Ep 46: Fractions for Young Learners

Pam Harris  00:05

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

Kim Montague  00:12

And I'm Kim Montague. 

Pam Harris  00:14

And we're here to suggest that mathematizing is not about mimicking or rote memorizing, but it's about thinking and reasoning, about creating and using mental relationships; that mathematics class can be less like it has been for so many of us, and much more like mathematicians working together. We answer the question: If algorithms are not your main goal, then what is? 

Kim Montague  00:42

Yeah, so last week, we talked a little bit and we started a new series on fractions. We talked about the five interpretations of rational numbers, and this week, we're going to dive into our young learners, right? This is for you "little" teachers.

Pam Harris  00:59

Haha! You just said "little teachers."

Kim Montague  01:00

Oh. Teachers of littles. 

Pam Harris  01:02

Okay, what's really funny y'all is that I'm the one that usually messes that up, and Kim doesn't. And so today for her to mess it up - I'm always the one that will say, "Okay, you young teachers out there, I mean, you teachers of young students." Which you could be young, but you're, you know, the fact is that you're teachers of young students. What's really bad is when I say"Okay, you old teachers, no." It's never about old teachers, right? It's only about teachers of older students. Today, we're going to talk about the littles. How can we introduce fractions to our youngest students, youngest kids? We're even going to suggest ages of kids who, how do I say this, that it's not in the standards to introduce fractions, which we agree with, by the way. We agree that the fractions should come later in the standards by far. But that doesn't mean that they're not naturally being used with our younger students, that we shouldn't be beginning to build a fraction understanding with those youngest, little students that we have. So today, we're going to start with one of my more humiliating, I mean, learning experiences. 

Kim Montague  02:05

Oh, really can't wait to hear. 

Pam Harris  02:07

So if you remember a little bit of my history, one of the ways that I became who I am today is that as my kids, I have four personal kids, were going to school, I was enamored, I was fascinated by the mathematizing they were doing at home, yet they were bored at school. So I dove into the research about how we can teach math better. And I also dove into my kids' classes at school. And I'm so grateful to this day for those teachers who let me come in and pilot stuff and try stuff and just, you know, get a feeling. And to be clear, I did a lot of dumb stuff at the beginning, not maybe a lot, but I did enough that you know - and so, here's an example of that one time. I had been doing PD with the teachers K-5. And a lot of the teachers were like, what does this look like? Like you're telling us to do stuff, but we can't picture it. Like, can you come do that. And so we kind of came up with this plan, where I actually went on campus, one day, it was kindergarten through fifth grade school. And I taught kindergarten through fifth grade at each grade level, a sample lesson, a model lesson. So I prepared this model lesson. And all of the teachers at that grade level sort of came into that one classroom, we were able to cover classes and subs and stuff. And so all the third grade teachers came in, and they stood in the back of the room. And I did this sample lesson, this model lesson to kind of try to give teachers a feel for how we could begin mathematizing with our students. I'll tell you what, Kim, the hardest part of that whole day, I had so much newfound respect for specials teachers. For PE teachers and music teachers. Because it might be like, yeah, sure, I just did this, you know, kindergarten, I can do it again. But they were not in order! So by, like, halfway through the day, kids would walk in the room and I would say, "Speak to me. I need to hear who you are a little bit. What animal am I working with in this moment? Okay, that's the third grades. Okay, I got it. Now I can talk to you." Because I hadn't like a sense or feel of - anyway, so that was kind of a funny outcome of me doing these model lessons. So on this one particular day, I was doing these model lessons in this one particular lesson it was in. I'm pretty sure it was kindergarten. I don't think it was first grade, but it might have been either one of them. kindergarten or first grade. We had planned this really super cool game to play with the kids. And I was going to show these teachers about how I could walk around while the kids were playing and as I was circulating I could ask these really cool questions. And then we can bring them together at the end and have this really neat conversation, by the way, which had nothing to do with fractions. It was a number lesson. Anyway, so we had it all planned. I had really thought through it well. I was ready to go. So I sat the kids down and I told them this beginning story. I said, "Okay, so now we're going to play this game." The kids were rapt attention. They were fascinated. Everything was going super. The teachers around the room were like, "Alright, we're ready to see this really cool teaching." I said, "Okay, we're gonna play this game, first thing you guys are gonna do is cut the deck in half. So split the deck in half. So each of you have half a deck." And the teachers looked at me like, "Are you crazy?" And I was like, "Go ahead everybody." All the kids, you know, they were in partners. And then they had the deck of cards. I was like, "Okay, like, you know, split the deck in half." Y'all, we spent the rest of that math time that day and the next week of math times in that class talking about splitting things in half. Like talking about what it means to have an equal, like, you have half the cards and I have half the cards, what half means. To be clear

Kim Montague  05:35

But there's 52 cards! 

Pam Harris  05:38

Well, we played a lot of cards at my house and in my family. And so I was really clear that they were just going to deal out. They were going to, you know, I give you one then I give me one, then I give you one, then I give me one. Or at a minimum, they were just gonna, like eyeball it and give you like, okay, here's half and here's half. Those kids just stared at me. And they would kind of, I don't know that they were eyeballing it, but they would like, give one kid two cards, the other kid gets to keep the rest. And I'm like, "That's, what?! That's not half." They just looked at me with this blank expression, kind of like, "What are we supposed to do, Miss?"  Oh, I'm sure. Teachers were cracking up. They were like, "I'm sure you had a really good lesson planned, Pam. But our kids can't, they don't know how to do that." So could we say, parents of younger students...

Kim Montague  06:20

Yeah. 

Pam Harris  06:20

...play card games with your kids. Like split the deck in half with your kids. You don't have to make a big lesson about it. But as you deal out, say even, I don't know you're playing Ten's Go Fish or you're playing just normal Go Fish or whatever and everybody gets seven cards. As they watch you deal - one for you, one for you, one for you, one for you back, another one for you another one for you another one for your another - as they watch you sort of deal out they're seeing you create these equal sets of cards. And that would be a thing to start with so that they get experience and knowing, "Ooh, if I'm going to give equal sets, or just in half, if it's just me and you playing, then I give you one I give me one, I give you one." That is a way to split things in half. So Kim, let me give you an example of what I said we spent the rest of the week literally. So give me some things, just name some things in a classroom.

Kim Montague  07:15

 Like staplers?

Pam Harris  07:18

No, like, Kim, there might be two staples.

Kim Montague  07:21

Oh, one for you, one for me.

Pam Harris  07:22

Something easy to split in half. Like, we're gonna split some things in half. Name some other things. 

Kim Montague  07:28

Oh ok, like snap cubes. 

Pam Harris  07:30

Okay, perfect. So I might give this these two students, I might give them a bag of or a tote of snap cubes. I might say, "Here are these snap cubes." Now, if they're kindergarten kids, I'm not going to give them like millions of snap cubes, right? 

Kim Montague  07:46

Yeah.

Pam Harris  07:46

I'm not going to give them all the snap cubes. I hopefully have a lot of snap cubes. I'm gonna give them a manageable amount of snap cubes. And I'm also going to be thinking about which set of students I'm giving the snap cube share versus, Kim, mention something else in the classroom.

Kim Montague  07:59

Teddy Bear counters.

Pam Harris  08:01

Nope, I have the same number of teddy bear counters as - well, maybe I have fewer, probably fewer teddy bear counters than have snap cubes. So I might give these kids who I know have some sense already of dividing by 2, not dividing by two, but like splitting numbers in half that they might deal out, I might give them the snap cubes. And I might give a group who's just kind of like on the verge of that, but they can count pretty high, I might give them the teddy bear counters. Now give me something where there's fewer of naturally in the classroom.

Kim Montague  08:28

Um, maybe books in a particular series that you're reading. 

Pam Harris  08:32

Okay, cool. Or maybe like books in this part of the shelf or books in this tote? I might hand them this tote of books and just say, "Hey, here's just this bucket of books. Split them up, you get half and she gets half. Or he gets half." What's something even fewer, not staplers, though. Not just two.

Kim Montague  08:52

I'm thinking pencils in a pack. Like there might be a pack of 10 pencils. 

Pam Harris  08:56

Oh, nice. Nice. So then I might give kids who are still kind of struggling to learn the counting sequence, I might say, "Hey, guys, let's split these up." And that might actually not even be the best criteria to determine who I'm going to give the small numbers to and the big numbers to. I might do my best at saying, "Oh, I think you may need a smaller set and you need a larger set." And then watch, right? I'm circulating and watching and I might go, "Oh, you're struggling with that. Let me back up. Let me give you a smaller - or you know, work with you a little bit, and give you a smaller number of things to split". And then as I see you being successful with that, then I might give you - what's something else we have in classes that we have a ton of? Paperclips, right like I might have a box of paper clips or what's another thing we have lots of?

Kim Montague  09:38

Erasers.

Pam Harris  09:40

Erasers, base 10 cubes, whatever kinds of things we have hanging around, I'm just gonna say, "Hey guys, today we get to have these, like we're gonna split these, we're gonna share them fairly." Teachers of younger students, a thing you could do is to just have kids divide things in 2, Just split them evenly among students. Then you can have fun conversations where there's odd numbers of things and decide kind of how you're going to do that. That gets really fun when the thing you're splitting is splittable. So for example, today's snack, we're going to have - okay, this is pre COVID where I'm at, we're in the middle of COVID. But I can only think of a snack right now, so help me Kim. If I had like graham crackers and so I say to them, "Hey guys for snack today I've got three graham crackers for each of the two kids." Three of those you know fose big, like whole crackers? And so I might say like, "How many graham crackers does each kid get it?" If I give two kids three graham crackers, well, they're gonna you know, you get one you get one. What do we do with this other one? Well, they can cut it in half. Right? That's a thing. But if I'm giving them pencils, or markers might be even better. Here two kids, here are three markers, what are you going to do? Like, then there's some conversation about oh, we can't really split that. So then we have to give this one away to another group or I don't know, today, you get two markers and I get one. And tomorrow I get - you know. There's a reason for sort of splitting things. If you can actually come up with a reason for splitting it would be even better. Like I just said, reason twice, meaning two different things. So one, thinking about splitting things that aren't splittable. And like reasoning how you're going to do that. But then also, as you say to students, "Hey, today, we're going to share fairly with these things. You guys get the snap cubes, you guys get the teddy bear counters, you guys get the pencils," as you do that, it'd be really cool if you can think of a context where that would make sense. Yeah, one context that I can think of that could make sense is that you might say, "Hey, we're going to have some visitors. And we're going to give them half of everything that I've given you guys to share today. And so you guys are going to have to split them all in half, because we're going to let this other class borrow our things." Like that could be a reason for like splitting them in half. And then you have to decide if you're actually gonna let anybody borrow him or whatever, but come up with something, some reason where kids are like, "Oh, all right. Well, we want to share them fairly. So we better do that." And then kids get right at doing a lot of counting, as they think about, oh, maybe I should be explicit about why they're counting. Like cuz some kids will take - woah, I'm just talking talking talking sorry Kim.

Kim Montague  12:13

No, I'm taking it in. I have a few things I want to ask you about, but I'm taking it in right now. 

Pam Harris  12:18

Excellent. Excellent. So say I give them the teddy bear counters. And I'm like, "Okay, let's share fairly, we got to make sure that we have the exact amount." Kids might just really go *moving sound*. If you can see my hands. I just like sort of split the teddy bears into 2 groups. And then they go, "Okay, we did it." Then I'm gonna say, "Oh, do you think it's the same? Like the same in that as here? Let's check." And so the kids count the one pile and they get 25. And the kids count the other pile and they get 27. And I go, "Huh, not the same. What are you gonna do?" Oh, Kim, the reason that we did five days of this is because sometimes the kids would like shove the two piles together completely. And like, "Okay, I better try again," and then randomly just split them apart. I'm like, "Okay."  At first I was like panicking. There was a little bit of me that was like, "Oh no, they're gonna have to," and I thought no, like, there's a whole opportunity for them to count again. 

Kim Montague  13:04

Yes. 

Pam Harris  13:04

So we do this with young kids who we want to give lots of experience with counting. So they count again, and this time they get 20. And what did I say before 25 and 27? So then they've got 52. So maybe then they have 20 and 32 in the other pile. And so then I'm like, "Well, that's not fair either. What are you gonna do?" Now, at some point, I might like, intervene, and go, "Woah before you just shove those two piles back together again, what if you like, which one's bigger? Which one has more? Well, I mean, like, could we do something besides shoving them all? Could you just like, give some...?" Like I do as little as possible to kind of suggest. Maybe I might say, "Oh, look at what the partnership over there did." And then they kind of watch them, just sort of take a couple from the big pile and move it over and recount. Take a couple from the big pile, move it over, and recount. Again, this is where the determination of what numbers are good for which pair of kids is important, because we want to be right on the edge of their proximal development, right? So they have the ability to think about it, they're struggling productively, but they're not struggling unproductively, they're not getting so frustrated that they're giving up. So that's why you're managing the numbers, that you're giving different pairs of kids to keep all kids learning. So it's right on that edge of their learning. Alright, Kim, I can't wait to hear the kinds of things that you were thinking about while I was going on and on and on. 

Kim Montague  14:27

Two things that I think that we hear a lot of times really young learners saying half this, half this, half this, right? And they don't mean 'half' they mean 'part'. 

Pam Harris  14:37

And to be clear, I hear teachers and adults saying, "Hey, just, you know, take your half over there." And I'm like, "Whoa, well, they didn't have an equal share." What you mean is the word 'part' is different than the word 'half'? 

Kim Montague  14:51

And so we want to say what we mean, right? If it's not actually half then we need to say part. The other thing I don't think I heard you mentioned that I think is noteworthy is that we can help kids develop the idea of half as we do these experiences, but also recording. I didn't hear you saying about recording. But if over and over again, I'm counting all of these objects, and then splitting them in half between me and you, then your amount, and my amount should be the same. So 10 on your side, 10 on my side, interesting, five on your side, five on my side, interesting. So they're gonna use these experiences to then see on my recording sheet, you get the same number as my number. And if those two numbers are not matching, then something must be off. 

Pam Harris  15:40

Nice. Nice, nice, nice. And can I take it just a little further? 

Kim Montague  15:44

Yeah. 

Pam Harris  15:44

And at some point, when we will then write the total down. So if you have 10, and I have 10, then we have 20. And if you have 5, and I have 5, we have 10. Those were two numbers that are way too...so if I have seven and you have seven, we have 14, then we might notice that the totals are always sort of these even numbers. And so that's noteworthy, then we can kind of make this connection between these even numbers. So if we've doubled something, do we always get an even number? Is that true? And I'm getting a little older at this point, right? We're not necessarily doing that with our youngest learners, but at some point, we can sort of notice that when we double whole numbers, we're always getting these even numbers, that's interesting. So then what happens when we try to split in half, the verb to halve has a 'v' in it, or take half of - Ooh, there's the operator, meaning - take half of an odd number, what's necessarily going to have to happen? We're gonna have to split something in half. Or like I said, you get the marker one day, and I get the marker the next day. Totally cool. 

Kim Montague  16:49

Yeah. 

Pam Harris  16:49

Nice. 

Kim Montague  16:50

So it's interesting to me that the standards often start with students considering the fraction of a continuous object. 

Pam Harris  16:59

Before we go there, though, Kim, can I just mention one quick story? 

Kim Montague  17:02

Sure. 

Pam Harris  17:04

You're going to go to standards. And I totally want you to talk about that. But I want to remind you of a story that the situation you were in, when you were doing a problem with a group of third graders. Do you remember the fracture thing that you were doing third graders that we videoed that one time? These are even older kids, right? Where I saw kids talk about halves, and thirds with - let's see, halfs? Yeah, no, when they talked about thirds, but they didn't really mean thirds. Do you remember the story I'm talking about? 

Kim Montague  17:35

No, say more.

Pam Harris  17:36

Okay, so you were having kids split these things evenly among kids, and I watched them take these rectangles, and first cut them in half, and they would give out a half, and then they were left with - So it's like they had three students they were sharing with, and they had like four rectangles. So they gave everybody a half, and then they were left with a half. And so they're like, okay, now we need to cut this half into pieces. And when they cut that half into pieces, then they counted - I'm not describing this very well. They had this whole rectangle cut in half. So if you picture that half line, but then they took one of those halves, and they cut it into three pieces. And when they cut it into three pieces, now all of a sudden, if you can picture that rectangle, they had a rectangle with - how many lines? 1, 2, 3, 4 lines. Is that right? No, 1, 2, 3 lines, because they had the half line. And then they had the two lines that were cutting it into three pieces. I don't know if I'm making this more complicated than it needs to be. But if you can picture the rectangle cut in half, so there's this whole, there are two slices that are the same size. But then in one of those halves, they cut it into thirds. My point is they were calling each of those pieces. 1, 2, 3, 4 fourths.  Yeah.  Because they had created four chunks, but the four chunks weren't equal. Right? Because they had the big half, and then they had the half cut into thirds. The part I remember is that they were like half, half, half third, third, third. And you were like, "Whoa, what is this piece right here?" And they said, "Oh, well, it must be a fourth." Then you said, "Why is that a fourth?" And they said, "Because there's four pieces." 

Kim Montague  19:19

Yeah, it's almost like as it got more complicated, they forgot a really important thing that they needed to be fair shares. Right? 

Pam Harris  19:27

Yeah. 

Kim Montague  19:28

I don't remember that, I'm glad you do. 

Pam Harris  19:29

Anyway, sorry. Continue. Go back to standards. 

Kim Montague  19:34

Well, I was just going to say that, um, particularly with our younger students, it's interesting that the standards talk about continuous objects. Like a particular shape. And we're suggesting here that we also need to have a set, right? To start dealing with and grappling with things like splitting this card deck in half, splitting a particular number of objects in half. 

Pam Harris  19:57

Absolutely. Yes. 

Kim Montague  19:57

All the while developing half.

Pam Harris  20:00

Yeah, so we might like take a brownie pan and we might split that up. And that's kind of what the standard suggests. But we want to have students have experience with both of those. Split up that brownie pan into fractional pieces, share that evenly. But we also want to do things like we've suggested today, like the deck of cards, or the markers or the cubes or the teddy bear counters. That's all important. We agree with the standards that we don't want to test that too early. So if you're like, "Oh Pam, my standards are not calling for that." Yeah, right, right. But we can still give students experience. Notice that with all of those splitting in half tasks that I just suggested with the younger learners. that it really became more about counting and less about one half. But we're building one half, it's like, we're using the opportunity to build one half really to get kids more experienced with counting, which is what we need at those younger grade levels. Totally cool. So y'all develop one half for a while, like we're actually suggesting, that of all of the unit fractions, the unit fractions is where there's a one in the numerator, that one half is the most important one to start with, like develop one half. Don't do one half 1/3, 1/4, 1/5, like don't do all those at the same time. No, no, start with one half and really do a good job developing one half and doubles. We'll just mention Cathy Fosnot, does an excellent job of helping young students do that. One example of how she does that is in her book called Beads, Shoes and Making Two. And she also does a really good job with some games in her book called Games for Early Number Sense. So we'll put those references in the shownotes, a couple places for you guys to look to really get some other kinds of things to help kids develop one half and doubling. Because we need kids to double and find half, those are two important things to do.

Kim Montague  21:43

Yeah, we've shared that we love the game I Have, You Need before in our podcasts and some other places. But remember that we also love to do a back and forth routine with kids of all ages with doubling number and halving numbers. So I might do a string like I say double five, half five. Now that wouldn't be with our super young learners, but double six, half six, double 10, half 10, double nine, half nine. So we're back and forth, getting kids to think about what's double of a number, what's half of a number.

Pam Harris  22:15

And in between there, we're asking kids what they think, we're modeling their thinking, or making that thinking visible. So yeah, really cool. Yeah.

Kim Montague  22:22

So what about beyond halves. Once I've got kids who've established a really good understanding of half, young students can do more than just half. What are the other fractions?

Pam Harris  22:31

Totally cool. Yeah. So let's make it all about fair sharing and my share...so if I'm sharing me and two other students are sharing, that's totally bad grammer, two other students and me are sharing, two other students and I are sharing, I don't even know how to do that...so if three of us are sharing then I get 1/3. If four of us are sharing, I get 1/4. If eight of us are sharing, I get 1/8. That's a unit fraction. So a unit fraction is where the numerator is one. And it's my share in a fair sharing. And so we want to give kids examples where they share fairly to develop unit fractions, and we want to name them, we want to write them. Name them first, for a long time before we write them. 

Kim Montague  23:17

Yeah. 

Pam Harris  23:18

But give kids this sort of sense and a feel for what's my share in a fair sharing situation.

Kim Montague  23:23

I think a lot of us also know too, that cooking is a really good context for giving kids examples of things like doubling or halving, right? My youngest bakes and makes things with me all the time. And he has his own measurement kit with spatulas and all the things. But what I love is that it came with a really nice set of measuring cups and spoons. And he is forever, particularly when he was younger, taking things like oil or water. And when I would say, "You need 1/4 cup of oil," he would - or you know water, whatever - he would take his 1/8 measuring and pour it into the fourth twice to see if it really was, double 1/8 would make 1/4. So he's constantly seeing what different versions of a fraction that he could do. We also talked about doubling recipes when people are coming over. We talked about halving recipes if it's something that we're making that not everybody in the house loves to eat.

Pam Harris  24:22

Yeah, we love to bake. And growing up we always made cookies on Sunday, measuring cups, spoons, ask all the questions. 

Kim Montague  24:29

Yeah. 

Pam Harris  24:29

I want to point out that you had the patience to let him do that. So we don't always have all the time in the world to let kids sort of pour things in and out of measuring cups. But the more you do, the better kids will get with that kind of stuff. Interesting, we han old set of plastic measuring spoons where the denominations, if it said one half teaspoon or whole tea spoon, it was rubbed off like you couldn't see. And so my kids were like, "Mom, we need a teaspoon of baking powder." And I'd be like, "Yeah, which one of those do you think it is?" We'd sort of reason by knowing what the smallest one was so we could kind of like, reason up. It must be this one. It can't be that one because that's the half tablespoon. So like whatever you can do to encourage kids to reason. So, how do you work with fractions with young learners? Share lots with young students and ask them to share fairly. Use the word 'part' when you mean part, use 'half' when you actually mean really half, not just part. So share with your kids. Do this fair sharing thing. Identify that kids share when they're sharing. Talk with your kids, wonder about fractional amounts when you share. Totally great ways to begin having your youngest think about fractions. 

Kim Montague  25:38

Hey, thanks for joining us today. Please remember to join us on hashtag MathStratChat on Facebook, Twitter, or Instagram on Wednesday evenings where we explore new problems with the world. 

Pam Harris  25:49

And if you find our podcasts helpful, please rate it and give us a review. That way more people can find it wherever they get podcasts. Also, don't forget that we're collecting your questions that you want answered because we're going to answer them in an upcoming episode. So send those questions to Kim at mathisFigureOutAble.com. So if you're interested to learn more math, you want to help yourself and your students develop as mathematicians, then don't miss the Math is Figure-Out-Able Podcast because Math is Figure-Out-Able!