Episode #23: They’re Engaged, NOW WHAT!? [Math Mentoring Moment]
In this Math Mentoring Moment episode, you’ll meet Bill Zbozny Del Percio, a Grade 3 teacher from Pittsburgh, Pennsylvania who always felt drawn to teaching, but feels like he’s struggling to squeeze as much mathematical thinking out of each task as possible.
Having built familiarity and confidence in sparking curiosity through 3 Act Math Tasks and delivering other rich tasks that leverage the natural curiosity of his students, Bill is seeking support from the Math Moment Maker Community to determine how he can best Fuel Sense Making on a daily basis.
- Explore how we might select worthwhile tasks to elicit mathematical thinking;
- Learn about efficiency and the fact that what is a fission for one student is not the same for all students;
- How we might approach pushing our students to be more clever with their mathematical strategies and thinking;
- Strategies to ensure we are anticipating how students at various levels of readiness might approach the task;
- How understanding how mathematics develops conceptually can help to fuel sense making for all students to move them along with the developmental continuum at their level;
Chicken Or The Egg: Area or Perimeter? [Blog – Graham Fletcher]
“Do We Have Enough Paint?” [Task – Robert Kaplinsky]
Cover It Up? [Task – Kyle Pearce
Airplane Problem [Task – Kyle Pearce]
Donut Delight [Task – Kyle Pearce]
Why The Unit Of Measure Matters? [Guide – Kyle Pearce]
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Jon Orr: Okay Math Moment Makers. We are on the line with Bill [Zbozny-Delpercio 00:20:08] who is from Pittsburgh and he’s calling in tonight. How are you doing tonight Bill?
Bill: I’m doing pretty well. How about you guys?
Jon Orr: We are great. We are great. We are pumped.
Kyle Pearce: Yeah, yeah. We were just coming off a weekend there and we are excited to chat with you tonight. So before we get going, do you mind telling us a little bit about yourself? We know where you’re from, we know you’re from Pittsburgh, but what’s your story?
Bill: Well started off, went to college for a couple of years on Area, started with my Education degree, got a little sidetracked, came back to it, in my mid to late 20’s, and then just been on like a wild ride here. Working around the Pittsburgh area is kind of hard to get a job around here sometimes and I was lucky enough to land one without having to move and just really enjoying being in a classroom. I teach third grade and it’s more fun than I ever thought I could have at work, but also three times more challenging and harder than I ever thought it would be. So, all those great things.
Jon Orr: Isn’t that the truth? I definitely relate to that and I have to get my hats off to you teaching third grade. I have two third graders in my house right now and I can just imagine what a classroom of third graders would look like. I’m a High school teacher, so it’s definitely daunting for me to think about teaching third grade, and maybe it’s the same for you to think about High school, but hats off to you for sure.
Bill: Oh yeah. I love it. It’s the perfect age. They can do enough on their own where you’re not constantly being interrupted or asked, but also at the same time, there’s still a lot of just a natural curiosity and joy that they still have for learning haven’t quite lost that yet.
Kyle Pearce: Yeah. You said at the natural curiosity, humans are curious beings, and that’s like that perfect age. My wife actually had spent most of her teaching career in grade three just came off a couple of years in kindergarten, which she enjoyed. Both of my children went through in the same school she enjoyed, but she definitely had missed, like you had mentioned just that they have just enough independence. They’re still like all geeked out about learning, they’re not at the too cool stage like the grade sevens. Yeah. So that’s kind of her niche. She’s in grade two right now, but I think hopefully one day she feels like she wants to hopefully get back to grade three so very cool.
Bill: It’d be great.
Kyle Pearce: Awesome. So why did you want to become a teacher? What inspired you to kind of head down this path especially when you’re saying in Pittsburgh that jobs weren’t necessarily super easy to come by a very similar story to us here in Windsor, Ontario over the past decade or so jobs we’re kind of hard to come by. So what’s your, I guess, passion that led you down that path?
Bill: I just kind of always have been drawn to it. Every other job I’ve done, I’ve ended up being the person that had to train people or do things along that lines or if it all was part of a team always like teaching the other players how to do this or how to do that. Baseball, Basketball or whatever it was and it just kind of is a part of me and one day I just decided to stop fighting it. And I had a tremendous group of teachers come in from Elementary all the way through High school. And I had a pretty special teacher in college and she was able to kind of get me back into it. We had kept the touch while I took the little break and she was like, “Bill, are you coming back to finish this?” And I was like, “Sure”, and she’s like, “All right be in my office tomorrow morning.” And I was like, “Oh, okay.” And I basically said like, “you’re doing this now or you’re never going to do it.” I had a couple people that kind of give me a little kick in the butt to do it, but it was definitely always something I was kind of trending towards.
Jon Orr: I’m always like, I just had a student come back and say that I taught that they were studying to become a Math teacher. And I always think of taking that two ways and it was the same way that I thought when I was becoming a Math teacher that, one way is you take it as a compliment that they, you think that you inspired them so much that they’re also going to become a Math teacher. But the other way, which is what my motivation was, was that I was gonna do it differently. So that student of mine said that I was like super pumped, but then in the back of my mind I’m like-
Kyle Pearce: Take a deep breathe-
Jon Orr: “… wait a minute. Are they are they doing this because they want to change things? They want to do things differently than they way they were taught.” So, yeah, I definitely relate there.
Bill: Yeah, that’s definitely a fear. I’m sure I have a couple of comeback, like, “No, I can do it better than you.” [crosstalk 00:04:13].
Kyle Pearce: And you know what? Nowadays, I try my best to keep that open mind and say, “You know what? I hope they do. I hope they come back and we want to keep getting better. So hopefully it’s not in spite of what we’re doing in our classroom, but hopefully just to keep pushing forward. So. Awesome. Awesome.
Jon Orr: Bill, there’s a question that if you’re a listener of this podcast or any listeners out there know this question is coming, but we definitely have to ask you, what is your most memorable Math moment from math class? This could be as a teacher, but it also could be thinking back to you as a student in the classroom. What’s popping into your mind soon as we say Math class?
Bill: Would have been my sophomore year, sitting in geometry with a teacher and he was a tremendous personality, extremely brilliant guy, but it was like he was kind of speaking another language most of the time. And I just remember I’d heard the word before, but he was really trying to explain to us what a Hypotenuse was. And that just kind of always sticks in my mind where it’s like you’re just talking a whole nother language than what I was hearing at that time or what I could comprehend. And just kind of like one of those little barriers to understanding. So that moment I keep in the back of my mind just as a, “Am I relating to my students and am I being overly complicated or am I using correct terms and explaining them.”
Kyle Pearce: It’s so funny when it comes to language and I heard a colleague of mine, Yvette, Yvette Lehman has said this a few times. I think it’s quoted from maybe Lucy West about discourse in general, but also mathematical discourse and talking about in Math class like we have, it’s very similar to science where sometimes the language can sound so complex and oftentimes it’s just our approach to how we use the language and making sure that we do so appropriately. Because know you’ve got kindergarten kids run around saying triceratops and that’s not an issue for them. It’s because they actually have a connection there.
Kyle Pearce: So I like how you have that awareness that you’re always thinking if I am introducing some complex language that students haven’t been exposed to, I’ve got to make sure that I sort of set the stage for that and make sure I use the language appropriately and often enough so that students can actually make those connections and actually start using it in their own daily language.
Bill: Yeah, exactly. It’s hard sometimes because you either fall off using the term that you need to be using or you interchange terms because as an adult it’s whether you’re talking about product or sum will often just say total and that’s incorrect, but then they see product on a Math quiz and they’re looking at you like, “you told me this but…” I’m like, “Oh yeah, it’s because I haven’t been using it. [crosstalk 00:06:48] And that’s my fault, not your fault.”
Kyle Pearce: Yeah, exactly. Exactly. Being so intentional around our language is really important. That’s great. So we’re wondering, Bill, what’s on your mind lately like as Math teachers? Jon and I know very well and we constantly have our own struggles and challenges going on. So we want to open the floor to you today. What sort of struggles or challenges seem to be on your mind lately and maybe we can get them out in the open and have a chat about it today?
Bill: The biggest thing has been on my mind then I’ve been reading a bunch of blogs and a bunch of email chains and things about, you guys call it sparking curiosity and I feel like I kind of have that part. I’ve been doing 3-Act Tasks for a year now. They’re kind of something, when I was first introduced to them I really was like, “This is a great way for me to reach all the learners in my classroom and they can all achieve the same goal.” Because there’s kind of like that open middle, so many different paths to go to it that whatever level they’re on, hopefully I can take them to a more, I call it sophisticated way of doing it in my classroom. Like, “Hey, we’re third graders now, let’s not just keep adding three together and skip counting. Let’s try multiplication or at least chunking it a little bit.”
Bill: But once I get them kind of interested in a task, I find that I get a little bit lost in there somewhere with them. And I’m not sure if I’m kind of getting the most out of it. I tend to find myself either with the high group trying to coach them up or with a low group sitting there and total confusion trying to figure out like, “okay guys, we’re exploring area here. Let’s get out the tiles, we’ve got some manipulatives. And I feel like, okay, while I’m doing that, everybody else has kind of gone and I feel like I’m missing some opportunities there. I don’t know if I’m not prepared for what they’re going to bring, if I didn’t anticipate it or if I’m not picking appropriate tasks. That makes sense.
Kyle Pearce: You know what? You used the word sophisticated. I know oftentimes we hear the words like efficient and there’s that desire in Math to try to kind of push down that developmental continuum and a word I really like. I hear Kathy [Fasano 00:08:55] use it quite often is she uses the word, “clever”. Like trying to get the kids to be more clever. It’s kind of a little bit for me anyway, I’ve noticed a difference. When I say it, it’s like seems less scary. It seems less, I don’t know, forced for me, because I used to try to work towards that efficiency and what I realized over time was that oftentimes many kids were working at what I would call their most efficient method thus far because that’s where they were. But kind of pushing them to kind of put a little bit of thought in that metacognitive piece in there and really try to think of like, “What’s going on here? How could I do this a little bit easier? Is there a way that maybe I don’t need to draw all of this out or I don’t need to, I could start moving from an array to an area model and that’s going to save me a little time?” Or whatever it might be depending on the stage for the students. So very cool. Very cool.
Bill: You are kind of saying like maybe they already are working at there, like maybe that’s a third option. I’m not thinking about that they might be working at their capacity at the moment.
Kyle Pearce: Yeah. It’s definitely something to consider and not to suggest we want to keep them there. We definitely don’t want that. But oftentimes, like what I used to do and what I often say when I do live workshops is, I used to take kids by the collar and I would drag them to where I wanted them to be. Now, I wouldn’t physically do this. Maybe back in the day you could, but not anymore. Not here anyway, but I used to sort of take them and I’d be like, “Oh, you’re doing…” I’d have some grade nine students who are not using multiplication, they were using repeated addition and I was trying to get them to think multiplicatively so I’d sort of be like, “no, don’t do it that way, come on down here.” And what a lot of Kathy Fasano’s work has helped me to realize is that, we want to help every student be a mathematician, they are mathematicians.
Kyle Pearce: We want to help them see through their solution strategy based on what they’ve selected and we want to help them get to that next place. But without it being like us, without it being us forcing it on them now, that’s definitely difficult to do. So it’s not something that, you just get to decide tomorrow and everything’s all hunky dory. But it really had me thinking more about like, what is the next step for this student? I know where I want my class to be at. We got this picture, we know our curriculum standards are Common Core standards or whatever standards we’re using. And I know that’s where I want them to be, but that’s by the end of a school year and there’s kids that are all over the map and the reality is, is that first many kids, they might not be there today and I’ve got to really think about like, what’s that little nudge I can push them in the right direction?
Bill: Yeah. An that where I think I kind of get lost sometimes. We just introduced that Area and Perimeter. And actually after reading some things I decided to introduce Area before Perimeter for the first time. Some of the things I read, it seems to be a more, it’s often confused. So I was like, “all right, let’s get secure in Area because been using Arrays and it seemed like a more natural progression than my curriculum introducing perimeter after doing all these arrays and kind of muddying it. And I did a 3-Act Task to try to get an idea to see how much they knew beforehand and it was the one where you have there, you’re painting the handball wall and all my kids were able to get the, most of the kids I guess should say and probably not all of them, but 16 out of 18 were able to grab the idea. “Hey I need to find the area, I need to make this array.” So they got that part of it but then translating into the work over the last week or two it’s like, Wait, Area…” Because I guess I don’t know if the context gone or they’re not connecting that experience to the work now because they’re like when it was a seven foot by 10 foot wall, I mean my lower students were yelling 70 at me before I even finished.
Kyle Pearce: Your idea to bring up area before perimeter. Actually I’m doing a quick Google search because I feel like there’s some research out there and I’ll do my best to add it to the show notes after, because I know I’m not going to be able to find it on the fly. But it’s a conversation I was having with a few Math teachers at a conference and it was just this idea that if we really think about measurement and we think about Perimeter, we think about Area, we think about Volume. If you think of what you actually come into contact with in your life first it would be like 3D objects, right? Like it’s the world around you and you think about Volume from the perspective of a child in the bathtub who’s dumping this container into that container.
Kyle Pearce: It’s like, it’s much more concrete, it’s real. And next in line would probably be this idea of Area. Even that’s kind of abstract. If I look at a… I’ve got a Kleenex box on the table here. It’s that time of year where the colds are going around. And I’m looking at this Kleenex box and I can see rectangles on the sides of this Kleenex box, but what young children, they just see this Kleenex box. They’re not even tending to the fact that there’s a bunch of different shapes that are making this Kleenex box. And then if you think about Perimeter and just this idea of Linear measure, that’s really abstract because for them it’s like measuring how long something is until they get to that non-standard of maybe using pencils to repeatedly iterate those pencils to count how many pencils long a room is or whatever it might be or the desk. That seems a little bit more abstract than just sort of covering stuff like we would with Area. So I think that’s kind of a cooler way for you to go and kind of build Perimeter in when you kind of get to that place.
Bill: Yeah. I guess after reading about it more, I thought about like, yes, we do area more often than we measure Linear because they cover things or you know if they’re playing with Legos or blocks like you said the three dimensional objects that seem to just make sense. You know you can look at that concretely and be like, “Yeah, there’s some amount that this has to cover.” But again, when I like kind of took that context away and we started doing some more procedural things, it was like, “Yeah, I can find Area but I’m not connecting that rack.”
Kyle Pearce: Right. Yeah. It becomes almost like something separate then you know maybe the 3-Act Math tasks that you had initially used.
Bill: Yeah, and that seems to be one of the issues is, like I can spark the curiosity. I can… 50% of the time I feel like it’s very successful and the kids have moved to a more, I like that clever way of doing things or have had their strategies reinforced or challenged and then it’s like back too, you can’t just do those all the time. You got to work some fluency and some procedural skills and then it’s kind of disconnected. And that’s a little frustrating sometimes because it’s like, “Hey guys, you successfully did this on this thing I graded and emailed mom and dad and sent it home and you’re the Prince of the world.” And now you can’t do a couple of problems.
Kyle Pearce: Yeah. So I’m wondering for this specific lesson, going back to, and I’m sorry the 3-Act Math task, you said it was finding the Area of the… Which?
Bill: Of a wall. I found it on some search for 3-Act Area Tasks and it was a wall that needed to be painted because it was going to be used for like handball or wall ball or some games. And then it’s like you had a can of paint and the paint covered a hundred square feet. Would you have enough to finish that, to paint the wall with one can? And the kids were, “Yes you would. Yes you would.” So quickly, I was like, “Okay, so you have some kind of-
Kyle Pearce: Beautiful.
Bill: Yeah, it was great. Absolutely. Okay, so you have some idea of this. But now it seems like guys we’re moving through the rest of the unit, that real life experience isn’t necessarily carrying over into a slightly more abstract way. I love using manipulatives but I’m like, “all of you did this, not all of you, but like a good number of you did this with like a number model or even just with mental Math.” And they’re like, “I know how to find Area, that’s 70 a 100 is more than 70.” Yes. Like you’ve got a real world answer quickly. But now as I’m trying to get you to like, “okay well, here’s some shapes, what would be the area of this one or that one?” Because we’re going to have to take PSAs in April and it’s like that’s how they’re going to ask them. You’re not going to have the context. Not that, that’s all that matters, but it matters.
Kyle Pearce: But that’s a challenge, right? Because that’s what you’re up against. We were talking with James Tanton a couple of weeks back and he mentioned that, that if that’s the end goal, it makes it really challenging for us to kind of fill in those gaps. So after you did the, how much paint for the Handball wall? And actually I did a quick check and it was tasked by Robert Kaplinsky. So, we’ll definitely include that task in the show notes. But then what would be, I guess like after that task, what was like the next, I guess set of questions that you might challenge them to do? What did that look like, sound like? Just to kind of paint us a bit of a picture.
Bill: First, for just about everybody in a class the next one was like, because the wall was freestanding so it was like, “Could you do the front and the back?” And they’re like, “no, we would need two cans.” Some the kids that were even more advanced, we looked up how wide a cinder block was. So they kind of cast like a three dimensional net over it to figure it out. But then like the subsequent lessons were involving working towards finding the Area of rectangles and squares and then if you had like a rectangle with like an extra rectangle sticking off of it to like non-traditional shapes working towards that. But I think because a lot of them, it’s the generic grid paper, shapes that every square on this one is to square inch on the next one it’s a square yard, then a meter.
Bill: And not like a concrete object or like you’re trying to figure out like, “Hey, I need a rug that will cover this area and this area.” What two size rugs would fit together to do that. You know, it seems that, I feel like they would have a better chance of answering that than just like, “Hey, I got a square that’s eight inches on each side, or I can tell you the Area of it.” But they’re not really making a connection that, that’s the same as the wall. I don’t think, at least just from my probing and as going on with it.
Jon Orr: I want to just maybe go back and think and just maybe make sure I grasp everything that you’re struggling with here. You’ve stated here using Robert Kaplinsky’s tasks and other tasks and 3-Act Tasks in your classroom, but you’re sparking that curiosity, but then something’s lost in between and you’re trying to balance, how do I keep good thinking happening in my classroom when I move to getting practice in and getting that procedural fluency in and how do I transition from all this curiosity we’ve built up and engagement we’ve built up to making all these other things happen that we know has to happen so our kids can be strong and successful on future problems? Does that sum up some of your struggles?
Bill: Yes, it is. And this task may be is, I’ve had probably more success than other ones just because ultimately my goal is, “Hey, can you find area?” Yes, they can. But as we’re moving into more complex Area questions, it’s not that they’re going back to earlier strategies, it’s more that they solve things. One way when it was presented it was context and now they’re doing something different without the context. Like this year is the number model, “Oh, I did a number model, I’m proficient with it.” Or like tiles it was squares or something in squares and they’ve now moved to, they’re doing something completely different that they had not done. It’s kind of made the like pre-assessment like, “Oh we have this.” And I’m like, “Wait, do we?”
Jon Orr: Can we think more generally right now, instead of this specific task, but let’s think generally about many tasks that you’ve tried. You do the tasks, you got all that curiosity built up, they’re trying these strategies. What are the moves that you’re doing next? Can you paint us that picture? Fill our listeners in on some of the moves that you’re making in your classroom. What does that look like for you now? And then we can hear that specifically and then maybe we can make some suggestions. What does it look like after you’ve built all this curiosity up, the kids are diving into the problem, they’re working there and all of a sudden they’ve got some stuff to show you and even go back to as far as like, “How is that set up?” Are they in groups, are they individual and what does that look like for us?
Bill: So in general, my students are always sitting at some kind of table or a pod. I really do not like anybody being completely alone. I think they learn way more from seeing each other and trying each other’s strategies and having a conversation than being isolated. So right now we’re in pods of three and there’s 18 students in the class and they’re roughly grouped by ability. I don’t want to have a student who things completely differently and have another student who’s just going to like piggyback on that. I want them to be with people who are all going to be working about the same pace as them. So they’re each trying something they could be doing different strategies. But I want them all working and not just sitting back and waiting for the smart kid at the table, quote unquote to do something and a little more specifically and I’m thinking about this task.
Bill: A student who has a lot of ability, this child went ahead and made a diagram of a wall but then tried to use inch tiles to cover it but it wasn’t like to scale. So my move there I was like, “okay, now you said you drew a picture and now you’re just filling it in with tiles. Is that….? So it was seven feet high. I was like, “well, are you going to try to make this seven feet tall or are you going to like try to represent it, kind of like we just got off of grass? I was like, “Are you going to use like a scale or are you going to have like each block represents some number?” Trying to kind of get her to, to come up with the idea that this was going to be, you weren’t going to get 70 inch tiles on your paper because you’re using a standard sheet of paper and you drew a very small square, you wrote seven, you wrote 10 and then you decided to tile it.
Bill: There was no correlation between that. With somewhat success, the child ended up not using the paper and just built the array. But this is a kid that’s extremely fluent with their multiplication facts. And I was kind of wondered, ring and puzzled, why you didn’t just multiply because everybody else in your group did, they immediately just wrote 70. Now they forgot it was a square unit but that was, that’s a teachable moment. And we worked on that and she got there. But it just was really surprising in a way because my push was like, “Okay, you want to build it because you want to see it for some reason. I guess you needed the concreteness of it but you were not going to be successful with the first couple of attempts because you’ve missed the fact that you couldn’t. Like you wrote the number seven and number 10 but then you were just putting tiles over it. It wasn’t going to work out.”
Kyle Pearce: Right.
Bill: Does that make sense?
Kyle Pearce: Because the scale was off, right?
Bill: Yeah, it was off. I don’t know how to say no but scale really other than on like a bar graph that you could… Or I’ll pictograph.
Jon Orr: I’m wondering if we go back to the learning goal for that lesson, obviously Area is very generally that’s kind of the learning goal. But how deep into Area, you’ve described like squares and rectangles and you’ve even described that even potentially some composite shapes involving rectangular sort of pieces put together and that sort of thing. So how involved does the standard go for, let’s say for this specific lesson? Where are you hoping to get students to by the end of that lesson?
Bill: By the end of that lesson I wanted them to have a pretty solid, because it was like an introduction to Area and to give them some reference points, a lot of contexts for when and why they would want to use this to measure Area, like painting a wall or putting rugs out. Those types of things were, it was the context I wanted them to get. But as far as my learning objective, I wanted to see how they were going about solving it so I could then try to group like-minded thinkers and get some other ideas and then push towards more clever, sophisticated way of doing it. Hopefully, most of the class could get to the point where they’re using a number of model, at least understand that. Because I do know one of the standards that we do have to get to is that multiplication using the Area model. So it was like we kind of did that and I wanted them to relate that to Area of playing shapes. We’re not really responsible for three dimensional shapes like Volume, although we do catch on it because it’s just kind of like you said back to beginning with the bathtub example. They’re aware of that and they know there’s some way to measure liquid or gases. But I don’t know if that’s clear enough to makes sense?
Kyle Pearce: No, that makes a ton of sense. And so if this is kind of like an introductory sort of activity, I wonder if in while some students are going to get clever quite quickly, I’m wondering if the next series of activities, one of the challenges I see with this particular task could even be, I pulled it up on the web here to get a good understanding. I see an image and for those who are listening I’ll try to help paint a picture. But you know there’s that, that handball wall as you described and there’s quite a few cinder block bricks that make this handball wall. But something, even though it seemed fairly obvious because the context was there that a lot of students actually ended up with 70 square feet right away. I’m wondering if they actually have that understanding, if they just knew the algorithm like they already, maybe someone had already told them like, “It’s length times width.
Kyle Pearce: So I see 10, I see seven, I multiply them. I’m wondering if they actually were to have, and I know in real life we can’t go to this wall. Who knows where it is? Probably in California with Robert, but if they actually had like a square foot and like sheets of paper that were a square foot in area because I’m curious as to whether maybe some students are kind of working on two different things here. Like this idea you’d referenced earlier that covering with tiles it’s almost like they’re not seeing the connection between the covering of tiles and multiplying length times width. Which makes me wonder if maybe having some activities in there where students are spending like a significant amount of time covering up different rectangles and squares with different unit squares as well. They could be non-standard, right? They could just be, like this square, it doesn’t have to actually, we could just call it a unit square or it could actually be something physical from the classroom or whatever it might be. But just to give them that hands on tangible experience to ensure that even some of our more, we’ll call them clever thinkers, aren’t actually kind of jumping the gun to the procedure and actually missing the conceptual underpinning that sort of underlying this pretty complex and really interesting area of mathematics.
Bill: That makes sense to me. As you were saying that, I was kind of thinking, yeah, more tiling and I know some other tasks I did last year with tiling, but even just more repetition, just tiling and not with necessarily trying to write a number model to go with it, like physically doing it. That makes sense. Because I think that’s probably where I’m missing it. If that kid couldn’t go to the number model, even though they know that multiplication fact and they know that maybe like just like the algorithm work, if they were probably told it in another way, they probably could have got that. But they wanted to tile that square because they didn’t have the next step yet.
Kyle Pearce: Right. And I’m even picturing like kids can get pretty clever even there… Even though like to… Bill, you and John and I were thinking the tiling, like physically tiling seems so obvious to us, but in reality for young children’s, oftentimes the things they do, they’re completely oblivious to what’s actually going on. And I had heard from someone somewhere discussing some research around students early on when you actually have them tile like a rectangle or a square. Oftentimes kids won’t even see the rows and columns until you explicitly call it out. They’ll actually spiral it. They’ll go around the outside all the way to the middle or vice versa. And it’s like they’re not even seeing it. But to you and I, it’s so obvious, it’s right there in front of us and by doing that tiling you might be able to get them to think cleverly and have them highlight by asking some of those questions like, “How many rows do you have?” And, “Oh I have four rows.”
Kyle Pearce: “Oh, and how many are in each row?” And really explicitly getting them to that connection that, “Wait a second, if I have four rows of three, that’s four groups of three. And I could also look at it as like three columns of four. Wait a second, if I multiply three times four, that’s 12.” And it’s like that student generated algorithm starts to emerge, but it doesn’t do so independently, right? It’s always through that questioning and those prompts and individual tasks that we really specifically pull out to really try to tease at that idea.
Jon Orr: Yeah, and the reverse would be true too is that the kid who went right to multiplying, having them do the tiling backwards would be getting them to make that connection. You guys said about how the tiling is connected to the multiplying.
Kyle Pearce: Yeah, I think that’s exactly what I was trying to get at then what I was looking for and where I kind of felt lost with my teacher move, my next step, my questioning and what tasks I was going to do next. And it’s making me kind of think like, “Hey, I want to maybe slow down and maybe spend another day or two or have a center.” When they’re rotating through center time that like this is something we need to do, tile thing, even the same squares with different objects.
Kyle Pearce: I love that idea of giving them that opportunity through centers and gradually moving towards the grid paper, and again, for all three of us, we’re thinking we’re adults and we’ve been on this planet for quite some time and it seems so obvious to us that the grid paper is the next place to go. But to the student, again, making sure that they are seeing that like, “Oh wait a second, we’re using these grid paper to represent these tiles? Oh.” For many children often it completely goes right over their head unless we really pull that in and we can let them explore it first and I would always advocate for that, is let them tinker and let them play and see what students come up with what.
Kyle Pearce: It’s great when a student comes up with a connection and then they can kind of do the teaching, but at some point I’ve got to make sure I go, “okay, so we all see that. Hey, look at this, this grid paper actually is the same size as this tile and so now we’re getting to that next little stage before we get to actually multiplying 10 times seven or whatever the dimensions might be.” Because as soon as we get to those measured dimensions where we’re starting to use numerals to represent a length in a numeral to represent a width, the tough part is it’s so easy for us to know to multiply the numerals, but we aren’t referencing the centimeters or for you folks it’d be inches or feet or whatever it might be. That for students can so easily be missed because this idea of a number with a unit next to it, is really abstract for them. They almost have to like paint that picture in their mind.
Bill: Yeah, exactly. That’s really hard for them to remember to do it. To get to that place where you’re like, “There’s something more than just a number. The number represents something. It’s not just the number.” That makes a lot of sense.
Jon Orr: And it’s important I think to note here if you think generally that, it’s so important to plan out that progression of representations of where they start and where they’re going to end. Like you have that kind of map in your mind or on paper as a teacher and so that when you recognize where a student is, you’re ready to move them to the next stage and then also to move along the stage back and forth. Ff you’re at the high end of the stage, like the high abstract and can you move back to the concrete and vice versa. I think going forward it’s so important to know that not just on this particular task but all your tasks when you’re doing 3-Act Tasks, it’s what are we looking for? What am I supposed to see come out of this task?
Jon Orr: Where do I want to go and where do I want to end up eventually? It might not be today, but it might be down the road. But I definitely need to plan out where that is eventually for my grade level, so that I can move that student along that pathway back and forth. And I think that can help a lot with knowing what to do next. It’s like, I saw this stuff and it’s all about knowing that one kid and what they’ve shown you in the past and what you can then say to them or what questions you can bring up, which are…? That’s a really hard thing to do is what are the questions? What are the prompts that I can say to the student to bring them along the path? What are those tasks that I can throw in? But I think the pathway is so important to know in advance where you’re going to go.
Bill: Yeah. So what I’m hearing then is maybe be as I pick my task, try to be a little more intentional with how I anticipate or how I’m seeing the progression of that? Maybe from tiling all the way to the number of model, but having checkpoints along that continuum that I’m kind of looking to see are anticipating where a kid might be and a strategy then the how I want to move that kid up or downward as they’re going.
Jon Orr: Exactly. Exactly. So for example, when I’m planning tasks it’s like we got to know where they are, like where they’re coming from, expectation wise, skill-wise. I definitely want to know where we need to go and when I pick a task I write out. Like what are the possible solutions I’m going to see? I map them all out on a piece of paper. I possibly could see this from Joey because I’ve seen him do stuff like that before or I might see this solution here. I might see this solution here that’s at a higher level. I definitely want to map those all out and be ready so that when I see them I can then provide a suggestion to that student to move to the next stage on that progression. You got to know what does that learning goal that we’re moving towards. I think that’s a really important thing to say. For me it’s in that one block of Math class. If I can get my students, majority of my students to that one learning goal and then I’m really happy with that in moving to… It might not be just that one lesson. I might move into the next lesson. But I think knowing that learning goal is one of the most important things.
Kyle Pearce: And I think one of the challenges you have, Bill, is just the simple that with the younger we go. When we’re teaching younger and younger children, our challenge becomes so much more complex because they are at such a different level of development, like just cognitively that we have so much that we know just through experience and just living every day we live on this earth. And when you think of how many lifetimes of grade three students we have, for a 10 year old student I’ve lived three and a half of their lifetime. And that is so hard for us to kind of unlearn what we already know to be true from the world around us. Like just the very obvious things, that we just know work. And for that I definitely commend all our early years teachers, our primary teachers. So grade three is no exception because when you’re heading into this idea of Area, something that I’m only recently been exposed to is this idea of approaching Area from accounting perspective, not even additive thinking yet, just counting.
Kyle Pearce: And that’s that covering it up, that tiling that you’ve mentioned that maybe you want to spend a little more time and do some centers with. And then eventually heading to this additive thinking. That I can start identifying, “Oh wait a second, I’ve got three rows with four in them and I could add four plus four plus four…” Start doing the skip skip-counting idea. And then we get into this multiplicative thinking, right? And that’s the early stages of proportional relationships and proportional reasoning. And it’s really complex stuff that to you and I, it doesn’t seem complex just because we just know it’s length times width and we’re good to go. But there’s so much nuance there and, and there’s no shame in us, having to kind of stop and reflect on that and go, “Huh, how am I going to do this?”
Kyle Pearce: But if I can look at every stage that in my mind, I’m picturing my students starting here and ending over here. I’ve got my arms out in the space in front of me here. And if I can find ways to find a step in between those two every single time and if I could then find a step in between those two and those two and if I can just break it down further and further and further. And really just by thinking and thoughtful planning and obviously collaborating, right? And that’s what I think the power of these conversations on the podcast star is, really just because three heads are better than one and just getting to chat it out I think is so important. So yeah, this has been an awesome conversation for me. I’m learning as we go, so thank you for that.
Bill: Oh yeah, thank you guys. I think that’s the piece I was missing is that, I was thinking about it abstractly as my lesson. Like, “okay, I’ll see some do this, some do this, but not maybe getting down into the minutia of how those things are connecting. How you just laid out like Areas counting and then as adding or skip counting and then multiplication. And I knew those looked like, but I wasn’t thinking about how to move the kid from one to the next as much as just knowing that I wanted to take the kid that was tiling and counting and get them to try to skip counter add rows and not thinking about the tasks that they would need to do or what would be appropriate for them to do to get there, I guess.
Jon Orr: Right. And as we wrap up, we don’t want to take too much of your time Bill, but it sounds like you are starting to think about some of the takeaways from this conversation, but is there anything else, if you think specifically about our conversation here today that you could try in your classroom or what would be a big takeaway from this? Sounds like you’re on that path already, but anything else you’d like to add to that?
Bill: Yeah, I guess the next thing is I’ve made a note for myself here that I want to try to look for more information on like learning progressions. I know I have my standard in my Algebra content. And the thousands of resources that you have just from being in a district or whatever. But trying to find progressions, I guess for the next topics I have to teach like Fractions, et cetera, et cetera. So that when I go into these tasks, I have a little bit of guidance on how to move the kids up and down that progression. And also I liked the idea of taking a hike and just making them work backwards a little bit to make sure they really have that concrete understanding of it too. Because it’s really easy to say, “Oh you got it.” And well, if the kid’s been tutored up or coached up on it, I miss there, that hole in their knowledge.
Jon Orr: Yeah. So that sounds like a great plan, I’d actually be interested in also following along with that plan. I’ve learned a ton of this conversation. I said at the beginning, I’m a High school Math teacher. So every time we start to talk about grade three Math and all the nuances, like Kyle said about even this one main example we’ve talked about during this episode, it amazes me. And like we had on our previous episode, James [Teane 00:40:25], talking about like how every level of math has these deep intellectual moments in deep intellectual thinking that we can go in and it’s opened my eyes just this conversation. So I’d like to thank you for that Bill, bringing this to us and having this conversation. Yeah, we’d like to thank you for being here. Kyle, you want to do a one more thanks.
Kyle Pearce: Yeah, one more thanks. And also if we’re able to stay in touch with us over the next little while, but also we’d love to have you back, maybe six to 12 months from now and kind of see where you are in that journey and checkup on your goal there to kind of look at some of the learning progressions and see where that learnings led to you. How does that sound?
Bill: Oh yeah, that sounds wonderful. I think that’s a… I’m online constantly stealing ideas from you guys and some of the other people you know with these 3-Act Tasks that you put out there and those other ideas of estimating an important Math thing. I want to be able to give back a little bit. So if I do come across some good stuff, I’ll make sure to send it your way and- [crosstalk 00:41:21].
Jon Orr: Yeah. Definitely, beautiful and we’ll stay in touch for sure.
Kyle Pearce: Awesome. Awesome. On that note, Bill, we’ll also put in the show notes a few task links. So not only the Handball task from Robert Kaplinsky, which is called, “Do we Have Enough Paint?” But also I was thinking as we’re chatting here and there’s a few tasks that people might consider using in a similar scenario, like really trying to explicitly pull together this idea of Area and Arrays and Multiplication. So a couple that come to mind is a cover it up tasks. So I’ll include that link. Also the airplane problem, if you haven’t tried that one out, that one is, it’s not Area, but you can approach it as Area and you’ll see what I mean when you check out that link.
Kyle Pearce: And then for folks who are working on double digit multiplication, so kind of moving up a little further, the Donut Delight Task could be a good one for folks as well. So we’ll make sure to put those all in the show notes and hopefully this conversation is going to help more than just us and Bill because I know, again, I always learn something new from these conversations. So thank you again, Bill for that.
Bill: Yeah. Thank you guys. This has been awesome. I got a lot to think about and I feel good knowing I have some things to work on. Now I know, “Hey, let’s try this.” And some of that anxiety of, “what do I do?” Has gone away.
Jon Orr: Awesome. Well, we’re glad to help you there.
Kyle Pearce: Well, thank you Bill. We hope you have a great night and definitely stay in touch. We will be in touch soon.
Bill: Sounds great.
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