Redesigning math communities for distance learning

About Kathy: Kathy Minas is a 5th grade teacher at a public school in Los Angeles. She is an enthusiastic mathematician who loves engaging in collaborative problem solving alongside her students. She began her teaching career in New York City eight years ago. She holds a M.A. in Childhood General and Special Education from Teachers College, Columbia University.  

Listen to Kara and Kathy’s conversation here

Kara: Well, welcome back, everybody. I am here in October with Kathy Minas who is our old friend from New York City. Kathy’s now in Los Angeles working at Gardner Street Elementary School in grade 5. And Kathy has been doing strings for many, many years. And I’m delighted to just talk with her today about, you know, doing strings in a variety of school communities and developing her practice and more recently learning how to do it remotely. So welcome, Kathy!

Kathy: Hi, thank you for having me. I’m so excited to talk math.

Kara: Yes, yes, me too. So tell tell the people listening or reading, “How did you first hear about strings? How did you first know what they were?”

Kathy: So I first learned about strings in my first teaching position, where I worked at a school in New York City, a public school on the Upper East Side. And we used Contexts for Learning and we had a partnership with Math in the City. So number strings were a part of our math routines that we use regularly. And there was, you know, professional development with you, and just through observing other teachers teaching strings. So it was a big part of our instruction.

Kara: Yeah, I remember being in your classroom back in the day. In fact, we’ve written about that on this blog. One of the things that you did then, which I’m sure you do, now, you built this really beautiful community of mathematicians, you were one of the first people who I remember referred to children as mathematicians, and you took took their ideas seriously. Can you say a little bit about like, it felt like you were at ease in the routine, that you knew what the purpose was, and that you really were having a lot of fun? Did you have other goals when you were leading strings?

Kathy: I think that the biggest part of leading strings for me, that that led to me feeling so comfortable was just doing the strings in advance, and really knowing what some of the different ideas that could come out of the strings were. In addition, something that I found really exciting about math strings is that the first couple of problems really engage everybody. So even if the concepts get more complicated as you go on through the numbers, the first one or two problems were problems that I would say almost 100% of the kids can answer. And that that excitement when you put a problem, you know, in fifth grade, that’s like two times five, and you see that look on that student’s face that’s just bright eyed and like the light bulb goes off, and they have that moment when they’re like, “I can participate in math today.” And without a doubt, always, like comment came out at the end of math for those students like, “I can’t believe I participated today.” You know, and that, that excitement about math, I think, really got me like, I have to be teaching number strings regularly.

Kara: That’s I love that idea of like, “I participated today.” Like, that’s a good one. I did that right? I changed the pattern. You said that you um, you got good at leading number strings, because you did them in advance. How you did? Like, do you have us? Did you sketch them out? Did you talk through them with a colleague? How did you prepare?

Kathy: So a lot of the number strings I did when I first started doing number strings came directly from the resources that we were using. So whether we were doing an investigation or using the number of string books, so if I had, if I knew we were working on multiplication, I would look at the progression of the skills, and then pick, you know, ones that I thought would be appropriate for my group of students. But also within those resources, they, they also highlight some of those big ideas for you. So I would definitely say that, in my first years of doing number strings, I really leaned on those resources until I felt more comfortable developing my own number strings eventually.

Kara: Hmm. And you you’ve probably been doing number strings long enough that, like me, there are certain strings that when I go into almost any group of of learners, I say, “Here’s the clunker.” It’s like, I know three strategies that are going to happen on this problem. Right? Do you find that to be true now?

Kathy: Yeah, yeah, absolutely. I think every once in a while, you know, you get a student who has a completely new way of doing that string, or to solving problems, but it’s almost like, if the first time you do it, two strategies come up, then you hold on to those two strategies. And the next time that you do that string, you already are familiar with what kids might say. That’s right. So the more that you do them, the more you have been exposed to other people’s thinking.

Kara: Yeah, that’s right. Recently, I’ve been working with lots of student teachers in New York and we’ve been thinking about how important kind of daily routines are. So we are simultaneously working on Read Alouds and creating a ritual — a class community —in which the teacher is kind of an expert reader who’s making his or her thinking visible through the read aloud. And, you know, we’ve been thinking about how the number string is similar and that we make it a daily routine. And we’re highlighting, like, “How do mathematicians think? And how do we develop strategies, right?” I’m wondering if if you can tell us a little bit about what your practice looks like now? How often are you able to do strings: What’s changed? What stayed the same?

Kathy: So I would definitely agree, I mean, always, the beginning of the school year is, you know, community building and the predictable things of establishing routines and expectations. And then this year, obviously, we had the curve-ball: we’re completely in remote learning. So I would say the first three, four weeks were really was about establishing technology, you know, help and expectations on technology. And so number strings and other math routines, were really helpful to get the students thinking mathematically, in a very low risk way. So the same, I would say that read alouds, and number strings and other math routines were among the first things that I began to integrate instructionally this year, because they’re like small chunks of experiences, and like literacy and math, but they just get the kids participating and engaged. And they allow you to sort of assess their knowledge before you really start to decide what you want to teach and how.

Kara: Yeah, so I like that idea that they’re small, kind of small “chunks,” small pieces of instruction, you’re not the first person to say that. But I think sometimes, that’s what makes them accessible. It’s like, you don’t have to throw out your entire curriculum, you just need to find find 15 minutes. Usually, there’s someplace right in the day where you can find 10, 15, 20 minutes like that.

You mentioned, a place where kids can be engaged. And I know some people are struggling with that. They’re saying things like, “You know, it’s hard to see all the kids on the screen. When I’m sharing my screen and showing strategies. I don’t know how to do a turn and talk.” Like I remember being in your classroom and you’d say, you know, “Mathematicians, turn to your partner” and the whole room would be like buzzing with conversation, right? Do you have any ideas about how do we how do we make strings really engaging for kids now?

Kathy: I think that whenever I launch strings, or I launch a math routine, I build a lot of excitement into it, before even doing it. I’ll say like, “Oh, gosh, mathematicians, you’re in for such a treat!” And we really just build it up so that they’re excited. And depending on whether or not they’ve had experience with strings before.

So fortunately, my fifth graders now came from a classroom where they were doing number strings last year, so I get to build off of that experience that they’ve had. And they really, again, like I said earlier that they get really excited about for those those first problems that really like are easy problems, and that they can all have a way of participating or sharing, specifically for the remote setting. And what I did was I put I mean, do you want me to go through like the actual like how I structured it?

Kara: Yes, walk us through one, that would be great.

Kathy: So logistically, how I set it up was that I, I had the kids all of their cameras are on and they’re all on mute, and I deactivated the chat box. So it [chat function] was just set up to “host only.” And I explained, “We’re going to do a number of strings, the problems are related.” And that I would show them the problem. And then they would have a chance to look at it and come up with their response. And then they type their response directly to me. So nobody else could see it. And that’s the same the equivalent of like put a thumb to your chest when you have your answer, which is what we would do in the classroom. And I love that because it took away the pressure of kids even looking around to see if someone else had their thumb up, or that one kid who couldn’t follow that rule would say like, I got this or this is easy.

I also then like I collected their answers [through the chat function]. And I love that because there was no risk of someone sharing an answer that was that they didn’t think was right, which might happen in the classroom, because it was coming to me anonymously, I could just record the different answers that I got from the chat box. So that’s the point where I got really excited about doing it remotely because someone might have a different answer. They don’t know what anyone else said. So it just takes away that pressure and it’s completely anonymous. And then there was the option for them to participate by raising a virtual hand and that would be the equivalent of like, “Okay, who wants to share how you got your answer?”

Kara: How did you that, that’s really lovely. You’re, in a way, you’re saying that the technology affords risk-taking…supports kids to take risks, and maybe you’re getting more kids to venture a strategy or even an answer, because it’s just you and that it’s like whispering in the teachers ear your idea? Don’t put it on the chart?

Kathy: Yeah, definitely. I think that part I thought was, was really interesting, because I saw a lot more kids just type in their answer and send it to me where I feel like in the classroom, I think that they would have also said the same thing that other students may have said. So it gave them that courage to know like, no one’s going to know what I said. Though, that being said, obviously encouraging them within math to be okay, sharing their thinking when we’re in our, in our community as well.

Kara: So, that’s lovely. And then I’m thinking there’s like a cool move where, let’s say you post a multiplication problem, and you get a series of answers like 40. And there’s a bunch of kids who think it’s 48. And then you think about it, say it’s 52, like potential, you say, “Oh, it looks like these are the three most popular answers. Let’s hear from you know, like, have each group of people kind of defend their choice.”

And I think that’s a nice moment where you can have kids say, “Did anyone change their thinking?” Hmm, you know, like, yeah, minute ago, I thought it was 52. And I was, I was in the 52 camp. And now that I’m seeing the array, or the number line or hearing logic, I’m pretty sure it’s 48. Yeah. Are there other things that you’ve learned to just use the technology to work for you and not against you?

Kathy: Um, I think one thing that sustains it not necessarily, for the better of the worse, was definitely using like, I have a document camera. So I have that set up for the kids, and I’m recording their thinking that way. I know, you know, other people prefer having a chart in the in the remote setting. In person, obviously, we would be charting the words that they were saying and the number string to capture their thinking. So I think I think that just having some way, for me, the document camera works the best.

I think another thing that works well is it kind of gets me to take pictures of the work and to post it so that the kids can access it afterwards. I would always do that with charts for families to see what we’re working on. But this way I can capture more of the learning so that the students can hold on to it more in a different time. So I might post later, you know, what would the next problem be in this in this string? Or do you have any big ideas? What was your big noticing and have them have a chance to like type in their own responses?

Kara: That’s cool. And then you have like a almost like a digital notebook of right like it was I remember being in your classroom. And certainly your school was really well versed in beautiful anchor charts where kids look back and, and have a memory not only, I think two things that were happening in your school.

One was that I remembered the chart being present for the entire string so that kids were not having to draw upon working memory to remember what the first problem was, it was always really visible. The space were like, we’re gonna make our thinking visible. And this is not a memorization activity. But then just the beautiful way in which you codified that into anchor charts, right? I didn’t walk in and see like 27 strings, I saw three solid anchor charts where kids could look around the room, look up at the screen be reminded of, what did we mean by doubling and halving? Hmm. And so it seems like you’re, you’re doing that now digitally making like a digital notebook?

Kathy: Yeah, I think having them be able to access the work that we do, because they don’t have the environment to reinforce the strategies and the learning that we’re doing. And then something that I learned from you and from my colleagues before was to name the students who are giving the different strategies, and that always works really nicely for engagement, that the first person who shares a big idea or notices something really big in the string, you just put their name, and then we start to kind of refer to it as that person’s, you know, idea. And then later on, the kids can come back and say, like, “Oh, that was Henry strategy that helped me remember to solve this problem using decomposition” or whatever strategy to use.

Kara: Yeah, I love the idea of local expertise, or you’re authoring your own ideas, and I’ve seen so many kids just kind of light up when they realize, wait, my name is on a poster, like that was important. And, you know, recently I’ve been, I’ve been helping teachers to think about how when there’s, you know, there’s like status differentials in classrooms. There’s, you know, kids have the perception, particularly in fifth grade already, of like, who’s “good” at math, and who’s “fast”? And so I think, when we author we give kids authority to like be experts. I feel like it starts to change the status or how they perceive themselves and how other kids perceive them. Do you find that to be true at all?

Kathy: Absolutely. I think especially in fifth grade, the students have such a strong sense of themselves as mathematicians that sometimes I’ll have a whole group of kids who have strong math identities, or I’ll have a group of kids who just feel really defeated when it comes to math. And then we have to go back. And I do a lot of work at the beginning of the year around growth mindset, and just the development of the brain. I’ve gotten really into reading and listening to Jo Boaler, I’m a big fan of hers, where I teach the kids about, like, what happens physiologically to the brain when you make a mistake, or people who are told that they’re smart all their lives, and what the implications of that is. So I found so far this year that it’s made a difference in the mathematicians approach to math and like, just everything we’re doing when they feel frustrated, just pointing back to that, like, you have to take risks, this is your brain growing, and it makes them feel more at least aware of what’s going on with themselves when they get frustrated or hit a challenge.

Kara: That’s cool. How do…how do you and the class, you know, given that work around growth mindset, and neuroplasticity, how does your class community this year talk about mistakes, their own and each other’s?

Kathy: I think that there, this particular group of students that I have is super supportive of one another. So they’re very encouraging. They’re really kind. So if somebody makes a mistake, they they ask questions to clarify what they meant, or how they got their answer. They’ll restate, “Oh, show you what you mean by this.” And they’ll restate in their own words, to see if they understood what their their classmate meant.

So there’s a lot of like, really productive collaboration and communication going on with them, which is really nice to see. I think just understanding what happens to your brain when it’s triggered in this like panic mode. It just being aware of that at 10, or 11 years old, makes such a difference in their willingness to take a risk. And last year was my first year doing doing that. And I found a big shift in even like my students who were, you know, the ones who always got in trouble, or the ones who had outbursts that they were like, “I’m in that place, you know, my part of my brain has shut down, I need to take a break. So sorry, I digress. But this is connected to math, right?”

Kara: Yes! It’s really important.

Let’s say people are listening to our conversation, or reading our conversation, and they’re just like, brand new to this. And maybe they’re a little skeptical. I mean, teachers are asked at this moment to just learn so many new things so quickly, and kind of keep so much in the air. What would you say to them about why this particular routine would be one worth learning, worth getting good at and worth doing particularly now?

Kathy: I would say that, definitely, in terms of a new teacher, just your own comfort level, listening to your students talk about math, and to get an idea of their own mathematical understandings. Number strings are super low risk, not only for the students, but for the teachers. I think that going through and having, you know, at the very minimum, like the correct answer written down for yourself before you go into it, so that you’ll know you know what the right answer is, but then just really making the floor open to the students to share how they got their answer.

And I would say that when I first began number strings, I probably got through maybe the first set like the half of the number string first, and then you can always come back to it another day. So not try trying to push through all of the problems necessarily in one sitting. But you know, coming back to it a different day, or if you’re working on it, and it’s not going the way that you want, you can always come back to it or if you’re not sure what the big idea is, you can just open it and then leave it for the students to kind of think about it and say we’re going to come back to it.

So I think there’s a lot of flexibility with number strings, but it’s not it’s not perfect. And I think letting go of that idea that something is going to be implemented perfectly is just so huge in relieving ourselves to that pressure that teachers, and new teachers, put on themselves. It’s it’s a work in progress. It’s been you know, I’ve been working on this. This is my eighth year of teaching so I am still developing my number strings practice myself.

Kara: Yeah. I mean, I feel like a lot of us are still kind of letting go of that need for it to be perfect or this idea that like, there is a perfect version of this lesson. Many of us are like, “Nope, there’s none.”

But given that, how do you know if a number string is going well? How do you know one didn’t work so well or that one was great?

Kathy: If the kids are listening to one another, or if their ideas that they’re building off of one another, then that’s great. I think that the times that I find personally most frustrating, doing numbers strings, is when the students aren’t seeing the connections between the problems. I think that I explained, if that happens, I’ll pause and say like, “Does anyone see how this, this problem relates to any of the other problems?” And I think that the time that I hit that wall is when when that when nobody sees the connection between the problems, and that’s when I know that, you know, I have to go back and we need to keep working on on that, depending on what the concept is.

Kara: Yeah, you said something really important that I want to highlight for people listening, which is, you know, you we design strings with this, like logic behind them, right, like I have chosen these problems, or someone has designed these, so that we can tackle this messy 19 with this very friendly 20. And all of us have been in the position of like, the kids are not taking up this relationship, they’re not seeing it, they’re not using it.

But what you said in that moment wasn’t what I sometimes hear, which is, “Arghh, they didn’t get it!” As if there’s one purpose of the string. And if the children don’t say it, then we’re, you know, it’s a mess. You said we need to keep working, which I think is just a very generous and lovely way to say like, if I put something in front of kids, and they don’t notice, we need to keep working on it. It’s might be new, it might be unfamiliar, they might not be ready, the numbers might be too tricky. Or just like we might just need some more experience and then, right and so it’s not about jumping at or not getting, it’s just like, okay, no biggie.

You also said basically that, like, you know, it’s almost like the conversation and a lot of strings ends with the right answer. Or in many math classrooms, it’s like when kids get the right answer it’s like, “Can I get some praise for this? Right answer, done!”

And I think I think for many of us in strings, it’s not that the conversation ends with the right answer. It’s like the conversation begins when we have an answer. Right? Then we get to explain what you call the how well how do we know it’s 48? Right? Let’s think about all the ways that we could convince ourselves. Yeah, that’s really neat. Tell us just a little bit about what you’re working on now, like, what are the strings happening this week in fifth grade?

Kathy: So we are just beginning our grouping problems, I think we spend a lot of time building our mathematical routines and just our community. And we actually also just finished doing a bunch of really difficult, like, mind puzzles in math for my class, just because I wanted to break down that idea that some people are good at math, and some people are not. And we kind of leveled the playing field with that. And that was kind of my first time doing that. And I loved it. And it was it was a very informative experience.

So we’re in grouping problems. And to launch strings, I did go into addition and subtraction work just because I felt like the students would be more comfortable just showing what they know about strings and the strategies that they’re using. And now we’re going to go into like multiplication, partial product strategies, things like that. So even in their work, that they’re doing the word problem work, I’m seeing different strategies popping up. So picking strings that can allow them to share those ideas with the whole class is kind of where we’re at.

Kara: Okay, well, I think we’ll pause there. I will give people a way to reach you if they heard anything or want to collaborate with you or need like, I think partner to, you know, design strings with or get some feedback. Thank you so much for taking the time, Kathy!

Kathy: Thank you for having me!

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