Carlos Cabana is a master teacher with a strong commitment to student engagement and a wide range of experience in East Bay secondary public schools. I’ve quoted him before on this blog: Technical Writing, Forward Design, Learning Tools. Last year, he mentored two student teachers. Today, he shares a conversation he had with them about their experience using the Lab Gear, the manipulative microworld I created for basic algebra. Carlos is a long-time Lab Gear user, who finds that it works well with his student-centered pedagogy.
As I read the transcript of this discussion, I was thrilled to see how some of my hopes for the benefits of Lab Gear use were realized in those beginning teacher classes. Specifically:
- The Lab Gear supports students’ intellectual engagement and creativity.
- It facilitates student-to-student, student-to-teacher, and teacher-to-student communication.
- It is helpful in different ways to students whether algebraic manipulation comes naturally to them or not. It is an equalizer among students with different prior experiences with math.
- It supports group work.
- It invites intellectual risk-taking.
- It provides a way to evaluate the correctness of work done on paper without a need for teacher intervention.
These benefits are more evident in a classroom that goes all in with the Lab Gear, but even more limited use of the blocks will provide some of them.
Many thanks to Carlos, Diego, and Shruthi for sharing this!
— Henri
PS: One point that Carlos made is one that would not be as obvious to a beginning teacher: the Lab Gear makes it possible to question traditional sequencing. With the Lab Gear, you can tackle factoring (“make a rectangle”) before the distributive law, making the latter much easier to understand. You can start with more complicated equations, which makes simple ones more accessible. I share a lot more about learning tools and curriculum sequencing in the book I co-authored with Robin Pemantle: There Is No One Way to Teach Math.
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Using the Lab Gear in Algebra 1
Carlos: Hi my friends, welcome back. Would you start by introducing yourselves?
Diego: My name is Diego Smet. For the past year, I was a student teacher in Carlos’s Algebra 1 classroom. It was my student teaching year, and now I’m going off to be a high school math teacher.
Shruthi: Hi, I’m Shruthi Aiyar. I also, with Diego, was one of Carlos’s student teachers last year. The two of us co-taught as student teachers, and three of us were a teaching team with ninth grade Algebra 1. And I’m sticking around to be Carlos’s classroom neighbor, also teaching Algebra. This year is my first year teaching as we leave our student teaching and our credential programs.
Carlos: Last year, as I have for many years, we used the Lab Gear to help students learn algebraic symbol manipulation: simplifying and solving. I’ve always been deeply grateful to Henri for teaching me about the Lab Gear the summer before my first year teaching 37 years ago. So I would love to pay that back a little bit and hear your thoughts about the learning process for kids and for you. Do you remember your first thoughts when you heard about us using Lab Gear? What were some things that you maybe weren’t sure about or had concerns about?
Diego: I’d never used or even seen algebra tiles before, so it was definitely very new to me. I remember being curious about how long it was going to take to introduce the pieces to students before we could use them to do algebra. I don’t think I had doubts that Lab Gear was going to work, probably because you had used it a lot before, so I knew that there had to be something working with it. I was just really curious because it was so different from the way I learned algebra. At the same time, my little sister was also going through factoring in her classes, so I remember getting excited when we talked about doing factoring with the Lab Gear and making generic rectangles. She was really struggling with factoring because for her, it was just trial and error, trying out all the different numbers and she was not very happy with that.
Shruthi: My immediate impression was that the Lab Gear felt much more real than the procedures I had interacted with before. I remember, when we were initially just playing with the blocks, seeing an x tile and wondering if its name was a three and a half. It pushed me to think about the visual representation of algebra differently than growing up. Like your sister, Diego, I also learned factoring in a way that I actually don’t know if I fully understood until last year. I think I was just kind of doing it, and crossing my fingers that it would work. But last year was when I saw I can understand factoring as length times width equals area – as something geometric. The experience of getting my hands dirty also gave me insight into what kids experience in their exploration.
Carlos: And what were the results? What happened with kids and their learning using Lab Gear?
Shruthi: I saw kids think and experiment so deeply and with a lot of fun and exciting risk taking. That was able to happen because they were working with tools that they really got to know. I think the way that it was structured from starting with the creation of a toolkit [naming the pieces and their dimensions] to naming a collection. The sequencing of it all, and the way that kids really got so much experience just being with these tools, and then becoming part of their language, gave them a lot of opportunity to get creative in how they could be used.
Cristal was a student who stood out in how the visual and tangible representation pushed her to think beyond two dimensions to see what needed for her to solve the problem. There was a headache and she needed aspirin, to quote Dan Meyer. She was like, “We need to go 3D.” I think the tools themselves let her be able to see that necessity, and then take the steps to make it happen. I think it’s so cool that it gives kids the opportunity to push themselves, and see what’s possible beyond just binomial multiplication, beyond the limits of FOIL.
Diego: Like Shruthi said, it was really cool seeing students use it as a tool.
Carlos: What do you mean by using it as a tool?
Diego: It was really cool to see the way students were able to try to explain their thinking and what they were doing using the tiles. As we kept working with Lab Gear, I think there were students where the algebra clicked really easily for them. But there were also students who took a little bit longer to get comfortable with manipulating the pieces and understanding how it all fit together. But even the kids who were struggling, they still were able to point to tiles or hold up a tile and talk about it. It was still a way for them to express what was in their heads, which was really cool.
Carlos: When I started teaching algebra, I frequently heard kids ask, what’s x? At times in the unit when I thought it would already be clear that x is the variable or an unknown number. I haven’t heard that since I started using Lab Gear for exactly the reason that you’re saying, that kids always know physically what x is. The fact that they can manipulate the blocks gives students something to try that isn’t wrestling with a procedure, wrestling with an abstraction.
For kids that you saw would otherwise struggle with number or other aspects of mathematics, did you see them making sense of algebra in unexpected ways because of the Lab Gear, or to an unexpected extent?
Shruthi: Yes. The Lab Gear was new to everybody, so it was completely new vocabulary for every single kid in the room that did feel like a fresh start around how we are talking about variables or numbers, and how we can use them. Vocabulary is what I feel like Lab Gear ended up becoming – a way to describe and understand the way these different pieces in algebra work together, that everyone was learning together from the baseline. It gave kids withvarying prior mathematical experiences and background knowledge a similar-ish starting point. They were able to manipulate what was in front of them and take risks. They came up with really interesting things because a lot of it also came from a place of play and exploration that can be really safe and exciting for kids.
Carlos: And hard to do with just numbers and variables.
Shruthi: Absolutely, absolutely.
Diego: Lab Gear gave kids a way to turn whatever was in their heads into something physical. Like Shruthi was saying, it gave everybody a similar starting point. I think it also gave a lot of the kids a good common language to talk about all the different algebra concepts. While they were explaining, they could hold up a tile or point to something or show it with the tiles. And since everyone knew the names of the pieces, it was a lot easier for students to understand each other while talking about algebra. Whereas I think if it’s just in their heads, you know, when it’s in your head and you are trying to explain a math concept, it can be really hard to get an idea across.
Carlos: Vocabulary can support our ability to explain or it can get in the way, depending on how abstract it is for the person listening. That’s a really good point. These are ideas I’ve never thought of before. You made me think of a class I taught years ago where one of the students, Jose, had an IEP and was a second language learner. He was very scared about his own understanding at any given moment. But he really latched on to the Lab Gear. One week he was sitting next to Lauren, who was full of charisma, very talkative, fearless about sharing, everything we would want in a courageous young woman in ninth grade. She at some point blurted out, “Mr. Cabana, Jose can solve equations in his head!!” I asked her what she meant, and she said, “He can visualize all the steps for solving equations with Lab Gear in his head and just come up with the answer.” Jose had no idea that he was doing anything really special. He was completely blown away by Lauren’s praise, and it completely altered how he showed up in class from that moment and also how kids perceived him. Maybe it was accentuated because of his IEP, but the fact that the procedural rules of the game were all visual and tactile, I think made all the difference for him.
That makes me want to circle back to Cristal because I think a common perception of manipulatives like Lab Gear is that they are just for kids who struggle or kids with special needs. I think your Cristal story is an example of how Lab Gear is a tool for sense making and has nothing to do with level of needs.
Shruthi: Lab Gear is very much a low floor high ceiling tool, in Jo Boaler’s language. It’s a sense making tool that is about how we can understand the algebra at so many levels, down to what is the meaning of each individual piece in isolation. What does it mean when we put them together in addition, in multiplication, solving? And we can just keep
getting bigger and bigger with what we use Lab Gear for. The Lab Gear gave Crystal an additional challenge to express and understand and work with the math visually and in a tactile way. She was really used to working mainly with numbers, which her prior math experience gave her a lot of comfort with. And I also think that having the opportunity to get her hands in this did give her the opportunity to do more than what she had done before, and translate it into what she also already knew. Lab Gear gives opportunities to every kid to learn more, and learn more about themselves and what they can do with math – and who they are as mathematicians.
Carlos: I think a lot of teachers might worry that high schoolers would find a manipulative childish. But you’re describing how, over time, kids see opportunities for exploration. But what about in the beginning? Do you remember kids having that reaction?
Shruthi: Reaction to being childish?
Carlos: Yeah, like, why are we playing with blocks?
Diego: I don’t know if I ever heard a kid say that or ask why are we playing with blocks, but I think one of the initial issues was them, just playing with it. I don’t know if I should call it an issue because even playing with it can be helpful for them to just get a feel for what the tiles are and how they fit together and what they look like. So I don’t know if I would have said that anybody had the reaction of Lab Gear is childish, but sometimes maybe they got a little distracted making a tall tower.
Carlos: There was that acculturation period where because it’s a thing in front of them that they can build with – that’s a temptation. But I think after a week or two, that went away, right? We didn’t have to tell kids to stop playing with the blocks when someone was presenting.
Shruthi: It loses its novelty of being a toy after a couple of weeks of using it as something mathematical. It became oh, this is my math class thing, you know? But I think it was helpful on day one, to give them some time to just play with the blocks for a little bit.
Carlos: That’s something I learned from elementary school teachers: that when you put out a manipulative, everyone’s impulse is to play, even ours, right? So we might as well give students that time because they’re going to take it anyway. Kids love nothing more than building tall towers and then knocking them down.
You guys started talking about this a little bit, but I’d love to hear more. How was teaching algebra using Lab Gear different from your own experiences learning it? Was it challenging for you to switch to a visual representation?
Diego: It was. It was very different. When I learned algebra, it was a lot of just, like for factoring, it was guess and check, just try out different numbers and see whether they work. And for balancing equations, something that was actually different in the visualization of Lab Gear was that, if you subtract a number to change the equation, in Lab Gear, you make zeroes, and so you’re adding new pieces in and then taking away what there is. When I visualize, you’re moving the piece to the other side rather than balancing it. In my head, that’s how I think about these things. But working with Lab Gear, I realized not only is that not what happens, it also doesn’t really make mathematical sense. You can’t just move something over. And so, I think there were challenges there. That was just an example of something that wasn’t quite lining up with what I had learned, what I’ve been thinking for the past eight years since I took algebra. It challenged my own understanding of algebra to make sure that I could use the same language as the kids. But once it worked out, it was super helpful because kids have a common language to talk to each other and now I also had a common language to work with the kids.
Carlos: I never really thought of that common language idea before – thank you! Now I’ll never forget it. Did you find yourself looking for a way to connect that experience of moving a term over with the Lab Gear so that you could explore it with kids?
Diego: I don’t think I ever really found the need or found the opportunity to say, you know, here’s another way to think about it, especially because that wouldn’t have felt mathematically sound, with everything else that we were telling kids about balancing the equation.
Carlos: What about you Shruthi?
Shruthi: I think I undoubtedly became a stronger mathematician in the process of teaching algebra with Lab Gear. I came into last year with such little conceptual understanding around factoring. I don’t think I really knew much about why it worked. I guess I knew why we did it – it’s helpful to make an equation look like that. But not really around the process. I think working with Lab Gear was what I needed as a student. As we were teaching it, and as we were exploring with it, I was like, oh, my gosh, the math makes sense to me. It’s really making sense to me. Factoring was a sticking point for me through all of high school math. I just never got it, and it always felt like something that was getting in my way. And so it was really exciting for me to get to work with Lab Gear, and to walk away from last year feeling like I really get this now. I’m right there with the kids on this one, you know?
Carlos: And you have a math degree in math, so it’s not because math has been a struggle. It’s because the way you are asked to think about factoring was not around sense making, it was around procedure.
Shruthi: Completely around procedure. It was just like a list of methods that never made any sense to me. I still could not tell you what the X method or the diamond method are. They are something that I was told to do and, and it was really hard for me.
Carlos: What lesson elements or pedagogical practices do you think are helpful in support of learning algebra with Lab Gear?
Shruthi: My instinct is to come back to what, Diego, you were saying around common language. I think the consistent language around, for example, making zeros and balancing as opposed to mathematically ambiguous language like canceling out is powerful and something that was important for me to stay really consistent on. I grew up using the word canceling out, and I see that really doesn’t mean anything. Or it can mean just too many things. With the Lab Gear, those things look different on the workmat, different ways that we describe them.
Diego: Yeah, and to keep on that idea of common language, I think that student presentations are super helpful to give the presenters practice in using the language and being very clear about the steps they were doing – what piece they were talking about, and how they were manipulating it. And also for the audience to hear the language and hear the explanations and see Lab Gear being moved around on the screen. And also groupwork. They have one set of blocks and they need to be showing each other, they need to be discussing how they use the Lab Gear. We need to be giving students as many chances as possible to have to explain their work and be very precise about what they’re doing.
Shruthi: An important bridge was our “taking notes in algebra day” [translating solving with Lab Gear to algebraic notation]. That gave students a next step of, we’re using this Lab Gear to understand the algebra, which can be written this way. We can represent the Lab Gear process in this written way – exactly what you’re doing here with the blocks looks like this in algebra. It felt more like an extension of what was just already happening mathematically with the Lab Gear. I think it was a necessary and helpful way to bridge what we do with the manipulatives and how we use what we learned here to write it when we don’t have the manipulatives in front of us.
Carlos: What about teacher checks? Say more about what those look like and how they supported learning with Lab Gear.
Diego: We gave students a problem, and when they feel like their whole team is ready, they call a teacher over to explain. We spin the pen or have some other random way to pick a student at the table to share their reasoning. Teams don’t know which student is going to get called on, so every student in the group needs to be ready to be able to explain. That means that, while they’re working, students need to make sure everybody is on the same page, and they all have, not just like the verbal language to explain it, but they can all show it with the Lab Gear. They have the verbal language, and they have the physical language moving blocks around They’re able to combine those into a strong explanation so that when they call the teacher over, no matter who gets picked, the group is ready to explain. They’re not only communicating with each other about their process, they can explain it in a way that makes sense to the teacher. And then if the teacher needs to ask questions, then we can also use that language to ask why they did a certain step or what was going on.
Shruthi: We ask students to be intentional about working in the middle of the table, and to rotate which person is leading each problem in the middle. That doesn’t mean they do the problem on their own – they’re just the one that has their hands on the Lab Gear for that problem and is moving the blocks as the whole group thinks and works together on it. As a teacher walking around the room, something that I was looking for was to keep kids together and make sure that no one was getting left out of the opportunity to engage with the materials in front of them.
Carlos: I think the only thing I would add to that is that Henri was the person who taught me the pedagogical value of sequencing problems that many teachers might find backwards. For example, traditionally kids are taught to distribute first and later multiply binomials. Those are taught as two distinct things. But if kids learn to multiply, not just binomials, but a binomial by a trinomial (or anything by anything), then distribution is just a special case. And because students have associated multiplication with building rectangles, they can see the relationships, they can see what happens, often mentally. At least they have intuition about what will happen in a way that makes it not just unnecessary to teach distribution separately, but unnecessary to even teach the word distribution. It’s just multiplication.
With combining like terms, because we started with a big mess of Lab Gear blocks to name, there was a lot to process there that made it a more interesting and informative problem than if we’d just started with 2x + 5x. With solving equations, because they were solving the Lab Gear first, they started with equations with lots of messiness on both sides. Then there is more to decide and do, and that means more emphasis on what moves are mathematically valid. Once students tackle those, then solving less complicated equations are easier for kids because they have mental models. They have had lots of decision-making experience because we started with geometric complexity.
Do you have final thoughts or advice for teachers who are considering using Lab Gear?
Shruthi: Besides do it? (laughs) I think there’s so much that makes Lab Gear worth it that we’ve already talked about, but it’s a different way to think about the math, which, I guess, means there was a learning curve for me. My learning curve helped me understand, the mathematics more deeply, and and also my kids’ understanding of the mathematics more deeply. Both of those things happened in my very first interactions with Lab Gear, and I think that’s going continue to grow as I meet and learn and get to know more kids, and get a deeper understanding of the power of Lab Gear. It is not something I would want to teach algebra without.
Diego: My advice would be, like Shruthi said, do it, but also let the kids do it. Let the kids play around with it. I had never used tiles before, so they were as new to me – not just were the tiles were new to me, but the learning process with them. Since it was all new to me, that meant that when a student was working something out in front of me, I would let them go a little bit longer, even if it didn’t initially make sense to me. I’d let them keep going. Students made mistakes sometimes, or went down the wrong path, but also a lot of the time they did do it right. It just wasn’t a way that I had thought of. Maybe it took them five extra steps to get there, but what mattered was the way that they thought about it how they showed it. That was really cool to see.
Going back to Cristal, she had a question that she wanted to work with more. Shruthi and I watched her figure it out and build it without us really needing to jump in much. I think just letting the students work with an idea and run with it. I’d suggest not pointing out a mistake as soon as you see it, don’t correct them. Just let them try it out and see where they go.
Carlos: Because the Lab Gear gives them ways of self-correcting that paper-pencil does not.
Shruthi: And different kids see patterns in the Lab Gear differently, just because our eyes go to different places. When we see a mess of tiles, our brains want to do different things first. The Lab Gear affords the opportunity to move things around without it feeling scary. There’s just many different storylines that it can take. It’s exciting seeing what kids come up with. It made those two months one of my favorite units of the year.
Carlos: Even after all these years, I never get bored of watching kids work with Lab Gear.
This conversation makes me even sadder that we don’t get to keep working together. Thank you both!
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