What role does terminology have in math teaching? How do we want students to approach problems? In this episode Pam and Kim discuss when to teach vocabulary how we can prepare students to tackle problems with confidence.
For more about Anchor Charts check out Episode 123
Check out our social media
Instagram: Pam Harris_math
Facebook: Pam Harris, author, mathematics education
Linkedin: Pam Harris Consulting LLC
Hey fellow mathematicians. Welcome to the podcast where Math is Figure-Out-Able! I'm Pam.
And I'm Kim.
And you are in a place where math is not about memorizing and mimicking, where you're waiting to be told or shown what to do. But it's about making sense of problems, noticing patterns, and reasoning using mathematical relationships. We can mentor students to think and reason like mathematicians. Not only are algorithms not particularly helpful in teaching mathematics. But when we just have students rotely repeat steps, that keeps them from being the mathematicians they can be. Alright, Kim, we have a quote from a listener. So, Stacy Stoker sent us a message, and she said, "I would love an episode about why key words aren't a good idea. Several teachers in my school have started making bulletin boards with key words for each operations. They're cute, but..." And then ,we also had a message from Autumn Wallace who said, "Or how about a whole episode on keywords?" So, let's dive in, Kim. Let's talk about keywords.
It's funny because those actually go together really well, right? Like, keywords always end up on a cute bulletin board with color, and picture, and all the things. We actually also had a Journey member ask something related as well. And she had a table where she had synthesized some information and some understanding to share with her teachers. And we do call the synthesis of information an anchor chart. So, can we start there? Let's talk about what anchor charts are, how we advocate for them, and then this keyword bulletin board idea.
Yeah. So, I would like to start by talking about something that you really helped me think about, where... Maybe we thought about it together. Where we talk about the difference between "just in time"
Versus "just in case?"
So, to be clear, the way I was taught was "just in case". My teachers did things like they said, "Hey, we're about to start this new unit. Here is the vocabulary that you need just in case you're going to need it in this unit. We're going to do it right now." So, just in...
Oh, my gosh. Yeah, it's the geometry unit workbook, right? Draw the picture in the box in the top, write, the definition, give 3 examples.
Yep, yep. And you do that at the very beginning, so that then we can... At least we... I want to say "pretend", but that sounds kind of rude.
We assume. We assume that then students, "Ah, now that you have written the definition. You've drawn the picture. Now, you can use it in the rest of this unit. So, we have to start with that vocabulary, so now students can use it as we go." We call that "just in case" vocabulary. We prefer "just in time". That means that we provide students with experiences, so that they want to use certain words. They're trying to describe their thinking. They're trying to describe what's happening. And as they're like, "You know? You know, the thing. The number on top." Then, we're like, "Oh, we call that the numerator." I'm trying to think of another example where kids are like, "You know, when the line is more... is more..." And I'm like, "Is it like steeper?" And they're like, "Yeah, steeper! Because the rate was, you know. It was like they were going faster."
"So, when the rate is higher, when the rate is faster, then the line is steeper." And so as they are grasping to describe what's happening in their head, what's happening in the problem, what's happening in the situation, then just in time we're like, "Oh, yeah, yeah. We call that horizontal." Oh, yeah, yeah. That's the denominator."
Yeah. I actually think...
"That's the ratio." Go ahead. Sorry.
I think that you actually share "just in time", "just in case" with me. Or at least those words about it. But I do want to add in that recently you talked about social knowledge versus logical. And so, vocabulary is a big social knowledge, right? It's one that we have to tell kids. And so, this is not this like big, let me stop what I'm doing, have this huge lesson on vocabulary. It's very much like you say. When a kid is grappling, and "Umm, I'm not really sure what to say," you're like, "Oh, that's the thing..." And you just casually use the word appropriately when it comes up like, "Oh, yeah. We just call that this."
Yeah. And let me dive into that a little bit. So, as you are developing a logical kind of knowledge as kids are experiencing, they're gaining connections, they're making mental, neural connections in their head. As they're really grappling with that logical knowledge, then handing them the social tag, the social part of it, and saying, "Oh, yeah. We call that blank." That's totally appropriate. And that is the right time for it because now they have this need. They're really grappling with this logical kind of knowledge. And so now, when you hand them that social tag, then they can just like "Oh, yeah. Okay. You call it that? Okay, great." And then, they can move on. So, that also might help out, parse out the difference between logical knowledge and social knowledge, convention. Convention, it's okay to just tell them. But don't tell them ahead of time. If you tell students ahead of time, that "just in case" kind of idea, they don't have anything to hang it on, so now it becomes a bunch of rote memory. That isn't as usable. It isn't as helpful. And kids who don't memorize well, it's not helpful at all. It's more efficacious. It's more useful. It will work better, we believe, if we do a lot of this stuff just in time, not just in case.
I think as we sometimes do, we've kind of gone off like a little bit off to the side on vocabulary. Which is fantastic. I appreciate all that conversation. But let's get back to the anchor charts and keywords.
Well, keywords or vocabulary, which is kind of why I went there,
Well, yes. But I think maybe what Autumn is talking about is the keywords like for operations.
Yeah, so let's talk about anchor charts for just a second.
Anchor charts, what we mean, versus maybe what some other people mean. Can we talk about those for just a second? We have a whole episode on anchor charts. And sometimes we get asked like, "What goes on anchor chart? How do you make the anchor chart? Why don't you make anchor charts and sell them on..."
Oh, yeah. People totally ask why we don't sell anchor charts. Yeah.
Yeah, like we have it on the Math is Figure-Out-Able website, there's shirts, and charts, and things like that. And people have asked, "Can you make us a poster of the major strategies?" And that's a hard no.
Yeah, we have said that that's a hard no. Like, we're not going to take. In fact, in some of my books, we have sample anchor charts. And they're like, "Hey, if you'll just, you know, make these in full color and everything, then will totally buy them, and we'll put those major strategies up on the wall." So, we don't advocate those pre-made, fancy anchor charts for a couple of reasons. But probably the biggest reason, or at least the biggest one in my head right now. We want to create them together because in the creation, learning occurs. Yeah, so it's not about buying a pre-made one, sticking it on the wall. That's like saying, "I can unzip your head and pour in the knowledge." We want to actually help kids understand the thing better by creating the anchor chart together. It helps in the sense making and everything. So, Kim, that thing that you helped me understand where you said, "Pam, Pam, can you agree that we can all do more than we can say, and we can say more than we can represent?" And so that Do, Say, Represent, I kind of took that and said "Yeah." And that also is a way of thinking about learning, in that we can do stuff with kids, give them a problem we're thinking about. They're thinking about it. They're grappling. And then, we can pull out words. We can help elicit their thinking. If you can see my hand right now, I'm like grabbing and like pulling toward. Can we help kids put words to their thinking? We know that that's going to be like halting, and they're going to be reaching for words, and it's kind of difficult for them to put words to their thinking. That's true for all of us. But as they do that, as they begin to put words to their thinking, and we help make sense of it, and we have students re-voice what someone else said, and we have students talk to each other. "Does everybody understand that? Can somebody ask him a question? Can you restate what that student said." All of that is toward students grappling, and putting words to, and making sense of those big ideas. And then, now that we have verbal words to it, then we want to represent what's happening. So, it's the Do. First, mess around with it in your heads. Then, the Say. Pull and elicit words out of kids. And then, let's Represent that on an anchor chart. So, the anchor chart is the last part of that sequence of helping kids learn. And as we create that anchor chart, we're asking kids, "What kind of words do you want to put up here? We've just had a whole discussion about this. What are words that are put-on-able?" I'm making words up here. If we're going to make an anchor chart, we're not going to put that whole discussion. We're going to the whole paragraph or the whole essay up here. So, now let's synthesize. We have to actually break it down a little bit. What's the word that means (unclear). We want to break make it more brief. We want to... Help me, Kim.
Summarize! Summarize! There's a good word. We want o summarize. Well, what are the key words that we could put on the anchor chart? Oh, check it out. Here's our conversation about keywords and anchor charts coming together. But notice, it's keywords that we're pulling out of students. Now, i's not all... To be clear, it's not all fluffy from kids. It's not all just whatever they say. No, no. We are carefully crafting the conversation.
We are suggesting the social terms, as they're having the conversation. We're helping craft and direct. In fact, Kim, you're a master at it. I remember one day, I was like. You and I were having a conversation. You said, "You know, Pam. I think teachers might find it interesting. That they might think that every time I do this, there's a different outcome. This Problem String, this Rich Task, this activity with kids. Every time I do this thing, it feels so organic to participants in workshops and kids in the classroom. That they might think that, you know, every time it really depends on what the kids say, and it's totally... You know, if you teach three different sections of fifth grade math, that it's a completely different conversation because it's completely different kids, and they come from different backgrounds and all that. When actually, it's the same math."
It was interesting that conversation that day. You're like, "You know, Pam, teachers might be interested to know it really ends up that we have about the same anchor chart coming out of all three sections. Enough, that when we..." And we suggested this in the episode on anchor charts, but I'll say it again. When you're creating an anchor chart with students. And again, you first do the thing, and then you pull the language out of them, and now you're summarizing it, and you're putting those keywords on an anchor chart. That when you do that with kids, first period, you might say, "Okay, you know, here, let's put it up on the whiteboard, and we'll kind of... Okay. Yeah, I'll put it on chart paper later." Second period, "Hey, let's put it on the whiteboard, and I'll put it on chart paper later." Third, fourth, fifth. However, many periods you have. That last period, you're like, "Hey, let's put this on chart paper." So, that the kids the next day, it's not going to look dramatically different enough. And they know that you're transferring it, and they know you have different periods. But the conversation is going to be similar enough because you know the math, right? Know your content, know your kids. You know the content enough that as you create that anchor chart, it's going to be similar enough to the other periods that when you finally put it on chart paper, it's going to look enough like the ones that the kids developed all through the day, that it won't be it be different.
Well, and it might be that it's not the exact problem that they chose in second period to be represented on the poster, but the big idea is there. And when you teach in the way that we're suggesting, it's less about the individual problem, and it's more about the big idea. And so, when they look at the poster that they helped create, the big ideas are still there.
And they should recognize those bigger ideas because those were the big ideas they were developing, as they were developing that anchor chart. Yeah, absolutely.
So, part of the anchor chart idea is that you're synthesizing learning from the students, right? Back to this keyword thing. There are keywords. It's the students' keywords, key ideas, key learnings, not mine. So, I'm a big fan of starting the year and my room's kind of blank. I mean, I imagine yours might have been too. Because throughout the year, as we're working, and growing, and learning together, that's when things go up on the walls. So, I imagine that some of my student's parents, when they would first come in would be like, "Oh, wow. She's not cutesy at all. There's nothing on the walls." But I would share with them that as the year goes on, we're doing things together, that that is the anchoring. Like, I want their work on the walls. So, it's less about my keywords or my understandings that I want to share with you. And we do see that pretty often. That teachers want to share, or leaders want to share, important pieces of information by just putting it out there. We're suggesting something quite different.
Yeah. And let me clarify. When you said, "It's the students keywords, not my keywords," that doesn't mean that it is student, cavemen-like strategies.
Because all through the discussion and creating the anchor chart, you're dropping those social words. You're dropping those keywords. As kids are grappling with the relationships, you're supplying, "Oh, yeah, we call that horizontal." Or, "Yeah, we call that slope." Or, "That's multiplication." Or, "Oh, it's almost like you're saying every time you added a bit too much, like you went Over?" "Yeah. Like, so if you Double on, and you Halve the other, you get the.." I'm just trying to mention a bunch of examples here. "So, when you divide, you can think about division two ways? It seems like you guys are talking about division in two, and one of them is all about like the number in the group. That's... Okay. Alright, and the other one is kind of the size of the group." And you walk away. Again, I'm not handing that out at first. I'm helping kids. As they are grappling with words, I'm saying back to them their ideas, but in kind of the precise mathematical language. So, I'm dropping that precise mathematical language, as they're grappling with making more precise mathematical language. So, by the time we get to the anchor charts, the keywords are not random, haphazard, "young kid" words. They're the words that we need, that we want because you've been sort of peppering them throughout the conversation. Did I say that well?
Yeah, I think so.
Does that makes sense?
Yeah. So, quite a different thing to co-create anchor charts together. Let's focus a little bit more on not just the creating of the anchor chart part of it. Because that's not quite all this conversation, right? We started out with the keyword thing. Let's go a little bit back to the keywords.
Yeah. You might find interesting that when I knew we were going to do some conversation about keywords, I just did a quick Google "keyword poster" when we got this question or the comment. So, I just... If you Google "keyword posters", you will probably be maybe a little surprised. Well, maybe not surprised. But the thing's you'll see, Pamela.
We were surprised.
Did you know that... Well, first of all, the problem with keywords is that people hang their hats on these words and want kids to circle the key word and say, "When you see this word, that means this particular operation." And it's just not true. Some words can indicate some operations in some problems and different operations in different problems. So...
..."in all". When I looked at these posters, the phrase "in all" is often on all four operations posters. So, how's that even helpful to be on the poster, right?" Like, so a kid sees "in all", and they're like, "Oh, I still have to guess." "Per." The word "per", is only found apparently on division posters.
The phrase "are not" is subtraction. I don't know if you knew that.
That was on a poster saying, if you see "are not" think subtraction?
Mmhmm. And listen, when I was like looking at these pictures. I'm blown away by this. Because you know I never really did anything with keywords. And I would have students come up kind of through the grade levels asking me about them, and I just... You know, not so much. But fifteen words per operation sometimes. So, memorize these fifteen words, instead of understanding what the operation really means.
Well, and so we can choose to spend our class time rote memorizing social things. Or we can choose to spend our class time building kids brains. And when I say that, I don't say that in a shaming, finger wagging way. We were all taught that way, so it is quite a shift to think about it a different way. But let's attempt. Let's at least give it our... You can only do better when you know better. But when you know better... So, let's consider. Like, what does it look like to not rote memorize fifteen words per operation? "Okay, when you see these words, make sure you do this thing." Instead, what would it be like to do to actually build kids brains?
Yeah. And I don't know that we've mentioned this yet. But what if there was a subtraction problem, and I actually want to think about it as a missing add-in? So, then what do I do? Or if it's a division problem as a missing factor? So, we want the numbers to drive us, not some keywords. Let me tell you one more thing. Cliff Notes. Do you know CliffsNotes?
I mean, I...
You don't want to read a book, you get the CliffsNotes version. It's like a summary of the book.
High school students might love that. Anyway, CliffsNotes says is, "You need to be able to translate words into mathematical symbols, focusing on keywords that indicate the mathematical procedures required to solve the problem." "Focusing on keywords that tell you the procedures required to solve that problem."
So, you're saying there's a book out there, CliffsNotes, that said that this is why we do keywords?
So, keywords are so important because we're going to translate them, those words?
They say that, "...the keywords indicate the mathematical procedures required."
Ah. And so, we don't do that. We don't do mathematical procedures. I mean, at least we don't... Yeah. We don't suggest that there is a procedure required to do this problem. Rather, we would like to reason through that problem using the mathematics inherent in it. So, no, CliffsNotes, we reject you. So, Kim, when I was just at CAMT, the Texas conference, I heard Dan Meyer speak. I've always liked Dan Meyer. I think he very nicely identifies some things that many of us should be thinking about in math education. And it was interesting to me. His talk was kind of... I'm not going to do it justice. But he was talking about... I think the title was, "Use your technology. Don't let it use you." Something like that. "Use your tech..." And he noted that technology has gotten to the point where you can... As a tutor. It's trying to teach you. So, (unclear) ask you a question, and you input your answer. And one of the things he noted is that the technology it has a nicer answer than it used to. So, it used to just like go [Pam makes buzzer sound], or you know like a big red X, or wrong, or something. And he showed several screens where some of the programs out there now, or apps out there now, will say things like, "Not quite. Try again." But he noted that that's the only. So, it's pleasanter. You know, it's like, "Oh, not quite." You know, "Not quite."
That's so helpful.
It's not like, "Absolutely wrong, sucker!" You know, it wasn't that. But it was, "Not quite. Try again." So, it was pleasant. But what it didn't do... And I just thought this was very nicely identified on his part. What it doesn't do is acknowledge that kids wrong ideas are worth ferreting out. They're worth discussing. It's worth building on the parts of their intuition that they got, right.
Oh, yeah. So good.
So, this point, that technology always has the same feedback, that that's super limiting. Super limited. Limiting and limited. And that teachers, one of the things that we can do best, is as we are pulling out kids ideas... You know we're talking a lot today about pulling out what they're thinking, and then helping and dropping some of those social terms, so that they can use those to tag to those big ideas that they're really wrestling with. And as we, then, summarize those, that actually helps learning occur because they are clarifying their own ideas as they have to put words to it. That all of that is involved as teachers respond to students thinking.
Right, right, right.
Not in ways that they're just like, "Not quite. Try again."
So, it's not about teachers listening to kids going, "Not quite. Try again." Like, AI can do that. What we are advocating is that we're really helping give good feedback to students.
Yeah. Well, because two different wrong answers from different students reveal completely different things.
Yeah. That's important, right?
Like, we want to know where. Know your content, know your kids. We know what the students... It reveals sort of what they're thinking, so we can build on that.
Both Pam and Kim 21:56
Okay. So, today, we're going to say that keywords? Not so helpful. But key ideas that we put into words? Absolutely, yes.
And we can take those key ideas, putting them into words, on helpful co-created anchor charts.
Bam! Alright, 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!