Keeping our math communities humming

In this post, Kara Imm shares a May 2020 interview with Kate Smallberg about how to transition our number strings practice to remote/distance learning. Their conversation was recorded and is available in two forms: as an audio interview or as written transcript. Kate also provides a five minute video montage of what her practice looks and sounds like with kids.

Kate Smallberg A former Math for America (MfA) Master Teacher and K-5 educator, Kate is an elementary math coordinator, coach, and teacher in NYC with a Masters in Leadership in Mathematics Education from Bank Street College of Education. She has a gift for helping children fall in love with math. Kate works to empower teachers and parents to foster a playfulness with math, inviting every child into the search for patterns in numbers and space. You can connect with her by tweeting @KateSmallberg and following her Facebook page, Playing with Numbers.


Listen to Kara and Kate’s conversation here. Or read it below.

Interview: Kara Imm and Kate Smallberg

Kara:              Okay, well hey, everybody. I’m here with Kate Smallberg, who is the grammar school math coordinator for Columbia Grammar and Prep (New York City). Hi, Kate.

Kate:              Hi, how are you? Thanks for having me.

Kara:              I’m great. Yeah, absolutely. I was so excited to learn that you have really masterfully figured out a way to take number strings—this routine that we love—and figure out how to bring it to distance learning. So I’m so happy that you’re willing to share what you’ve been doing and what you’ve been learning with all of us. Tell everybody a little bit about your role at Columbia Grammar.

Kate:              Sure. So I just started this year as the grammar school math coordinator, so that means I’m working with grades Pre-K through fourth grade. And the work entails a little bit of everything. I’m doing math with students, I’m doing math with teachers, I’m doing math with parents and faculty. I pretty much do math all day every day, which is the dream job for me. And right now I’m also covering some teaching positions, so I’m actually getting to teach as well, even in distance learning.

Kara:              That’s so cool. Great. Where did you, as a teacher, first learn about strings? How did you come to know them?

Kate:              So when I started teaching I knew I always loved math as a kid and I had a feeling I would love teaching math. So even as an associate teacher I asked my head teachers could I try this math piece? Could I do it? And I discovered that what was really fun was just having discussions — rich math discourse with kids. I was always doing a version of either a number talk or a number string without even knowing what it was called. I was just getting kids to talk about numbers and equations.

And then when I met Kara at Bank Street Math Leadership Program, she came and did some number strings with us as part of our coursework, I discovered that there was a name for something that I knew would be wonderful and then I discovered the blog and then I discovered the books. And then I kind of just poured into number strings. What are they? How many different versions? What can you do? What’s the point? Oh, there’s a different goal for each one. How do we embed them into our curriculum? And I just started really diving head into number strings. So now for about five or six years I’ve been working on number strings with all different grade levels.

Kara:              That is really cool. And you’re referring to the time at Bank Street I put everybody on the floor? All these adults in July?

Kate:              Yes, yes, yes. Yeah, we all sat on the floor and we all did a number string together as grownups. And even when I lead number strings with grownups, that’s the best thing. Whether you’re a teacher, a student, a parent, anything — number strings is all about just playing around with math and playing around with numbers and seeing what happens. So it’s even more fun when you’re on the floor.

Kara:              I agree. Well, we can’t be on the floor now and I think you and I appreciate something about number strings, which is like the beauty of having close physical proximity with kids, allows us to hear little ideas of strategies, things that you might whisper to your turn and talk partner. We can’t replicate that right now, but you’ve done a really beautiful job of replicating so many aspects of the routine. So I wondered if you just talked through the technical part of how are you doing it with classes of first through fourth graders, so that maybe somebody could follow in your footsteps and replicate for their own kids.

Kate:              Well thank you for that. I think you’re absolutely right. This idea that we don’t have the physical presence of children in front of us where we can use our Vulcan hearing when they’re doing that turn and talk. Ooh, I heard you say that. Can you say more about that? And that was the first thing that I really felt like how do I replicate that? So when I do these strings I am signed in on two devices. I’ve got my computer in front of me so I go into gallery view or grid mode and I can see everyone in the class. And then I also sign in on the secondary tablet device. I am finding that having a pen or like an Apple pen or a stylus is critical for writing on boards.

My iPad becomes my whiteboard screen so I use Notability or any kind of whiteboard app sometimes in tandem with the various virtual tools — virtual math tools that you can find online, like the Math Learning Center has tons of them. So I am able to recreate a whiteboard. I am able to recreate virtual tools.

But how do I recreate those moments of discourse that are so powerful? And something that I’ve tried is getting kids to respond to questions with physical gestures. So I may say, “Show me on your fingers what the difference is between these two numbers,” and they’ll hold up their fingers to the camera. Or I might ask some yes or no questions and they can use their thumb to respond this could be greater than or less than. Any kind of question that forces everybody to answer in that way is the best mode that I can use for checking for understanding as we go.

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Kara:              That’s nice.

Kate:              The chat window is also a nice feature. I think you do have to support students in terms of understanding how to appropriately use the chat. That takes a little bit of modeling in the beginning, especially in elementary grades where kids are so excited about the chat and they want to say hi to their friends. But we can actually use the chat to respond to questions as well. So…”Type in what you’re thinking.” What do you notice about this picture of dots before we even dive into the string and then they’re typing in their answers?

Kara:              That’s beautiful.

Kate:              So I have been able to use those techniques to somewhat replicate the turn and talk. I know that the breakout feature room is another technique that can be used, if you want to send kids into a breakout room and then bring them back. What I’m discovering though is we’re moving a lot slower in distance learning than we might be able to move in a classroom. So if it means covering half as much of that string or pulling out the really important ones and then honing in on those because the act of going into a breakout room, coming back or the technical difficulties that come with all the — you have to account for that time.

But I’m okay with it going a little slower. I think that’s okay because the most important thing to me is maintaining the community of mathematicians in a way that we can have a discussion and discourse. And it’s been working and the kids are building off of each other’s ideas in a way that I would have been asking them to do in the classroom. Can you respond to so and so’s ideas? Can you say more about that? And so those techniques I can still use. I’m coming to you in three hands. Be ready. You can paraphrase, you can ask a question or you can share your own thinking.

So if a reluctant learner is not raising their hand, rather than cold-calling which is a technique I can use in the classroom, that is a lot more intimidating for students in this setting. So I’m trying to find ways to make it more – to feel more comfortable for students. And I’m finding that I’m asking a lot of leading questions in order to make that happen, and those are not the kinds of questions I would normally want to ask or if I was really being reflective of my practice like I did at the Bank Street Math Leadership Program when I thought about questioning techniques, and I noticed those are so many leading questions you’re asking, Kate.

How do you re-frame the question? I’m now noticing that those questions actually work to get everyone engaged in these strings so students have their cameras off on my call. And I would say hey, put your camera on for a second or just chime in if you want to leave your camera off and tell me about this. Do you notice anything doubling in this image? And that’s a very leading question, but it’s one that they can say yes or no to. And so that engagement is more important to me right now in this setting…

Kara:              That part that you just mentioned was really interesting that kind of in the distance learning modality you’re noticing that your questions are changing and that you’re allowing yourself to use some leading questions if it then leads kids to engage in a way that they weren’t before. That’s kind of the permission you’re giving yourself.

Kate:              Yeah, I think it is. And hopefully with time I’ll even find more ways to do it so that I can keep asking those open questions. I think for me the golden one is what do you notice, letting that question be the guiding question for everything that I do. And that’s not one I’m willing to ever let go of, but yes, you’re right. I do have to give more permission. And something else that’s really challenging is the wait time, the silence in the classroom is okay. ‘Cause we know that we’re all thinking.

In a classroom you can feel the thinking. It’s palpable. You can see kids looking, you can feel the brain power working. But silence in a distance learning classroom doesn’t feel that same way. Everyone wants to know is the audio working or should I fill the silence or whatnot. So that’s challenging but giving students the space and the time to think is certainly needed, but also just to be aware of that silence doesn’t feel as golden in this setting. Just that awareness of it might help us to be okay with that.

Kara:              That’s right, that’s right. So it sounds like you’re trying to stay true to the routine. The routine is all about kids’ strategies, kids’ thinking, you’re continuing to as much as possible get kids to build on each other’s ideas in distance learning. And you mentioned our questions have changed a bit and wait time and silence has a different kind of effect in distance learning. Are there any other ways that you’ve made modifications to kind of make it work in this environment? Are there kids that are just really hard to engage in and you’re still working on that as a teacher?

Kate:              The modeling that often goes into strings and the ways that we model things, I’m finding the need for the visual to be almost the starting point sometimes rather than the numbers. Just like this idea of virtual tools, ’cause I mentioned sometimes I use virtual tools on the iPad. Those are not the same as manipulatives that kids can hold in their hands and do stuff with. And the fact that they can’t do any of that right now unless they’re using their Legos to build ten sticks or whatever, but it’s certainly a challenge to try to do manipulatives virtually. However, I think the visual model creates starting point that I’m using a lot more than I did in the classroom.

And I think that is a way of engaging all students as well as just start with the picture and tell me what you notice and then build the numbers into it. Certainly as we’re doing fractions that’s been a big help as well. But that’s certainly — it’s really sweet as the kids talk about the picture they might point to their screen and say this right here. And so there’s actually a nice silver lining to this is oh, now we’ve rediscovered the importance of precision in our math language. And I don’t know what you’re pointing at. I know you’re pointing at something and it’s gotta be good, but can you tell me specifically what you’re pointing at? And so it’s actually forced us to use precise math language in a way that we might have gotten away with in the classroom because we can point to the chart in front of us or we can point to the board and we all know what we’re pointing at.

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Kara:              That’s really cool. Great. Your school is an independent school with some degree of resources. So in terms of getting your kids had digital devices and that wasn’t a problem, are there any other — certainly people are raising equity issues in general with distance learning. Is that true for the population you’re working with in terms of access or equity?

Kate:              And I think equity can take the form in terms of physical resources, but also this idea that this math is hard in a number string, and by the time we get to that last, a “zinger” of a problem, as my colleague calls it, it’s tough and I can certainly see that some kids might want to tune out when we get to those challenge problems. But all the techniques that I might have used in the classroom are not always going to translate for them here. And so it is more challenging and that’s kind of where those leading questions certainly come in for me and I feel like that’s where I can grab them a little bit and lead them somewhere to get that engagement.

But I also think that it’s important not to not do the hardest ones. I think that we still have to provide these really big questions and to be okay with time running out and not getting to feel completely confident. Living with that disequilibrium of not knowing until we come back to this the next time is okay, and that’s a feeling that I love. When kids can leave feeling like I think I got it up to there but we’ll come back to it next week. So that’s okay. So we’re actually finding that we’re putting ourselves in that position a lot more now than we would have in that classroom where we might have had more time to dive into it later that afternoon.

So now we have to live with it until the next day and we come back to it and we really wrestle with it and we see oh, understanding can come from misunderstanding. So that’s a nice outcome of that as well. But in terms of the physical resources we were able to provide every student with that. In terms of teachers though, I think one of the things that I mentioned having the writing tool so that I can write on the iPad to replicate the board, that’s part of this as well. Do teachers have all the right tools to do that?

And how can schools help teachers get this, and what do you do if you don’t have those tools? Could you actually use a whiteboard behind you? I’m sure you could. I’m sure you could — and see how it works. Could you put the numbers on a Google slide — typed versions of it and just switch from slide to slide and do all the work ahead of time to populate those slides with what you anticipate students might say and the models you anticipate they might create you could do. It just removes – you can’t write it in the moment. So there’s lots of workarounds to replicate it in terms of accessing technology, but it’s certainly a challenge and one that I don’t think we even have all the answers to at this point. But I’m excited to engage in more learning over the summer as to all the possibilities that technology affords us.

Kara:              That’s really cool. You beautifully are offering our community like a montage video so we can spy and sneak in on you in the classroom. Can you just let people know what you put in the video? What will we see in that five-minute video that you made? (see video below)

Kate:              Absolutely. So it’s like a montage of the selection of strings from fourth grade, third grade and second graders, and you’ll see it in that order as the fourth graders working through fractions is offer grade…and the third graders are starting to think about factors and multiplication and division and then even the second graders learning about constant difference and what happens when we’re making pairs to 10 and do those replicate in pairs to 20 and on to 100. So all of the big ideas from the landscapes are coming out in these strings, and you’ll hear it in their noticings, their strategies and how they describe the models.

Kara:              And so it sounds like you’re working with a group of kids who have already been doing strings before. So you’ve moved the practice into a distance learning. Can you imagine trying strings brand new? Do you think for a teacher who’s never done numerous year routines, is this the time to start?

Kate:              You know, I think it’s never a wrong time to start a number [string] — because the whole point of an instructional routine like number strings is it has to become routine. That means you have to start somewhere. And that means when you start it doesn’t have to work the first time and that’s okay. In fact it doesn’t have to work the first five times and I think the first year that I was doing number strings I don’t think I’d do any of the things now that I did on my first year. I’ve completely transformed the way I do it. So start any time because every time you do a string, even now every time I do a string I learn something new.

And whether it’s about how kids think about the numbers, what kids notice, what I say in response — I’m always learning about the teacher moves in strings. And so I guarantee even if you started it now for the first time you’re gonna learn something. You’re gonna take something out of it. And the kids will learn mathematics from it no matter what. There’s always some math involved that they’re getting from it. Even if it’s different from what you had hoped or planned.

So I’d say yeah, absolutely go for it. And yes, like you said we had been doing strings all year so this was a continuation of a routine. And in fact having that be something that we continued in distance learning was that comfort of school is still school. I’m still doing the routines I did at school and so in many ways it felt very comforting to the students. In fact, one signed on and realized they signed into number strings and said yes, number string time. It’s something that everyone loves.

Kara:              That’s really cool. All right, well we thank you for doing this interview with us and sharing a bit of your practice. Can people reach out to you if they have questions about —

Kate:              Absolutely.

Kara:              Great. That would be really cool. So, I will provide an email so you all can reach Kate. Anything more to say? Are you having fun? It seems like you are.

Kate:              So much fun. So much fun. Yeah, you know what I have to say is the silver lining to the crisis here is not only are we engaging students in this thinking, we’re actually engaging families as well. ‘Cause now we’re looking into each other’s homes. We’re not looking, but we’re in each other’s homes doing math. And so we’re kind of — the beautiful thing about this is we’re all doing it together. And I’ve noticed that some families are sitting by their kids and that’s wonderful and we welcome that and we want them to be a part of this discussion with them.

Even after the string ends they’re still talking about the numbers at home, so I think that that’s a nice takeaway point is how do we bring these strings and how do we continue to do strings in the classroom, but how do we also keep this transparency of what we’re doing in math and school so that it can also happen at home and this playfulness with numbers can also translate to home. And I think that’s a beautiful kind of circumstance that this is allowing for.

Kara:              That’s beautiful. We should tell people about — is your site called Play with Numbers? Playing with Numbers?

Kate:              Yes, Playing with Numbers. I’m starting to build the site with lots of different – I’m hoping to populate it with some other videos and maybe even some video versions of number strings, like digital number strings that you could use as prerecorded content for upcoming school years if we do find ourselves in this position again. And as soon as summer vacation hits I plan to have a lot more time to populate that with tons of content.

Kara:              That’s really cool. Kate, thank you for this interview and for this contribution and I have a feeling lots of people will reach out to you after today.

Kate:              Thank you so much.

Kara:              You’re welcome.

Kate:              This has been my pleasure as always.

Kara:              Thanks.

Here’s a video from Kate showing a variety of school math communities doing strings together.
Thanks to Kate, kids and their families for allowing us to study their practice up close!

Let’s keep collaborating: Reach Kate by tweeting @KateSmallberg and/or following her Facebook page, Playing with Numbers and Kara at @KaraLouiseImm.

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