Uncovering Student Thinking

In this episode of Room to Grow, Curtis and Joanie consider ways to uncover how students are actually thinking about the mathematics they are learning. Using a real-life, recent incident between Curtis and his sixth grade son, our hosts consider the challenging fact that many students think that success in math class means figuring out what answer the teacher (or the computer program/app, or the back of the book) is looking for. 

They posit that when educators are always focused on the mathematics of the moment – what is being learned in a single lesson, week, or unit – we can focus students on the smaller grain size ideas instead of helping them to place their learning in the bigger picture of mathematics as a whole. As always, the episode recognizes that teachers work very hard at a very complex task: teaching young minds to deeply understand important mathematics!

We encourage you to explore the resources below, referenced in this episode:

• “Listening to and Learning from Student Thinking,” by Elham Kazemi, Lynsey K. Gibbons, Kendra Lomax, and Megan L. Franke from Teaching Children Mathematics, October 2016.
• “Making Student Thinking Public,” by Shari Stockero and Laura R. Van Zoest from The Mathematics Teacher, May 2011.
• “Attending to Evidence of Students’ Thinking during Instruction,” by Miriam Gamoran Sherin and James Lynn, from Mathematics Teaching in the Middle School, May 2019.
• The Five Practices for Orchestrating Productive Mathematical Discussions by Margaret Smith and Mary Kay Stein, ISBN: 978-1-68054-016-1

Did you enjoy this episode of Room to Grow? Please leave a review and share the episode with others. Share your feedback, comments, and suggestions for future episode topics by emailing roomtogrowmath@gmail.com. Be sure to connect with your hosts on Twitter and Instagram: @JoanieFun and @cbmathguy. 

Transcript

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SPEAKER_01 0:00

I think it might be worth us exploring a little bit how we are considering the difference between student thinking and student understanding. Like how are those related to one another? How are they different? In our first episode of season four, Curtis and I talk about how we understand what students understand about mathematics. We start with some personal stories about how students sometimes think that math is about finding the answer the teacher wants. Although most teachers' intentions are that students make sense of the mathematics for themselves so they are able to apply what they learn to new situations. We think it is important that we engage students in the work and conversation about big ideas of mathematics from time to time rather than always being focused on the math topic of the day. And we acknowledge that while building conceptual understanding is important, so is finding answers. We hope you'll hear some practical ways to shift instructions to ensure that student thinking is uncovered in your classroom. We're glad you're with us today, so let's get growing.

SPEAKER_00 1:03

Well, hello, Joni. I am very excited to be starting a new year uh with you today, talking about uh our new topic this year, our first episode of 2024. Uh it's where we're recording this in 2023. But uh yeah, yeah, really excited about uh about this. So today we're gonna be chatting uh a little bit about this idea of understanding student understanding, not just the assessments, but really thinking about what students are thinking and and ways that we can uh uh think about what students are thinking uh about. Um so really cool topic. Uh, excited to start the year here. Um and actually it's funny, we were talking about this yesterday, and um it made me think about uh an experience I had with my son uh the other day. So I talk about my son, my he's now sixth grader, um, and he's taking uh a class, uh an advanced uh math class or whatever, and trying to kind of accelerate um through two classes in one year. And we were talking about uh a multiple representations problem that he had been kind of given uh by his teacher. And I'm I'm really excited about this problem. I thought it I thought it was kind of a cool thing, just you know, seeing uh all the representations on this one uh piece of paper. And they were asked to, you know, given one of those was filled out with all the information. I think it was the verbal description. And if I remember right, it was something about a baker making bagels or something like that, and he makes 40 bagels every hour or something like that, is is kind of the verbal description. Uh, and then of course he had a prompt for uh you know a place to put the graph and a place to write an equation and a place to make a table. Um, and so this is really kind of a nice little organizational tool um and getting him to think about these multiple representations. But there was a follow-up problem uh that was in there that he had left blank. He wasn't really sure what to do with it, and it says uh uh I don't remember the exact wording, but it said something to the effect of uh explain the independent and dependent variables. And he had he had left it blank, and so I kind of asked him a little bit about it, and his response to me was uh well, they're just represented by by random number, random letters. This is this was how he responded to me with that. And I thought I I started to try to probe him a little bit about about, you know, well, how did you come to that? What do you think? How do you think those letters come to be? You know, all of the little questions that I was trying to get at. And he started to uh try to think about what he was thinking, except that he started to think about what he was thinking and what he thought that I was thinking he should be thinking. Right.

unknown 4:02

Right.

SPEAKER_00 4:02

So in other words, he was trying to read your mind a little bit, right? He was trying to read my mind. Okay, dad's asking me this question. Apparently, what I wrote down wasn't what I was supposed to write down. Dad's asking me this question about what I'm thinking. What does he want me to be thinking so I can tell him that? Uh as I think the process that was going on in his mind. And I when we were thinking about student understanding and this idea of understanding student understanding, um that story and that kind of process kind of came to mind because what I was trying to get at literally was what do you know, Tegan, about independent and dependent variables? And how can I how can I understand what you know and understand about independent and dependent variables such that we can kind of work towards what are independent and dependent variables and then apply that to this particular situation uh later on? I was trying to kind of scaffold him up, if you will, uh to that. And I thought this was a a good kind of a place to begin thinking about well, how do we work at understanding what students actually understand about topics?

SPEAKER_01 5:17

Right. Okay, I I totally love this story, and I have about a hundred things spinning through my mind that I want to say. But first I want to talk about our poor children, the children of math educators trying to go through school. It it's very reminiscent of, you know, my kids are grown and married and live out of state now, but it's reminiscent of when they were kids. And, you know, my one of my sons would do their homework and I would look over it. You know, we that's that was just the pattern. And I would ask questions and they'd be like, ah, mom, the teacher doesn't care about that. I just I have the answer. It matches what's in the back of the box. Right. Um, just acknowledging that that's a challenge for our children, but also a blessing for our children, right? And I think I I love this story that you shared about Tegan, and I think it sets the stage for this conversation so nicely because a a student's classroom experience is often about, you know, what what math am I learning in the moment? And what answers are we looking for in the moment? And as a, you know, reflecting back on my own time as an educator, that was often the focus of my lessons. Like, okay, we're working on, you know, in my head, I knew we were working on a bigger concept, but for the students, it was, you know, a series of answers to questions. Um, so thinking about, you know, Tegan's specific example of you've got this real world scenario where there's probably a linear relationship between these two um quantities and coming up with multiple ways of representing that relationship through a table of values, through a graph, through an equation, all of that stuff, just kind of going back to, you know, sixth, seventh grade math. And I suspect that the Hallmark question that he was answering, you know, what are the independent and independent variables was meant specific to that question, right? For sure. But knowing you and knowing the ways that you're trying to support your student, your child, is you're trying to help him think more broadly about what independent independent variables are. You're thinking long-term. You're not thinking this homework assignment and filling in, you know, question 2b on this homework assignment. You're thinking independent and dependent variables are a way important concept that this is the first time he's engaging with them and he's gonna engage with them through the rest of middle school, all through high school. Like that it's such a key concept, right? And you're trying to build that conceptual understanding now so that he has that to build on as his as his instruction progresses. So I loved this story as an introduction to this episode, because I think what you know, we what we might be trying to get at in this idea of understanding student understanding is um just being more intentional about giving kids the opportunity to engage in that level of thinking, the the concept level of thinking, not the what's the answer for today. The other thing that I'd like us to kind of flesh out here early on, Curtis, is you know, we went back and forth a little bit in our planning conversation about is this understanding student understanding or is this understanding student thinking? Um, and I don't think it's clearly one or the other, but I I think it might be worth us exploring a little bit how we are considering the difference between student thinking and student understanding. Yeah. Like how are those related to one another? How are they different? Just share some of your thoughts and then I'll kind of chime in on mine.

SPEAKER_00 9:01

Yeah, no, I think I think that's exactly I think that's an interesting question. And, you know, as we were thinking about this topic, uh to me, the way I'm viewing this or thinking the way I'm thinking uh about this, the way I'm understanding. We could get into lots of loops of these uh thinking about thinking about thinking. Uh, but the way that my understanding is here um is this when I ask a student, I can uh when I was asking Tegan uh about how he was thinking about this, it was something that he could explain. Now, admittedly, in the moment uh he kind of got flustered thinking about what I was asking him to think about, he kind of got frustrated trying to answer the the question the the right way. Um and and so there is a there is a conversation to be had, perhaps at a different time or maybe later in our podcast, uh, about the the idea of you you really can't do this well uh until you've got real rapport. And even though my son and I have great rapport, even then uh he he got a little bit flustered uh by me pushing on that. That's a different topic, that's a different topic. What I'm really trying to get at is the idea that you know it's hard uh for me to measure someone's understanding, uh because that's uh to me that's a a hard thing to express, my understanding. I can, however, talk to you about uh how I'm thinking about something. And to to elaborate on that or to to show what I'm meaning, I I'll tell you a different story about my other son uh briefly. So last night, uh and I believe it or not, folks, I I'm not making these things up. It's not it's not just because I needed an example for today's for today's uh topic. But last night, uh my son Truett was uh was showing me uh a bag of our wiffle balls, and it just so happened that in that bag of wiffle balls uh that we had, uh, there were 12 of them in the bag. Uh he was wanting to go out in the front yard and and uh practice baseball and wanted to use the wiffle balls, and I was trying to tell him he needed to use a different type and he was arguing with me. And so then we started talking about how many were in the bag. And uh and he told me there were 12 in the bag. Uh and then he told, I said, Well, how do you know there's 12 in the bag? And because I wanted to know how he was thinking about this 12. Yeah. Was he seeing it as you know, four copies of three? Was he seeing it? And his first response was, Well, I see six and I see six more. I see two groups of six. Um, and he kind of put his hands on them and showed me that. And I said, and then he's then he went on to explain, or I could see it like this. I didn't unpromp it, or I could see it like this. I see that there's a nine square, and then I add three more. And what's a I said, well, what's a nine square? What are you talking about with a nine square? He was well, there's there's nine of them in a in a square that's three on each side. And he does actually say three squared, and he's um, but uh mostly because he's offered up the idea of you that's a square. Yeah, uh and so he talks about square numbers, and we but anyway, he showed me all these different ways he could think about how there were 12 in uh in the box, and it and it made me think about what he was thinking. I could see how he was thinking about the problem, which lets me know how he understands multiplication, repeated addition, the fact that he could even show a square and a square number, and thinking about the idea of numbers squared uh are things that you can visualize as squares. All of those things were understandings that I can know about him because he explained his thinking. So to me, we're really talking about measuring or thinking about how students think about things, and from that being able to make assumptions or or make or jump to conclusions about what students understand about things.

SPEAKER_01 13:21

Yeah, I I I love how you differentiated that. Um, and I do think there is some nuance here that maybe isn't um, you know, make or break in terms of our conversation, but I think it's an interesting thing for our listeners to think about and talk with their colleagues about, or even just wrestle with in their own minds about the difference between understanding what students are thinking, which is a huge important um component of effective teaching, I would argue. Um, and then and then trying to understand what they understand, which I think for me flags the idea of assessment. So um I want to talk about these ideas both through like an uh an instructional lens and just through a like moving student learning forward. That's always our ultimate goal, but but also maybe challenging a little bit about how we assess or how we um interpret the results of assessment um and shifting the attitude toward assessment from assessing um what students know or don't know to understanding what students understand to help to make that responsive for instruction. I also kind of want to circle back to this, and you sort of said this uh early in your Tegan story that you felt like he was trying to give you the answer he thought you were looking for. True. And that just that just so resonates with me, especially I think about, you know, my husband does not consider himself a strong understander of mathematics. And when he talks about his classroom experiences growing up as a student in math class, he would say that all the time. Like I just felt like I could never figure out what the teacher wanted. Um, and I, you know, you talk about Tegan and Truitt all the time. I talk about my nephew Andrew all the time. He's in seventh grade, and I get to work with him once or twice a week on math. And it's it's a similar thing. I hear him say, like, I, you know, he's used to me saying, Well, how are you thinking about that? Or why does that make sense? Or what if this or what if that to try to get to his own understanding? That's the whole purpose of him working with me in addition to the instruction that he's getting in school. But I I've heard him say multiple times too, like, but but what do they want? What is what does the book want? Or what is this, you know, this quiz app that I'm on my Chromebook answering? Like, what do they want? It's not what, it's not what they want, right? So that idea of what the teacher wants, I think is a really common belief from students that that's what math class is, is thinking about how do I, like my goal in math class is to provide the answer the teacher wants. Right. And that's the thing that I would love for us to challenge in today's discussion is how do we say, no, math class is about understanding how students think and what students understand and guiding their thinking, using their thinking to guide them to deeper learning of important math concepts. So um I I just I I think that's maybe the nuance that matters in terms of our conversation. So I think that's a really kind of that foundational.

SPEAKER_00 16:52

I think that's a really important thing that you bring up, the idea that um w what we're trying to accomplish in mathematics uh in math class um is related to being able to uh get answers to questions that are out there. I I don't wanna I don't wanna say that there aren't right answers or that we're not interested in getting correct. Or that they don't matter. Or that they don't matter.

SPEAKER_01 17:23

I totally agree.

SPEAKER_00 17:24

I do want to make sure that that is a statement that we make. That answers do matter, and we should be trying to help students get to those things. But uh in the process of getting to those things, and maybe even uh more important than getting those things are are making sure that students learn about this thing called mathematics and all of the different ways you can look at it and how it explains the things that we uh learn about in the world. I mean, uh without getting too ethereal here, um, there's this there's this thing and this area of study called mathematics that has uh seemingly infinite uh possibilities within it. And what we are trying to do is trying to give our our we're trying to give students foundational understandings of concepts that they can build on. And we have to find out where they are in order to burn build on firm foundations uh for future learning, right?

SPEAKER_01 18:27

Right, yes. And thank you for thank you for calling out the specificity that we're not talking about answers don't matter or right answers don't matter. They definitely do. But kind of circling back to the assessment piece, like we've talked often, and we both know from our own classroom experience and the work that we do with with teachers who are in the classroom right now that there's this great pressure to teach all of the content that students are responsible for knowing for especially those high-stakes summative assessments that come at the, you know, the end-ish of the school year, whether it's a state assessment or whether it's um, you know, a college entrance exam or even a uh you know a district wide midterm or school-wide midterm exam, like there is this pressure to get through content. And I think there is an easy default practice to say, well, the fastest way for me to teach this is just to tell them. Um and and when we default to that very I mean, I did that a hundred times, a hundred I can't name how many times I did that. Thousands, yes, for sure, lots of times, because that's just sort of the natural instinct when you're feeling that time pressure. And I guess that what I want to get at here is by thinking about zooming out more often, by thinking about not focusing on how can students get the answer to this question, but how are students thinking about this bigger concept and how can I ask a question or provide a learning opportunity in class that helps them connect what they already know to this new idea and build true understanding of it, because I firmly believe that if we teach to big understanding, then the test questions are figure outable, right? Like students, if they if they understand the bigger concept of what multiplication is all about or what um you know scaling means, then they can reason their way through a problem that might look like a unique experience to them. Similar with your independent, independent variable, right? Like let's understand what those things really are. If I understand what they really are, I have more adaptability in unique situations and the ability to apply what I know to a different scenario that might be different than I've seen before. I think about how many times I would have students go, uh, I couldn't get this right on the test because you never taught it to us, right? Or how many times have we heard kids say that? That wasn't, you never taught me how to do this one. Well, less focus on teaching how to do the problems and more focus on teaching how to think about it. Um, I think goes a long way in terms of those high stakes assessments. So it's this counterintuitive um our our instinct as educators is to go to let me just tell you, um, because that feels faster. But we all also have had that experience of like, yeah, then a week later they don't remember it at all. You know, they act like they've never even seen the content before. So I guess what I'm trying to get at is it's one of those situations where it's worth slowing down and it's worth maybe fighting that instinct to just tell, to zoom out and try to give students the opportunity to engage in the bigger ideas, to ask them the more understanding oriented questions rather than. The answer getting questions um as as part of their learning experiences. So then when they get to the answer getting situation where the answer is the high stakes thing, they have the knowledge and the understanding to build out their answers.

SPEAKER_00 22:14

And I think what yes, and I think what you're kind of uh what you're talking about is the flexibility to apply what I know about said topic to a new context and a new situation. And uh again, as the proponent of of some of the things that uh you know we argue about uh in in certain cases, I I also want to say that as we do these kind of opportunities to understand the bigger ideas and apply them to uh broader context, students will need practice with that. Uh and and seeing the ability to take uh an understanding uh of multiplication, for example, with the wiffle balls and and getting student, you know, truett to kind of tell me what he was thinking about that, to take that and apply it to another scenario where maybe it isn't wiffle balls we're talking about, and it isn't just that we replace wiffle balls with baseballs, but we're we're looking at you know tiles on the floor and asking about how large the square footage is the room, or we're taking you we're taking that same uh model of thinking about multiplication and understanding multiplication and applying it to contexts that are significantly potentially different than what he saw with uh the wiffle balls as we were looking about and looking at those on the counter. Um you know, so so I think that you know, we we talk about these these ideas of building conceptual understanding and then being able to assess it, ask questions and be able to do this uh to understand how students understand certain things. Um so we talk about a lot a lot about those kinds of things. And the tendency for people then is well, do you just not want me to ever practice? No, no, no, no. I I I think we do want to practice and we should practice these kinds of things. Um and we should do them, you know, often enough that students get exposure to those. And I know you know, time ends up being a crunch. Um, but I think it's worth it um to build that understanding and give students the ability to apply it in different situations.

SPEAKER_01 24:53

Let's as we um, you know, come towards wrapping up this episode, let's could we maybe toss around some ideas for what this would actually look like in practicality for a teacher trying to do this more? And a couple of things are popping into mind for me, maybe for us to build from. I'm thinking about um NCTM's principles to actions, and one of their mathematical teaching practices is to elicit and make use of student thinking. So we know that um drawing out how students are thinking about the mathematics of you know, whatever we're we're doing, and then actually making use of that as a teacher. So thinking about how do I not only get kids to talk about how they're thinking, but actually build on student thinking, um, creating structures for students to respond to one another, to tap into one another as resources of knowledge, not just the teacher as resources of knowledge. Um, so I'm thinking about that. And I would love for you to maybe uh give your thoughts a little bit about what that looks like in action. And then the other thing to connect to are the mathematical practices, the standards for math practice, particularly, I think the ones that lend themselves to this idea will be um math practice two, where we're asking students to reason abstractly and quantitatively. Like if I'm putting math practice two at the forefront of my mind, that's where I'm saying, you know, I'm thinking about my work with Andrew last weekend on percent word problems. And it's like, okay, 20, what does that 20 mean in this problem? You know, 0.25, what does that mean in terms of this problem? So getting them to contextualize and decontextualize the numbers. And then of course, um, math practices seven and eight around using structure and repeated reasoning. So those are just a couple of things that I'm thinking about. What does this look like in your mind, you know, practically in the classroom? What are what are things that keep teachers could do as part of their instruction to ensure that, you know, what we're talking about today, those opportunities are provided for students.

SPEAKER_00 26:59

So I think, you know, one of the things as we were preparing uh this and thinking about this, one of the things that came to my mind was um that ability, you know, we talk about teachers floating around the room while students are are working and listening to the discussions. And and you know, in my mind, I'm visualizing the group, you know, conversation. They're working on some uh project, some task, something that's going on, uh, and and teacher monitoring that. And we do that, right? That's that's what teachers we we monitor, we think about, uh, but as we go about monitoring, uh, not just listening, but then asking questions about how did you come to this conclusion and listening to what the response was and and seeing what the student actually is thinking uh and and hearing their explanation, right? To dig in and dive in further. How are they on this particular uh topic that we're that we're talking about? Do they understand it uh deeply or are they just performing the procedures and they're not really sure why they're performing the procedures because somebody in the group told them this is how they're supposed to do it. So those kinds of things are are the ideal, right? The uh the individual one-on-one conversation. Is that practical in the in the large case? Probably not, right? When you think about um on the whole, can you do that all the time uh with every student, with every question? No, we'd be teaching for for years on the one topic. Um, so alternatives to that, thinking about if you use multiple choice in your assessments and your quizzes and those kinds of things, thinking about the distractors themselves. Do they tell me something about the students' understanding where how the student got there? Of course, students can guess. So you do have to kind of understand that you're gonna lose a little information there, or that you are at least making an assumption that, hey, if a student answered B, this is how they got there.

SPEAKER_01 29:04

Uh and that's probably making the common error they made is this, yeah.

SPEAKER_00 29:08

Making sure that there's common errors or common, you know, they stopped too early, right? They got a partial answer. So they understood the first part of the question, but they they forgot to finish out the question, right? On your discount problem that you were talking about uh with me about uh earlier. So so those kinds of things, thinking about our assessments, how do our assessments inform what we know about student understanding? If it's open response, the work that they show is how they explain what their thinking is. Um and so training students, thinking about and talking about uh how you explain your thinking in your assessments, I think is is a big uh a big deal uh in all of this. Yeah. And that's not new information. I'm just calling that out.

SPEAKER_01 30:01

Um no, it's a good connection though to the points that we're trying to make for for this episode. And I would just add, like, I'm sort of thinking through the lens of okay, what if I've been a very traditional teacher, lecture focused, you know, kind of back to my like feeling the pressures of getting through content and my strategy is just to tell them everything. How do I like ease into this? Because, you know, early on you said, Curtis, like this has to be a classroom culture thing. Like you can't just like tomorrow go in and expect kids to start talking about their thinking if that's not something they've ever been asked in my class before. So I I just kind of wanted to name a couple of resources that folks might tie into. I'm thinking a lot about um our conversation with Grace Kalamannick and Amy Lucenta and their routines for reasoning, particularly the three R's routine and even um the idea of using sentence frames or sentence starters to help kids, you know, kind of scaffold that idea of helping them start to talk about their thinking if it's not something they've been used to before. Um, and then the other thing around elicit and make use of student thinking, I think, and I was hearing you say this when you were talking about, you know, the teacher kind of wandering the room as students are working in small groups and um, you know, noticing what they're thinking about. It made me think of the five practices for orchestrating productive mathematical discussions. Um and we'll put a link in the show notes to that text if people haven't heard of that before, but um really a way of focusing on how do I as a teacher refocus on student thinking and then how do I elevate student thinking through a structured classroom routine. So I think those are great. What would you add?

SPEAKER_00 31:48

I just was thinking about you know some of the other things that uh Grace and Amy highlighted. Um, for example, the routine around uh uh annotation, right, uh of work. And I I just all of these are things that we can create as structures and practices and routines that we have in our classroom for regular instruction. This is about being uh mindful. And it's I guess it's kind of a cool way to start the year thinking about being mindful about our practice. Um, because it is very easy, especially uh in the months of February and March, as we start to really think about end-of-the-year assessments and state assessments and you know, accountability and all of those things, uh, that the pressure kind of the dial starts to get turned up a little bit, and it can be very easy to uh kind of bounce into uh sort of just the the tell routines um where we can see high value in continuing to go for uh really the idea, these other understanding routines, these routines in our classroom that really focus on student thinking. So I this was a really great conversation. Uh lots of things to think about, gave me some things to think about um as I go back and continue to work with both of my boys uh in their own respective math uh journeys. And so, Joni, I I I really appreciated uh today's conversation. Uh thanks for thanks for having me.

SPEAKER_01 33:25

Same, same. And I know this this idea of student understanding and student thinking is something I anticipate we're gonna circle back to over and over again in this new season. So uh thanks for the conversation today, Kurt, and happy new year, everyone. Happy New Year.