The open number line is not just a powerful tool for computation, but it's an amazing model for developing all sorts of mathematical concepts! In this episode Pam and Kim discuss how to help students develop the open number line in a natural and meaningful way.
What is an open number line vs a closed number line?
Are number lines, closed or open, discrete models and why does it matter?
Why shift to a measurement model vs a discrete model?
What are some common errors students make when we use the traditional way of teaching measurement?
We suggest a rich task by Cathy Fosnot to help students develop an understanding of the open number line through measurement
What can you model on an open number line? Elapsed time, addition and subtraction of whole numbers and integers and more
Is an open number line helpful for higher math concepts?
Check out Cathy Fosnot's amazing rich task: https://www.heinemann.com/products/e01010.aspx
Hey fellow mathematicians, you're listening to the podcast where math is Figure-Out-Able. I'm Pam.Kim Montague:
And I'm Kim.Pam Harris:
And you found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But y'all it's about making sense of problems, noticing patterns and reasoning using mathematical relationships. We can mentor mathematicians as we co-create meaning together. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keep students from being the mathematicians they can be.Kim Montague:
So in this episode, we want to focus on kind of one of our favorite things, the open number line model. You've heard us mention it before. But today, we're going to dial in to how to develop it, how we use it, and why we love it so much.Pam Harris:
Because we do. The open number line model is amazing. And I'm a little embarrassed to say how long it was, before I realized that the open number line model really is the precursor to all of the coordinate axis stuff that we do in higher math, because I think I've only ever seen it with numeracy in addition, subtraction, and so it hadn't really occurred to me that it is such an important model across the curriculum across the grades. And so we thought we'd dial in today and really talk about some details of the open number line model, especially maybe how to develop it. Yeah, so pretty cool. So let's just get at some definitions a little bit. Sometimes people call the open number line, an empty number line. So a closed number line is a number lines that you might have seen above the blackboard when you were growing up where all of the tick marks are there. So if you see me right now, my hands are like karate chopping, because like, I'm like, there's a tick mark, for every number. Every whole number has a tick mark, and they're all there. And there's no whole number missing. And so as a second grader I had on my desk, I had a laminated number line. It was this laminated piece of thing that was stuck on my desk, and it went from zero and it had 1, 2, 3, 4, 5, 6. And I think it went to 20. I actually don't remember, I didn't find it all that useful, or we didn't ever do anything with it. I think it also had my name on that strip. So it was just I don't know, it was something, maybe it was like the background to my name that was kind of like the picture that was there. It wasn't very useful. That's a closed number line, a closed number line has all the tick marks. An open number line, or an empty number line is where we get to choose where the tick marks go. We get to choose the numbers that we want to use, and then use those in ways that are helpful. So yeah, that's an open number line. It is a definite shift, to help students go from the beginning counting that they might be doing, or that they are doing when they're younger, which is discrete. So when students are counting objects, beads, teddy bears, fingers, those objects are sort of contained, they're kind of one object each. And we call that discrete. And so as they're counting it's one number for every object. And that is the beginning of the Development of Mathematical Reasoning, we have students using counting strategies to do things. The open number line, or a number line in general, is not discrete. Now it could feel discrete, especially if we're talking about a closed number line. Because each of those tick marks very clearly, I'm snapping, represents a number. And so if I just look at the tick marks, then those are sort of discrete. But the open number line or number line in general is continuous because it's a measurement model. And measurement, by definition is continuous. You can get in and further. I can measure something that is a footlong. But I can get inside that measurement. And I can have something that is an inch long. I can get inside that measurement and have something that is a fraction of an inch. Now that was sort of customary measurements. If I was using the descent, the real measurement system, I could have something that was a meter long, but I can get inside that measurement. And I could look at 10 decimeters. And I can get inside of that. I can look at 100 centimeters per that meter or 10 centimeters for that decimeter. I can get inside of those and I can look at millimeters. And we can continue to just get smaller and smaller, more precise measurements, you know down to electron microscopes and everything and then we can get bigger and zoom out. And measurement by its definition is this continuous model. That is a big shift for students. One of the ways that we see that as a shift, maybe not made so well with some students is, as students are, might be looking at a closed number line, they're not sure whether to count the tick marks, or to count the span in between the tick marks. They're not sure how to, looking at this, say, for example, a ruler is a closed number line. They're not sure as I line that ruler up, they might even line it up correctly to see how long something is. But then what do I count? Because if I count tick marks, I count the beginning tick mark. So if you can picture a ruler lined up, say, against my keyboard right here, I'm sort of looking at my keyboard and I line up a ruler against that keyboard, I might be tempted to count the tick mark at zero. And then count the tick mark at the one and a two and three. And if I get all the way up to say, 11, I think my keyboard is 11 inches long, then I will have counted 12 tick marks because when I get to that end, I've counted the zero. So I've counted too many tick marks because if I'm that discreet mode of counting objects, then I'm counting the tick marks. What I'm not counting is that span of an inch. And as I'm saying, Kim's like, her eyes or eyebrows are like, "Really Pam?" as a span because she, I bet she can see right now because we've done this enough in person where my hand is going, my hands are together. And I'm kind of pointing at like the beginning of the inch. And then I'm moving one hand out because I'm measuring the span of that inch or the span of that foot. Now I just measured much further because it's a foot long. That span, oh, go ahead.Kim Montague:
I was just gonna say I would bet a lot of third grade teachers know exactly what you're talking about as we start introducing non standard, I mean, sorry, standard measurement measure.Pam Harris:
Uh huh.Kim Montague:
Yeah. And especially when you start with something not at the zero. So like when you do activities, where there's a broken ruler or something like that. And it's really helpful to help kids make sense of talking about that span as the measure.Pam Harris:
Yeah, absolutely. You reminded me of the old NAEP question, the NAEP test, that had a pencil lined up with what like you said a broken ruler. And so there was sort of this broken ruler, you couldn't see the beginning of it, you couldn't see the end of it. And the pencil was sort of lined up. And the end of the pencil was at the three and the other end of the pencil was at the seventh. And then we asked, that it may ask students in third grade and in eighth grade, how long was the pencil? And one of the most chosen incorrect answers was, so let's see, three to seven. I have to think for a second. So it should be four. Right? That's the correct answer. But if I count the tick marks, I'll count five tick marks. And so one of the most incorrect answers that was often answered was five. So it's five long because it can't count the tick marks. One of the other most chosen wrong answers was seven. Yes. Because if students were only taught this very procedural way of measuring, then you line it up, and you read off the last answer. And so it was already lined up for them. And when they read off the last answer, it was the seven but they sort of didn't pay attention to the fact that it started at the three. Yeah, so. Exactly. So that's a thing that we need to help students make the transition. How do we help students make the transition? How do we develop with them this idea of a continuous measurement system that then can turn into this very helpful open number line? Well, Kin and I thought that we would share with you one of our favorite rich tasks ever written by a master, a master rich task and problem string writer, Cathy Fosnot. We are, my hat is off to Cathy. She has done an amazing job of creating some interesting things. And we thought we'd promo that a little bit today, tell you a little bit about one of her rich tasks, that again, I find... Kim and I were talking about this. We were like, "Should we do an episode on this?" And she's like, "You really liked that." I was like, "Yes," like it for several reasons. We won't get to all of them today, because it's so involved. It's so well written. There's so many details. But maybe that's one of the reasons I like it, because it's so well planned, that it really helps students develop the open number line well, and they will tell you where you can find it. And you can check that resource out. Or maybe I'll tell you now. So in a book by Cathy Fosnot. In fact, let me look. Yep. Catherine Twomey, I may have just said that completely wrong. Cathy, I'm not sure if I ever said your middle name, Fostnot. She's the author of the book. It's called Measuring for the Art Show. And I'm not going to read the second title yet, because it will give away something that we're going to talk about in a minute. So in Measuring for the Art Show, the task for students is the teacher launches - Now, I'm going to be honest with you, I have seen, Cathy had created, Cathy Fosnot had created a professional development series a while back. I believe it's out of print now. Well, except it's out of print. But she also has it in her new perspectives online resource. But I had seen it a long time ago. And so I may inadvertently, I've watched those videos a lot. I may inadvertently sort of talk about what's in the video and less in the book. So forgive me for that. Just because my, I've done a lot with the video in the past. But it's the same idea. So in rich tasks, the teacher announces that there's this art show that they're going to have. The students have already done, oh, by the way, this is in, could be in a first grade, second grade, beginning of third grade classroom. Because again, we're helping students transition from that discrete counting to this continuous measurement model. So this teacher introduces, "Hey, we've been doing a lot of artwork, and you know, at the end of the year, we're gonna have an art show. And so here's some artwork that you've created. And for this artwork, we want to be able to put labels on them." And so she holds up kind of an adding machine, tape size things sort of, if you don't know what that is, like an index card kind of width. And she says, but adding machine tape because it's kind of long, so there's a long strip of this white paper. And she says, "So if I were to put a label on this artwork, you know, it has your name or the name of the artwork, the name of the piece, we're going to label all the artwork we need to do, we're going to cut all these labels, and we have some certain papers around the class that we're going to make artwork on all year long. We've already made some (she could point out our work). So for all this artwork, we're gonna want labels." And then in the video, the teacher says, which by the way, the teachers Hilde Martin in the video, Hilde Martin, if you ever hear this, my hat is off. You are brilliant. I learned so much about teaching from watching this particular teacher and her teacher moves. And now I teach people about those teacher moves. It is amazing work. Hilde Martin says, "I have a friend, my father's friend, my father has a friend who will cut these labels for us if we can give him a plan. So we need to create a plan to tell him the length of these labels, these white paper strips, and then he'll cut the labels and then we can put them on-" And she says it completely without missing a beat, you know. I want to meet the father's friend someday. So she says, "We need to make the plan. And I'm too busy to do the measuring so I've set out the paper that we are going to use all year long. And I'm gonna ask you guys to measure, ask children to measure because I'm too busy to do the measuring." I was cracking up about that a little bit in there. And so she sends them out to measure. And at each of these stations where there's a different paper that there'll be creating artwork on, there are some cubes. And she says, "We're gonna use cubes, we have a lot of them. So we might as well use the cubes. And we'll measure how long the length of the paper is the width of the paper. And you can write that down on your paper as you go around. And your partners, the partners, you're gonna measure with the cubes, you write down the length and the width of the paper." And then you watch students kind of go off with their partner, and they start to stick cubes together, in sort of a train of cubes, and they line it up against the edge of the paper and you kind of watch what students do. One of my favorite things to do with my pre-service students at the university is to ask them at this point, "What do you think the main point of this rich task is?" And, or I've done it with in service teachers as well. And to a teacher, including me, the first time I'd seen it, teachers will say, "Well, it's obviously measurement, you know, we're trying to help students learn to measure and we're using non standard units, we're using these cubes as non standard units." Often teachers at that point will say, "We don't really understand why you would use non standard units, you know, we could just hand kids rulers and they can just measure and it'll be fine." To which then I will give them those NAEP results, that item that we talked about with a broken ruler. And I'll say, "Or does that work well?" Like really, how's that working for us? The sort of traditional way that we kind of help kids in learning the procedure of measurement doesn't typically get a lot of really good measurement results. So teachers are very convinced that this rich task is all about measurement. Now, is it about measurement? Absolutely. But Cathy has a much deeper and so cool way of taking it further than that. So then the students are measuring and after the students have sort of recorded these measurements and everything, then she holds a class Congress on this rich task where the students come together. And she says, "Okay, so we're gonna make this plan." And on the board, she has hanging 100 cubes with every other five is a color. So white five. green five, white five, green five, and there's 100 of them of these fives. And below that she has that measuring strip or excuse me that white strip of paper that they're going to make the plan on. And then she asks kids, "So what measurement did you find for Oh random..." And then she you know, picks not so randomly one of the papers. And the student says, "It's 10 wide." And so then, "How would I? Where would I mark 10?" And the students begin to use the cubes that are hanging above the strip to help find these lengths of the paper. And then they literally mark them on the strip. And as they mark these measurements on the strip with very purposefully planned measurements, like what I just said, "She randomly chooses a paper," it's not random at all. She says, you know, "How long was the purple paper?" And it was 10. So they stick the 10 up there. And then, "How long was the length of the purple? That was the width? And how long was the length? It was 14? How can I find 14?" And the students start discussing how they can use those groups of five cubes to find where the 14 would be or find where the 44 would be, or the 36. And then she even sticks a measurement in there that's not, that wasn't one of the papers. She's like, "So what if I found a paper that was, hmm, random, like 66 cubes long? Where would that be? And then where would I make the mark?" And as the children begin to discuss how they could use these five white, five green, five white, five green cubes, where they could find numbers based on the structure of five and 10. And as they discuss where to put the mark. Is the mark at the edge of the cube, in the middle of the cube? Like where would we mark 66 cubes? Students, in a sense, construct the open number line. It is amazing work. And it's brilliantly done. And so again, I would highly recommend that you check out Measuring for the Art Show. But then, then in the video, Hilde, the next day, stands up and says, "So I went home, and I found some more paper."Kim Montague:
And then says, "What if I had this paper that was 19 cubes long? And what if I then wanted to, what if we were going to all year long create artwork that were two papers together? So what if I had this paper that was 19 cubes long, and I had a different paper that was 21 cubes long?" and she literally writes on the board 19 plus 21. "How long of a strip, a label would we need for that?" And students start to use this burgeoning, this beginning sense of what the open number line is to find those on the open number line and use. Now at this point, it's a bit of a closed number line, because they have all the cubes up there. And she's sort of using the strip where they've put some landmark numbers. It's the beginning of an open number line, and then it transitions over time, to our students are literally using relationships. And as they use those relationships, Hilde is modeling their thinking on an open number line. And then students begin to make the transition for themselves to be able to use that open number line as a tool for them to solve addition, and then later, subtraction problems.Kim Montague:
Yeah, it's so brilliant. And I think that there were so many bits and pieces that Cathy embedded along the way, and you mentioned a couple of them, but not all of them, right? toPam Harris:
For sure.Kim Montague:
Don't want to steal all the thunder. But I remember you being so excited about them using that strip as an open number line. And as you were describing all the things about it, I you know, I've seen a video multiple times as well. And I was remembering what Hilde looks like, and her working with her students, and like, I was picturing the whole thing again, as you were saying it. And I remember being blown away that second day when she came back and said, "We're gonna put the two papers together." And that was the moment for me where I was like, "Whoa, that is brilliant." Yeah, absolutely.Pam Harris:
And when you say, "That is brilliant." It's the entire rich task.Kim Montague:
That starts with this scenario where are the kids learning about measurement? Absolutely.Kim Montague:
Which they need, right?Pam Harris:
Which they need? Yeah. But it gets to where students are not only building the open number line, but they're really building addition.Kim Montague:
And earlier when I was talking about the name of the book, so it's Measuring for the Art Show. And the subtitle is Addition on the Open Number Line.Kim Montague:
Because then at that point, we're all like, with our jaws open, "Oh, my gosh." This is how we get there. This is how we help students transition from that discrete one by one model, to this continuous model that is brilliant for modeling, addition, subtraction. So stinckin cool.Kim Montague:
We would recommend, right, that at any point, that teachers have an opportunity to develop the open number line in such a way that that would be so much more meaningful to their students, and really help them understand the open number line, much more than just starting work on an open number line.Pam Harris:
And it's about measurement. And so wherever you are, if you have older students and you're like, "Pam, we're not going to..." then hang back, lean on, emphasize the idea that you're measuring. In fact, at one point when they're talking about where to put the mark Hilde says, "If we wanted a paper that was 66 cubes long, where would we put the mark?" And you just see kids like light bulbs go off. And they're like, "Well, you would have to put it at the edge of the cube, it would have to, you know, or otherwise, you'd have 65 and a half cubes long, it wouldn't go to the whole length of the paper." And so that idea helps students really understand that it's that span, it's not the, I'm gonna put them tick mark at the edge of the cube. So that I get that entire length of the cube are all 66 cubes. It really helps students like, own this idea of measurement, which means, but which we need in order to use an open number line for lots of different things. So Kim, what are some of our favorite things that once we've developed this open number line as a model as a tool for thinking, what are some of our favorite things that we'd like to use it with?Kim Montague:
For sure, one of my favorite things is for elapsed time, right? Three-five grade level is, you know kids are still wrapping their heads around time. And so elapsed time in third grade, I remember being a little bit of a struggle. And then -Pam Harris:
A little bit?Kim Montague:
Oh, my gosh, you know, you could put a start and end time or start and it's been this much time or an end time and how much was it previous to that. We could stick those amounts on a number line. And kids had had experience with addition, subtraction of whole numbers on an open number line, and it just made so much more sense to them.Pam Harris:
And to be clear, you're actually putting the 24 hour clock on a number line. So it's not a traditional typical open number line. But we're literally, I guess I shouldn't say you're putting a clock on the number line, but you're representing times.Kim Montague:
On the number line. Yeah, so you just like stick them down there. And then you can figure out what's in between them. And all of a sudden, bam, you've got elapsed time. Brilliant.Kim Montague:
It's also very helpful for subtraction because we want to consider subtraction as both difference and removal. And you can model that so well on the open number line, not to mention the connection between addition and subtraction. We can do integer addition, subtraction, or even integer multiplication work as we think about something like five times negative six as five jumps have six in towards negative six. So we're going to the left on an open number line. Ssort of jump of negative six is a jump to the left and we have five of those jumps. Where would we land? Oh, we would land on negative 30. All the way, like I mentioned earlier to where it literally becomes a coordinate axis and all the time and higher math, we choose where to put the tick marks, I'm thinking of calculus problems where we have a function and we're trying to find the area underneath the curve, or we're trying to rotate a section of a curve and find the volume of that of the thing that we're creating. What am I trying to say? The surface area of the surface or the volume of the solid, there's the word I wanted. Volume in the solid that we create as we rotate things. Every time we do that, we only put tick marks where it's helpful, where it's handy. And that is the definition of an open number line. We're just putting two of them together, we have the x axis and the Y axis. And then we wanted to go 3d. We get the Z axis. I mean all the things that really is the precursor that all too often we just start the moment that we start graphing points, but we could start it in this way as a model. It is a measurement model started early to help students make that transition and really learn measurement at the same time learning addition, it's so so cool.Kim Montague:
It's for everyone. Right? So you just mentioned-Pam Harris:
I mentioned elementary, you mentioned middle school in high school. The open number line is for all Right?Pam Harris:
So if you are actually interested to learn more about how to help students learn, counting and counting strategies, and then transition to use an open number line, we have a super cool workshop called Building Addition for Young Learners. And this workshop is for teachers of young students, and for leaders pre K through second grade.Pam Harris:
You will love it. So we do a lot of work in that workshop with young kids and you get to like notice video and all. It is an amazing thing. And we only open up registration for workshops three times a year, and that's coming soon. So if you're listening to this podcast when it first drops, that registration is opening soon.So hop on our email list so you'll know when that's going to happen. Keep listening to the podcast. We'll let you know when registration opens, but be planning that that's happening soon. Thank you for tuning in and teaching more and more Real Math. To find out more about the Math is Figure-Out-Able movement. Visit mathisFigureOu-Able.com. Let's keep spreading the word that math is Figure-Out-Able