Pam  00:00

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

Kim  00:06

And I'm Kim. 

Pam  00:07

And you found a place where math is not about memorizing and mimicking, where you're 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 believe we can mentor students to think and reason like mathematicians. Not only are algorithms, not really helpful in teaching mathematics, but rotely repeating steps actually keeps students from being the mathematicians they can be. 

Kim  00:33

You know, when you don't say that exactly the same way. I always make note. Like, "Hey, that was  something different to listen to."

Pam  00:41

Trying to change it up a little bit. I was talking to some people at a recent workshop, and they were like, "You never say that the same." And I was like, "Is that thing?" And they're like, "Yeah, please don't." I was like, "Okay." 

Kim  00:49

Yeah. 

Pam  00:50

I'll keep trying to say it different every time.

Kim  00:52

You know what I'm super glad for too is it's not a two minute introduction. Nobody (unclear). 

Pam  00:56

Yeah, yeah. 

Kim  00:57

Okay. Anyway, so this week as a special treat? 

Pam  01:02

Whoa!

Kim  01:02

Yes? I don't know what you want to call it. We've decided to bring back one of our favorite episodes. So, Pam, I'm going to have you tell our listeners why you chose (unclear).

Pam  01:13

So, yeah. We have been really working hard to put the final touches on our Building Powerful Fractions workshop that launches January of 2024. We are so excited about it. You are not going to want to miss this.

Kim  01:29

Hey, this one's been asked for for a very long time. And we finally are able to make it happen. I'm particularly excited about it.

Pam  01:38

You know what. and it was so fun to film. And it's been so fun to work on the rest of the parts of it and the behind the scenes. And it's been really on my mind, one of the previous episodes that we did. So much so that I was like, "Kim, I think..." Let's talk a little bit about how this impacts fractions and send our listeners back to listen to episode 138. Which is called "To Include Or Not? An Important Subtraction Pitfall." And you might be like, "What does subtraction have to do with fractions?" Well, there's something so important. In that episode, we discuss the "fence post" problem. What is that? Well, it's so important in measurement. In measurement, there's this idea of fence posts versus rails. If I were to say something like, "I'm on the 38th fence post, and we have to replace from the 38th fence post to the 45th fence post. How many fence posts are you replacing? And how many rails between those fence posts are you replacing?" It's so important in measurement. Well, ya'll, fractions are so important in measurement. So, we need to get the things right that we discuss in this episode, so that we can get them right with fractions as well. So, keep that in mind as you listen to this previously aired episode. Think about fractions. Which we don't really talk about in the episode. So, this is your chance to go back and listen to that episode. As you're listening to it, think about fractions. How are you labeling the tick mark one-half? When you share a candy bar with someone, where do you cut the candy bar? And how do you label the piece? How do you label where the cut is? How do you label the piece itself, the chunk itself? Which one, in that candy bar, when you're sharing. I get a fourth of a candy bar. I'm going to cut the candy bar. In that candy bar situation, which is the fence post and which is the rail?

Kim  03:44

We hope you enjoy a revisit to Episode 138.

Pam  03:47

Yeah, ya'll are going to enjoy it. Remember, think fractions and measurement. Go. Hey, fellow mathematicians! Welcome to the podcast where Math is Figure-Out-Able! I'm Pam. 

Kim  04:01

And I'm Kim. 

Pam  04:02

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 are 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  04:32

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  04:59

(laughs) 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 Moraine...I hope I'm 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, @dmoraine. 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 1 and the last number you say is how many. However, both children and adults fall into the 'off by 1' 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, that will come up between us and everything. And so he asked, "What are some of the favorite ways to convey the concept?" Well, listeners, I'm going to probably write a Problem String. Like, chances are, if you ask 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, that 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  07:16

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 a number line. And you know I like Over. So, they might subtract 20...

Pam  07:35

Nice. 

Kim  07:35

...to get to 3, and then realize they removed too much. So, then, they add 3 back on to get 6 pages.

Pam  07:44

Mmhm. 

Kim  07:46

Somebody else might say what is the distance between 17 and 23. And so, I have to read 6 pages because that's the kind of the gap between 17 and 23.

Pam  08:00

So, you think 17 up to 20. 20 up to 23. That's two 3s. That 6. And the context kind of calls for that, kind of looking between 17 and 23. Cool. So, as a teacher, we would probably represent 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 would 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 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  08:54

What I expect students to do?

Pam  08:56

Mmhm. 

Kim  08:57

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

Pam  09:07

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

Kim  09:11

7 spots.

Pam  09:12

But I thought 23 minus 17 was 6? Didn't we just decide that the distance between 17 and 23 was 6? 

Kim  09:19

Yeah. 

Pam  09:21

But you just got 7? There's seven spots? 

Kim  09:23

Yep. 

Pam  09:24

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 on 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 and a 23 in them, and for one of them we got the answer 6, and for one of them we got the answer 7. Hmm." Next question. What's the distance between the 8 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 7 centimeter mark and the 8 centimeter mark. And then, 9, 10, 11, 12, 13 centimeter mark. Maybe the 14 centimeter mark. So, if I'm asking between 8 and 13, I might have a centimeter on either side. And I might, then, draw a line on top of the ruler to kind of show I'm looking between 8 and 13, the 8 centimeter mark and 13 centimeter mark. So, the question would be, "What is the distance between the 8 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. N-A-E-P exam. We call the broken ruler question. So, Kim, what would you expect students to do with that question?

Kim  10:47

It was also a very common question in third grade state exams for Texas years back. So, I would think many students would say, "Okay, between 8 and 13, I might figure out between 8 and 10 would be 2 centimeters. And between 10 and 3 would be 3 more centimeters.

Pam  11:06

10 and 13. You said, 10 and 3. Between 10 and 13.

Kim  11:11

(unclear). So, the total distance between 8 and 13, would be 5 centimeters.

Pam  11:17

Those 5 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. Alright, 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 just aware that we have international listeners. And so, like signs at 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 8 through 13 got blown over in the storm.

Kim  11:56

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

Pam  12:05

Okay. 

Kim  12:06

And so, they might say, 8, 9, 10, 11, 12, 13. So, that's 6 mile markers.

Pam  12:14

Because you have to replace mile marker 8. You have to replace my marker 9, 10. So, when you were saying... Like, you said "counting by ones". They're actually counting each of those mile markers. So, you're saying that even though we had 8 and 13 in both of those questions, that the distance between the 8 centimeter mark and the 13 centimeter mark was 5. But when it was the 8 mile marker to the 13 mile marker, then we had to replace 6?

Kim  12:39

Mmhm.

Pam  12:40

Ah! 

Kim  12:41

Yeah. 

Pam  12:42

Like, why do we have 5 one time and 6 another time? That's confusing. Alright, one more question. How many fence posts do you need to paint, if you're painting fence posts number 34 through fence posts number 40. 

Kim  13:00

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  13:10

Okay.

Kim  13:11

So, I'm going to paint each one of those 35, 36, 37, 38, 39, 40. Did I say that? 34, 35, 36, 37, 38, 39.

Pam  13:22

Yeah, you said it right. 

Kim  13:23

I was like, "Wait, why did I get 6 and not 7. That's weird." So, there's going to be 7 fence posts.

Pam  13:29

So, when you recounted you got 7.  7 fence posts. Even though 40 minus 34 is clearly 6. 

Kim  13:32

Yeah.  Yeah. 

Pam  13:37

34 plus 6 is clearly 40. But you're saying the answer is 7. So, listeners, you might want to actually like draw the fence posts. Like, count those fence posts. And are they're really 7 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.

Kim  14:00

I don't know how much wire you need. But  the distance between them from 34 to 40 is 6. The distance between the 6, whatever 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 40, it's 7 fence posts. And we call this the 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 all go, "Oh! This is a fence post 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... Oh, go ahead. Well, I was going to 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  15:46

Yeah. Oh, 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 6." Well, it depends, right? It depends on the context and what you're asking. Oh, that's well said. Shoot. Kim, we may have just given high stakes test writers a way to trick kids. Not trick. Not 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." 

Kim  16:17

Yeah. 

Pam  16:18

Not just pluck the numbers, throw in an operation that makes 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'll be clear, I'm a little embarrassed how long it took me to be willing to do enough fence post problems to make sense of what was happening, to come up with a strategy. Kim, I was... I'm not going to tell you how old. But it was quite a while. That when I realized it was a fence post problem, I would literally count. 

Kim  16:53

Yeah. 

Pam  16:54

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.

Kim  17:02

Okay. 

Pam  17:02

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 at one point, they have this brilliant conversation and some video that I saw of students doing this task. Video she produced of students doing 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. And I think, Kim, we did an episode where we talked about? Yeah, with Measuring for the Art Show. 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 back up. So, if we give kids, "Hey, I got 5 puppies. Show me 5 puppies." They're going to literally grab 5 counters. They're going to put 5 fingers up. They're going to move 5 beads on a number rack. They're counting the fence posts. But then, when we say what is 5 minus 3, now they have to think about the distance between 5 and 3. 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  19:51

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

Pam  19:56

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, "Well, don't just pause and let me hang there." And I'm like, "Okay. I don't know."

Kim  20:08

It's all good. I'll jump in.

Pam  20:13

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 we'll 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 that 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. What do I mean? Do I mean the span between the fence posts or the tick marks? Or do I mean that 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 of 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  21:28

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 is a fence post problem." And I'm wondering if people might find themselves kind of reverting back and counting in those situations?

Pam  21:47

I mean, I did. 

Kim  21:48

Yeah.

Pam  21:48

I did so often. 

Kim  21:49

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  21:57

Oh, and I got really clear that once I realized it was a fence post problem, that if I didn't count, I was getting it wrong. And I was started getting it wrong. And so, I was like, "I'm just going to 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  22:23

6.

Pam  22:24

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

Kim  22:35

7.

Pam  22:36

7. So, if you're just subtracting and looking at the span, the answer was 6. If you're counting objects, counting spots, the answer was 7. Is it always 1 off?

Kim  22:51

Are you wanting me to say something?

Pam  22:53

I kind of am. Yeah.

Kim  22:54

Okay, yeah. So, yeah. And I don't know when I generalized this, but I think about it like it's the distance plus 1. And so, I'm going to go to your fence posts. 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. The space and the fence posts. The space and the fence posts. And keep doing that till you get to the last fence post. So, there's that kind of span. But then, you've got to remember to come back and pick up that first fence post. So, you can think about it like the span of the distance between them. But oh yeah, don't forget that first one. 

Pam  23:46

And the reason you can think about it that it's a span or distance because as you said, as you're walking down the fence posts. And I'm going to 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 post 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 1 more because I would have missed the first fence post. 

Kim  24:38

Yeah. 

Pam  24:38

So, ya'll, 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 8 and 13, what would you really have done?

Kim  24:57

I would have thought about the distance between 8 and 13, which is 5.

Pam  25:02

Yep. 

Kim  25:02

And then, added 1 more.

Pam  25:04

Added 1 more. Bam, that was 6. That's your strategy.

Kim  25:07

Yep.

Pam  25:08

Listeners, I wonder what your strategy is. Hey, ya'll, 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!