We think Dr. Peter Liljedahl has some great ideas about how to keep students thinking and reasoning in math class. In this episode Pam and Kim continue to discuss some of his ideas and how they align with the Math is Figure-Out-Able mindset.
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Hey, fellow mathematicians. Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris.
And I'm Kim.
And you found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But ya'll it's about making sense of problems, noticing patterns, and reasoning using mathematical relationships. We believe we can mentor students to think and reason like mathematicians. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keep students from being the mathematicians they can be.
Hey, Pam, so we got a comment from a Journey leader recently. You know her, Susan Smith.
Ah, yes. Mmhmm, Mmhmm.
And she actually said, "I heard an interview with Peter in which he talked about number talks, or jumpstarts, or Problem Strings, and how they could be a whole class portion, followed by a Rich Task at vertical boards." And she said, she's been doing 10 minute Problem strings as a class, and then she introduces Rich Tasks at a vertical board board for the majority of the class time. And then, she says she's ending with consolidation. "And the strategies they learned in number strings are highlighted when they naturally appeared and used in tasks by students."
Nice, which is really what we talked about in the last episode. We kind of suggested that we like a lot of what Peter Liljedahl says in Building Thinking Classrooms, but that we wouldn't suggest that Problem Strings... So Susan, they're called Problem Strings, not number strings. It's fine. That Problem Strings are not, the ones I write especially, are not intended to be at vertical spaces. They're intended to be whole group, just like she said, whole group conversations. And then, after you've done a Problem String, send the kids to those vertical, nonpermanent surfaces and do those Rich Tasks. And then, you might consider that a follow up to that Rich Task could also be a Problem String to cinch something that happened in that Rich Task. So, you could actually change up your order a little bit and start the day with the Rich Task at the vertical, nonpermanent surfaces, consolidate, and then...we would call that a Math Congress...and then finish it with a Problem String to kind of cinch something that came out in that Rich Task. So, yeah, Susan. That's great. And super cool to hear that Peter's starting to talk about that too. He's a great guy. I mean, one of the things that I think he's proven over and over again, is that he listens to teachers and that he is, you know, super thoughtful. And he goes with the evidence. And so if he's finding that people are having success with that, and he starting to talk about it, so do we. And so that's awesome. I think we agree there. Cool. In the last episode, Kim, we kind of focus...last two episodes...we kind of focused on part one of his four part set of 14 teaching practices. Today, Kim, let's focus on more on part two. At least some of the parts. I think when we focused on part one, we talked about kind of all of the parts. In Part Two, we're just going to focus on a couple of things that we find really noteworthy. So, in part two, one of the things that he says is to answer only "keep thinking" questions. What's that all about?
Yeah, and well, so it's super interesting because teachers we talk with all the time, right? We're talking, talking, talking. And one of the things that Liljedahl says is that he came to the startling conclusion that a typical teacher will answer between 200 and 400 questions in a day. That's probably not surprising to many of us. With some people answering as many as 600 questions. No wonder we're tired. The problem is that with answering all these questions, and it's antithetical to the goal of getting students to think, right? So, it's interesting because we're teachers, and we answer questions, and we want to be helpful.
I was going to say, I mean, we're helpful, right? Teachers, of course, we're going to answer student questions. That would be mean not to answer their questions.
Yeah. Well, so this actually made me think about kind of the stage of parenting I am right now, where it's lots of guiding. And so, I'm like, giving kind of instructions and teaching lots of things. But they ask a bunch of questions. And sometimes we need to just set them forth, and try and maybe fail a little bit. If I direct every single move and every single thing that they do, are they really learning anything? So, you know, we've heard teachers say, "My kids can't do this."
Before you go back to teaching, if I may. You know, my kids are older than yours.
They're adults now. And looking back, I wish I maybe was a little bit less of a helicopter parent. A little. I wish I would have answered a few less questions.
Like, I can see more clearly, now, the benefit in having my kids feel empowered to fail early. Rather than get a little older before they start taking risks. And then, let's be clear, then the failures are bigger. And hear me right. They're doing great.
But in hindsight, I see the value and the wisdom in... We're saying "fail". It's not like we're going to leave kids hanging, right? It's not like we're going to hanging over the precipice, they're going to... Not that, but.
But be okay in trying in things, not knowing.
Wait, say that again.
Being okay with trying things, not knowing the exact outcome. So, we hear this a lot with teachers who say, "My kids can't do this thing that you're talking about." You know, "They can't think, and they can't..."
Yeah. "Pam, you don't understand my kids..."
"My kids they can't. Working memory."
Right. Yeah. But sometimes when they describe the classroom dynamic, looking from the outside, we can see that they're directing every move, and they're not giving the space for their kids to even attempt to think. We're doing it all for them.
Mmhmm. Which is tricky and subtle, right? Like, we're not trying to be ornery. There's this thing about the balance maybe between how helpful we are, or at least how helpful we think we're being, versus how helpful we can be when we actually empower students to think. And I don't think it's... I don't know. It's trivial. It's not... I don't think it's trivial. Where's the line? What's the balance? So, let's talk about that today. Like, and I think Peter gets helpful with that, Dr. Liljedahl, in his Building Thinking Classroom book because he identifies three main types of questions that students ask, and he then suggests that we focus and only really answer one of those types. So, three types of questions. One of them is "proximity" questions. Those are kinds of questions that students only ask when you're in proximity. So, when you are close to them, then those questions get asked. "Stop thinking" questions is what he calls this next set of questions. And it's cute that he... Or, cute, I don't know. It's it's well named, that he calls them "stop thinking" questions because he's suggesting this category of questions actually stopped students thinking. And so, of course, we don't want to answer those. So, don't answer questions if they're just "proximity" questions. Don't answer questions if they're just "stop thinking" questions. And then he says, there's a third kind, "keep thinking" questions. So, let's dive into these three types a little bit more, Kim.
Yeah. So, "proximity" questions, you and I've actually talked about a little bit before, because we recognize that this happens. And so we kind of jokingly say, "Get in, get out", right? Get in to the situation, say the thing, give directions, do, you know, whatever you're attempting to communicate with your students, and then leave, get out. We don't want to hang out, you know, in a small group for too long when we go visit with kids. Because what we hear are things like, "Do we have to learn this? Is this going to be on the test? Is this right?" And those aren't furthering the math in any way. They're just saying it because you happen to be standing right beside them. So, I have a couple of other quotes from the book. It says, "These questions are motivated by the reality that, for students, thinking is difficult, and it's hard to decide for themselves, that what they're doing is correct. If they can just get you to do that for them, their life would be so much easier. So, students ask questions with the hope that you will answer it and they can stop thinking."
And let's be clear, it is easier not to think.
Where was I just? Was it when we were filming... Excuse me. Was it when we were filming, Building Powerful Fractions when a teacher said, "Oh, I love what we're doing, Pam. This is so much easier." And I paused, and I was like, "Don't say that. What you mean is, it's so much more figure-out-able." And the teacher kind of thought for a second and was like, "Yeah, that is actually what I mean. It's not easier. Like, my brains really working hard. But woah, it's like so figure-out-able." So, there's a difference between like things being figure-out-able, and therefore you feel like it's doable. That's brilliant. We want to really advocate that. But thinking, you know like, really diving in, and fussing, and fumbling, and grappling...that's probably the best word I want to use there...grappling with relationships, grappling with what's happening, so that then working out and making sense of. That's hard work, right? And so, yeah. He's suggesting that when you are in proximity, when the teacher is nearby, and the student looks up and is like, "Oh. Hey, save me. Is this right?" And then, if the teacher goes, "Oh, yeah," or "No, no like that one," or "Ooh, not add, subtract," or "The number up there." Like, as soon as the teacher dives into quote, unquote, "help", bam. Yeah, that's absolutely easier, and then the student is doing less thinking. So, if we're trying to build a thinking classroom, which we are, I think we might maybe say we want to build a reasoning classroom. Not to say that thinking is not right. But maybe I would add reasoning. And we've been talking about that for a long time. Then absolutely, those proximity questions (unclear).
And I think there's kind of a connection between the "stop thinking" and the "proximity" questions because they, if you're there, they're going to ask you something. And typically, it's in order to stop their thinking. So, here's another quote, it says, "What's important is that you do not answer 'stop thinking' or 'proximity' questions. And that you read the situation..." I love that, "You read the situation, so as to be able to give the best response when such questions are asked." Right.
I know why you like that so much.
"Read the situation." Because you are the one who said, "Know your content, know your kids." Right? So, is that what you're thinking right there "read the situation" means "know your kids"? Yeah. So,what do you mean by that? Like, if you walk up to a group of kids, and they're asking either a "stop thinking" or "proximity" question, and how does that "know your kids"? How does that impact how you might handle that situation?
Well, I mean, if we know our students, and we know kind of what their hang up is, I'm not going to answer questions because I happen to be there that stops their thinking about the math. If I have a student that is not going to move forward because they're not understanding something, I'm absolutely going to answer that for them. If it's a clarifying question, for sure.
You don't mean math though, right? Right there. You mean, Like instructions or...
Like, their role at the moment, if they're supposed to be writing on paper or not. In other words, know your kid. I can picture the kid who's kind of like, fussing around to do anything but dig into the work. And when I walk by, they might go, "Hey, are we supposed to do this in our notebook? Is this supposed to be..." That kind of a question is a kid trying to just waste time. But if I walk by either...now I'm going to make a guess here, so you can correct me...either your oldest or my second, and they say, "Hey, is this supposed to go in the notebook?" Then, I would absolutely answer that question because that kid, it's a real question for that kid. It's not it's not a time waster. We just know our kids.
Did I guess that right?
And what will happen is the time will be wasted because they're so focused on that question that then they won't move on. They will absolutely sit there thinking about it, wondering about it, worrying about it. And if the goal is to maximize the amount of thinking and reasoning time in the classroom, I'm going to answer that question to get them back into the math.
Exactly, yeah. Yeah. So, you're not going to leave... How do I say this? You're not going to answer a mathematical question to starve them or keep them from the learning of the math. But you are going to answer a procedural kind of, "Yes, you're supposed to be in your vertical, nonpermanent surface, randomly chosen group right now." Or, "Yes, we're working on this problem right now." Or, you know, like, those kinds of, "let's keep you doing the work" questions.
If you know the student, and that student's not just popping it off to waste time, then you would absolutely do that. Can you think of any other examples of times that you would know your student, and so it would impact whether you would answer a question like that? Yeah, so am I. I thought I had an example when I was thinking of this the other day, and now I can't come up with anything.
I mean, it's so interesting, because right now I can picture specific kid's faces in my mind, you know? Like, the kids that...
I don't know. I don't know that I have a specific example. I'm sure I will as soon as we're done talking.
Yeah, we'll be thinking about it. Well, and part of maybe what's hard for both of us right now is, we don't... Early, quickly, we develop a thinking classroom. We develop a reasoning community of mathematicians.
So, we don't get a lot of "stop thinking" questions. So, if a kid asks a question that might be quoted here as a "stop thinking" question, then we're going to probably know that it's not a time waster, and that we're actually going to dive in and answer it to keep that student going, right? Does that make sense?
(unclear). So, in his book, he suggests that 90% of those hundreds and hundreds of questions are "proximity" or "stop thinking" questions.
It's a high. Mmhmm, mmhmm. So, let's talk about the other kind.
Okay, so if we're not going to answer "stop thinking" questions, and we're not going to answer "proximity" questions, either of those, unless we know the kid, then what are the other ones? He calls them, "keep thinking" questions. Nice name. You know, I like names that aren't just names for silly sake.
Totally describes. "Keep thinking" questions. You know, it's funny. He doesn't actually say a lot about those in his book. And as you and I have kind of been brainstorming what those "keep thinking" questions are, we think they're kind of the questions that are... Let me say it. If they're "keep thinking" questions, kids want... They have... How do you say have dived in? Dove in? Diven in?
They dove in.
(unclear) that word in English. They dove in. There we go. And so, they're in the math. And it's almost like they're wondering aloud. It's almost like they're saying things like, "Will that work all the time?" But they don't actually expect you to answer it. They're like wondering if it will work all the time. "Would that work with negative numbers?" And they're looking at you not as if...again, when we've developed this classroom atmosphere, this community of learners...not because they're expecting you to say yes or no, it's because they're like, "I'm wondering that. Wonder that with me." Right? It's like an invitation that the questions are more mathematical. What they're not is permission. Like, "Why are we learning this? Is this going to be on the test? Do I have to do this in my notebook? Do we have homework tonight?" Like, all of those kind of sort of like procedure or permission questions, he suggests with those, just smile and walk away. In the section where he talks about "keeping thinking" questions, he does mention a few that feel kind of permission based to me, but it's permission based about, "Can I go further?"
And I suspect that what we've seen is true. And that the more that you create a situation where it's a thinking classroom, like kids aren't really asking permission to do that. They know that they can, and so as you get further into the year, they stop asking permission to go deeper because you've already established that that's what you're okay with. They can ask themselves questions. They can dig deeper into what's happening. So, it's less (unclear) "can you?" and then it's this wondering about that you're talking about.
So, I think Peter would agree with you. I think he said something about that when you've developed thinking classroom. I think he actually said something, students will eventually stop asking those (unclear).
Oh there is that. Yeah, okay, cool.
Can we go further, deeper kinds of questions because yeah, they will have built in themselves the confidence to know, "Yeah, that is what we do here." Like, what we do here is go deeper, further, wonder, pursue the questions that we have, and see what we sense we can make of the math, right? Because that's the whole situation that we're working on. Yeah. Nice.
So, I mean, I think it's really cool that you can envision a situation where the kids are, you know, if you're ignoring "proximity" questions, and you're not answering those, and you're not answering "stop thinking" questions, and the "keep thinking" questions kind of go away from questioning and more wondering, what a great atmosphere where kids might be calling you over, but it's to share generalizations or things that they've learned. And it's less less asking, and more, "Here's what we've discovered. Here's what we figured out." It's pretty cool.
Yeah, and maybe even still, "And here's what we're wondering."
Right. But it's less asking you to solve that for them. Right? And more...
...I want to continue this. So, he suggests some things to do. And one is that Peter suggests that you can tell students about the three types of questions, which I find it really interesting.
And to be clear. Yeah, as I was reading that part, I found it noteworthy that he said, "But maybe don't do it at first." Like, don't front load, "Hey, I'm about to do. I'm about to only answer these questions." Maybe start only answering the questions, and then when students start like, "Wait, what's going on?" Then, you can be like, "Oh, yeah, let me explain what's happening." I thought that was kind of interesting. He lists some reasons why, that if you do it at the front, students will hear you saying, "So, therefore behave." And then, they'll be like, "Oh, what if I don't want to behave?" And so, instead of doing it that way, like he does say it can be helpful. And I think there are... Again, know your kids. I think there are some students who it might be helpful sooner than later. You know, I can picture some kids maybe on the spectrum or that it will stop their thinking to have something shift. "Wait, this is different than it's been." Like, "Help me out here." But to start kind of doing only answering "keep thinking" questions, and not answering the "proximity" and "stop thinking" questions. Do that first for a while, and then say to them, "Oh, you might have noticed. And let me tell you what's happening." And then, students don't maybe react poorly, in that case. So, how you tell them seem to matter a little bit.
Yeah, it's funny, because I actually, with my oldest last night said, "Oh, you're only asking me because I'm standing here," and I walked away. (laughs) I did. And, you know, I can do that because he's my kid. But there's so many applications. Okay, so three strategies. One...
When you say you can only do that because it's your kid, I think you can say that very same thing when you know the students.
Yeah, maybe that. Yeah.
I think if a student says something that's it's just to kind of look like they're engaged. I mean, we all know those students that as soon as you walk by, they do something to look like they're doing what they're supposed to be doing. I think in that moment, you can go, "Look, I saw what you were... You know, you were on your phone playing around. Like, you can't hide it from me. No, I'm not going to answer that question because you're only asking it because I'm here, and you're trying to look like you're doing what you're supposed to be doing. How about if you actually do what you're supposed to be doing." In a polite way. You can say that in a polite way, right?
But again, know your kids.
So, three strategies. One, reduce proximity during the first three to four minutes of the tasks. That's when lots of proximity questions happen. So, as we said earlier, get in, get out. And just don't be around for those questions. Another really good strategy is don't answer any questions from individuals.
Oh, yeah. I thought this was a good one.
Yeah, if an individual comes back, then you redirect them back to the group and ask, "What did your group decide? What did your group say?" And then, individuals ask the group, and then the group can ask the teacher. Found that interesting.
Yeah, that's a nice one. I like that a lot.
And then probably my favorite is, lead with your own questions. And this is exactly something that we do when we're, you know, in a Rich Task and circulating with small groups. Instead of just walking up and listening. We do listen in, but usually we walk in with our question, rather than just wait for them to bombard us with questions.
Yeah, I have seen you do this very thing where you'll walk up to a group, but they don't notice you yet, and you'll look at what's on their vertical, nonpermanent surface, and you'll kind of then walk up and go, "Oh, I'm really interested in this. Could you guys tell me about this? What's happening here? What's this thinking going on here?" I've totally seen you do that. Yeah, when I read that, I was like, "Oh, yeah, we do that." We absolutely walk in with something that we're going to ask them about. Tell me about your thinking here. Yeah.
Something that you mentioned earlier, that I'm going to mention, again, is that he also says to smile and walk away. And we have a little bit of a different take on that, right? Will you share about that?
So, we would say... And I think we would kind of agree with the smile and walk away. But we would add to that maybe that often a super good response to any question from students is a very neutral response. Whether it is a question from students or a response. Like, if the students... You know, you walk up and they say, "Hey, is this right?" Or, "Look what we did here." I might be inclined, or teacher might be inclined, to be super helpful and go, "Yeah, yeah, that's really right." Or "Oh, no, we need to work on that. That's wrong." Or, "Mmm..." you know, with it that look of like, the look where kids are clearly like, "Oh, dear, it's wrong." We like a neutral response. We like to go, "You think? Tell me more about that."
WAIT! I'm going to interrupt you real quick. Because my favorite thing that I see you do in workshops, and when you're working with teachers and students, is a student will say, "I think it's 13." And almost immediately you say, "You think it's 13?" And they're like, "Uh huh."
Even if they don't say, "I think it's 13." Even if they say. You know, like, "Hey, so and so what did you get for that one?" And they were like, "43." I'll say, "You think it's 43." And I'll write 43 on the board. And then, sometimes I'll be like, "Did anybody get another answer?" So, this sort of real neutral response, we love that. We have a whole video montages of different teachers, where the response is just this kind of neutral, or super interested, kind of a concentrated look like, "Oh, you think it's 43?" "You think you think it's 50?" "So you think the line is y equals x plus 2?" Like, whatever it is. "That's what you're thinking?" It's this response back of not right, not wrong, but tell me more. Like, I'm super interested. I'm curious. And I'll never forget, when I said to you, "Kim, how do you get kids to share their thinking and not just focus on the answer?" And that day when you said,
"Pam, it's what you celebrate." And I was like, "What do you mean by that?" And you're like, "It's what you talk about?" So, we just said, you know, 43. "Oh, you think it's 43? How? Why? Tell me more about that? Did anybode else think it's 43? How are you thinking?" And then, the rest of the conversation focuses not on the 43. It focuses on how you got the 43 or the y equals x plus 1. Or whatever the answer is, that becomes a given. Alright, we've got the right answer now. Like, tell me how? What are you thinking about. What you celebrate. And if you celebrate the thinking, then students are going to dive in with their thinking and less, "Is this right?" "Is it wrong?" "Am I doing it correctly?" "Did I get the next step?" "What's the next step?" Less of those "stop thinking" questions and more of "Oh, what we do here? What we do here is think, and reasoning, and share that with everybody, and then sharpen our thinking as we bounce it off of the rest of the class.
Yeah. Yeah, actually.
Is that what you were hoping (unclear).
So, one thing that we're hoping the listeners think about are the kinds of questions that your students are asking, so that you are only answering "keep thinking" questions. That's the goal.
Yeah. And, Kim, I will never forget the day when I was in a classroom of high school students. And we were going to film the next day, and so we always go in that first day and we make the name tents, so I can pronounce their names correctly because the names are super important. And we do a short Problem String, so they kind of get a feel for what we're going to do on camera the next day. And I will never forget this group of high school students. I'm in the middle of this short Problem String. It's not really a time to ask a question. This kid right in the middle of the room, second row, right in the middle of the room raises his hand, and he says... I'm thinking in my head, "It's not really time to ask a question, but okay, what do you got?" And he goes, "It's almost... It's almost like you want us to use what we know, to solve the problem." And what was super was the entire rest of the room was like, "Yeah, that's true. Yeah." So, he didn't really expect an answer for me, but I just smiled and said, "Mmhmm, yep. Yep. Just use what you know." We had already, in that short period of time, started to build the community where he was like, "Huh. Like, here. The task here, the job here is to just use what you know. Okay. Alright. I'm going to dive in and do that."
Yeah. So good.
And then, we get thinking and reasoning classrooms. Super cool. Ya'll, thank you for tuning in and teaching more and more Real Math. To find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutable.com. Let's keep spreading the word that Math is Figure-Out-Able!