of a listening stance. So, a listening stance describes what you're listening for. It describes how you're listening. Are you listening for the right answer? Are you listening to understand students' thinking? Are you listening to respond or are you listening to hear more—and asking for more information from your student or really from any listener? So, one of the ways we could think about that, and perhaps this sounds familiar to you, is you could have what we call an evaluative listening stance. 

An evaluative listening stance is listening for the right answer. As you listen to what students say, you're listening for the student who gives you the answer that you're looking for. So, here's an example of something you might see. Perhaps a student covers their space and has dimensions for the rectangle of 7 times 6, and they tell you that this is a space that has an area of 48 square units. There's something right about that. They are really close. Because you can look at their paper and you can see squares on their paper and they're arranged in an array. And you can see the dimensions on this side and the dimensions on that side, and you can see that there's almost 48 square units. 

I know we all can see what's wrong about that answer, but that's not what we're thinking about right now. We're thinking about what's right. And what's right is they covered that space with an area that is something by 6. This is a great place to start with this student to figure out where they got that answer. If you're listening evaluatively, that's a wrong answer and there's nowhere else to go. So, when we look at what is right in student work, we're looking for the starting point. We're looking for what they know so that we can begin there and make a plan to move forward with them. You can't change where students are unless you meet them where they are and help them move forward. 

Mike: So, the second question that you posed was, “Can you cite evidence for what you're saying?” So again, talk us through what you're asking, when you ask teachers to pose this question to themselves or to their colleagues. 

Kim: Think about ways that you might be listening to a student's answer and very quickly say, “Oh, they got it,” and you move on. And you grab the next student's paper or the next student comes up to your desk and you take their work and you say, “Tell me what you're thinking.” And they tell you something. You say, “That's good,” and you move to the next one. 

Sometimes you can take the time to linger and listen and ask for more and ask for more and ask for more information. Teachers are very good at gathering information at a glance. We can look at a stack of papers and in 30 seconds get a good snapshot of what's happening in that classroom. But in that efficiency we lose some details. We lose information about specifics, about what students understand, that we can only get by digging in and asking more questions.

Someone once told me that every time a student gives an answer, you should follow it with, “How do you know?” And somebody raised their hand and said, “Well, what if it's the right answer?” And the presenter said, “Oh, you still ask it. As a matter of fact, that's the best one to ask. When you ask, ‘How do you know?’ you don't know what you're going to hear. You have no idea what's going to happen.” And sometimes those are the most delightful surprises, is to hear some fantastical, creative way to solve a problem that you never would've thought about. Unless you ask, you won't hear these wonderful things. 

Sometimes you find out that a correct answer has some flawed reasoning behind it. Maybe it's reasoning that only works for that particular problem, but it won't work for something else in the future. You definitely want to know that information so that you can help that student rethink their reasoning so that the next time it always works. Sometimes you find out the wrong answers are accidents. They're just a wrong computation. Everything was perfect up until the last moment and they said, “3 times 2 is 5,” and then they have a wrong answer. If you don't ask more, either in writing or verbally, you have incorrect information about that student's progress, their understandings, their conceptual development, and even their skills. That kind of thing happens to everyone because we're human. By asking for more information, you're really getting at what is important in terms of student errors and what is not important, what is just easily fixable. 

I worked with a group of teachers once to create some open-ended tasks that require extended answers, and we sat down one time to create rubrics. And we did this with student work, so we laid them all out and someone held up a paper and said, “This is it! This student gets it.” And so, we all took a copy of this work and we looked at it. And we were trying to figure out what exactly does this answer communicate that makes sense to us that seems to be an exemplar. And so, what we did was we focused on exactly what the student said. We focused on the evidence in front of us. This one was placing decimal numbers on a number line. We noted that the representation was accurate, that the position of the point on the number line was correct. We noticed that the label on the point matched the numbers in the problem, so that made sense. But then all of a sudden somebody said, “Well, wait a minute. There's an answer here, but I don't know how this answer got here.” Something happened, and there's no evidence on the page that this student added this or subtracted this, but magically the right answer was there. And it really drove home for this group—and for me, it really stuck with me—the idea that you can see a correct answer but not know the thinking behind it. And so, we learned from that point on to always focus on the evidence in front of us and to make declarative statements about what we saw, what we observed, and to hold off on making inferences. We saved our inferences for the end. 

After I had this experience with the rubric grading and with this group of teachers and coaches, I read something about over-attribution and under-attribution. And it really resonated with me. Over-attribution is when you make the claim that a student understands something when there really isn't enough evidence to make that statement. It doesn't mean that's true or not true; it means that you don't have enough information in front of you. You don't have enough evidence to make that statement. You over-attribute what it is they understand based on what's in front of you. Similarly, you get under-attribution. You have a student who brings to you a drawing or a sketch or a representation of some sort that you don't understand because you've never seen anybody solve a problem this way before. You might come to the assumption that this student doesn't understand the math task at hand. That could be under attribution. It could be that you have never seen this before and you have not yet made sense of it. 

And so, focusing on evidence really gets us to stop short of making broad, general claims about what students understand, making broad inferences about what we see. It asks us to cite evidence to be grounded in what the student actually put on the paper. For some students, this is challenging because they mechanically have difficulties putting things on paper. But we call a student up to our desk and say, “Can you tell me more about what you've done here? I'm not following your logic.” And that's really the solution is to ask more questions. I know, you can't do this all the time. But you can do it once in a while, and you can check yourself if you are assigning too much credit for understanding to a student without evidence. And you can also check yourself and say, “Hmm, am I not asking enough questions of this student? Is there something here that I don't understand that I need to ask more about?”

Mike: This is really an interesting point because what I'm finding myself thinking about is my own practice. What I feel like you're offering is this caution, which says, “You may have a set of cumulative experiences with children that have led you to a set of beliefs about their understanding or how they come to understanding. But if we're not careful—and sometimes even if we are careful—we can bring that in a way that's actually less helpful, less productive,” right? It's important to look at things and actually say, “What's the evidence?” rather than, “What's the body of my memory of this child's previous work?” It's not to say that that might not have value, but at this particular point in time, what's the evidence that I see in front of me?

Kim: That's a good point, and it reminds me of a practice that we used to have when we got together and assessed these open-ended tasks. The first thing we would do is we put them all in the middle of the table and we would not look at our own students' work. That's a good strategy if you work with a team of people, to use these extended assessments or extended tasks to understand student thinking, is to share the load. You put them all out there. And the other thing we would do is we would take the papers, turn them over and put a Post-it note on the back. And we would take our own notes on what we saw, the evidence that we saw. We put them on a Post-it note, turn them over and then stick the Post-it note to the back of the work. There are benefits to looking at work fresh without any preconceived notions that you bring to this work. There are other times when you want all that background knowledge. My suggestion is that you try it differently, that you look at students' work for students you don't know and that you not share what you're seeing with your colleagues immediately, is that you hold your opinions on a Post-it to yourself, and then you can share it afterwards. You can bring the whole conversation to the whole table and look at the data in front of you and discuss it as a team afterwards. But to take your initial look as an individual with an unknown student. 

Mike: Hmm. I'm going to jump to the third shift that you suggest, which is less of a question and more of a challenge. You talk about the idea of moving from anticipating to targeting a learning trajectory, and I'm wondering if you could talk about what that means and why you think it's important?

Kim: Earlier we talked about how important it is to understand and unpack our standards that we're teaching so that we know what to look for. And I think the thing that's often missed, particularly in standards in the older grades, is that there are a lot of developmental steps between, for example, a third-grade standard and a fourth-grade standard. There are skills and concepts that need to grow and develop, but we don't talk about those as much as perhaps we should. Each one of those conceptual ideas we talked about with the area problem we discussed may come at different times. It may not come during the unit where you are teaching area versus perimeter versus multiplication. That student may not come to all of those conceptual understandings or acquire all of the skills they need at the same time, even though we are diligently teaching it at the same time.

So, it helps to look at third grade to understand, what are these pieces that make up this particular skill? What are the pieces that make up the standard that you're trying to unpack and to understand? So, the third shift in our thinking is to let go of the standard as our goal, but to break apart the standard into manageable pieces that are trackable because really our standards mean “by the end of the year.” They don't mean by December; they mean by the end of the year. So that gives you the opportunity to make choices. What are you going to do with the information you gather? You've asked what is right about student work. You've gathered evidence about what they understand. What are you going to do with that information? That perhaps is the hardest part. 

There's something out there called a learning trajectory that you've mentioned. A learning trajectory comes out of people who really dig in and understand student thinking on a fine-grain level, how students will learn developmentally. What are some ideas they will develop before they develop other ideas? That's the nature of a learning trajectory. And sometimes those are reflected in our standards. The way that kindergartners are asked to rote count before they're asked to really understand one-to-one correspondence. We only expect one-to-one correspondence up to 20 in kindergarten, but we expect rote counting up to 100 because we acknowledge that that doesn't come at the same time. So, a learning trajectory, to some degree, is built into your standards. But as we talked about earlier, there are pieces and parts that aren't outlined in your standards.

One of the things we know about students and their interactions with grids and arrays is that a student might be able to recognize an array that is 6-by-8, but they may not yet be able to draw it. The spatial structuring that's required to create a certain number of lines going vertically and a certain number of lines going horizontally may not be in place. At the same time, they are reading arrays and understanding what they mean. So, the skill of structuring the space around you takes time. The task where we ask them to draw these arrays is asking something that some kids may not yet be able to do, to draw these grids out. If we know that, we can give them practice making arrays, we can give them tools to make arrays, we can give them blocks to make arrays, and we can scaffold this and help them move forward. What we don't want to assume is that a student who cannot yet make a 6-by-8 array can't do any of it because that's not true. There's parts they can do. So, our job as teachers is to look at what they do, look carefully at the evidence of what they do, and then make a plan. Use all of that skill and experience that's on our teams. Even if you're a new teacher, all those people on your teams know a lot more than they're letting on, and then you can make a plan to move forward and help that student make these smaller steps so they can reach the standard. 

Mike: When we talked earlier, one of the things that you really shifted for me was some of the language that I found myself using. So, I know I have been in the habit of using the word “misconception” when we're talking about student work. And the part of the conversation that we had that really has never left me is this idea of, what do we actually mean when we say “misconception”? Because I found that the more I reflected on it, I used that language to describe a whole array [laughs] of things that kids were doing, and not all of them were what I think a misconception actually is. 

Can you just talk about this language of misconception and how we use it and perhaps what we might use instead to be a little bit more precise? 

Kim: I have stopped using the word “misconception” myself. Students understand what they understand. It's our job to figure out what they do understand. And if it's not at that mature level we need it to be for them to understand the concept, what disequilibrium do I need to introduce to them? I'm borrowing from Piaget there. You have to introduce some sort of challenge so that they have the opportunity to restructure what it is they understand. They need to take their current conception, change it with new learning to become a new conception. That's our teaching opportunity right there. That's where I have to start. 

Mike: Before we close, I have to say one of the big takeaways from this conversation is the extent to which the language that I use—and I mean, like, literally what I say to myself internally or what I say to my colleagues when we're interpreting student work or student thinking—that that language has major implications for my instruction. And that the language that surrounds my assessing, my interpreting and my planning habits really matters. 

Kim: It does. You are what you practice. You are what you put forth into the world. And to see a truly student-centered point of view requires a degree of empathy that we have to learn. 

Mike: So, before you go, Kim, I'm wondering, can you share two or three resources that have really shaped your thinking on the interpretation of student learning? 

Kim: Yes, I could. And one of them is the book, Children's Mathematics[: Cognitively Guided Instruction]. There's a lot of information in this book, and if you've ever engaged with the work of cognitively guided instruction, you're familiar with the work in this book. There's plenty of content knowledge; there's plenty of pedagogical content knowledge in this book. But the message that I think is the most important is that everything they learn, they learn by listening. They listened to what students were saying. 

And the second piece is called “Warning Signs!” And this one is one of my favorites. And in this [article], they give three warning signs that you as the teacher are taking over students' learning. And one example that comes to mind for me is you take the pencil from the student. It's such a simple thing that we would just take it and to quickly get something out, but to them, they expressed that that's a warning sign that you're about to take over their thinking. So, I highly recommend that one. 

And there's another one that I always recommend. It was published in Mathematics Teaching in the Middle School. It's called “Never Say Anything a Kid Can Say!” That's a classic. I highly recommend it if you've never read it. 

Mike: Kim Morrow-Leong, thank you for joining us. It's really been a pleasure.

Kim: Mike, thank you for having me. This has been delightful.

Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability.