Featured Speakers:
José Franco, Senior Research Associate, Quality Schools and Districts at WestEd
Bob Rosenfeld, Senior Engagement Manager, Quality Schools and Districts at WestEd
Host:
Danny Torres, Associate Director, Events and Digital Media at WestEd
Danny Torres:
Hello, everyone, and welcome to the fourth session of our Leading Together series. In these 30-minute learning webinars, WestEd experts are sharing research and evidence-based practices that help bridge opportunity gaps, support positive outcomes for children and adults, and help build thriving communities. Our topic today, Mathematical Conversations That Engage Students in Enhanced Learning. Thank you all very much for joining us. My name is Danny Torres. I’m Associate Director of Events and Digital Media for WestEd. I’ll be your host. Our featured speakers today are Jose Franco, Director of Math Pathways and Pitfalls, and Senior research Associate with WestEd’s Quality Schools and Districts team.
We also have Bob Rosenfeld, Senior Engagement Manager, also with WestEd’s Quality Schools and Districts team. Now, before we move into the contents of today’s webinar, I’d like to take a brief moment to introduce WestEd. WestEd is a nonpartisan organization that aims to improve the lives of children and adults at all ages of learning and development. We do this by addressing challenges in education and human development, reducing opportunity gaps, and helping build communities where all can thrive. Now, I’d like to pass the mic over to Bob. Bob will be leading today’s session. Bob, take it away.
Bob Rosenfeld:
Thank you so much. I’ll be co-leading along with Jose, who you’ll be meeting in a moment. We’re just so excited to have you here today because this is an opportunity for us to share, and that’s what we love doing here, is just sharing what we’ve learned in the field, and we all hear about mathematical conversations, getting students talking about math, but we have an opportunity here to actually show you it in action because it’s much easier said than done. Now, all of this is built on research and time that we put into this. So it’s, over the last more than a decade, we’ve had the opportunity to conduct research into what mathematical conversations really look like in a classroom with a diverse population of students from across the country, and we’ve created some materials that we’re excited to share with you here today, that we hope will make this come to life in your classroom.
And again, we’ve engaged with teachers, and as we’ve engaged with teachers in this work, we keep going back into the research, and NCTM is one of our main resources, and we’ve worked to make sure that everything that we have designed is built on these five principles. And again, these five principles sound great on paper, but really seeing what they look like is what it comes down to. So let’s talk a little bit about these, and then I’m gonna hand it over to Jose who’s going to give us some concrete examples that will show you what this looks like, sounds like, and feels like in a classroom. So building mathematical discussions, getting students to talk to each other about math.
Students with different language backgrounds as well, students with different educational backgrounds, students with different comfort levels with math, and helping them to make sense of what the math is saying, and we hear a lot about these days around growth mindset. What we’ve discovered is, yes, this is true. Even before the word was a buzzword out there, this idea of, we use the word Pitfalls, that helping students feel comfortable with having a pitfall, that is being comfortable with making a mistake in math, we all do. And so helping them see that that’s part of the learning process, and through discussions and not only discussions with peers, but self-reflection, they can lift themselves outta that pitfall and really make connections between the math and the real world, and that’s the other piece here, is helping students visualize the math and make connections to other areas of their life.
And then finally, taking all of that learning and being able to summarize it, put into their own words to capture those ideas is central to what we’ve learned. And you’re gonna see all of this in action here, so we’re gonna share with you some quick little videos of what this looks like and give you an opportunity to reflect on and think about how this might look like in your classroom or your school. So I’m gonna hand things over to Jose now, who is gonna share with us why we’re sharing these particular videos and what you’re about to see.
José Franco:
Thanks, Bob. Student talk is a core tenet of math Pathways and Pitfalls. It’s about teachers asking probing questions to help the kids clarify their thinking, to justify their ideas, or to help them get unstuck. Math Pathways and Pitfalls is about creating a classroom where every student is engaged and has their voices heard. It’s about creating opportunities for students to talk like mathematicians, to share what they think they understand about the mathematics, and as Bob said, the problem of practice is easier said than done. So fortunately over the years, we’ve worked with lots of teachers who are interested in making that vision of student talk possible. So today, we’re going to show you two different videos.
The first video is Katie Arrillaga. She’s a second grade bilingual teacher. She’s working with her students in the video on a subtraction problem, 11 minus eight. As you watch the video, I want you to focus on some effective teacher or student moves that you see, actions that you would like to see in all classrooms. So without further ado, just see Katie at work.
Katie Arrillaga:
Now, we’re going to look at Olivia and what Olivia did. And here it says, oops,
Class:
I counted up from eight to 11. There are four numbers, eight, nine, 10, 11. The difference is four.
Katie Arrillaga:
I want you to think about what Olivia did, okay? I’d like you to turn and talk to your partner
Student 1:
And Olivia got it wrong cause 11, you, uh… you subtract eight and equals, it’s not four…
Student 2:
You want help?
Student 1:
Yeah.
Student 2:
Eight plus three equals 11, but she said, you’re not supposed to go eight, nine, 10, 11. You’re supposed to put the eight in your head and then go nine, 10, 11.
Student 3:
You need to go past the number. You need to go past the friendly number, and then go to add the number of friends.
Student 1:
You get to a friendly number.
José Franco:
That was Katie. Now, I wanna hear from you what you saw that worked. So here’s how we’ll do that, let’s do a waterfall. And on Zoom, our Zoom life, we probably have heard about waterfall. Let me explain, I’m going to give you about 30 seconds to write into chat something that you saw that was powerful, that was effective from the teachers and the kids, but don’t display your comment until I say waterfall. And when I say waterfall, then I want everyone to post your comment in a chat, and Bob and I will then read and highlight them. So I’m gonna give you 30 seconds, but remember, don’t post your comment until I say waterfall. So let me put on my timer, you got 30 seconds to write something in chat, but don’t hit the return yet. Okay, that was my 30 seconds of wait time. On the count of three, one, two, three, waterfall. Post it. Boy, Bob, there’s lots of great comments here already.
Bob Rosenfeld:
Oh boy, there are.
José Franco:
Listening. Mark Feliz, I see rich, authentic, low-effective filter, discourage, right, discourse, right? I love it.
Bob Rosenfeld:
Students helping each other, students having dialogue back and forth.
José Franco:
Yeah, Eileen, thank you very much for that comment. Michael, turn and talk. We see that all the time. There’s a lot of interaction, right, Jennifer? Susie, welcome. Kids are given the opportunity to engage in math conversations with each other. And that’s one of the basic, like I said, one of the core tenet of math path wisdom Pitfalls. A couple, Bob, do you want to point out any other ones before I-
Bob Rosenfeld:
Oh yeah. This idea, Brynn says the teacher prompted there might be a mistake or an oops, but did not specify or give hints, right? Let the conversation go where it’s go. Let the students have that discovery moment. Yes, and one thing I wanna point out, Jose, is that oops is a fictional student. That was in a lesson. It’s important to note that that wasn’t a student in the classroom who was being called out. It was a fictional student in the lesson around which students were having the conversation, so it was totally safe.
José Franco:
Right, before I turn over to Bob, I just wanted to point out a couple other things that I notice whenever I watch this video. I noticed that the kids were justifying their thinking on that subtraction problem, and that goes back to the standard of mathematical practice number one, making sense of problems. I also saw the kids constructing viable arguments. That goes back to the standard of mathematical practice number three, if you’re familiar with the book from NCTM, “Principles to Actions,” practice number four is about teachers facilitating meaningful math discourse.
And then the last thing I wanna share before I turn over to Bob is that this particular video, we chose it because it would address two of our foundational principles that Bob introduced at the beginning, building mathematical discussions and making sense. So there you have it. Bob, why don’t you go ahead and take over?
Bob Rosenfeld:
Thank you. Yeah, yeah. The other thing I wanna say is that, and I put this in the chat, it was obvious that this didn’t happen overnight, that students had had these kind of conversations, it takes practice, it takes a concerted effort, it takes intentionality to create a classroom climate and culture that gets to this point. So let’s not create any false expectations that, oh, we’re just gonna, overnight, have some student turn and talk to each other and have these sorts of conversations. We need time, we need practice, and we need tools.
And one of the things that we’re gonna be sharing with you here today, Danny’s gonna put a box note link at one point for you to get access to, you might’ve seen in the background in the video, there was a poster on the wall with what we call discussion builders, and that’s a poster that gives the teacher and the students some scaffolding. So it shows the students the kinds of sentences and questions they might use with each other so that they have something concrete around which to build that discourse and to make it safe. And all the language is safe that they can use, and it provides opportunities for students to not only have partner and whole group discussions, but some of the questions actually lend themselves to self-reflection, which we know is so key in math.
Students asking themselves, “How did I know that? Or what other ways might there be to think about that?” Yes, you can ask another student that, but ask yourself first. And so those discussion builder posters, let’s see what those look like quickly ’cause they went by very quickly in the video. So they look like this, and they’re available right now in English and Spanish. And as I said, Danny’s gonna make sure you’re gonna get 8.5×11 versions of these that you can print at home or in school. And then we have our books do come with a full poster-sized version of this, but we wanna make sure you can start using these right away. So we’re gonna provide those.
And so as you can see, and there’s different kinds of questions for presenting ideas, for adding to other ideas, for asking more questions, and again, they can be used as self-reflection with a partner or whole class. So I just wanna make sure that you are aware of those. Now, scaffolding for students is one important piece of this, but we all know that scaffolding for teachers, and so we also have a tool where we help teachers scaffold. And so let’s take a look at what that tool looks like, and that tool, we call it the clipboard prompts. And again, these were developed over years of talking to teachers.
So this is, teachers walk around actually with a clipboard or you could have it electronically on an iPad or you could have them memorized, but the idea here is that teachers have scaffolding on the kinds of questions they can ask of students as they’re mingling around the classroom to help students understand a problem or to help them process or to help them reflect. And so these questions help to move the conversation forward based upon what the teacher is hearing or not hearing coming from a student group. And again, Danny’s gonna provide these for you in a folder that you’ll have available to you as we conclude this session, but I’d love to share some more. So I’m gonna hand it back over to Jose, who’s gonna share some more video and some more examples I believe from another grade level, right, Jose?
Jose Franco:
Yes, yes. Thanks a lot Bob. So Math Pathways and Pitfalls is about looking at correct solutions and helping the kids make sense of them. Why do those answers make sense? And Bob mentioned earlier about oops, so in math Pathways and Pitfalls lessons, we will always show a oops problem, a pitfall, because we want the kids to see and be able to explain why that doesn’t make sense. So you saw Katie. Now let me introduce Kim Kean. And Kim is a sixth-grade teacher, and in this video you’re gonna see her students working on a ratio problem. Again, as you watch the video, focus on some effective teacher or student moves that you see. Danny, go ahead and show us the video.
Kim Kean:
So we have our first imaginary student. We have Erica, she’s our imaginary student dor this one, I would like you to follow along as I read out loud to you. If he uses twice as much red and white, it’s 12 red to 16 white. If he uses half as much red and white, it’s three red to four white. They’re all three red for every four white. This is how Erica solved the starter problem that we just worked on. So right now, I just want you to take a couple minutes, look at Erica’s work, look at what she labeled, look at why you think she labeled them that way. She has some connecting lines that are connecting on her ratios, see if you can figure out why she did that, what she’s identifying. And I’m gonna give you about 30 seconds, 45 seconds to look at it yourself, and then I’m gonna ask you to talk to your partner.
Student 4:
She made 20 boxes, cut ’em in half, and then made four, so then when she was trying to show was that two times as much was like the same thing as what you times, what you multiply six over eight by two by two.
Student 5:
So what I did, I draw some squares. 20 squares, and from the top is six, I cut six and I’m gonna cut eight, and I simplify to the third, six times two equals 12 and eight times two equals 16, and I color, and then you can turn it to the simplify, and I simplify again. And then 12 times four, 12 divided by four equals three, and 18 divided by four equals… 16 divided by four equals four, so 10 squares here.
Duwan:
You got three-quarters for the answer? Me too.
Kim Kean:
Okay, great job discussing. I heard a lot of really good in insight to a lot of the different things you thought Erica was thinking. And the first thing I’d like to do is I start it off with a drawing here that you have in front of you. I was wondering if someone can come up here and explain what Erica was doing, how she got the ratio 12:16.
Duwan:
So she did 20 squares and shaded in six reds and eight whites, so that was six to eight. I think she divided two by six-eighths to get three-fourths because it’s a equal ratio to six-eighths, and that’s why I think she got three-fourths.
Kim Kean:
So Duwan, you did a great job explaining what she did here. I’m wondering if you can identify the simplest terms of this equal ratio here.
Duwan:
How I found that out was because when I found out she either divided by two or just halved it, it was three-fourths, so I know two of three is six, so I found out she did it by three and four because it’s half, and that’s the equal given ratio.
Kim Kean:
Okay, so what about on your 12th to 16th? Can you identify your pieces here? I’m gonna kind of cross over a little bit if that’s okay. So we talked earlier about our vocabulary words, and one of the vocabulary words we talked about was scale factor, so can you tell me what scale factor she used in this problem here? So you told me that our original ratio was six to eight, and then she got to go to the cafeteria to make it a bigger room, she got 12 to 16. So what was the scale factor she used here to get our number, 12-16ths.
Duwan:
Two.
Kim Kean:
Two, very nice job. So in other words, what does a scale factor tell you to do with the number two?
Duwan:
Times or divide.
Kim Kean:
Okay, so did we multiply in this problem or did we divide in this problem?
Duwan:
We multiply.
Kim Kean:
Okay, so then let’s go to the third problem.
José Franco:
Great. So now, once again, we’re gonna do the waterfall. Remember, I’m gonna give you 30 seconds to write into chat something that you saw that you’d like to see in all of the classrooms from teachers or students, but don’t display your comment until I say waterfall. So I’m gonna give you 30 seconds, but remember, don’t post your comment until I say waterfall. You have 30 seconds to do this. Go. That’s our wait time. One, two, three, waterfall. Bob, these come like a waterfall. Cherie, thank you for saying multiply, time is given for understanding the visual, the auditory, and the hands-on.
Bob Rosenfeld:
Yeah, I was there when I saw that. I see the emphasis on the precise language, right? And making and having those anchor charts to show that.
José Franco:
And Francine, you commented about the teacher facilitating the learning through questioning. That goes back to a question that was posed earlier after the first video. That’s part of the process in order to get the kids prepared to talk about mathematics.
Bob Rosenfeld:
And we saw those people making the connection to the visual, seeing the visuals there.
José Franco:
And Jason, you comment about one of the students needed to visually see the scaling factor to explain it while the other students were able to explain into words. That’s one of the nice things about Math Pathways and Pitfall is you address different modalities, different strengths, all of us have five different learning styles. Luana, you praise the conversations and the students’ insight. These are all good effective practices that I’d like to see, and again, going back to what I saw, again, standard mathematical practice one, making sense of problems and persevering, solving them, constructive, viable arguments. The boy constructed that viable argument and explained it very eloquently.
Teaching practice number four about facilitating those discourse, and then also the teaching practice five from NCTM about posing purposeful questions. That was Kim straight to get the kids to that point. Bob, I’m gonna turn it back over to you.
Bob Rosenfeld:
Yeah, I mean, I see somebody talked about the idea here that clearly a climate had been created to facilitate students not feeling hesitant. And how was that created? And again, created over time, it’s created through concerted effort. This is not something that happens overnight, and it’s created by using the scaffolding that we’ve talked about and having tools, and teachers feeling comfortable using those tools as well. So the teacher was perfectly comfortable holding a clipboard there and looking down at her notes and thinking about what questions to ask. And that’s wonderful. Teachers are acting as learners themselves in front of their students, and that’s something we really wanna see.
The other thing I wanna talk about is this whole philosophy of introducing these sorts of Pitfalls and having these sorts of discussions around key mathematical concepts is something that we’ve found can be very flexible in the way that it’s used. In some schools we’ve worked with, it is used before a difficult concept is coming in the core curriculum, in the core textbook. So I know this is gonna be very difficult, this concept of the ratios, and so I’m gonna present this to students, this pitfall, and have students discuss it ahead of time before presenting the lesson. In other schools, it’s used as the lesson.
And in some schools, we’ve seen it used afterwards when we discover if students have had some difficulty with the lesson from the core program, it’s a way to go back and rethink this and to see it through another lens. And in some schools, this is used again as part of the regular core curriculum, and in others, it’s used more as an intervention in a separate setting. It works in all of these settings, because again, it’s all about being aware of what are the key concepts in math, how to get kids to visualize it, how to get them to talk about it, and as many of our participants said in the chat, to be comfortable having these conversations. So it’s very flexible in how it can work, and even beyond that, I know, Jose, you were sharing with me, you’ve seen teachers start to use these discussion builders outside of math.
So yes, it’s fantastic in math, but these clipboard prompts and discussion builders work great in science. They work great in ELA, in social studies. It’s really the same kind same mindset, is what I’m getting at.
José Franco:
We have been working with teachers from different parts of the country over the years, and I’m sure all of us have attended workshops or participated in projects that provided us with great ideas to help address our professional needs. So I wanted to share with you a comment that Ellen said about her involvement in a three-year initiative that we had at Tucson Unified School District. Ellen says, “Math was the yearly struggle to get certain students to and above grade level, and the stress of teaching math was growing, not only for the teachers, but also for the students at times. Math Pathways and Pitfalls made the math being taught accessible to all students. Give it a chance, find a way to use it in your classroom. The proof is in the research.
Once you make it a version for your classroom, the explosion of learning begins.” Ellen left the classroom a year after participating in the project, but she still introduces Math Pathways and Pitfalls to her colleagues because now she’s an instructional coach. Katie and Kim had similar reactions to their work with Math Pathways and Pitfalls. As a matter of fact, when we worked with Katie and Kim, we did three two-hour sessions of PD with them before they ran with it. Kim told us that, after the project ended, she told us that the boy who presented his thinking to the class, before she started introducing Math Pathways and Pitfalls to her students, he just sort of sat back, in the back of classroom and didn’t really participate with mathematics, and for whatever reason, because of the structures or the techniques or the opportunity to talk with one of his partners, by the end of the year, he was as engaged as you saw him on the video. And by the end of the year, he was voted class leader by his peers. He found his voice. Bob, I’m gonna turn it over to you.
Bob Rosenfeld:
Thank you for sharing that, Jose. So again, we have all these resources. We provided some to you today. There’s other resources, and Danny, I believe, is gonna provide a link to our website where we’re excited to say we just came out with a second edition of our books that have all of these lessons with all the oops and pitfalls in them. We call them oops, for younger grades, pitfalls for the older grades, all kinds of lessons all built around the key mathematical concepts for each grade span. So there’s not lessons for everything, but lessons for the key pieces. But the real point here is about the mindset and the philosophy and getting students talking. That’s really what it’s all about. And we hope that that’s what you leave with, is the desire to really embrace this philosophy and this approach.
But Danny will make sure that you’ve got access too. If you wanna read more and see more of our resources and the research that went into all of this, we’d be happy to share that with you, or feel free to reach out to any of us, and we’d be happy to have a further conversation with you about what we’ve learned over the years doing this. This is our passion. This is what excites us.
José Franco:
And Bob, I just wanna add one quick comment before Danny closes today’s session. It’s about talking, but it’s also about learning from our mistakes. That’s what the pitfalls is all about. Let’s talk about what can we learn from our mistakes? And with that, Danny, I’m gonna turn it back over to you.
Danny Torres:
Well, thank you, Jose and Bob, for a great session today, and thank you to all our participants for joining us. We really, really appreciate you being here. I’ve dropped links into the chat for the clipboard prompts and the discussion builders in English and in Spanish. I’ll also share them via email and post them on the webinar archive page on WestEd.org. For those of you interested in learning more about the Math Pathways and Pitfalls work at WestEd, visit us online at mpp.WestEd.org. Please feel free to reach out to Jose and Bob via email if you have any questions about the work we discussed today. You can reach Jose at jfranco, that’s J-F-R-A-N-C-O, @wested.org and Bob at [email protected]. And you can also sign up for WestEd’s email newsletter to receive updates. Subscribe online at WestEd.org/subscribe. With that, thank you all very, very much. We’ll see you next time.