Zandra de Araujo:  I’m Zandra de Araujo. I am the Director of Mathematics at the Lastinger Center for Learning here at the University of Florida. Our center does a lot of different things, one of which is mathematics, and we do professional learning, we do research and all kinds of things around the state of Florida and beyond. 

And I’m joined by some comms support, Maxine and Hope on the call, who, if you have any challenges, they can help. And then Amy, who does partnerships in our math team, so some of you might have interacted with her before. And then Jose, who is the manager of the math team here at the Center. And if you’ve taken the math micro-credential, the math matrix micro-credential, he probably looks familiar because he’s in a lot of the videos. 

So, welcome, nice to see everyone. If you didn’t catch part one of this series, it’s okay. They are distinct and separate, and you can watch Part 1 later if you wish. 

Today, we’re really going to focus in on math discussions, which are one of the things that I always enjoy, but one of the hardest things about teaching mathematics is to have a really good math discussion. So hopefully we share some insights and ideas that you can take away, maybe even apply right after lunch today in your own classrooms.​

So, we’re going to talk about the quality of math discussions, we’re going to talk about the teacher’s role in this, highlight some math talk moves that you could use (and you probably already are using, maybe you just haven’t heard them called that before,) we’re going to talk about questioning, and then I’m going to share some instructional nudges for discourse that I think are a lot of fun and also helpful for getting students talking about mathematics. So here we go!​

I’ll come back to that survey later. Quality of math discussions. So, this is an opportunity for you to share in the chat before we get started. So, you’ve all been participants in or facilitators of, math discussions. So, what does it look like, or sound like, or what are the characteristics of a high-quality math discussion? Like, how do you know that this math discussion is good, and it’s high quality? So what are you kind of listening for or hearing in your own classrooms?​

So, you can put it in the chat. So, what are the kinds of things that you listen for or notice about a high-quality math discussion?​

Thank you, Marcia. So, math vocabulary. Maybe you hear some precise language and some good use of math terms that is accurately used. Specific vocabulary, okay. 

Students defending their answer. I love that one so much. That is so important. Melissa, thank you. 

Students questioning each other as well as agreeing and disagreeing with each other, that is hard to do, but so meaningful when that happens.​

Participation, justification, math talk, lovely! So, it sounds like it’s not just kids saying answers to us. There’s more than that, right? It’s kind of like this interaction, there’s justifications going on. Probably a lot of the MTRs, the math thinking and reasoning routines, or mathematical practices, are coming out in the conversation. Students are using vocabulary purposefully and accurately. Great!​

And I think I already alluded to this, but low-quality math discussions, often you might just hear ping-ponging, I would say, like teacher to student, and then teacher to next student, and then teacher to next student, and most of the responses are pretty limited. They might just be the solution, an answer, a missing word and there is a time and place for that, but maybe that’s not the highest quality math discussion that we can have. 

Maybe a higher quality one would have more open-ended questioning, as Michelle put in the chat. Exactly. Alright, so we’re going to look at some strategies on how to do that today.

​So, we’re gonna kind of, and if you’ve taken the math matrix micro-credential, some of this will look familiar, but we’re going to go deeper. So, math discourse kind of involves the teacher a lot, right? We’re usually facilitating these conversations with our students. We kind of know where we want it to go. It’s not just organically happening. There’s stuff that we are doing consciously to foster this discourse.

​But, when it’s high quality and a good discussion, it just looks like magic. So I’m not a basketball player, and I don’t know much about basketball, but to me, it always just looks like basketball just happens.

​But it doesn’t just happen. There’s playbooks, there’s moves, people have positions, there’s all kinds of practice involved. The coach is facilitating a lot of these interactions during practice and stuff. There’s a lot of planning and careful thought that goes into a basketball, I was gonna say performance, I don’t think that’s the right word, game.

​And similarly, in teaching, when you’re having a really good conversation, it’s not like it’s just spur of the moment. Probably a lot of really thoughtful planning went into that. So the teacher’s role is central, because it’s so important to having a strong math discussion in your classroom.

​And then, you know, the students are engaging in questioning with you, you’re engaging with questioning with them. Somebody said this in the chat: “if the kids’ students are questioning each other,” that is amazing, and so rare to see, but so important.

​And then the kids are also explaining their mathematical thinking. So there’s a lot happening, and the teacher is responsible for a lot of that, even though it doesn’t look like it when it’s going well, because the kids are talking to each other more than they are just talking to the teacher.

​I’m gonna start by just saying this whole series is built upon our aim of increasing math proficiency. So I’m gonna very briefly revisit what I mean by math proficiency. I think it’s really important that we’re always on the same page of what do we mean by being good at math, being proficient at math.

​It is not just being fast and accurate at computation.

​There’s these five strands of math proficiency, and if you want to catch webinar one, we kind of went over these, but it’s not just procedural fluency, it’s not just being able to calculate answers. There’s more to it. That is certainly one part of it. We want to be efficient, be flexible and know our procedures.

​But we also want to understand why things are working, and understand math as patterns and connections, not just, this is a rule, this is a rule, this is a rule, but how does it all work together?

It’s also about strategic competence, which I think I said was my favorite one last webinar, and I think it still is. I might have said adaptive reasoning is my favorite, but I like this one a lot as well. So it’s the ability to formulate, represent, and solve math problems. Like, how do I actually ask questions that I can mathematize and understand and investigate the solution to. That’s really cool when kids start to ask questions, and then we explore that as the focus of our lesson.

So we want kids to start posing problems. We don’t always want to be the ones that hold all the knowledge and ask all the questions. We want our students to take some ownership as well.

​And they should be able to translate into real-world solutions and useful representations, etc.

​Productive disposition, we want kids to like math and see it as useful. I think this affective feature is one we kind of don’t concentrate on because it’s kind of hard to measure.

​But we want people to like mathematics. We probably like mathematics, we’ve chosen to teach it or be educators, and we want our students to see, like, math is actually useful and kind of cool and creative.

​And finally, adaptive reasoning is very key to mathematical discussions, as many of you mentioned in the chat. It’s explaining why something is true, justifying your solutions, seeing errors in reasoning, which is really important and making logical arguments, which is one of the most fun parts about mathematics to me.

​So that’s the kind of math we’re trying to foster through rich mathematical discussions.

​Okay, so I’m going to come back to this every webinar, because I think it’s so important that we all see mathematics in a similar way.

​Alright, teacher’s role in math discussions.

​So, what is the teacher’s role? What do you think if you were coaching a new teacher, maybe you are a novice teacher and somebody coached you… What did they say? What is your role in the discussion? What do you see as your main role in the discussion? And you can put it in the chat if you’d like.

​Thank you, Melissa. So, facilitator or guide? I’m seeing a lot of support for that response. Asking, guiding, open-ended questions, intentional problems, yes.

​Guide their conversations, prompting, drop a question to push them forward, kind of like invest in an instigating kind of discussion. You’re kind of saying, like, what if this, or have you thought about this, or how did that sound to you? Yeah, you’re kind of moving it forward.

​So that all seems great, but I think it’s all a little nebulous still, just to say, you’re a facilitator. I think it’s helpful to say, like, but what does that actually mean? What am I actually doing to facilitate? And tasks are definitely part of this. I’m just gonna put a disclaimer out, the next webinar is about tasks.

​If you don’t have a good task, there’s not a lot to talk about. If I just say, add three and five, what are you gonna talk about? You could talk about what you got for an answer, you could talk about maybe how you got there, but there’s not a lot to talk about.

If you said, huh, what are different ways I can add numbers to make five? There’s so many more things we can talk about as a class. So, I think tasks matter a lot. We’re gonna talk about that next time.

​For the rest of the day, we’re going to kind of assume we have decent tasks that will facilitate some conversation.

​But given that, the teacher still has to do some stuff. So here’s some specific things that you, as a teacher, probably are already doing, but I’m going to give names to it.

​So, the math talk moves. Probably some of you have heard about these already, and they’re super helpful and I use them all the time. And they’re still not easy to do. I don’t want to say, because I can name them simply; they’re easy. Doing this is difficult work. It takes professional knowledge, expertise, and skills, so give yourself a lot of grace as you work on this and support your students, because having a really good math discussion is hard, and kids don’t get a lot of opportunities to practice it. 

So, as you come into the classroom and you’re working with your students, they need to learn what this sounds like, because it’s new to them often, and so we want to give a lot of grace to ourselves and to our students as we learn how to have these deep conversations.

​So the first one, and I think a lot of us do this, is repeating. So, ask students to restate another student’s reasoning in their own words.

​So Jose says something to me, and I say, “Okay, Jose, what I think I hear you say is this,” or I say, “Oh, Paul, can you tell me what you think you heard Jose say in that situation?” So it’s just kind of validating that they had an idea that was worth listening to, and that we’re all gonna take some time to listen to right now. And we’re gonna actually restate it, repeat it, to make sure we got it right.

​So you’re not just saying, like, “Jose” says five. You want somebody to say it in their own words. I heard Jose saying that he thinks the answer is five. Did I get that right? He might even check in with Jose. 

So that’s the first one, repeating. So, a lot of times you’ll say something, like, I do this in meetings a lot, I’ll say something and nobody picks up on what I just said. And it’s like, oh, that must not have been a very interesting or meaningful contribution that I had. But if somebody takes the time to repeat it, suddenly I feel like my idea might have been worth something, or at least somebody was listening to it, because they listened enough to be able to repeat it in their own words, which is helpful.

​Okay, so in this instance, if you’ve not seen a “which one doesn’t belong,” this is an opportunity where you can see one. Basically, there are quadrants, like you see here on the right-hand side of this slide, and each image in the quadrant can be in or out of the set, depending on how you define that in or outness. And there’s a great website, just Google or search which one it belongs to, you’ll find it and they have lots of these.

​But you could have the situation where somebody says, I think the top left one doesn’t belong because it’s a gray circle.

​And so, I might ask, as the teacher, okay, does somebody want to restate or repeat what you heard Zandra saying? And they might say, she doesn’t think it belongs because it’s a gray circle. And then you say, like, okay, is that what you meant, Zandra? So it’s really just kind of reaffirming the answer, making sure people heard it.

​Getting everybody on the same page about that strategy. So that’s the first one.

​The second one, reasoning. So ask students to apply their own thinking to another student’s reasoning.

​For example, I’m going to do a lot of elementary examples, just because they’re on the top of my mind at the moment, but say that I multiplied two fractions. So, one-half and three-fourths, those are mean ones to multiply in your head. But I multiplied them, and I drew an area model to multiply them, to get my answer.

​So, I might have explained it to the class, and then I might say, “Hey, Jose, do you think you could use Zandra’s strategy to solve this other problem?” That’s all it takes. So you’re saying, like, oh, she had a strategy or some kind of reasoning, could we apply it to this different task?

​That is hard to do, actually, right? Because you have to understand their reasoning on this problem, and then transfer it to something else. So you’re trying to see if they understand it and can also apply it in another situation.

​Which is pretty interesting to do, and pretty hard to do, but it’s a great move to use with students. “Hey, we heard you doing this. Can you apply it to this situation?”

​I’m gonna skip some of the examples, because I really want to get to the dentist.

The third one, adding on. I see this one a lot, and it’s very valuable. So, Jose says something and I say, as the teacher, does anybody want to add on to what Jose just said?

​Now, I have seen it, especially in kindergarten and first-grade classrooms, where the kids will say, “Yes, I agree. Also, last night, my sister did this thing,” and you just keep going on and on, and it’s not relevant to math.

​But the idea is, you want them to get the experience of practicing and say, like, “Oh, I like that you did add on, but let’s keep it about math,” and you start to get these norms in your classroom around this.

​So the idea is you just add on to what they said, and keep it related to what they said, not going out into wonderland with your responses, which is apt to happen.

​So, for example, you could give a problem like this. Would you rather have 312 Sour Patch Kids and give 187 to friends? Or, have 291 Sour Patch Kids and give 139 to friends.

​So, maybe a student just raises their hand and says, I’d rather have 312. Often, we might just leave it there and not do anything.

​But instead, what we could say is, “Amy, would you be willing to add on to what Zandra just said” and see where they go with it. If they carry it through to another place, do they kind of understand what’s happening? You can also have students add on to their own thinking if they don’t give you sufficient reasoning to go off of. So that’s adding on.

​Again, I see this one a lot, and it usually works pretty well to get students to collaborate across different people.

​Alright, now, this one is a hard one, especially for teachers, because most of us like to talk, but it’s wait time.

​I talk fast, as you probably noticed, and I also kind of talk as I’m thinking, but that’s not everybody. Everybody is not a Zandra, right? Some people are really thoughtful, and they just like time to process first.

​And if you have a room full of Zandras that just are blurting out really quickly, those folks feel like they can’t contribute to the conversation because somebody else has already taken the floor. So, a strategy that many of us have learned and we forget to do sometimes is wait time.

​So you ask a question, or you pose a wondering, or a prompt, and then you just wait.

​And wait, there’ll be kids like me that will just shoot their hands up, because we like to talk, but you kind of, like, acknowledge them by a look, and then just, like, go, “Wait. I’ll get to you. Let’s let everybody think first.” So, wait time is really important to discourse, because students don’t all process at the same time, and their comfort levels in reaction. Sometimes people like to rehearse things in their brains first.

​For example, multilingual students, if you were asking me to answer in Portuguese, which is not a language I feel confident in, though it’s my parents’ first language, I need time to think about what I want to say in English, then translate it to Portuguese, and then try to say it. Right? So I need wait time to contribute to the classroom, otherwise I wouldn’t say anything at all. And I think some of your students may fall into that category as well, regardless of their language status.

​And these are all kinds of good things to think about, like, how does this change the conversation? Maybe it opens it up to more students than just your quick, eager-to-answer and eager-to-please students.

​And how long do you decide to wait? Like, longer than you would think. I always, like, I get uncomfortable, and then I try to wait a little bit longer than I’m comfortable, and that’s usually the right amount of time. 

Some people say I, like, count to 10 in my brain, or things like that, but I don’t think you have to prescribe it that much. But it’s okay to be a little uncomfortable with the silence, and then wait a little bit more, and then sometimes students will get uncomfortable, and the ones that usually don’t talk may even contribute, because they just want the silence to break, which is also a strategy, I suppose.

​And then during wait time, some students might just be like, “Gosh, I just want to say the answer, I know the answer.” Some students might be practicing what they want to say, and other students might be still thinking, and that’s all okay. We don’t know, but the wait time opens the possibility for all those scenarios.

​And the final one is re-voice it. So, paraphrase what a student stated, and then ask for clarification. So this is very similar to repeating.

​Repeating is usually you have another kid, kind of say it and check in with the student. This one, you can kind of use them interchangeably. I do. There is some nuance there, but I think it’s mostly not super important. Revoicing is kind of like you’re restating in your own words and then asking for clarification.

​As a teacher, this one’s important when students are using informal language, and you want to bring in more formal language. So, for example, if I’m in a first grade classroom and the students are calling a square or a rhombus a diamond. I don’t stop them and say, “That’s not a diamond.” I might just say, “I heard you say that you saw a rhombus,” or I might interject the right vocabulary. Not that you can’t say it’s not a diamond, you can definitely say that, but sometimes it’s nice to just restate it in the proper words, because you know what they meant, they just aren’t using the proper words yet.

​And so that’s a nice strategy that you can use, and then check in with them, and see if they’re, like, perturbed that you called it a rhombus and not a diamond, and then talk about that.

​Okay, so before I go on, if you have any questions about the math talk moves, so these are specific things that you could do every day in your classroom to kind of get the conversation going. Put them in the chat or in the Q&A, and I’ll make sure to get to those questions if you have them. If you have other Math Talk moves that you use, and you want to put into the chat, I invite you to do that as well, because it’s important that we share amongst each other as well.

​Questioning! So, what precipitates all these math talk news and the discussion is, like, how the teacher is questioning the students, because usually a lot of the conversation is starting with the teacher. I’m posing a question, I’m asking students for input, etc.

​So, why do we pose questions? NCTM, the National Council of Teachers and Mathematics, kind of summarized four main purposes of questions.

​Sometimes you want to just gather information, what do the students know? Where are we at as a class? 

Sometimes you want to understand their thinking, so you probe their thinking, like, oh, I think this is what they’re saying, I might ask more to understand what they’re saying more fully.

​You sometimes want to make the math more visible to the student or to the class.

​Sometimes you just want them to think through something again, or reflect, or justify their reasoning. So there’s lots of different reasons why we ask questions.

​So, teachers should consider the task they’ll carry out and draft a handful of questions that address these. 

So, this is that basketball analogy, metaphor, analogy, I forget which is which, but, that I posed earlier, is that it looks like it’s a great natural conversation, but really, when it’s going well, the game is going well in your classroom, you probably had some good questions in your brain ahead of time. So, taking some time to jot down questions you might ask is actually quite beneficial to your lesson and to the quality of the discourse that happens afterwards.

​In the Math Matrix, we kind of simplified questioning to think about it in just two different ways. Whether or not a question is closed or open is one of those ways.

​So, closed questions have a small number of limited responses, typically. Did you solve the problem? Yes or no? What is the square root of 16, plus or minus four, right? Like, what is the name of this shape? There’s not a lot that’s going to come from that answer. It’s a pretty simple, short response answer. There’s, like, a correct answer you’re trying to get at.

​Open questions invite more discourse. And we know there’s been lots of studies on questioning in the classroom, and almost always, the vast majority of questions are closed questions, without a doubt.

And in my observations of classrooms, that is also the case. Most of the questions teachers ask and students ask are closed questions. So if you can start to open up your questioning a bit, you’ll get more quality discourse.

​So, how can you prove that the answer was correct? Like, that’s inviting you to kind of explain your reasoning in more than just one word.

​What have you tried? It’s not like a yes or no, or like, one… maybe they said the calculator, right? That’s very brief, but it does offer more opportunities, even if the students are kind of conditioned to answer very briefly. So we want to, like, kind of be conscious of closed and open questions.

​And try to move toward more open questions, if possible, because you’ll get more discourse out of your students.

​The other way we think about it is product questions or processes. Are you asking just about the answer, the solution and what you made, or are you asking about the process that got you there? Again, most questions in math classrooms are product questions.

​Did you get five? What is the answer? You know, like, what’s your picture look like? That kind of thing.

​We want to ask more process, like, how did you decide what strategy to use? Why did you decide to use a visual strategy? Did you consider thinking about manipulatives, things like that. So, kind of questions around the process that got them to a product are really helpful.

​So, instead of just closed product questions, we kind of want to move toward more open process questions, and it’ll help you in the long run in your classroom.

​If you have a math coach, I know there’s a couple of math coaches on the call, you might invite them one day just to sit in on your class and track, what are they seeing? Are they seeing, like, teacher-student interactions, mainly? It goes teacher-student, teacher-student, teacher-student. 

Or, are they seeing student-to-student interactions? Or are they seeing teacher to the whole class? Like, teacher oral response?

​And then you can ask to kind of get the questions cataloged, so you can kind of see what your tendencies are. As teachers, we develop patterns. I bet your kids know your patterns. If you ask what kind of questions, or if you say, “Hey, if you’re going to pretend to be me,” which I don’t recommend, but if you did, and they started to play teacher in your voice, they would ask the type of questions they typically hear you do, or the interactions, and you’ll get a good insight into, like, what your patterns are, even when you’re not fully aware of what they are.

Asking a colleague or somebody to track some of these things for you is actually really illuminating, and it’s not for evaluative purposes or anything, it’s just to say, like, I wonder what my patterns are.

I know when I listen to myself, if I were to listen to this webinar later, I know there’s speech patterns that I have. I use “so” a lot.

​There’s some others that I catch myself doing and I try to break, but they’re hard to break, because they’re just patterns I get into. Questioning is kind of similar. So the more you know, the more you can be conscious of it and try to break those patterns.

Ruth, we can give you a link to this document, yes, I believe, and maybe Amy or Jose will be able to find it and link it in the chat.

​Okay, I just did a lot. So, we talked about questioning, the teacher’s role, explaining math thinking. We’re gonna get at a little bit, but not as deeply, but the math talk moves help to get students explaining.

​Now we get to my favorite part, which is, what are some things we can do as teachers to encourage more talk on tasks?

And so, we’re gonna get to some nudges here.

​Okay, so we have instructional nudges. If you’ve not seen them before, we have a website that we’ve made for this project, and we also share them in the math matrix at times. This is an example, it’s called “Say What.”

And all it is asking, at the end of a lesson or a set of problems, you have students clarify the main ideas, either in writing or by discussing, with their partner. Sorry, there’s a typo there. So, for example, we’re gonna show a video, and you’re going to kind of see what happens in this video. And these are…

​I think kindergarteners in this classroom. So, it’s a way of getting students to talk about what they learned, because often we just go through things in the classroom, we never pause to reflect, what did we just do this about? Can you tell somebody else what this lesson was about?

​In this instance, we are going to look at a video where you can kind of see how one teacher took up this idea in their classroom. So this is a quick strategy to just have kids kind of share what they thought the purpose of the lesson is, or the main things, takeaways that they were to get out of it.

And I thank this teacher for sharing with us. It’s very nice that teachers open their doors to us and allow us to film their classrooms.

​[Outside video played on Zoom call. Watch the video HERE.]

​​Zandra de Araujo: So, that is a quick routine, a quick strategy.

You’re trying to get the students to articulate what they did that day. Because I can tell you, I will visit a lot of classrooms, and then the students do all these great things, they’re doing all these tasks, and then you say, okay, like, “What did you learn today?” And they can’t really articulate, like, they’re like, we did problems, we learned math, but they don’t really get the same main idea that they hoped for in the lesson. 

This is kind of one way that you can formatively assess, and also get the students talking. With older kids, or as you get more comfortable with this routine, you might have them build on, and you might have already been thinking as you watch that, “Oh, she could have had that kid add on or repeat their reasoning.” So once you know about these math talk moves, you can kind of embed them into situations like this and get even more discourse and discussion out of your students.

​Okay, another one is, “In Other Words,” and again, this is to get students talking about mathematics.

​Not all of these are gonna result in a deep conversation, it’s just starting to get kids to talk about mathematics in some kind of way.

​For example, you could have a problem like this, or a situation like this, a prompt, and say, here’s a topic, or a term, a concept–reword this definition using more common language or a different formal definition, or another representation. 

So, if I said to my students, area, and maybe I even give them the definition, the formal definition of area. You want to ask them, like, can you say in your own words what area means to you? What do you think of when you’re thinking about area? So you’re kind of opening it up to them to make sense of this concept and explain it in their own words. When they do so, you might learn, oh, their understanding of area might be a little different than mine, or not as deep or nuanced.

​And that gives you opportunities to talk more about it, invite them to talk more and say more. So this is another quick way that when you have a definition, you might ask students, “could you say in your own words, can you write it in your notes in your own words? Can we explain it in a different way, so that we’re kind of getting a stronger concept of what this idea is?”

Three reads. I imagine many of you are familiar with three reads. If you are, you can give a thumbs up in the chat or, you know, say a yes or something, but the three-read strategy is one that helps us make sense of word problems or story problems.

And story problems are really important to support discourse. Thanks, Stacy. 

So, three reads is literally, you read a problem three times. Each time you read it, you have a different focus.

​In the traditional three-reads format, kind of read one is you’re just reading to make sense, like, what is this situation here? What’s going on in this problem? That’s the first read. Now, again, you can have a whole conversation about what the situation is about and use your math talk moves during that time.

​So that’s just the first time, because we really want kids to slow down, understand the task before they just start pulling numbers and calculating. So the first read is just, what is this situation about? And you can have a whole conversation about it.

​I seldom see teachers having conversations about the problem before kids start, and we know it’s very important to do so for students’ comprehension and also engagement with the task.

​The second read, you might focus on, what values or information do you notice? What relationships do you notice? So you’re kind of focusing in on some of the specifics of the test. So kind of read one as, like, broadly, what is this about? What’s happening? Read two is, like, what quantities do we notice, perhaps, or, you know, things that are going on in the task? Are there relationships, etc. So you’re kind of getting at a finer grain size.

​And the third one, you’re kind of anticipating some strategies. Oh, what are some strategies that might help us answer this task? So that’s a classic three reads. We’re going to remix it a couple of different ways, because students get into this too many times in the same process, and they get bored with it. I get bored with it, so we’re going to remix it a couple different ways.

​Remix number one!

​You can still do three reads, you can also just do this one read if you’d like, but you do the task, you present the task, but you withhold the question. You don’t actually show what they’re trying to get at. 

So, for example, Manny brought  a cookie tray with 48 cookies to a party. On the tray label, it said a serving size is 1/12th of the tray. So there’s a question, probably, that exists on this initially, but this strategy is you take out the question.

​Part of it is to try to slow down students and have them think before they act, but in the chat, if you want to, what is a question that you might assume or think would follow this prompt. So, if Manny brought some cookies to the party, on the tray, it says a serving is 1/12th of the tray. What question do you think might be asked of this task?

​Any ideas? Marcia, how many cookies is a serving? That’s probably where I would have gone too. Any other ones that you think?

​You can put them in the chat if somebody had a different question. How many people can you feed?

​Yeah, and then that would be a good conversation, because you’d say, like, could we answer that question with the information we have? What information will we need to be able to answer that question? Which would be a good one?

​Yeah, great. And that’s assuming everybody eats the same amount. Do we need to know if there’s a different number of people or something? That’s an interesting one, yeah.

​So, math problems follow a genre. There is a genre of math story writing problem. I’m convinced that curriculum writers go through some course that teaches them, because they all kind of look the same. It’s like information, information and question. So kids are kind of conditioned to look for that, and they will likely come up with a question that was the original intent.

​But there’s a lot of reasoning that occurs in just making sense of the quantities and anticipating what the question could be. So that’s the first strategy.

​The second strategy is, you could just remove the quantities. We know students like to pull the numbers and start calculating. If they learn about multiplication today, they will multiply on this problem, even if it’s not a multiplication problem, because they’re just so conditioned to being like, well, we’re in multiplication, so I bet it’s a multiplication problem.

​So, to eliminate a little bit of that, you could do something like this. You just remove the quantities.

​So, Adriana has some pieces of candy to share. When she went to the park, she shared some pieces of gum. When she left the park, Adriana shared a number of pieces of bubble gum. How many pieces of gum does she have now?

​It’s allowing you to then talk about what is the situation. What do you think is happening? What do you need to know? What do you think’s gonna go in those boxes? What kind of numbers? You could say, like, Do you think it’s gonna be decimal numbers, or fractions, or whole numbers, things like that?

​And then, do you have a strategy that you might approach? And they still can’t pull numbers yet, right? So we’re still talking in generals, which is nice, because once you give them the quantities, they’ll start calculating.

​Yeah, like a numberless word problem, exactly, exactly right, for sure. Yep, they’re great. There’s, like, so many resources on those as well.

​The last one is anticipating what the answer will be.

​So, we want students to look at a problem, and to say, “When I solve this, I think I’m gonna get an integer, or I think it’s gonna be a graph, or I think it’s gonna be a sentence, because I have to explain my thinking, with a number.” We don’t often ask students to anticipate what kind of answer they’re gonna get or what they’re aiming for.

​Even more problematic is we have units in the original problem, and then they just kind of disappear. Like, it could be a whole nice task about bubblegum and things like that, and then at the end, the kids are just, like, 12.

​Well, then the context doesn’t really matter, we’re not using it meaningfully, we just kind of abandon it. So if you have kids kind of anticipate ahead of time, what is the form, what kind of answer are we going to get? Money? Like, decimals, fractions, graphs, equations, expressions?

​And then, what kind of units does it have, if any? That’s a really valuable strategy. There’s some nice research behind this, too, that says it, like, increases the problem-solving capabilities of the subsequent problem, which is nice.

​Here’s an example of this. Rosa has 12 cups of condensed milk for her grandmother to make many batches of Tres lechees, my favorite kind of cake. So far, her grandmother has used 8 and 1/4 cups of condensed milk. How much condensed milk does she have remaining?

​So, if you just be very clear with your students, we’re not solving it yet, I just want to know, what kind of answer are we gonna get?

What is that answer referring to? Like, what unit is that answer going to be in? It’s super valuable, because you do a lot of thinking, and you’re trying to make sense of the context to get to what kind of answer you’re going to get. So it’s a good strategy.

​Alright. Now, moving on from three reads.

​Say your students just did a worksheet with a lot of problems, which is common, right? Kids have to practice math. That’s not a problem.

​What if you wanted to get more talk out of your students, though? So they just did all these problems, and you’re like, well, we could just talk about the answers, I could just give them the answers.

​Or, you could have them partner up and say, hey, you did all these problems. Can you tell me the two that you did that you thought were the most similar?

​And then, do you think your partner could guess why you thought they were the most similar? So, it’s kind of encouraging students to think about what they just did, and have a conversation about what they just did. So, I know you’re going to give your students problems to practice, like, it’s part of math, it’s part of that procedural fluency.

​But how can we get them to talk about it a little bit with their students, their peers, and also do some reasoning and justifying and sense-making? So this is a strategy we like.

​So, if you want to put in the chat which two you think are the most similar, you can.

​For me. Let’s see, which ones do I think are the most similar? I’m not sure. Maybe 3 and 5 would be the most similar to me. So I might say to my partner, Jose.

​I thought three and five were the most similar. And so then, Jose would have to try to understand and anticipate or guess why Zandra thought that three and five were the most similar.

​And he would have some kind of reason for saying that, and he might say… because they’re exactly equivalent, because one is the simplification of five over five, or something like that, 5/5. So, that’s a way to get students to talk across problems. They might not all say the same thing, in fact, they usually don’t, and that’s good, because then there’s more conversation to be had.

​You could also do the analog of this and say, like, which two did you think are the most different and why? And have them proceed in that way as well. So the point is to get students talking about mathematics in a meaningful way. And we call that one pair and compare.

​Alright, this is the last little set, I believe, and I hope this is the video I’m thinking of, so I’m gonna play it, and then I’ll talk about it, because I think this is the video that I love.

​[Outside video played on Zoom call. Watch the video HERE]

​Zandra de Araujo: We talked about vocabulary earlier, and how vocabulary is key to a good conversation, the students are using it.

This is a way, a fun way, and I’ve seen teachers use this before, so it’s, like, basically like the game Headbands, and you put the terms or representations on a card, and you have the students describe it, or like, you know, guess it based on descriptions or whatever. So they’re having practice talking about the concept to try to guess it, right? So they’re describing it to their partner, and their partner’s trying to guess what number is on, or deduce, I guess, really, because it’s not a guess, right? They have some information they’re getting from the process.

​Kids love this! It always hits in the classroom. The students like to talk to each other. This was kindergarten students doing it. The teacher facilitated it in small groups. They’re very small children. They were doing it for one of the first times when we joined them on this day. And then we’re going to see some older kids do it next. I think. Yes.

​[Outside video played on Zoom call. Watch the video HERE]

​Zandra de Araujo: So, you can see that those are older kids, they were high school kids, but they’re still engaging with it, in a meaningful way. 

So, you can kind of see what language they use to describe these things, too, and what you might push on, or think about, like pushing further on in subsequent things, but they do a great job kind of communicating what’s important to them and making sense of mathematics collaboratively. 

I think the key is, I like to use fun, engaging activities to get kids comfortable talking to one another about math.

​Students talk to each other all the time, right? Especially in phone-free schools, which are increasing in popularity right now. The students are talking to each other, but they’re not necessarily used to talking about math, so when you say, explain your thinking, what you have in your brain as a high-quality explanation may not be what they have.

When you say, partner up and talk about math, you might not get a lot of valuable, high-quality discourse at first, because they need strategies just like we need strategies to facilitate. So I find activities like Say My Name, or some of those, making sense of the three reads, story problem through three reads, or variations on it are helpful because it lets kids start to use language publicly to talk about mathematics. They get comfortable with it, and then the conversations get better. But it’s like a skill. They have to learn how to do it, and these strategies kind of help students build those skills over time.

​Okay, well, we got to the end. If there’s any reflections or questions you have, you can put it in the chat.

​I find it helpful when I go to webinars. I have little field notebooks that I take notes on, when I do online classes or webinars, but you might want to jot down what you want to bring back to your colleagues or your classroom, or maybe after lunch you want to try something in your own classroom based on what you heard here today. 

​So, if you want to write down, kind of, your ideas.

​That you have from this webinar, and this will be posted on our website in the coming weeks, so you’ll be able to see the first one, like I said, and they will be on the website.

​If you have capacity, we are gathering survey data right now for a project we have on high-quality professional learning. We’re trying to understand more about how decisions about professional learning are made.

​If you’re interested in participating, we’d love for you to either take it yourself or share it with a colleague. We will put the link to the survey in the chat.

​But any help you can do on taking the survey or sharing it is greatly appreciated. We will always share back what we learn from people that share their time and expertise with us, so thank you ahead of time, if you’re willing to do that.

​Our next webinar will be on math tests, my favorite topic, and that will be on March 24th. I did change the time, because there were requests to change the time around, so I’m hoping this still works for those of you that were enjoying the lunch time.

​If not, like I said, I promise I’ll post them on the website. I won’t, but Maxine and Hope will from our comms team. And so, you will have access to it, but we love to have you join live. It’s more fun to talk to people than just black boxes or just Jose, which Jose’s a lot of fun to talk to, but I talk to him every day.

​So thank you for joining. If you have any questions, you can put them in the chat.

​We love to hear questions and answer them, and I’ll stick around for a few more minutes. If you have to go, thank you for spending your lunch hour with us, we greatly appreciate it. So, take care, and if you have questions, put them in the chat, because I will answer them.