Pam Harris: 0:01

Hey fellow mathematicians. Welcome to the podcast where Math is Figure-Out-Able. I'm Pam.

Kim Montague: 0:07

And I'm Kim.

Pam Harris: 0:08

And you found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But it's about making sense of problems, noticing patterns and reasoning using mathematical relationships. We can mentor students to think and reason like mathematicians, not only our algorithms and step by step procedures, not particularly helpful in teaching mathematics. But rotely repeating those steps actually keep students from being the mathematicians they can be.

Kim Montague: 0:38

So we get some great ideas from all over the place about what to chat about on the podcast. Right? And so this week, we thought we would talk about a tweet that we actually saw quite a while back, but it's popped up in conversation between us several times, as we talk about subtraction. So Pam, we've actually joked about this a few times, because sometimes as we talk about this particular type of problem, one of us will go, it's this right?

Pam Harris: 1:07

Yeah, let's be clear, it's usually me. It's usually me going, "Hey, is that, is that..." and we'll tell you, we actually have a name that we've tossed around, but we don't want to give it away right now. So, David Marain, I hope I am saying your last name correctly, David, is a guy that I've gone back and forth on Twitter with and post some excellent questions. So@dmarain, feel free to follow him. He's got some good things that he posts. But quite a while ago, he posted, "Children learn at an early age that counting a number of objects starts from one and the last number you say is how many. However, both children and adults fall into the 'off by one trap', when presented with a problem, like how many whole numbers are there from 13 through 23? With the instinctive response being 10. What are some of your favorite ways to convey the concept?" Now, you may or may not have caught that. I read that kind of fast, and I didn't really stop and explain or anything. But when I read that tweet, it was one of those times where I was like, "Kim, Kim, look, somebody's talking about it!" this thing that we keep, you know, come up between us and everything. And so he asked,"What are some of the favorite ways to convey the concept?" Wow, listeners, I'm gonna probably write a Problem String, like chances are, if you asked me for a way to convey the concept, chances are high, I'm going to write a Problem String to do that. And so I did. So let's do a Problem String today. Again, don't feel like you have to go listen to that, again, to understand what it is because we're going to do a Problem String. And by the end of it, I think you'll understand what David was talking about. Yeah. So usually in the podcast, when we do a Problem String, I'll ask the questions. And then Kim will talk about her thinking. And we'll sort of, you know, kind of go back and forth and discuss how she's, what her strategy is, or vice versa. But that won't work so well today, because both Kim and I have constructed a strategy for this, for these problems. It won't be super helpful to just tell you that. We want to help construct that strategy. Here's the string. Kim, first problem is you are on page 17 in the booklet. There are 23 pages in this small booklet. How many pages do you have left to read?

Kim Montague: 3:22

Okay, so I would expect that some students would think about that like a subtraction problem and say 23 minus 17. And then, you know, they could solve it a couple of different ways. So they might start with 23 on the number line. And you know, I like Over, so they might subtract 20 to get to three, and then realize they removed too much. So then they add three back on to get six, six pages. Somebody else might say. "What is the distance between 17 and 23?" And so I have to read six pages, because that's the kind of the gap between 17 and 23.

Pam Harris: 4:07

So you think 17 up to 20? Twenty up to 23? That's the two threes and that's six. That's kind of, and the context kind of calls for that kind of looking between 17 and 23. Cool. So as a teacher, we would probably represent one or at least the distance one, maybe the removal one, but we would expect students to kind of be thinking about one of those two, and we'd say, "Great." Okay, next question. There are some parking spots in the senior parking lot left to paint. So I don't know if you guys have that at your school. But sometimes schools will like allow senior students to 'rent' a parking spot for the year and they can go paint it and make it their own or whatever. So there are some parking spots left to paint in the parking lot. How many more kids can choose a spot if spots 17 through 23 are open?

Kim Montague: 4:59

Hmm. What should I expect students to do?

Pam Harris: 5:02

Umhum.

Kim Montague: 5:03

I would expect that many students will count by ones and say, "17, 18, 19, 20, 21, 22, 23."

Pam Harris: 5:13

And when you just counted that, how many did you get?

Kim Montague: 5:17

Seven spots.

Pam Harris: 5:19

But I thought 23 minus 17 was six. Didn't we just decide that the distance between 17 and 23 was six?

Kim Montague: 5:26

Yeah.

Pam Harris: 5:27

But you just got seven. There's seven spots.

Kim Montague: 5:30

Yep.

Pam Harris: 5:31

That's interesting. And that's literally what I would say in class. I would be like, "That's interesting. Wait, what?" And I might ask students,"How are you making sense of that?" But I don't want to kill it too much in this first, these first two problems. But I do want to encourage students, you know, "How are you making sense of the fact that both of these problems have a 17 to 23 in them? And for one of them we got the answer six, and for one of them, we get the answer seven." Next question. What's the distance between the eight centimeter mark and the 13 centimeter mark? Now, ideally, I would maybe show a picture where I would have like, we call it the broken ruler problem, where you kind of see like this jaggedy edge where you don't see the rest of the ruler, and you kind of see maybe the seven centimeter mark and eight centimeter mark, and then 9, 10, 11, 12, 13 centimeter mark, maybe the 14 centimeter mark. So if I'm asking between eight and 13, I might have a centimeter on either side. And I might then draw a line on top of the ruler that kind of shows I'm looking between eight and 13, the eight centimeter mark and 13 centimeter mark. So the question would be, what is the distance between the eight centimeter mark and the 13 centimeter mark? This is a typical question that showed up on the NAEP exam. It's one of the things that we talk about a NAEP exam, we call the broken ruler question. So Kim, what would you expect students to do with that question?

Kim Montague: 6:54

It was also a very common question in third grade state exams for Texas, years back. So between, I would think many students would say, "Okay, between eight and 13, I might figure out between eight and 10 will be two centimeters. And between 10 and three would be three more centimeters...

Pam Harris: 7:12

Ten and 13. You said 10 and three. Between 10 and 13.

Kim Montague: 7:20

So the total distance between eight and 13, would be five centimeters.

Pam Harris: 7:23

Those five centimeters. Yeah. And hopefully, you know, like, you can picture that. We would probably draw an open number line on the board and maybe do those jumps, the 8 to 10 and the 10 to 13. Cool. All right. Next question. How many mile markers, so we're driving down the road, and there's these mile markers. How many mile markers, those are signs. So I'm aware that we have international listeners and so like signs that every mile on highways often will have markers of how many miles you are from the beginning of that whatever the beginning is, often it's the state line. So how many mile markers do you need to replace, if mile markers eight through 13 got blown over in the storm?

Kim Montague: 8:02

Yeah. So again, I would expect that many students would think about each one of those individually.

Pam Harris: 8:11

Okay.

Kim Montague: 8:12

And so they might say, "Eight, 9,10, 11, 12, 13. So that's six mile markers."

Pam Harris: 8:20

Because you have to replace mile marker eight. You have to replace my marker 9, 10. So when you were like saying, like you said, counting by ones are actually counting each of those mile markers. So you're saying that even though we had eight and 13 in both of those questions that the distance between the eight centimeter mark and the 13 centimeter mark was five, but when it was the eight mile marker to the 13 mile marker, then we had to replace six?

Kim Montague: 8:47

Yeah.

Pam Harris: 8:48

Like why do we have five one time and six another time? That's confusing. I have one more question. How many fence posts do you need to paint, if you're painting fence post number 34 through fence post number fourty.

Kim Montague: 9:06

Okay, so this is going to be one like the one you just asked me where I have to consider that I'm painting the 34th fencepost.

Pam Harris: 9:16

Okay.

Kim Montague: 9:17

So, I'm gonna paint each one of those 35, 36, 37, 38, 39, 40. What did I just say, 34, 35, 36, 37, 38, 39, 40. I was like wait, why did I get 6 not 7. That's weird. So there's gonna be seven fence posts.

Pam Harris: 9:36

So when you recounted you got seven?

Kim Montague: 9:38

Yeah.

Pam Harris: 9:39

Even though 40 minus 34 is clearly six.

Kim Montague: 9:43

Yeah.

Pam Harris: 9:43

Thirty four plus six is clearly 40. But you're saying the answer is seven. So, listeners, you might want to actually like draw the fence posts, like countless fence posts, are there really seven fence posts. Okay, okay. Hey, what's the distance, how much wire do I need to put how many feet of wire if they're a foot apart, that's dumb a meter apart. I don't know how much fence posts or how much wire you need, but the distance between them from 34 to 40 is six. The distance between the six, whatever that unit is. So if the fence posts are, say this is a really skinny fence, I don't know, the fence posts are very close together. So 34 to 40 is 6. But if I'm painting fence posts, 34 to 47, fence posts, and we call this fence post problem. At least I do, I don't know why. And so I wanted to end this string with fence posts, so that I could start calling it the fence post problem. This is what Kim and I will refer to, we'll get to some sort of problem in lots of different contexts, lots of different situations, higher math, lower math, you know, like all over the place where all of a sudden, I'll go, "Ah, this is a fencepost problem." Where all of a sudden, I realize it's not just a distance between problem. But it's a fence post problem. I can't just look at how far apart the posts are, I have to actually count the posts. So a big takeaway, a big question I would ask students is, "Are you counting objects? Or are you finding the distance between objects?" And this is a... go ahead.

Kim Montague: 11:28

Well, I was gonna say I'm struck by how this is exactly the scenario or the situation, why it's so important that we don't have kids just pluck numbers out of a problem. Because clearly the context of the situation is changing your solution.

Pam Harris: 11:53

Yeah, well, that's a really good point. Because we might have kids look at a problem and go, it's clearly subtraction, grab the, pick some numbers, grab the 34 and the 40. Bam, it's six. Right? It depends, right? It depends on the context in what you're asking. Oh, that's, that's well said. Kim, we may have just given high stakes test writers a way to trick kids. Not trick, no trick, right, because high stakes test writers do it, do it because then that will help encourage teachers to go, "Oh, we actually have to help students understand what's happening. Not just pluck the numbers, throw in an operation that make sense and go from there." Yeah. And I want to pay homage a little bit to Cathy Fosnot, because she's one of the people that helped me begin to make sense of this. Like, I'm, I'll be clear, I'm a little embarrassed how long it took me to be willing to do enough post problems to make sense of what was happening to come up with a strategy. Kim, I was, I'm not gonna tell you how old, but it was quite a while that when I realized it was a fence post problem, I would literally count. Yeah. That's the only strategy I had was to count. And I want to hear from you what your strategy is in just a second. But first, I want to pay homage to Cathy Fosnot. So she has a replacement unit. In her Addition, Subtraction collection of units, that is called Measuring For the Art Show. And in Measuring For the Art Show, she has this amazing task for kids to do, where they measure strips of paper so that they can label their artwork. And it's at one point, they have this brilliant conversation and some video that I saw of students doing this task, video she produced the students in this task, where the question is, where would you put if the paper was 66 cubes long, 66 units long, where would you put 66? And she's able to use the idea of well, if we want to make the paper 66 cubes long, then the 66 has to go at the edge of the 66th cube. You're not counting cubes, it wouldn't be in the middle of the 66th cube that might be like if you're labeling, here's cube 66. So we're not counting cubes. We're measuring a span of distance. And I may not have just given that enough credit. But if you're building the open number line as a model, I highly recommend considering that. I think, Kim, we did an episode where we talked about, yeah, with Measuring For the Art Show. So so we'll put that episode in the show notes. So you go check that out where we talk more about that. But it is a big deal. Because what do kids do initially? The very first thing kids do when we're teaching them to count is they count the fence posts. And I would then take that to a ruler, they count the tick marks. So if we say, if we just throw a ruler at well, let me start, let me backup. So if we give kids hey, I got five puppies, show me five puppies, they're gonna literally grab five counters, they're gonna put five fingers up, they're gonna move five beads on a number rack. They're counting the fence posts. But then when we say what is five minus three? Now they have to think about the distance between five and three. And that's one of the reasons why a ruler can be so difficult for kids is because they have to make sense of am I counting the tick marks? Or am I looking at the span between the tick marks? It's one of the reasons why measurements can be so difficult for kids. And we have to give them experience making sense of the difference between the tick marks. And the span between the tick marks. It's a huge deal when we are moving to the open number line. That's me pausing in case you wanted to say anything.

Kim Montague: 15:57

No, I was thinking if there is anything I wanted to say. But I don't think anything I want to add.

Pam Harris: 16:02

No problem. So listeners, Kim and I were talking and she's like,"Sometimes I don't feel like interrupting you, Pam." And I was like, "Oh, then I'll pause more often." She's like, "We'll just pause and let me hang there." Okay, I don't know. [laughs]

Kim Montague: 16:16

[laughs] I'll jump in.

Pam Harris: 16:19

So I was actually talking about this with one of my sons the other day, and he said, "Oh, yeah, like, this is a thing in computer science." And I was like, "Tell me more." And he goes, "Oh, it's often that will have strings in computer science." A string is like a list of characters. So it could be numbers. It could be letters, it could be words, but you know, it could be names. So you have this list of characters, and then there are times you need to do things with those. And you have to decide, are you going to be inclusive? In other words, are you counting fence posts? Or exclusive where, no, you're leaving out that first fence post? And you're just looking at the span or the distance between them. And you have to make sense of that. And it was interesting for me that that's kind of where it came up for him was in computer science. All of a sudden, he had to make sense. Do I, what do I mean? Do I mean the span between the fence posts or the tick marks? Or do I mean that no, no, no, I actually have to count that first fence post as well. And so he said, like, sometimes you'll have to order lists, or edit lists and string and you have to know the number of objects, not the distance between them. And so that's a huge, that's a specific instance. But then Kim, you were like, oh, but Pam, it shows up everywhere.

Kim Montague: 17:34

Yeah, I think it shows up in our daily life more than we probably realize. And now that we're mentioning what we call fence post problems. I think listeners will probably be able to go, "Oh, wait, that that is a fence post problem." And I'm wondering if people might find themselves kind of reverting back and counting in those situations?

Pam Harris: 17:53

I mean, I did.

Kim Montague: 17:54

Yeah.

Pam Harris: 17:54

I did so often.

Kim Montague: 17:56

Yeah, because it's just so it's just like a little twist on a subtraction problem that maybe shakes you up a little bit.

Pam Harris: 18:03

Oh, and I got really clear that if I, once I realized it was a fencepost problem, if I didn't count, I was getting it wrong. And I was trying to get it wrong. And so I was like, I'm just gonna count. Then I know, I'm right. And what I didn't do was take the time to compare. What I didn't do was take the time to compare. So let's look back at the problem string that we just did a minute ago. So I said, "We're on page 17. There's 23 pages in the book." You said, how many pages were there left to read?

Kim Montague: 18:30

Six.

Pam Harris: 18:30

Six. So that's like 23 minus 17. Yeah? Then I said,"Okay, but they're senior parking spots, number of parking spots 17 through 23." And you said the answer was?

Kim Montague: 18:42

Seven.

Pam Harris: 18:42

Seven. So if you're just subtracting and looking at span, the answer was six. If you're counting objects, counting spots, the answer was seven. Is it always one off?

Kim Montague: 18:58

You wanting me to say something?

Pam Harris: 18:59

I kind of am, yeah.

Kim Montague: 19:00

Okay. Yeah. So yeah, it's one because, and I don't know when I generalize this, but I think about it like, it's the distance plus one. And so I'm going to go to your fence post, I'm going to jump ahead to your fence post problem and give kind of a picture in that context. So if I'm standing at the first fence post, or the 17th, or whatever number you gave us, then you can think about the span of it, like the space and the fence posts, this space and the fence posts, the space and the fence posts and keep doing that till you get to the last fence posts. So there's that kind of span. But then you've got to remember to come back and pick up that first fence posts. So you can think about it like the span or the distance between them. But oh, yeah, don't forget that first one.

Pam Harris: 19:52

And the reason you can think about it, this is span or distance because as you said, as you're walking down the fence plus and I'm gonna actually use the numbers we used in the string so 34 to 40. So if I start at the 34th fence post, but I'm not counting it, because I'm thinking about distance, then I'm going to count that span from 34 to 35. That's also equivalent to counting the 35th fence post. Then I count the span from 35 to 36. That's like I've grabbed the 36th fence post, 36 to 37. So as I count each of the spans, I'm also kind of grabbing the fence posts that goes with it. But I've missed the first fence post if I do it that way. So you're saying I can just count the number of spans because that's equal to the number of fence posts. But then I'm going to add one more, because I would have missed the first fence post.

Kim Montague: 20:44

Yeah.

Pam Harris: 20:45

So y'all, we can help students realize that in fence post problems, we got to go back and grab that first fence post. So if I would have asked you, Kim, a question like, how many mile markers do you need to replace between eight and 13? What would you really have done?

Kim Montague: 21:03

I would have thought about the distance between eight and 13, which is five and then added one more. So it's six.

Pam Harris: 21:11

Bam if it was six. That's your strategy. Listeners, I wonder what your strategy is? Hey, y'all, thanks for tuning in and teaching more and more Real Math. To find out more about the math is Figure-Out-Able movement visit mathisFigureOutAble.com. Let's keep spreading the word that Math is Figure-Out-Able.