Podcasts about fractions

Survivor: 46 - Recaps from Rob has a Podcast | RHAP

Play Episode Listen Later Jan 20, 2026 21:49 Transcription Available

There is much more to fractions than "cross multiply and divide". In this episode, Pam and Kim facilitate a Problem String to explore the partitive meaning of fractions.Talking Points:Fractions and divisionThe partitive meaning of fractionsEquivalent fractionsRelational reasoning in understanding fractionsScaling up or down to find equivalent, simpler problemsCheck out our social mediaTwitter: @PWHarrisInstagram: Pam Harris_mathFacebook: Pam Harris, author, mathematics educationLinkedin: Pam Harris Consulting LLC

playing fractions

Purple Pants Podcast | Survivor 49 Episode 10 Recap: Fractions

Play Episode Listen Later Nov 28, 2025 87:34

The baby boys are linking back up for another round of Survivor News as Brice, Wendell, and Jack dive into Survivor 49 Episode 10, “Huge Dose of Bamboozle.” With the game heating up and the beach getting messier by the minute, the trio breaks down all the wild maneuvering — from shaky alliances trying to solidify, to players tossing out misdirection like confetti, to the strategic fireworks that had everybody side-eyeing each other. The energy is high, the chaos is premium, and the boys are serving jokes, insight, and pure baby-boy brilliance as they unpack the episode's twists, turns, and near-miss moves that kept the whole tribe on edge from jump.

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Purple Pants Podcast | Survivor 49 Episode 10 Recap: Fractions

Reality TV RHAP-ups: Reality TV Podcasts

Play Episode Listen Later Nov 28, 2025 87:34

survivors brice wendell fractions bamboozle purple pants podcast

Purple Pants Podcast | Survivor 49 Episode 10 Recap: Fractions

Purple Pants Podcast

Play Episode Listen Later Nov 28, 2025 91:48

Purple Pants Podcast | Survivor 49 Episode 10 Recap: Fractions The baby boys are linking back up for another round of Survivor News as Brice, Wendell, and Jack dive into Survivor 49 Episode 10, “Huge Dose of Bamboozle.” With the game heating up and the beach getting messier by the minute, the trio breaks down all the wild maneuvering — from shaky alliances trying to solidify, to players tossing out misdirection like confetti, to the strategic fireworks that had everybody side-eyeing each other. The energy is high, the chaos is premium, and the boys are serving jokes, insight, and pure baby-boy brilliance as they unpack the episode's twists, turns, and near-miss moves that kept the whole tribe on edge from jump. Tickets are now available for Brice and Wen 49 Survivor Watch Party Tour! Grab your tickets here:https://briceandwenpresent.flite.city You can also watch along on Brice Izyah's YouTube channel to watch us break it all down https://youtube.com/channel/UCFlglGPPamVHaNAb0tL_s7g Previously on the Purple Pants Podcast Feed:Purple Pants Podcast Archives LISTEN: Subscribe to the Purple Pants podcast feed WATCH: Watch and subscribe to the podcast on YouTubeSUPPORT: Become a RHAP Patron for bonus content, access to Facebook and Discord groups plus more great perks! Learn more about your ad choices. Visit megaphone.fm/adchoices

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Season 4 | Episode 6 - Christy Pettis & Terry Wyberg, The Case for Choral Counting with Fractions

Mysteries About True Histories (M.A.T.H.)

Play Episode Listen Later Nov 20, 2025 37:00

Christy Pettis & Terry Wyberg, The Case for Choral Counting with Fractions ROUNDING UP: SEASON 4 | EPISODE 6 How can educators help students recognize similarities in the way whole numbers and fractions behave? And are there ways educators can build on students' understanding of whole numbers to support their understanding of fractions? The answer from today's guests is an emphatic yes. Today we're talking with Terry Wyberg and Christy Pettis about the ways choral counting can support students' understanding of fractions. BIOGRAPHIES Terry Wyberg is a senior lecturer in the Department of Curriculum and Instruction at the University of Minnesota. His interests include teacher education and development, exploring how teachers' content knowledge is related to their teaching approaches. Christy Pettis is an assistant professor of teacher education at the University of Wisconsin-River Falls. RESOURCES Choral Counting & Counting Collections: Transforming the PreK-5 Math Classroom by Megan L. Franke, Elham Kazemi, and Angela Chan Turrou Teacher Education by Design Number Chart app by The Math Learning Center TRANSCRIPT Mike Wallus: Welcome to the podcast, Terry and Christy. I'm excited to talk with you both today. Christy Pettis: Thanks for having us. Terry Wyberg: Thank you. Mike: So, for listeners who don't have prior knowledge, I'm wondering if we could just offer them some background. I'm wondering if one of you could briefly describe the choral counting routine. So, how does it work? How would you describe the roles of the teacher and the students when they're engaging with this routine? Christy: Yeah, so I can describe it. The way that we usually would say is that it's a whole-class routine for, often done in kind of the middle grades. The teachers and the students are going to count aloud by a particular number. So maybe you're going to start at 5 and skip-count by 10s or start at 24 and skip-count by 100 or start at two-thirds and skip-count by two-thirds. So you're going to start at some number, and you're going to skip-count by some number. And the students are all saying those numbers aloud. And while the students are saying them, the teacher is writing those numbers on the board, creating essentially what looks like an array of numbers. And then at certain points along with that talk, the teacher will stop and ask students to look at the numbers and talk about things they're noticing. And they'll kind of unpack some of that. Often they'll make predictions about things. They'll come next, continue the count to see where those go. Mike: So you already pivoted to my next question, which was to ask if you could share an example of a choral count with the audience. And I'm happy to play the part of a student if you'd like me to. Christy: So I think it helps a little bit to hear what it would sound like. So let's start at 3 and skip-count by 3s. The way that I would usually tell my teachers to start this out is I like to call it the runway. So usually I would write the first three numbers. So I would write "3, 6, 9" on the board, and then I would say, "OK, so today we're going to start at 3 and we're going to skip-count by 3s. Give me a thumbs-up or give me the number 2 when you know the next two numbers in that count." So I'm just giving students a little time to kind of think about what those next two things are before we start the count together. And then when I see most people kind of have those next two numbers, then we're going to start at that 3 and we're going to skip-count together. Are you ready? Mike: I am. Christy: OK. So we're going to go 3… Mike & Christy: 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36. Christy: Keep going. Mike & Christy: 39, 42, 45, 48, 51. Christy: Let's stop there. So we would go for a while like that until we have an array of numbers on the board. In this case, I might've been recording them, like where there were five in each row. So it would be 3, 6, 9, 12, 15 would be the first row, and the second row would say 18, 21, 24, 27, 30, and so on. So we would go that far and then I would stop and I would say to the class, "OK, take a minute, let your brains take it in. Give me a number 1 when your brain notices one thing. Show me 2 if your brain notices two things, 3 if your brain notices three things." And just let students have a moment to just take it in and think about what they notice. And once we've seen them have some time, then I would say, "Turn and talk to your neighbor, and tell them some things that you notice." So they would do that. They would talk back and forth. And then I would usually warm-call someone from that and say something like, "Terry, why don't you tell me what you and Mike talked about?" So Terry, do you have something that you would notice? Terry: Yeah, I noticed that the last column goes up by 15, Christy: The last column goes up by 15. OK, so you're saying that you see this 15, 30, 45? Terry: Yes. Christy: In that last column. And you're thinking that 15 plus 15 is 30 and 30 plus 15 is 45. Is that right? Terry: Yes. Christy: Yeah. And so then usually what I would say to the students is say, "OK, so if you also noticed that last column is increasing by 15, give me a 'me too' sign. And if you didn't notice it, show an 'open mind' sign." So I like to give everybody something they can do. And then we'd say, "Let's hear from somebody else. So how about you, Mike? What's something that you would notice?" Mike: So one of the things that I was noticing is that there's patterns in the digits that are in the ones place. And I can definitely see that because the first number 3 [is] in the first row. In the next row, the first number is 18 and the 8 is in the ones place. And then when I look at the next row, 33 is the first number in that row, and there's a 3 again. So I see this column pattern of 3 in the ones place, 8 in the ones place, 3 in the ones place, 8 in the ones place. And it looks like that same kind of a number, a different number. The same number is repeating again, where there's kind of like a number and then another number. And then it repeats in that kind of double, like two numbers and then it repeats the same two numbers. Christy: So, what I would say in that one is try to revoice it, and I'd probably be gesturing, where I'd do this. But I'd say, "OK, so Mike's noticing in this ones place, in this first column, he's saying he notices it's '3, 8, 3, 8.' And then in other columns he's noticing that they do something similar. So the next column, or whatever, is like '6, 1, 6, 1' in the ones place. Why don't you give, again, give me a 'me too' [sign] if you also noticed that pattern or an 'open mind' [sign] if you didn't." So, that's what we would do. So, we would let people share some things. We would get a bunch of noticings while students are noticing those things. I would be, like I said, revoicing and annotating on the board. So typically I would revoice it and point it out with gestures, and then I would annotate that to take a record of this thing that they've noticed on the board. Once we've gotten several students' noticings on the board, then we're going to stop and we're going to unpack some of those. So I might do something like, "Oh, so Terry noticed this really interesting thing where he said that the last column increases by 15 because he saw 15, 30, 45, and he recognized that. I'm wondering if the other columns do something like that too. Do they also increase by the same kind of number? Hmm, why don't you take a minute and look at it and then turn and talk to your neighbor and see what you notice." And we're going to get them to notice then that these other ones also increase by 15. So if that hadn't already come out, I could use it as a press move to go in and unpack that one further. And then we would ask the question, in this case, "Why do they always increase by 15?" And we might then use that question and that conversation to go and talk about Mike's observation, and to say, like, "Huh, I wonder if we could use what we just noticed here to figure out about why this idea that [the numbers in the] ones places are going back and forth between 3, 8, 3, 8. I wonder if that has something to do with this." Right? So we might use them to unpack it. They'll notice these patterns. And while the students were talking about these things, I'd be taking opportunities to both orient them to each other with linking moves to say, "Hey, what do you notice? What can you add on to what Mike said, or could you revoice it?" And also to annotate those things to make them available for conversation. Mike: There was a lot in your description, Christy, and I think that provides a useful way to understand what's happening because there's the choice of numbers, there's the choice of how big the array is when you're recording initially, there are the moves that the teacher's making. What you've set up is a really cool conversation that comes forward. We did this with whole numbers just now, and I'm wondering if we could take a step forward and think about, OK, if we're imagining a choral count with fractions, what would that look and sound like? Christy: Yeah, so one of the ones I really like to do is to do these ones that are just straight multiples, like start at 3 and skip-count by 3s. And then to either that same day or the very next day—so very, very close in time in proximity—do one where we're going to do something similar but with fractions. So one of my favorites is for the parallel of the whole number of skip-counting by 3s is we'll start at 3 fourths and we'll skip-count by 3 fourths. And when we write those numbers, we're not going to put them in simplest form; we're just going to write 3 fourths, 6 fourths, 9 fourths. So in this case, I would probably set it up in the exact same very parallel structure to that other one that we just did with the whole numbers. And I would put the numbers 3 fourths, 6 fourths, 9 fourths on the board. I would say, "OK, here's our first numbers. We're going to start starting at 4 fourths. We're going to skip-count by 3 fourths. And give me a thumbs-up or the show me a 2 when you know the next two numbers." And then we would skip-count them together, and we would write them on the board. And so we'd end up—and in this case I would probably arrange them again in five columns just to have them and be a parallel structure to that one that we did before with the whole numbers. So it would look like 3 fourths, 6 fourths, 9 fourths, 12 fourths, 15 fourths on the first row. And then the next row, I would say 18 fourths, 21 fourths, 24 fourths, 27 fourths, 30 fourths. And again, I'd probably go all the way up until I got to 51 fourths before we'd stop and we'd look for patterns. Mike: So I think what's cool about that—it was unsaid, but it kind of implied—is that you're making a choice there. So that students had just had this experience where they were counting in increments of 3, and 3, 6, 9, 12, 15, and then you start another row and you get to 30, and in this case, 3 fourths, 6 fourths, 9 fourths, 12 fourths, 15 fourths. So they are likely to notice that there's something similar that's going on here. And I suspect that's on purpose. Christy: Right, that's precisely the thing that we want right here is to be able to say that fractions aren't something entirely new, something that you—just very different than anything that you've ever seen before in numbers. But to allow them to have an opportunity to really see the ways that numerators enumerate, they act like the counting numbers that they've always known, and the denominator names, and tells you what you're counting. And so it's just a nice space where, when they can see these in these parallel ways and experience counting with fractions, they have this opportunity to see some of the ways that both fraction notation works, what it's talking about, and also how the different parts of the fraction relate to things they already know with whole numbers. Mike: Well, let's dig into that a little bit more. So the question I was going to ask Terry was: Can we talk a bit more about the ways the choral counting routine can help students make sense of the mathematics of fractions? So what are some of the ideas or the features of fractions that you found choral counting really allows you to draw out and make sense of with students? Terry: Well, we know from our work with the rational number project how important language is when kids are developing an understanding of the role of the numerator and the denominator. And the choral counts really just show, like what Christy was just saying, how the numerator just enumerates and changes just like whole numbers. And then the denominator stays the same and names something. And so it's been a really good opportunity to develop language together as a class. Christy: Yeah. I think that something that's really important in these ones that you get to see when you have them. So when they're doing that language, they're also—a really important part of a choral count is that it's not just that they're hearing those things, they're also seeing the notation on the board. And because of the way that we're both making this choice to repeatedly add the same amount, right? So we're creating something that's going to have a pattern that's going to have some mathematical relationships we can really unpack. But they're also seeing the notation on there that's arranged in a very intentional way to allow them to see those patterns in rows and columns as they get to talk about them. So because those things are there, we're creating this chance now, right? So they see both the numerator and denominator. If we're doing them in parallel to things with whole numbers, they can see how both fractions are alike, things that they know with whole numbers, but also how some things are different. And instead of it being something that we're just telling them as rules, it invites them to make these observations. So in the example that I just gave you of the skip-counting, starting at 3 fourths and skip-counting by 3 fourths, every time I have done this, someone always observes that the right-hand column, they will always say it goes up by 15. And what they're observing right there is they're paying attention to the numerator and thinking, "Well, I don't really need to talk about the denominator," and it buys me this opportunity as a teacher to say, "Yes, I see that too. I see that these 15 fourths and then you get another, then you get 30 fourths and you get 45 fourths. And I see in those numerators that 15, 30, 45—just like we had with the whole numbers—and here's how I would write that as a mathematician: I would write 15 fourths plus 15 fourths equals 30 fourths." Because I'm trying to be clear about what I'm counting right now. So instead of telling it like it's a rule that you have to remember, you have to keep the same denominators when you're going to add, it instead becomes something where we get to talk about it. It's just something that we get to be clear about. And that in fractions, we also do this other piece where we both enumerate and we name, and we keep track of that when we write things down to be clear. And so it usually invites this very nice parallel conversation and opportunity just to set up the idea that when we're doing things like adding and thinking about them, that we're trying to be clear and we're trying to communicate something in the same way that we always have been. Mike: Well, Terry, it strikes me that this does set the foundation for some important things, correct? Terry: Yeah, it sets the foundation for adding and subtracting fractions and how that numerator counts things and the denominator tells you the size of the pieces. It also sets up multiplication. The last column, we can think of it as 5 groups of 3 fourths. And the next number underneath there might be 10 groups of 3 fourths. And as we start to describe or record what students' noticings are, we get a chance to highlight those features of adding fractions, subtracting fractions, multiplying fractions. Mike: We've played around the edges of a big idea here. And one of the things that I want to bring back is something we talked about when we were preparing for the interview. This idea that learners of any age, generally speaking, they want to make use of their understanding of the way that whole numbers work as they're learning about fractions. And I'm wondering if one or both of you want to say a little bit more about this. Terry: I think a mistake that we made previously in fraction teaching is we kind of stayed under 1. We just stayed and worked within 0 and 1 and we didn't go past it. And if you're going to make 1 a benchmark or 2 a benchmark or any whole number a benchmark, when you're counting by 3 fourths or 2 thirds or whatever, you have to go past it. So what choral counting has allowed us to do is to really get past these benchmarks, and kids saw patterns around those benchmarks, and they see them. And then I think we also saw a whole-number thinking get in the way. So if you ask, for example, somebody to compare 3 seventeenths and 3 twenty-thirds, they might say that 3 twenty-thirds are bigger because 23 is bigger than 17. And instead of embracing their whole-number knowledge, we kind of moved away from it. And so I think now with the choral counting, they're seeing that fractions behave like whole numbers. They can leverage that knowledge, and instead of trying to make it go away, they're using it as an asset. Mike: So the parallel that I'm drawing is, if you're trying to teach kids about the structure of numbers in whole number, if you can yourself to thinking about the whole numbers between 0 and 10, and you never worked in the teens or larger numbers, that structure's really hard to see. Am I thinking about that properly? Terry: Yes, you are. Christy: I think there's two things here to highlight. So one of them that I think Terry would say more about here is just the idea that, around the idea of benchmarks. So you're right that there's things that come out as the patterns and notation that happen because of how we write them. And when we're talking about place value notation, we really need to get into tens and really into hundreds before a lot of those things become really available to us as something we talk about, that structure of how 10 plays a special role. In fractions, a very parallel idea of these things that become friendly to us because of the notation and things we know, whole numbers act very much like that. When we're talking about rational numbers, right? So they become these nice benchmarks because they're really friendly to us, there's things that we know about them, so when we can get to them, they help us. And the choral count that we were just talking about, there's something that's a little bit different that's happening though because we're not highlighting the whole numbers in the way that we're choosing to count right there. So we're not—we're using those, I guess, improper fractions. In that case, what we're doing is we're allowing students to have an opportunity to play with this idea, the numerator and denominator or the numerator is the piece that's acting like whole numbers that they know. So when Terry was first talking about how oftentimes when we first teach fractions and we were thinking about them, we were think a lot about the denominator. The denominator is something that's new that we're putting in with fractions that we weren't ever doing before with whole numbers. And we have that denominator. We focus a lot on like, "Look, you could take a unit and you can cut it up and you can cut it up in eight pieces, and those are called eighths, or you could cut it up in 10 pieces, and those are called tenths." And we focus a lot on that because it's something that's new. But the thing that allows them to bridge from whole numbers is the thing that's the same as whole numbers. That's the numerator. And so when we want them to have chances to be able to make those connections back to the things they know and see that yes, there is something here that's new, it's the denominator, but connecting back to the things they know from whole numbers, we really do need to focus some on the numerator and letting them have a chance to play with what the numerator is, to see how it's acting, and to do things. It's not very interesting to say—to look at a bunch of things and say, like, "2 thirds plus 4 thirds equals 6 thirds," right? Because they'll just start to say, "Well, you can ignore the denominator." But when you play with it and counting and doing things like we was talking about—setting up a whole-number count and a fraction count in parallel to each other—now they get to notice things like that. [It] invites them to say things like, "Oh, so adding 15 in the whole numbers is kind of adding 15 fourths in the fourths." So they get to say this because you've kind of set it up as low-hanging fruit for them, but it's allowing them really to play with that notion of the numerator and a common denominator setting. And then later we can do other kinds of things that let them play with the denominator and what that means in those kinds of pieces. So one of the things I really like about choral counts and choral counts with fractions is it's setting up this space where the numerator becomes something that's interesting and something worth talking about in some way to be able to draw parallels and allow them to see it. And then of course, equivalency starts to come into play too. We can talk about how things like 12 fourths is equivalent to 3 wholes, and then we get to see where those play their role inside of this count too. But it's just something that I really like about choral counting with fractions that I think comes out here. And it's not quite the idea of benchmarks, but it is important. Mike: Well, let's talk a little bit about equivalency then. Terry. I'm wondering if you could say a little bit about how this routine can potentially set up a conversation around ideas related to equivalency. Terry: We could do this choral count—instead of just writing improper fractions all the way through, we could write them with mixed numbers. And as you start writing mixed numbers, the pattern becomes "3 fourths, 1 and a half, 2 and a quarter," and we can start bringing in equivalent fractions. And you still do the same five columns and make parallel connections between the whole numbers, the fractions that are written as improper fractions and the fractions with mixed numbers. And so you get many conversations about equivalencies. And this has happened almost every time I do a choral count with fractions is, the kids will comment that they stop thinking. They go, "I'm just writing these numbers down." Part of it is they're seeing equivalency, but they're also seeing patterns and letting the patterns take over for them. And we think that's a good thing rather than a bad thing. It's not that they're stopped thinking, they're just, they're just— Christy: They're experiencing the moment that patterns start to help, that pattern recognition starts to become an aid in their ability to make predictions. All of a sudden you can feel it kick online. So if you said it in the context, then what happens is even in the mixed-number version or in the improper-number version, that students will then have a way of talking about that 12 fourths is equivalent to 3, and then you're going to see that whole-number diagonal sort of pop in, and then you'll see those other ones, even in the original version of it. Terry: Yeah, as we started to play around with this and talk with people, we started using the context of sandwiches, fourths of sandwiches. And so when they would start looking at that, the sandwiches gave them language around wholes. So the equivalence that they saw, they had language to talk about. That's 12 fourths of a sandwich, which would be 3 full sandwiches. And then we started using paper strips with the choral counts and putting paper strips on each piece so kids could see that when it fills up they can see a full sandwich. And so we get both equivalencies, we get language, we get connections between images, symbols, and context. Mike: One of the questions that I've been asking folks is: At the broadest level, regardless of the number being counted or whether it's a whole number or a rational number, what do you think the choral counting routine is good for? Christy: So I would say that I think of these routines, like a choral count or a number talk or other routines like that that you would be doing frequently in a classroom, they really serve as a way of building mathematical language. So they serve as a language routine. And then one of the things that's really important about it is that it's not just that there's skip-counting, but that count. So you're hearing the way that patterns happen in language, but they're seeing it at the same time. And then they're having chances, once that static set of representations on the board, those visuals of the numbers has been created and set up in this structured way, it's allowing them to unpack those things. So they get to first engage in language and hearing it in this multimodal way. So they hear it and they see it, but then they get to unpack it and they get to engage in language in this other way where they get to say, "Well, here's things that stand out to me." So they make these observations and they will do it using informal language. And then it's buying the teacher an opportunity then to not only highlight that, but then to also help formalize that language. So they might say, "Oh, I saw a column goes up by 5." And I would get to say, "Oh, so you're saying that you add each time to this column, and here's how a mathematician would write that." And we would write that with those symbols. And so now they're getting chances to see how their ideas are mathematical ideas and they're being expressed using the language and tools of math. "Here's the way you said it; here's what your brain was thinking about. And here's what that looks like when a mathematician writes it." So they're getting this chance to see this very deeply authentic way and just also buying this opportunity not only to do it for yourself, but then to take up ideas of others. "Oh, who else saw this column?" Or, "Do you think that we could extend that? Do you think it's anywhere else?" And they get to then immediately pick up that language and practice it and try it. So I look at these as a really important opportunity, not just for building curiosity around mathematics, but for building language. Mike: Let's shift a little bit to teacher moves, to teacher practice, which I think y'all were kind of already doing there when you were talking about opportunities. What are some of the teacher moves that you think are really critical to bringing choral counting with fractions particularly to life? Terry: I think just using the strips to help them visualize it, and it gave them some language. I think the context of sandwiches, or whatever it happens to be, gives them some ways to name what the unit is. We found starting with that runway, it really helps to have something that they can start to kind of take off and start the counting routine. We also found that the move where you ask them, "What do you notice? What patterns do you notice?," we really reserve for three and a half rows. So we try to go three full rows and a half and it gives everybody a chance to see something. If I go and do it too quick, I find that I don't get everybody participating in that, noticing as well, as doing three and a half rows. It just seems to be a magic part of the array is about three and a half rows in. Mike: I want to restate and mark a couple things that you said, Terry. One is this notion of a runway that you want to give kids. And that functions as a way to help them start to think about, again, "What might come next?" And then I really wanted to pause and talk about this idea of, you want to go at least three rows, or at least—is it three or three and a half? Terry: Three and a half. Christy: When you have three of something, then you can start to use patterns. You need at least those three for even to think there could be a pattern. So when you get those, at least three of them, and they have that pattern to do—and like Terry was saying, when you have a partial row, then what happens is those predictions can come from two directions. You could keep going in the row, so you could keep going horizontally, or you could come down a column. And so now it kind of invites people to do things in more than one way when you stop mid-row. Mike: So let me ask a follow-up question. When a teacher stops or pauses the count, what are some of the first things you'd love to see them do to spark some of the pattern recognition or the pattern seeking that you just talked about? Christy: Teacher moves? Mike: Yeah. Christy: OK. So we do get to work with preservice teachers all the time. So this is one of my favorite parts of this piece of it. So what do you do as a teacher that you want? So we're going to want an array up there that has enough, at least three of things in some different ways people can start to see some patterns. You can also, when you do one of these counts, you'll hear the moment—what Terry described earlier as "stop thinking." You can hear a moment where people, it just gets easier to start, the pattern starts to help you find what comes next, and you'll hear it. The voices will get louder and more confident as you do it. So you want a little of that. Once you're into that kind of space, then you can stop. You know because you've just heard them get a little more confident that their brains are going. So you're kind of looking for that moment. Then you're going to stop in there again partway through a row so that you've got a little bit of runway in both directions. So they can keep going horizontally, they can come down vertically. And you say, "OK," and you're going to give them now a moment to think. And so that stopping for a second before they just talk, creating space for people to formulate some language, to notice some things is really, really important. So we're going to create some thinking space, but we know there's some thinking happening, so you just give them a way to do it. Our favorite way to do it is to, instead of just doing a thumbs-up and thumbs-down in front of the chest, we just do a silent count at the chest rather than hands going up. We just keep those hands out of the air, and I say, "Give me a 1 at your chest"—so a silent number 1 right at your chest—"when you've noticed one thing. And if you notice two things, give me a 2. And if you notice three things, give me a 3." They will absolutely extrapolate from there. And you'll definitely see some very anxious person who definitely wants to say something with a 10 at their chest. But what you're doing at that moment is you're buying people time to think, and you're buying yourself as a teacher some insight into where they are. So you now get to look out and you can see who's kind of taking a while for that 1 to come up and who has immediately five things, and other things. And you can use that along with your knowledge of the students now to think about how you want to bring people into that discussion. Somebody with 10 things, they do not need to be the first person you call on. They are desperate to share something, and they will share something no matter when you call on them. So you want to use this information now to be able to get yourself some ideas of, like, "OK, I want to make sure that I'm creating equitable experiences, that I want to bring a lot of voices in." And so the first thing we do is we have now a sense of that because we just watched, we gave ourselves away into some of the thinking that's happening. And then we're going to partner that immediately with a turn and talk. So first they're going to think and then they're going to have a chance to practice that language in a partnership. And then, again, you're buying yourself a chance to listen into those conversations and to know that they have something to share. And to bring it in, I will pretty much always make that a warm call. I won't say, "Who wants to share?" I will say, "Terry or Mike, let's hear." And then I won't just say, "Terry, what was your idea?" I would say, "Terry, tell me something that either you or Mike shared that you noticed." So we'll give a choice. So now they've got a couple ways in. You know they just said something. So you're creating this space where you're really lowering the temperature of how nerve-racking it is to share something. They have something to say, and they have something to do. So I want all of those moves. And then I kind of alluded to it when we were doing the practice one, but the other one I really like is to have all-class gestures so that everyone constantly has a way they need to engage and listen. And so I like to use ones not just the "me too" gesture, but we do the "open mind" gesture as well so that everyone has one of the two. Either it's something that you were thinking or they've just opened your mind to a new idea. And it looks, we use it kind of like an open book at your forehead. So, the best way I can describe it to you, you put both hands at your forehead and you touch them like they're opening up, opening doors. And so everyone does one of those, right? And then as a teacher, you now have some more information because you could say, "Oh, Terry, you just said that was open mind. You hadn't noticed it. Well, tell us something different you noticed." So you get that choice of what you're doing. So you're going to use these things as a teacher to not just get ideas out but to really be able to pull people in ways they've sort of communicated something to you that they have something to share. So I love it for all the ways we get to practice these teacher moves that don't just then work in just this choral count, but that do a really great job in all these other spaces that we want to work on with students too, in terms of equitably and creating talk, orienting students to one another, asking them to listen to and build on each other's ideas. Terry: When you first start doing this, you want to just stop and listen. So I think some of my mistakes early on was trying to annotate too quickly. But I found that a really good teacher move is just to listen. And I get to listen when they're think-pair-sharing, I get a chance to listen when they're just thinking together, I get a chance to listen when they describe it to the whole class. And then I get to think about how I'm going to write and record what they said so that it amplifies what they're saying to the whole class. And that's the annotation piece. And getting better at annotating is practicing what you're going to write first and then they always say something a little different than what you anticipate, but you've already practiced. So you can get your colors down, you can get how you're going to write it without overlapping too much with your annotations. Mike: I think that feels like a really important point for someone who is listening to the podcast and thinking about their own practice. Because if I examine my own places where I sometimes jump before I need to, it often is to take in some ideas but maybe not enough and then start to immediately annotate. And I'm really drawn to this idea that there's something to, I want to listen enough to kind of hear the body of ideas that are coming out of the group before I get to annotation. Is that a fair kind of summary of the piece that you think is really important about that? Terry: Yes. And as I'm getting better with it, I'm listening more and then writing after I think I know what they're saying. And I check with them as I'm writing. Mike: So you started to already go to my next question, which is about annotation. I heard you mention color, so I'm curious: What are some of the ideas about annotation that you think are particularly important when you are doing it in the context of a choral count? Christy: Well, yeah, I think a choral count. So color helps just to distinguish different ideas. So that's a useful tool for that piece of it. What we typically want, people will notice patterns usually in lines. And so you're going to get vertical lines and horizontal lines, but you'll also get diagonals. That's usually where those will be. And they will also notice things that are recognizable. So like the 15, 30, 45 being a number sequence that is a well-known one is typically wouldn't going to be the first one we notice. Another one that happens along a diagonal, and the examples we gave will be 12, 24, 36, it comes on a diagonal. People will often notice it because it's there. So then what you want is you're going to want to draw in those lines that help draw students' eyes, other students' eyes, not the ones who are seeing it, but the ones who weren't seeing it to that space so they can start to see that pattern too. So you're going to use a little bit of lines or underlining that sort of thing. These definitely do over time get messier and messier as you add more stuff to them. So color helps just distinguish some of those pieces. And then what you want is to leave yourself some room to write things. So if you have fractions, for example, you're going to need some space between things because fractions take up a little bit more room to write. And you definitely want to be able to write "plus 15 fourths," not just, "plus 15." And so you need to make sure you're leaving yourself enough room and practicing and thinking. You also have to leave enough room for if you want to continue the count, because one of the beautiful things you get to do here is to make predictions once you've noticed patterns. And so you're going to probably want to ask at some point, "Well, what number do you think comes in some box further down the road?" So you need to leave yourself enough room then to continue that count to get there. So it's just some of the things you have to kind of think about as a teacher as you do it, and then as you annotate, so you're kind of thinking about trying to keep [the numbers] pretty straight so that those lines are available to students and then maybe drawing them in so students can see them. And then probably off to the side writing things like, if there's addition or multiplication sentences that are coming out of it, you probably want to leave yourself some room to be able to sometimes write those. In a fraction one, which Terry talked about a little bit, because equivalency is something that's available now where we can talk about, for example, the really common one that would come out in our example would be that 12 fourths is equivalent to 3 wholes. Somehow you're going to have to ask this question of, "Well, why is that? Where could we see it?" And so in that case, usually we would draw the picture of the sandwiches, which will be rectangles all cut up in the same way. So not like grilled cheese sandwiches in fourth, but like a subway sandwich in fourths. And then you're going to need some space to be able to draw those above it and below it. So again, you're kind of thinking about what's going to make this visible to students in a way that's meaningful to them. So you're going to need some space to be left for those things. What I find is that I typically end up having to write some things, and then sometimes after the new idea comes in, I might have to erase a little bit of what's there to make some more room for the writing. But I would say with fractions, it's going to be important to think about leaving enough space between, because you're probably going to need a little bit of pictures sometimes to help make sense of that equivalency. That's a really useful one. And leaving enough space for the notation itself, it takes a little bit of room. Mike: Every time I do a podcast, I get to this point where I say to the guest or guests, "We could probably talk for an hour or more, and we're out of time." So I want to extend the offer that I often share with guests, which is if someone wanted to keep learning about choral counting or more generally about some of the ideas about fractions that we're talking about, are there any particular resources that the two of you would recommend? Terry: We started our work with the Choral Counting & Counting Collections book by Megan Franke[, Elham Kazemi, and Angela Chan Turrou], and it really is transformational, both routines. Christy: And it has fractions and decimals and ideas in it too. So you can see it across many things. Well, it's just, even just big numbers, small numbers, all kinds of different things. So teachers at different grade levels could use it. The Teacher Education by Design [website], at tedd.org, has a beautiful unit on counting collections for teachers. So if you're interested in learning more about it, it has videos, it has planning guides, things like that to really help you get started. Terry: And we found you just have to do them. And so as we just started to do them, writing it on paper was really helpful. And then The Math Learning Center has an app that you can use—the Number Chart app—and you can write [the choral counts] in so many different ways and check your timing out. And it's been a very helpful tool in preparing for quality choral counts with fractions and whole numbers. Mike: I think that's a great place to stop. Christy and Terry, I want to thank you both so much for joining us. It has really just absolutely been a pleasure chatting with you both. Christy: So much fun getting to talk to you. Terry: Thank you. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2025 The Math Learning Center | www.mathlearningcenter.org

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Pourquoi peut-on utiliser des fractions de billets de banque aux Etats-Unis ?

Choses à Savoir

Play Episode Listen Later Oct 15, 2025 2:02

À la fin du XIXᵉ siècle, les États-Unis ont connu une situation monétaire paradoxale : un pays riche… mais à court de petite monnaie. La guerre de Sécession (1861-1865) avait provoqué une pénurie de métaux précieux. Les Américains, inquiets, thésaurisaient leurs pièces d'or et d'argent. Résultat : plus de monnaie pour rendre la monnaie. Pour y remédier, le gouvernement eut une idée étonnante : imprimer des billets fractionnaires, des coupures de papier valant moins d'un dollar.Ces billets, officiellement appelés Fractional Currency, furent émis entre 1862 et 1876 par le Trésor américain. Ils remplaçaient temporairement les pièces métalliques devenues rares. Leur valeur allait de 3 à 50 cents, avec des coupures intermédiaires de 5, 10, 15 et 25 cents. Ils mesuraient à peine quelques centimètres — certains à peine plus grands qu'un timbre postal — et étaient imprimés sur un papier renforcé pour limiter la contrefaçon.L'idée venait du secrétaire au Trésor Salmon P. Chase, qui proposa ces billets pour faciliter le commerce quotidien. Sans eux, acheter un journal, un repas ou un billet de tramway devenait presque impossible. Les premières séries, surnommées Postage Currency, portaient même l'image de timbres-poste, pour rappeler leur petite valeur et encourager la confiance du public.Au fil des années, cinq séries différentes furent imprimées, avec des portraits de figures historiques américaines comme George Washington, Spencer Clark ou William Meredith. Mais leur petite taille et leur fragilité en firent aussi un cauchemar pour les utilisateurs : ils se froissaient, se déchiraient et se perdaient facilement.Lorsque la production de pièces reprit dans les années 1870, les billets fractionnaires furent retirés de la circulation. Mais juridiquement, ils n'ont jamais été démonétisés. Autrement dit, ils ont encore cours légal aujourd'hui — même si leur valeur réelle dépasse largement leur valeur faciale. Un billet de 25 cents peut valoir plusieurs centaines de dollars chez les collectionneurs.Ces billets racontent une page étonnante de l'histoire économique américaine : un moment où le pays dut remplacer le métal par du papier, et où chaque centime comptait. Symbole d'un pragmatisme typiquement américain, ils témoignent aussi de la confiance que les citoyens étaient prêts à accorder à une promesse imprimée : celle du Trésor des États-Unis. Hébergé par Acast. Visitez acast.com/privacy pour plus d'informations.

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Synthentral 20251014 New Tunesday (ep. 693)

Synthentral

Play Episode Listen Later Oct 13, 2025 80:59

It's New Tunesday: new releases from the past week! Give the bands a listen. If you like what you hear, support the bands! Today's episode features new releases by JXNS, , Lights Of Euphoria, Fractions, Error Enter Exit, Ein Sir, God Module, Massenhysterie, Miss Construction, Chem, Hocico, Potochkine, Kyd Barrett, Madeline Goldstein, Ghostbells, Calaverx, The Stave Church, Bats Is Dead, Nouveau Arcade, Ductape, Balduvian Bears, Culture Hex, and The City Gates!

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A Wrinkle in Time Travel

Play Episode Listen Later Oct 9, 2025 18:35

Episode Description: Max and Molly suspect that Mr. Avogadro's student book club may be a front for Mole recruitment. Using Math to calculate reading goals, averages, and even a new “Suspicion Number” system inspired by mathematician Paul Erdős, they try to narrow down suspects. But when they overhear Mr. A and Mr. Z talking about “burrows” and “alternate history,” the mystery deepens. Math Concepts: Division with remainders; Multiplication to find daily/weekly totals;; Application of averages;Fractions/percentages; Graph theory-inspired reasoningHistory/Geography Concepts: Misconceptions about animals (bats and moles' eyesight/echolocation); Introduction to mathematician Paul Erdős and the Erdős Number System; Literature reference: A Wrinkle in Time by Madeleine L'Engle

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SBRT for Metastatic Prostate Cancer with Dr. Ron Chen

The Dr. Geo Podcast

Play Episode Listen Later Oct 5, 2025 61:39

Can high-precision radiation change how we treat metastatic prostate cancer? In this episode, I'm joined by Ronald C. Chen, MD, MPH—radiation oncologist, national guideline author (AUA/ASCO), and clinical-trial leader with 170+ publications—to unpack stereotactic body radiation therapy (SBRT) for disease that has spread to lymph nodes, bones, and beyond. We get practical about who benefits, where SBRT shines, and how to balance treatment intensity with quality of life.SBRT offers highly focused, short-course radiation that can control limited (“oligo-”) metastatic prostate cancer and delay systemic therapy for many men. Dr. Chen explains when to treat individual nodes/bone lesions versus comprehensive nodal fields, how anatomy determines dose/fraction choices (often 3–5 treatments), and why modern SBRT sometimes reduces the need for concurrent hormone therapy. We cover salvage options after prior radiation (brachytherapy seeds, HIFU, cryo, repeat SBRT, or salvage prostatectomy), the role and limits of PSMA PET, fracture risk and bone health (DEXA), and the evolving data—including the large NRG-GU013 trial—for higher-risk disease. Throughout, we emphasize shared decision-making, realistic expectations, and considering clinical trials when data are evolving.00:00 – Can SBRT change metastatic prostate cancer care? Meet Dr. Ron Chen.01:00 – Disclaimer: Views are Dr. Geo's and guests'—independent of NYU Langone.07:00 – Recurrence scenarios: prostate-only, nodal, or bone/other; why catching early matters.12:00 – Five salvage options after prostate radiation: seeds (brachytherapy), HIFU, cryo, SBRT (focal or whole-gland), or salvage prostatectomy.19:00 – Nodal relapse: treat all pelvic nodes + ADT ± abiraterone vs. SBRT to a few nodes only—how patient priorities drive the plan.26:30 – Oligometastasis: SBRT alone can control disease for many men ~2+ years on average, delaying hormones.30:00 – Fractions: why 3–5 treatments is typical and how adjacent bowel/organ anatomy sets the pace.31:00 – SBRT in 2 fractions for select primary cases looks promising; high-risk SBRT under study (NRG-GU013).37:00 – Bone mets: SBRT preferred; understanding fracture risk (tumor size, dose, shrinkage).40:00 – DEXA before ADT; spine SBRT can spare the spinal cord with modern planning.48:00 – Clavicle/hilar nodes: SBRT near lung/heart/esophagus—safe with careful dose constraints.56:00 – Why clinical trials matter for “how long on hormones?” and other open questions.57:00 – Soft-tissue mets (liver/brain): SBRT can help, often alongside systemic therapy.59:00 – Parting advice: early detection, close follow-up, and hopeful trajectory of care.___________________________________

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Colloque - Edward Hubbard : Illuminating Fractions Learning: Neuronal Recycling of Non-Symbolic Ratios for Symbolic Fractions

Colloques du CollÃ¨ge de France - CollÃ¨ge de France

Play Episode Listen Later Oct 2, 2025 39:07

Stanislas DehaeneChaire Psychologie cognitive expérimentaleAnnée 2025-2026Collège de FranceColloque : Seeing the Mind, Educating the BrainTheme: Numerical and Mathematical DevelopmentIlluminating Fractions Learning: Neuronal Recycling of Non-Symbolic Ratios for Symbolic FractionsColloque - Edward Hubbard : Illuminating Fractions Learning: Neuronal Recycling of Non-Symbolic Ratios for Symbolic FractionsEdward HubbardRésuméWithin mathematics, fractions hold a special place. They present perennial difficulties to students, and yet, mastering fractions is a critical stepping stone towards algebra and higher-order mathematics. More than 20 years ago, Stanislas Dehaene suggested that fractions are difficult because they lack the intuitive perceptual foundation that permits us to readily comprehend whole numbers and instead may depend on formal and symbolic processes. Here, I will present research from my lab showing that fractions may indeed have a perceptual foundation, and that this perceptual foundation may be recycled to allow us to understand symbolic fractions. Behaviorally, we have shown that symbolic fractions do not need to be processed componentially and instead can be represented on a coherent mental number. We show that wholistic fraction comparisons (and translation to decimals) does not require time consuming computations, and that non-symbolic ratio perception in college students and American elementary school children predicts formal fractions skills. Using fMRI, we have further shown that non-symbolic ratio perception reliable recruits right parietal cortex, even before the onset of formal schooling, and these parietal systems become tuned to symbolic fractions after as little as two years of formal education. Despite this evidence that fractions do, indeed, have a perceptual foundation, they still present significant difficulties. I will close by arguing that fractions (and other domains) may be difficult not due to a lack of foundational systems, but rather, due to educational methods that fail to align with these perceptual foundations. Furthermore, I will argue that research in numerical cognition can (and should!) provide new pedagogical approaches that better align with the foundational systems we have discovered to help students better grasp higher-order mathematical concepts.

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Molecraft!

Mysteries About True Histories (M.A.T.H.)

Play Episode Listen Later Sep 25, 2025 17:47

Episode Description: Max and Molly attempt to infiltrate the POGs by applying for a mysterious “Mole job” that sends them 2 million years back to Olduvai Gorge. There, they discover Homo habilis, early human tools, and the dangers of altering history—better known as “The Butterfly Effect.” Using Math to calculate volume, they dig a hole to bury a strange box. But could this simple action end up changing the future forever? Math Concepts: Volume formula: length × width × height (to determine hole depth);Large number comparisons (quadrillions vs trillions); Fractions, decimals, and multiplication; Time conversion (millions of years → human timeline).History/Geography Concepts: Olduvai Gorge in Tanzania as a key anthropological site; Discovery of early human fossils (Mary Leakey, Homo habilis). The Butterfly Effect in relation to history and consequences. Fossilization and preservation conditions (lava and ash deposits).

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GETCHO | FRACTIONS AS TV SITCOMS | WINS & LS OF WRESTLING WEEK

GET CHO! Podcast

Play Episode Listen Later Sep 4, 2025 93:00

Welcome back The Getcho Crew is joined by Katie | SHElite Showcase and Kyle | Apron Bump to give our take on Fractions as Tv Sitcoms Kyle Apron Bump link https://linktr.ee/ApronBump?utm_source=linktree_profile_share<sid=63ec10e3-3fa9-4143-89d0-67f7c18e0775Katie SHELITE SHOWCASEhttps://linktr.ee/ShEliteShowcase?utm_source=linktree_profile_share<sid=a4d3c170-4459-47b0-bebc-27440a6d2c0b

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Rendez-vous avec la maîtresse : Additionner les fractions sans mélanger les torchons et les serviettes

Un bonbon sur la langue

Play Episode Listen Later Aug 30, 2025 3:31

Dans cet épisode, Lisa Kamen et Christophe discutent de l'importance de ne pas mélanger les unités, illustrée par l'addition des fractions. Lisa explique comment trouver un dénominateur commun pour additionner des fractions, tout en utilisant des métaphores amusantes comme les torchons et les serviettes. L'épisode se termine par une question ludique sur l'orthographe des homophones "sceau".Hébergé par Audiomeans. Visitez audiomeans.fr/politique-de-confidentialite pour plus d'informations.

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Fractions

Tales in Two Minutes- Jay Stetzer, Storyteller

Play Episode Listen Later Aug 22, 2025 2:38

Davis just hung his head and mumbled, “Fractions.”

fractions

The Engines of Our Ingenuity 2493: Child’s Play

Engines of Our Ingenuity

Play Episode Listen Later Jul 24, 2025 3:48

Episode: 2493 Child's Play: the role of play in education. Today, child's play.

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A New Old School (Part 1)

Mysteries About True Histories (M.A.T.H.)

Play Episode Listen Later Jul 17, 2025 20:14

Episode Description: Max and Molly are shocked to discover their school has mysteriously time-warped back to 1970—and so has Aunt Murgatroyd, who's now a teenager again. As they navigate outdated vending machines, retro slang, and payphones, they realize something more sinister may be at play. For one thing, there is a rocket emergency in outer space – while closer to home, the POGs seem to be targeting a young Aunt M – and Charlene may be helping them! Math Concepts: Elapsed time calculation (i.e. - 2025 minus 1970 = 55 years); Basic probability concepts; Fractions and percentages; Pattern recognition; Estimation and coin math. History/Geography Concepts: The Apollo 13 crisis – a real-world 1970 event involving NASA's near-catastrophic moon mission. Cultural context of 1970s America: decor, fashion, technology, and retro candy

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Fine Fractions & Big Discoveries: A Deep Dive into Drainage Geochem with Mary Doherty

GeocHemiSTea

Play Episode Listen Later Jul 9, 2025 53:27

This month on GeOCHemISTea, we are talking about one of exploration geochemistry's most foundational, and rapidly evolving, methods: stream sediment sampling. Sam is joined by geochemical consultant and educator Mary Doherty, lead author of a 2023 review on stream sediment geochemistry, published in Geochemistry: Exploration, Environment, Analysis.Mary shares a career's worth of field, lab, and leadership experience, from carrying samples for her USGS geologist father as a child to building geochem training programs for Newmont and ALS, and recently teaching at the Colorado School of Mines. We talk about key developments in stream sediment methods, from fine fraction and BLEG to HMC and indicator mineral chemistry, as well as how to choose the right tool for the job.The episode also covers the practicalities of QA/QC, field planning, and how geomorphology and hydrology shape interpretation. Mary emphasizes the importance of training, knowledge-sharing, and the growing integration of mineralogical data, and machine learning into modern workflows.Whether you are designing your first drainage survey or revisiting legacy data, this episode is a reminder that good geochemistry begins with solid fundamentals and that the future of exploration is already here.For this episode we read: Stream sediment geochemistry in mineral exploration: a review of fine-fraction, clay-fraction, bulk leach gold, heavy mineral concentrate and indicator mineral chemistry (Doherty et al., 2023)

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Spring Nearly Broke You? Here's Why That's a Good Sign

Staging Sips

Play Episode Listen Later Jun 14, 2025 17:08

Have things felt… messy lately? You're booked out, your client list is growing, and you should be celebrating—but instead, you're overwhelmed, behind on admin, and wondering how long you can keep this up. If that's you, hit pause on the panic because this episode is your permission slip to see that chaos as progress. In today's episode, we're unpacking what most business owners never talk about: how growth often looks like disorganization before it looks like success. I'll show you why a chaotic season, especially in spring, when staging demand spikes—isn't a failure of planning. It's a visibility milestone. It means your business is asking you to lead differently. This episode is part pep talk, part strategic wake-up call. You'll leave with tangible ways to simplify your workflows, spot the systems that are breaking under pressure, and take back control of your time as the CEO, not just the doer. WHAT YOU'LL LEARN FROM THIS EPISODE: Why operational friction means you're doing something right How to spot and fix decision-making bottlenecks in your staging business The “Visibility Threshold” concept—and how to know you've hit it A 3-step mini assignment to immediately regain clarity and control RESOURCES: Apply for Private Coaching: www.rethinkhomeinteriors.com/privatecoachingapp Enroll in Staging Business School Accelerate Track: www.rethinkhomeinteriors.com/accelerate Join the Staging Business School Growth Track Waitlist: www.rethinkhomeinteriors.com/growth Follow the Staging Business School on Instagram: www.instagram.com/stagingbusinessschool Follow Lori on Instagram: www.instagram.com/rethinkhome CEO Fractions of Actions List If you want to learn how to streamline your operations so you can grow with less stress and burnout in your staging business, enrollment is open for Staging Business School Accelerate Track. I'd love to see you in the classroom! ENJOY THE SHOW? Leave a 5-star review on Apple Podcasts so that more Staging CEOs find it. Also, include links to your socials so that more Staging CEOs can find you. Follow over on Spotify, Stitcher, Amazon Music, or Audible.

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551. The Math Mindset and how to be Math-ish feat. Jo Boaler

unSILOed with Greg LaBlanc

Play Episode Listen Later Jun 9, 2025 51:35

What is the role of active versus passive learning for math? How would data science become an avenue of math study for high school students and why isn't it already? Where does change in math education start? At the college level or before?Jo Boaler is a professor of mathematics education at Stanford University and also the author of a number of books, including Math-ish: Finding Creativity, Diversity, and Meaning in Mathematics, Limitless Mind: Learn, Lead, and Live Without Barriers, and Mathematical Mindsets: Unleashing Students' Potential Through Creative Math, Inspiring Messages and Innovative Teaching.Greg and Jo discuss creativity, diversity, and meaning in math education. Their conversation identifies certain flaws in current math teaching methods, the resistance to educational change, and the importance of metacognition, visual learning, and collaborative problem-solving. Jo shares insights from her journey as a math educator, including her experiences with educational reform and the implications of neuroscience on learning math. They also examine the role of active versus passive learning, the potential of data science in education, and the impact of AI on future teaching practices.*unSILOed Podcast is produced by University FM.*Episode Quotes:How conjectures ignite mathematical thinking17:00: When we ask kids to reason about maths and to come up with their own conjectures, we like to share that word with kids. This is a word that all mathematicians use—a conjecture for an idea they have that you need to test out. It's like a hypothesis in science, but kids have never heard of that word, which is, you know, means there's a reason for that. But anyway, we teach our kids to come up with conjectures and then to reason about them and prove it to each other. And they get these great discussions where they're reasoning and being skeptical with each other. And that's what sparks their interest. They actually feel like they're discovering new things. And it's, like, really engaging for the kids to get into these discussions about the meanings of why these things work in maths. So it's a great route in, not only to engage kids, but have them understand what they're doing. Yeah, it's not that common.Why every kid should learn data science31:02: Data science is really something all kids should be learning in school, before they leave school, and developing a data literacy and a comfort with data and being able to read and analyze data, to some extent, is an important life skill. And it probably is really important to say, if a democracy, as a lot of misinformation is shared now, and if kids aren't leaving able to make sense of and separate fact and fiction, they will be left vulnerable to those misinformation campaigns. So, it's important just to be an everyday citizen.Why estimation is really important34:48: The idea of Math-ish is, estimation is really important. There's a lot of research evidence that we should be getting kids to estimate, but I know that kids in schools hate to estimate, and they resist it, and they will work things out precisely and round them up to make them look like an estimate. But you ask them, what's your ish number? And something magical happens. Like, suddenly they're willing to share their thinking, but it doesn't happen enough.The problem with teaching everything every year14:28: In the US, we have this system of teaching everything every year. So, you start learning fractions in maybe grade three, but you also learn them again in grade four and grade five and grade six. And at the end of that, kids don't understand fractions and everything else. Everything is taught every year. Whereas if you look at very successful countries like Japan, they don't teach in that way. Fractions is taught in one year—one year group—deeply, well, conceptually. So this is why you see kids going around in these massive textbooks that they can hardly carry, because it has all this content. And, of course, when you try and teach everything every year, often kids don't learn any of it well.Show Links:Recommended Resources:Randomized Controlled TrialMetacognitionCompression as a unifying principle in human learningCarol DweckGuest Profile:Faculty Profile at Stanford GSEProfile on WikipediaYouCubedSocial Profile on InstagramSocial Profile on XHer Work:Amazon Author PageMath-ish: Finding Creativity, Diversity, and Meaning in MathematicsLimitless Mind: Learn, Lead, and Live Without BarriersWhat's Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire SuccessData Minds: How Today's Teachers Can Prepare Students for Tomorrow's WorldMathematical Mindsets: Unleashing Students' Potential Through Creative Math, Inspiring Messages and Innovative Teaching

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Three (3) Critical Math Topics To Avoid Rushing To The Algorithm

Play Episode Listen Later May 12, 2025 19:14

In the push to cover content and keep pace, it's easy to jump straight to the algorithm. But what if that focus on efficiency is doing more harm than good? In this episode, we explore how rushing to procedures can rob students of opportunities to reason, make connections, and develop true mathematical understanding in three(3) critical math topics. Together, we'll unpack what's really at stake—and how a shift in mindset can help you cultivate curious, confident math thinkers. Slowing down might just be the most powerful move you make.Key Takeaways:Why students fall back on tricks—and what to do instead.The impact of jumping to algorithms too soon.How to create space for reasoning and connection.Practical strategies for building conceptual understanding.Shifting your math block from procedural to purposeful.Not sure what matters most when designing math improvement plans? Take this assessment and get a free customized report: https://makemathmoments.com/grow/ Math coordinators and leaders – Ready to design your math improvement plan with guidance, support and using structure? Learn how to follow our 4 stage process. https://growyourmathprogram.com Looking to supplement your curriculum with problem based lessons and units? Make Math Moments Problem Based Lessons & UnitsShow NotesLove the show? Text us your big takeaway!Get a Customized Math Improvement Plan For Your District.Are you district leader for mathematics? Take the 12 minute assessment and you'll get a free, customized improvement plan to shape and grow the 6 parts of any strong mathematics program.Take the assessmentAre you wondering how to create K-12 math lesson plans that leave students so engaged they don't want to stop exploring your math curriculum when the bell rings? In their podcast, Kyle Pearce and Jon Orr—founders of MakeMathMoments.com—share over 19 years of experience inspiring K-12 math students, teachers, and district leaders with effective math activities, engaging resources, and innovative math leadership strategies. Through a 6-step framework, they guide K-12 classroom teachers and district math coordinators on building a strong, balanced math program that grows student and teacher impact. Each week, gain fresh ideas, feedback, and practical strategies to feel more confident and motivate students to see the beauty in math. Start making math moments today by listening to Episode #139: "Making Math Moments From Day 1 to 180.

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Instructional Recipes For Teaching Math

Play Episode Listen Later Mar 20, 2025 39:10

Not sure what matters most when designing math improvement plans? Take this assessment and get a free customized report: https://makemathmoments.com/grow/ Math coordinators and leaders – Ready to design your math improvement plan with guidance, support and using structure? Learn how to follow our 4 stage process. https://growyourmathprogram.com Looking to supplement your curriculum with problem based lessons and units? Make Math Moments Problem Based Lessons & UnitsHave you ever wished for a simple yet effective way to improve math instruction without overwhelming teachers?Many educators struggle with making math lessons engaging, equitable, and effective. Without clear guidance, teaching methods can vary widely, leading to inconsistent student experiences. But what if there was a structured, research-backed approach that empowers teachers while ensuring high-quality instruction for all students?You'll learn: Discover how instructional recipes provide clear, research-based strategies that simplify lesson planning while enhancing student engagement.Learn how small, high-leverage instructional changes can lead to significant improvements in student understanding and classroom equity.Gain insights into practical teaching techniques, including effective task launches, student discourse strategies, and how to provide hints and extensions without lowering cognitive demand.Tune in now to explore how instructional recipes can transform your math teaching approach—giving both you and your students a more rewarding experience!Show NotesLove the show? Text us your big takeaway!Get a Customized Math Improvement Plan For Your District.Are you district leader for mathematics? Take the 12 minute assessment and you'll get a free, customized improvement plan to shape and grow the 6 parts of any strong mathematics program.Take the assessmentAre you wondering how to create K-12 math lesson plans that leave students so engaged they don't want to stop exploring your math curriculum when the bell rings? In their podcast, Kyle Pearce and Jon Orr—founders of MakeMathMoments.com—share over 19 years of experience inspiring K-12 math students, teachers, and district leaders with effective math activities, engaging resources, and innovative math leadership strategies. Through a 6-step framework, they guide K-12 classroom teachers and district math coordinators on building a strong, balanced math program that grows student and teacher impact. Each week, gain fresh ideas, feedback, and practical strategies to feel more confident and motivate students to see the beauty in math. Start making math moments today by listening to Episode #139: "Making Math Moments From Day 1 to 180.

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The Key to Understanding Fractions and Algebraic Thinking

The Teaching Toolbox - A Podcast for Middle School Teachers

Play Episode Listen Later Mar 17, 2025 25:53

Not sure what matters most when designing math improvement plans? Take this assessment and get a free customized report: https://makemathmoments.com/grow/ Math coordinators and leaders – Ready to design your math improvement plan with guidance, support and using structure? Learn how to follow our 4 stage process. https://growyourmathprogram.com Looking to supplement your curriculum with problem based lessons and units? Make Math Moments Problem Based Lessons & Units comparison is a crucial yet often overlooked concept in elementary mathematics. Many students in grades 4-6 struggle with fractions and multiplication, while those in grades 7-8 need a strong foundation to think algebraically. In this episode, we explore how understanding multiplicative comparison can unlock deeper mathematical reasoning and support students' progression. When should we introduce it? How do we leverage it effectively? Join us as we break it down with real-world examples!Key Takeaways:Understanding how it differs from additive comparison.A bridge between multiplication, fractions, and algebraic thinking.How a strong grasp of multiplicative comparison supports algebraic reasoning.Key moments to reinforce the concept in elementary math.Practical ways to help students develop this understanding through rich tasks and discussion.Show NotesLove the show? Text us your big takeaway!Get a Customized Math Improvement Plan For Your District.Are you district leader for mathematics? Take the 12 minute assessment and you'll get a free, customized improvement plan to shape and grow the 6 parts of any strong mathematics program.Take the assessmentAre you wondering how to create K-12 math lesson plans that leave students so engaged they don't want to stop exploring your math curriculum when the bell rings? In their podcast, Kyle Pearce and Jon Orr—founders of MakeMathMoments.com—share over 19 years of experience inspiring K-12 math students, teachers, and district leaders with effective math activities, engaging resources, and innovative math leadership strategies. Through a 6-step framework, they guide K-12 classroom teachers and district math coordinators on building a strong, balanced math program that grows student and teacher impact. Each week, gain fresh ideas, feedback, and practical strategies to feel more confident and motivate students to see the beauty in math. Start making math moments today by listening to Episode #139: "Making Math Moments From Day 1 to 180.

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73. Ellie's Passion for Fractions Taught Right

Play Episode Listen Later Feb 25, 2025 13:52 Transcription Available

Fraction operations can be one of the most challenging topics to teach. In today's episode Ellie shares about her journey in supporting teachers in the area of fractions.ResourcesEllie's Fraction Course - https://cognitive-cardio-math.thinkific.com/courses/teaching-fraction-operationsBlog Post - https://cognitivecardiomath.com/cognitive-cardio-blog/why-not-the-butterfly-method-when-adding-and-subtracting-fractions/Please subscribe on your favorite platform so you don't miss an episode. Whether it's Spotify, Apple Podcasts, Google Podcasts, or some other listening app, we encourage you to take a moment to subscribe to The Teaching Toolbox. And if you feel so inclined, we would love a review at Apple or Spotify to help other listeners find us just like you did.This episode may contain affiliate links.Amazon links are affiliate links from Brittany Naujok and The Colorado Classroom, LLC®. I earn a small amount from your clicks on these links.Let's ConnectTo stay up to date with episodes, check out our Facebook page or follow us on Instagram.Join Brittany's 6th Grade Teacher Success group on Facebook.Join Ellie's Middle School Math Chats group on Facebook.Brittany's resources can be found on her website or on TPT.Ellie's resources can be found on her website or on TPT.Reach out to share your ideas for future episodes on our podcast website.https://teachingtoolboxpodcast.com/contact/

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Practice with 5-syllable words.

Wadjasay? American English Pronunciation Practice

Play Episode Listen Later Jan 30, 2025 49:41

Send me a text message. Suggestions? Subjects for future podcasts? Let me know--thanks!1- I have never seen or met an abominable snowman, also known as a Yeti.2- I.R.S. is the abbreviation for the Internal Revenue Service.3- My uncle is an aeronautical engineer.4- I put the books in alphabetical order. 5- My son doesn't like to bathe, and as a result, dirt is accumulating in his ears.6- The doctor told me to take an antidepressant, but I was too depressed to follow his advice. 7- She lives approximately seven kilometers from the office where she works.8- My uncle does biomedical research. 9- In the cafeteria I sat next to a cardiologist. 10- My fiancee is a computational engineer in New York.11- He communicates his curiosity about criminology at every opportunity. 12- Have you ever heard of the cosmological constant?13- Fractions have a numerator and a denominator. 14- Dermatologists study dermatology. (But perhaps their interest is only skin deep…).15- She was disinterested in the documentary about snails.16- Mr. Johnson is working on a project involving the eradication of mosquitoes. 17- The doctor gave me an exceptionally thorough examination.18- The police erroneously accused me of stealing a double decker bus.19- I am trying to achieve a state of emotional equilibrium. 20- My allergies were exacerbated by a bumper crop of pollen.Intro & Outro Music: La Pompe Du Trompe by Shane Ivers - https://www.silvermansound.com Support the showEmail me: swift.water3883@fastmail.comYou can now support my podcasts and classes:Help Barry pay for podcast expenses--thank you!

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Season 3 | Episode 04 - Making Sense of Unitizing: The Theme That Runs Through Elementary Mathematics - Guest: Beth Hulbert

Play Episode Listen Later Oct 24, 2024 30:27

Rounding Up Season 3 | Episode 4 – Making Sense of Unitizing: The Theme That Runs Through Elementary Mathematics Guest: Beth Hulbert Mike Wallus: During their elementary years, students grapple with many topics that involve relationships between different units. This concept, called “unitizing,” serves as a foundation for much of the mathematics that students encounter during their elementary years. Today, we're talking with Beth Hulbert from the Ongoing Assessment Project (OGAP) about the ways educators can encourage unitizing in their classrooms. Welcome to the podcast, Beth. We are really excited to talk with you today. Beth Hulbert: Thanks. I'm really excited to be here. Mike: I'm wondering if we can start with a fairly basic question: Can you explain OGAP and the mission of the organization? Beth: Sure. So, OGAP stands for the Ongoing Assessment Project, and it started with a grant from the National Science Foundation to develop tools and resources for teachers to use in their classroom during math that were formative in nature. And we began with fractions. And the primary goal was to read, distill, and make the research accessible to classroom teachers, and at the same time develop tools and strategies that we could share with teachers that they could use to enhance whatever math program materials they were using. Essentially, we started by developing materials, but it turned into professional development because we realized teachers didn't have a lot of opportunity to think deeply about the content at the level they teach. The more we dug into that content, the more it became clear to us that content was complicated. It was complicated to understand, it was complicated to teach, and it was complicated to learn. So, we started with fractions, and we expanded to do work in multiplicative reasoning and then additive reasoning and proportional reasoning. And those cover the vast majority of the critical content in K–8. And our professional development is really focused on helping teachers understand how to use formative assessment effectively in their classroom. But also, our other goals are to give teachers a deep understanding of the content and an understanding of the math ed research, and then some support and strategies for using whatever program materials they want to use. And we say all the time that we're a program blind—we don't have any skin in the game about what program people are using. We are more interested in making people really effective users of their math program. Mike: I want to ask a quick follow-up to that. When you think about the lived experience that educators have when they go through OGAP's training, what are the features that you think have an impact on teachers when they go back into their classrooms? Beth: Well, we have learning progressions in each of those four content strands. And learning progressions are maps of how students acquire the concepts related to, say, multiplicative reasoning or additive reasoning. And we use those to sort, analyze, and decide how we're going to respond to evidence in student work. They're really maps for equity and access, and they help teachers understand that there are multiple right ways to do some mathematics, but they're not all equal in efficiency and sophistication. Another piece they take away of significant value is we have an item bank full of hundreds of short tasks that are meant to add value to, say, a lesson you taught in your math program. So, you teach a lesson, and you decide what is the primary goal of this lesson. And we all know no matter what the program is you're using that every lesson has multiple goals, and they're all in varying degrees of importance. So partly, picking an item in our item bank is about helping yourself think about what was the most critical piece of that lesson that I want to know about that's critical for my students to understand for success tomorrow. Mike: So, one big idea that runs through your work with teachers is this concept called “unitizing.” And it struck me that whether we're talking about addition, subtraction, multiplication, fractions, that this idea just keeps coming back and keeps coming up. I'm wondering if you could offer a brief definition of unitizing for folks who may not have heard that term before. Beth: Sure. It became really clear as we read the research and thought about where the struggles kids have, that unitizing is at the core of a lot of struggles that students have. So, unitizing is the ability to call something 1, say, but know it's worth maybe 1 or 100 or a 1,000, or even one-tenth. So, think about your numbers in a place value system. In our base 10 system, 1 of 1 is in the tenths place. It's not worth 1 anymore, it's worth 1 of 10. And so that idea that the 1 isn't the value of its face value, but it's the value of its place in that system. So, base 10 is one of the first big ways that kids have to understand unitizing. Another kind of unitizing would be money. Money's a really nice example of unitizing. So, I can see one thing, it's called a nickel, but it's worth 5. And I can see one thing that's smaller, and it's called a dime, and it's worth 10. And so, the idea that 1 would be worth 5 and 1 would be worth 10, that's unitizing. And it's an abstract idea, but it provides the foundation for pretty much everything kids are going to learn from first grade on. And when you hear that kids are struggling, say, in third and fourth grade, I promise you that one of their fundamental struggles is a unitizing struggle. Mike: Well, let's start where you all started when you began this work in OGAP. Let's start with multiplication. Can you talk a little bit about how this notion of unitizing plays out in the context of multiplication? Beth: Sure. In multiplication, one of the first ways you think about unitizing is, say, in the example of 3 times 4. One of those numbers is a unit or a composite unit, and the other number is how many times you copy or iterate that unit. So, your composite unit in that case could be 3, and you're going to repeat or iterate it four times. Or your composite unit could be 4, and you're going to repeat or iterate it three times. When I was in school, the teacher wrote 3 times 4 up on the board and she said, “Three tells you how many groups you have, and 4 tells you how many you put in each group.” But if you think about the process you go through when you draw that in that definition, you draw 1, 2, 3 circles, then you go 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4. And in creating that model, you never once thought about a unit, you thought about single items in a group. So, you counted 1, 2, 3, 4, three times, and there was never really any thought about the unit. In a composite unit way of thinking about it, you would say, “I have a composite unit of 3, and I'm going to replicate it four times.” And in that case, every time, say, you stamped that—you had this stamp that was 3—every time you stamped it, that one action would mean 3, right? One to 3, 1 to 3, 1 to 3, 1 to 3. So, in really early number work, kids think 1 to 1. When little kids are counting a small quantity, they'll count 1, 2, 3, 4. But what we want them to think about in multiplication is a many-to-1 action. When each of those quantities happens, it's not one thing, even though you make one action, it's four things or three things, depending upon what your unit is. If you needed 3 times 8, you could take your 3 times 4 and add 4 more, 3 times 4s to that. So, you have your four 3s and now you need four more 3s. And that allows you to use a fact to get a fact you don't know because you've got that unit and that understanding that it's not by 1, but by a unit. When gets to larger multiplication, we don't really want to be working by drawing by 1s, and we don't even want to be stamping 27 19 times. But it's a first step into multiplication. This idea that you have a composite unit, and in the case of 3 times 4 and 3 times 7, seeing that 3 is common. So, there's your common composite unit. You needed four of them for 3 times 4, and you need seven of them for 3 times 7. So, it allows you to see those relationships, which if you look at the standards, the relationships are the glue. So, it's not enough to memorize your multiplication facts. If you don't have a strong relationship understanding there, it does fall short of a depth of understanding. Mike: I think it was interesting to hear you talk about that, Beth, because one of the things that struck me is some of the language that you used, and I was comparing it in my head to some of the language that I've used in the past. So, I know I've talked about 3 times 4, but I thought it was really interesting how you used iterations of or duplicated … Beth: Copies. Mike: … or copies, right? What you make me think is that those language choices are a little bit clearer. I can visualize them in a way that 3 times 4 is a little bit more abstract or obscure. I may be thinking of that wrong, but I'm curious how you think the language that you use when you're trying to get kids to think about composite numbers matters. Beth: Well, I'll say this, that when you draw your 3 circles and count 4 dots in each circle, the result is the same model than if you thought of it as a unit of 3 stamped four times. In the end, the model looks the same, but the physical and mental process you went through is significantly different. So, you thought when you drew every dot, you were thinking about 1, 1, 1, 1, 1, 1, 1. When you thought about your composite unit copied or iterated, you thought about this unit being repeated over and over. And that changes the way you're even thinking about what those numbers mean. And one of those big, significant things that makes addition different than multiplication when you look at equations is, in addition, those numbers mean the same thing. You have 3 things, and you have 4 things, and you're going to put them together. If you had 3 plus 4, and you changed that 4 to a 5, you're going to change one of your quantities by 1, impacting your answer by 1. In multiplication, if you had 3 times 4, and you change that 4 to a 5, your factor increases by 1, but your product increases by the value of your composite unit. So, it's a change of the other factor. And that is significant change in how you think about multiplication, and it allows you to pave the way, essentially, to proportional reasoning, which is that replicating your unit. Mike: One of the things I'd appreciated about what you said was it's a change in how you're thinking. Because when I think back to Mike Wallus, classroom teacher, I don't know that I understood that as my work. What I thought of my work at that point in time was I need to teach kids how to use an algorithm or how to get an answer. But I think where you're really leading is we really need to be attending to, “What's the thinking that underlies whatever is happening?” Beth: Yes. And that's what our work is all about, is how do you give teachers a sort of lens into or a look into how kids are thinking and how that impacts whether they can employ more efficient and sophisticated relationships and strategies in their thinking. And it's not enough to know your multiplication facts. And the research is pretty clear on the fact that memorizing is difficult. If you're memorizing a hundred single facts just by memory, the likelihood you're not going to remember some is high. But if you understand the relationship between those numbers, then you can use your 3 times 4 to get your 3 times 5 or your 3 times 8. So, the language that you use is important, and the way you leave kids thinking about something is important. And this idea of the composite unit, it's thematic, right? It goes through fractions and additive and proportional, but it's not the only definition of multiplication. So, you've got to also think of multiplication as scaling that comes later, but you also have to think of multiplication as area and as dimensions. But that first experience with multiplication has to be that composite-unit experience. Mike: You've got me thinking already about how these ideas around unitizing that students can start to make sense of when they're multiplying whole numbers, that that would have a significant impact when they started to think about fractions or rational numbers. Can you talk a little bit about unitizing in the context of fractions, Beth? Beth: Sure. The fraction standards have been most difficult for teachers to get their heads around because the way that the standards promote thinking about fractions is significantly different than the way most of us were taught fractions. So, in the standards and in the research, you come across the term “unit fraction,” and you can probably recognize the unitizing piece in the unit fraction. So, a unit fraction is a fraction where 1 is in the numerator, it's one unit of a fraction. So, in the case of three-fourths, you have three of the one-fourths. Now, this is a bit of a shift in how we were taught. Most of us were taught, “Oh, we have three-fourths. It means you have four things, but you only keep three of them,” right? We learned about the name “numerator” and the name “denominator.” And, of course, we know in fractions, in particular, kids really struggle. Adults really struggle. Fractions are difficult because they seem to be a set of numbers that don't have anything in common with any other numbers. But once you start to think about unitizing and that composite unit, there's a standard in third grade that talks about “decompose any fraction into the sum of unit fractions.” So, in the case of five-sixths, you would identify the unit fraction as one-sixth, and you would have 5 of those one-sixths. So, your unit fraction is one-sixth, and you're going to iterate it or copy it or repeat it five times. Mike: I can hear the parallels between the way you described this work with whole numbers. I have one-fourth, and I've duplicated or copied that five times, and that's what five-fourths is. It feels really helpful to see the through line between how we think about helping kids think about composite numbers and multiplying with whole numbers, to what you just described with unit fractions. Beth: Yeah, and even the language that language infractions is similar, too. So, you talk about that 5 one-fourths. You decompose the five-fourths into 5 of the one-fourths, or you recompose those 5 one-fourths. This is a fourth-grade standard. You recompose those 5 one-fourths into 3 one-fourths or three-fourths and 2 one-fourths or two-fourths. So, even reading a fraction like seven-eighths says 7 one-eighths, helps to really understand what that seven-eighths means, and it keeps you from reading it as seven out of eight. Because when you read a fraction as seven out of eight, it sounds like you're talking about a whole number over another whole number. And so again, that connection to the composite unit in multiplication extends to that composite unit or that unit fraction or unitizing in multiplication. And really, even when we talk about multiplying fractions, we talk about multiplying, say, a whole number times a fraction “5 times one-fourth.” That would be the same as saying, “I'm going to repeat one-fourth five times,” as opposed to, we were told, “Put a 1 under the 5 and multiply across the numerator and multiply across the denominator.” But that didn't help kids really understand what was happening. Mike: So, this progression of ideas that we've talked about from multiplication to fractions, what you've got me thinking about is, what does it mean to think about unitizing with younger kids, particularly perhaps, kids in kindergarten, first or second grade? I'm wondering how or what you think educators could do to draw out the big ideas about units and unitizing with students in those grade levels? Beth: Well, really we don't expect kindergartners to strictly unitize because it's a relatively abstract idea. The big focus in kindergarten is for a student to understand four means 4, four 1s, and 7 means seven 1s. But where we do unitize is in the use of our models in early grades. In kindergarten, the use of a five-frame or a ten-frame. So, let's use the ten-frame to count by tens: 10, 20, 30. And then, how many ten-frames did it take us to count to 30? It took 3. There's the beginning of your unitizing idea. The idea that we would say, “It took 3 of the ten-frames to make 30” is really starting to plant that idea of unitizing 3 can mean 30. And in first grade, when we start to expose kids to coin values, time, telling time, one of the examples we use is, “Whenever was 1 minus 1, 59?” And that was, “When you read for one hour and your friend read for 1 minute less than you, how long did they read?” So, all time is really a unitizing idea. So, all measures, measure conversion, time, money, and the big one in first grade is base 10. And first grade and second grade [have] the opportunity to solidify strong base 10 so that when kids enter third grade, they've already developed a concept of unitizing within the base 10 system. In first grade, the idea that in a number like 78, the 7 is actually worth more than the 8, even though at face value, the 7 seems less than the 8. The idea that 7 is greater than the 8 in a number like 78 is unitizing. In second grade, when we have a number like 378, we can unitize that into 307 tens and 8 ones, or 37 tens and 8 ones, and there's your re-unitizing. And that's actually a standard in second grade. Or 378 ones. So, in first and second grade, really what teachers have to commit to is developing really strong, flexible base 10 understanding. Because that's the first place kids have to struggle with this idea of the face value of a number isn't the same as the place value of a number. Mike: Yeah, yeah. So, my question is, would you describe that as the seeds of unitizing? Like conserving? That's the thing that popped into my head, is maybe that's what I'm actually starting to do when I'm trying to get kids to go from counting each individual 1 and naming the total when they say the last 1. Beth: So, there are some early number concepts that need to be solidified for kids to be able to unitize, right? So, conservation is certainly one of them. And we work on conservation all throughout elementary school. As numbers get larger, as they have different features to them, they're more complex. Conservation doesn't get fixed in kindergarten. It's just pre-K and K are the places where we start to build that really early understanding with small quantities. There's cardinality, hierarchical inclusion, those are all concepts that we focus on and develop in the earliest grades that feed into a child's ability to unitize. So, the thing about unitizing that happens in the earliest grades is it's pretty informal. In pre-K and K, you might make piles of 10, you might count quantities. Counting collections is something we talk a lot about, and we talk a lot about the importance of counting in early math instruction actually all the way up through, but particularly in early math. And let's say you had a group of kids, and they were counting out piles of, say, 45 things, and they put them in piles of 10 and then a pile of 5, and they were able to go back and say, “Ten, 20, 30, 40 and 5.” So, there's a lot that's happening there. So, one is, they're able to make those piles of 10, so they could count to 10. But the other one is, they have conservation. And the other one is, they have a rope-count sequence that got developed outside of this use of that rope-count sequence, and now they're applying that. So, there's so many balls in the air when a student can do something like that. The unitizing question would be, “You counted 45 things. How many piles of 10 did you have?” There's your unitizing question. In kindergarten, there are students—even though we say it's not something we work on in kindergarten—there are certainly students who could look at that and say, “Forty-give is 4 piles of 10 and 5 extra.” So, when I say we don't really do it in kindergarten, we have exposure, but it's very relaxed. It becomes a lot more significant in first and second grade. Mike: You said earlier that teachers in first and second grade really have to commit to building a flexible understanding of base 10. What I wanted to ask you is, how would you describe that? And the reason I ask is, I also think it's possible to build an inflexible understanding of base 10. So, I wonder how you would differentiate between the kind of practices that might lead to a relatively inflexible understanding of base 10 versus the kind of practices that lead to a more flexible understanding. Beth: So, I think counting collections. I already said we talk a lot about counting collections and the primary training. Having kids count things and make groups of 10, focus on your 10 and your 5. We tell kindergarten teachers that the first month or two of school, the most important number you learn is 5. It's not 10, because our brain likes 5, and we can manage 5 easily. Our hand is very helpful. So, building that unit of 5 toward putting two 5s together to make a 10. I mean, I have a 3-year-old granddaughter, and she knows 5, and she knows that she can hold up both her hands and show me 10. But if she had to show me 7, she would actually start back at 1 and count up to 7. So, taking advantage of those units that are baked in already and focusing on them helps in the earliest grades. And then really, I like materials to go into kids' hands where they're doing the building. I feel like second grade is a great time to hand kids base 10 blocks, but first grade is not. And first-grade kids should be snapping cubes together and building their own units, because the more they build their own units of 5 or 10, the more it's meaningful and useful for them. The other thing I'm going to say, and Bridges has this as a tool, which I really like, is they have dark lines at their 5s and 10s on their base 10 blocks. And that helps, even though people are going to say, “Kids can tell you it's a hundred,” they didn't build it. And so, there's a leap of faith there that is an abstraction that we take for granted. So, what we want is kids using those manipulatives in ways that they constructed those groupings, and that helps a lot. Also, no operations for addition and subtraction. You shouldn't be adding and subtracting without using base 10. So, adding and subtracting on a number line helps you practice not just addition and subtraction, but also base 10. So, because base 10's so important, it could be taught all year long in second grade with everything you do. We call second grade the sweet spot of math because all the most important math can be taught together in second grade. Mike: One of the things that you made me think about is something that a colleague said, which is this idea that 10 is simultaneously 10 ones and one unit of 10. And I really connect that with what you said about the need for kids to actually, physically build the units in first grade. Beth: What you just said, that's unitizing. I can call this 10 ones, and I can call this 1, worth 10. And it's more in face in the earliest grades because we often are very comfortable having kids make piles of 10 things or seeing the marks on a base 10 block, say. Or snapping 10 Unifix cubes together, 5 red and 5 yellow Unifix cubes or something to see those two 5s inside that unit of 10. And then also there's your math hand, your fives and your tens and your ten-frames are your fives and your tens. So, we take full advantage of that. But as kids get older, the math that's going to happen is going to rely on kids already coming wired with that concept. And if we don't push it in those early grades by putting your hands on things and building them and sketching what you've just built and transferring it to the pictorial and the abstract in very strategic ways, then you could go a long way and look like you know what you're doing—but don't really. Base 10 is one of those ways we think, because kids can tell you the 7 is in the tens place, they really understand. But the reality is that's a low bar, and it probably isn't an indication a student really understands. There's a lot more to ask. Mike: Well, I think that's a good place for my next question, which is to ask you what resources OGAP has available, either for someone who might participate in the training, other kinds of resources. Could you just unpack the resources, the training, the other things that OGAP has available, and perhaps how people could learn more about it or be in touch if they were interested in training? Beth: Sure. Well, if they want to be in touch, they can go to ogapmathllc.com, and that's our website. And there's a link there to send us a message, and we are really good at getting back to people. We've written books on each of our four content strands. The titles of all those books are “A Focus on … .” So, we have “A Focus on Addition and Subtraction,” “A Focus on Multiplication and Division,” “A Focus on Fractions,” “A Focus on Ratios and Proportions,” and you can buy them on Amazon. Our progressions are readily available on our website. You can look around on our website, and all our progressions are there so people can have access to those. We do training all over. We don't do any open training. In other words, we only do training with districts who want to do the work with more than just one person. So, we contract with districts and work with them directly. We help districts use their math program. Some of the follow-up work we've done is help them see the possibilities within their program, help them look at their program and see how they might need to add more. And once people come to training, they have access to all our resources, the item bank, the progressions, the training, the book, all that stuff. Mike: So, listeners, know that we're going to add links to the resources that Beth is referencing to the show notes for this particular episode. And, Beth, I want to just say thank you so much for this really interesting conversation. I'm so glad we had a chance to talk with you today. Beth: Well, I'm really happy to talk to you, so it was a good time. Mike: Fantastic. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2024 The Math Learning Center | www.mathlearningcenter.org

Rethinking How We Teach & Learn Fractions

Play Episode Listen Later Oct 21, 2024 35:07

Why do so many students struggle with fractions and how do we make math concepts stick beyond rote memorization?This episode is for educators looking to move beyond the frustration and confusion of teaching and learning fractions. We speak with Tara Flynn and Shelley Yearley, two co-authors of the book Rethinking Fractions. Tune in now and learn how a deeper understanding of fractions and a focus on student ownership can make math more meaningful and engaging.You'll: Gain insights from educators who've experienced both the pitfalls and rewards of teaching fractions in new and transformative ways.Discover practical strategies to shift from memorization to conceptual understanding, especially with unit fractions and counting strategies.Learn about upcoming opportunities to engage with like-minded educators and access resources that make teaching fractions approachable and effective.Tune in to this episode of Making Math Moments That Matter to revolutionize your approach to teaching fractions and inspire deeper mathematical understanding in your students!Show Notes PageHow are you ensuring that you support those educators who need a nudge to spark a focus on growing their pedagogical-content knowledge? What about opportunities for those who are eager and willing to elevate their practice, but do not have the support? Book a call with our District Improvement Program Team to learn how we can not only help you craft, refine and implement your district math learning goals, but also provide all of the professional learning supports your educators need to grow at the speed of their learning. Book a short conversation with our team now. Love the show? Text us your big takeaway!

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S02E19 - Professor Nancy Jordan on Number Sense, Early Numeracy Screeners and Fractions

Knowledge for Teachers

Play Episode Listen Later Oct 12, 2024 69:23

In this episode, Brendan Lee speaks with Professor Nancy Jordan. She has been at the forefront of all things to do with early numeracy research including looking at screeners and intervention. Many of you come from the science of reading world and are fans of Scarborough's Reading Rope, Nancy and colleagues have put together a Number Sense one! I've popped the citation in the show notes. Throughout this conversation she delves into what number sense is, why it's important, how we can develop it and how to assess it. Nancy also covers the role of manipulatives and the transition to understanding fractions. Resources mentioned: SENS: Screener for Early Number Sense Number Sense Interventions Jordan, N. C., Devlin, B. L., & Botello, M. (2022). Core foundations of early mathematics: refining the number sense framework. Current Opinion in Behavioral Sciences, 46, 101181. ICME Ginsburg, Greenes, & Balfanz - Big Math for Little Kids Bob Siegler Nora Newcomb Chelsea Cutting. What Works Clearinghouse You can connect with Nancy: Twitter: @Dr_nancyjordan Email: njordan@udel.edu Website: https://sites.google.com/a/udel.edu/nancy-jordan/ You can connect with Brendan: Twitter: @learnwithmrlee Facebook: @learningwithmrlee Website: learnwithlee.net Support the Knowledge for Teachers Podcast: https://www.patreon.com/KnowledgeforTeachersPodcast

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Understanding Multiplication with Whole Numbers, Fractions, and Decimals - An Interview with Brittany Hege

The Wizard of iPhone Speaks (20-22)

Play Episode Listen Later Oct 7, 2024 31:44

Are you struggling to help your students truly understand multiplication, beyond just memorizing math facts?In this episode Brittany Hege from Mix and Math is here to share insights into how you can strengthen students' multiplication skills. This episode dives deep into practical strategies for teaching multiplication in grades 3-6 using models that connect concrete and abstract thinking. Brittany will share the details of her upcoming Make Math Moments Summit session on multiplication. Whether you're dealing with students' struggles or aiming to strengthen your own approach, this discussion offers insights into creating "light bulb moments" in math.Learn how to use the area model effectively to help students visualize and understand multiplication concepts.Discover strategies that connect procedural knowledge with conceptual understanding to foster deeper learning.Gain insights into how multiplication forms the foundation for higher-level math and how to teach it across grade levels.Tune in now to get a sneak peek into Brittany's upcoming virtual summit and transform your multiplication lessons!Resources For Buzzsprout Show Notes Page. District Math Leaders: How are you ensuring that you support those educators who need a nudge to spark a focus on growing their pedagogical-content knowledge? What about opportunities for those who are eager and willing to elevate their practice, but do not have the support? Book a call with our District Improvement Program Team to learn how we can not only help you craft, refine and implement your district math learning goals, but also provide all of the professional learning supports your educators need to grow at the speed of their learning. Book a short conversation with our team now. Love the show? Text us your big takeaway!

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Episode 2: Wizard Of iPhone/Special -- Nashville Writes -- The Sleep Lady

Play Episode Listen Later Oct 5, 2024 12:40

Opening piano music courtesy of Harpeth Presbyterian Church -- closing banjo music courtesy of Banjo HangOut -- William Tell overeater (used with permission)An Open Letter to Senator Marsha BlackburnI know you are busy doing the people's work and running for office, but when the dust settles TVA's recent increase needs a look-see!Their recent increase was bad math and violates the 10% ceiling. Any increase of 5 1/2 percent on top of 4 1/2% amounts to 10.1%When your banking partner pays you interest on the interest you have already earned, it's called “compound” interest. Fractions have rules of their own. I know Miss Cornwall told me so. We are reaping the benefit of 50 years of outcome-based education. Two plus Two is never Five, no matter how good you feel about it! PS To illustrate my point, three weeks ago, a Media pudent (supposedly financial) said, “Lowering the Fed rate effects credit card interest”! And that's not true either! My description of credit card interest rates would be usurious'Or Just as in your local supermarket labeling 1/3 smaller than 1/4!

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Zakariah Hour - A's final game in Oakland, Fractions of concern for the rest of Utah's schedule + more

The Sean O'Connell Show

Play Episode Listen Later Oct 2, 2024 51:24

Zakariah gives his memories of watching the A's in Oakland, Going to the final game, Fractions of concern + more

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Full Sean O'Connell Show 10-2-24

The Sean O'Connell Show

Play Episode Listen Later Oct 2, 2024 152:26

OC talks with Utah Cross Country Coach Kyle Kepler, He talks Utah FB in the bye week with Scott Mitchell, MLB Playoffs with Rob Bradford, & a little of everything including Fractions of Concern in the Zakariah Hour

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Help with fractions, EV sales up, a $50K Lucid, and solar is bigger in Texas

Quick Charge

Play Episode Listen Later Sep 10, 2024

On today's spectacular episode of Quick Charge, we bust the myth of slowing EV sales by teaching journalists how to do math. We also check out the new, $50,000 mainstream Lucid and break the news to California that they're not #1 anymore. We also mark Greenlane's groundbreaking (literally!) flagship EV charging station for big trucks, and talk up Rivian's Top Safety Pick+ status, making it unique among little trucks. All this and more – enjoy! Source Links EV sales have not fallen, cooled, slowed or slumped. Stop lying in headlines. BMW tops Tesla in EV sales for the first time as gap narrows in Europe Tesla deletes its blog post stating all cars have self-driving hardware Ford F-150 Lightning sales surged 160% in August, but gas cars still dominate the total Lucid teases its new midsize electric SUV: Here's our best look yet at the sub-$50K EV Greenlane breaks ground on its flagship electric truck charging stop The only pickup truck awarded an IIHS Top Safety+ rating is the all-electric Rivian R1T Texas just became No 1 in the US for most utility-scale solar Prefer listening to your podcasts? Audio-only versions of Quick Charge are now available on Apple Podcasts, Spotify, TuneIn, and our RSS feed for Overcast and other podcast players. New episodes of Quick Charge are recorded Monday through Thursday (and sometimes Sunday). We'll be posting bonus audio content there as well, so be sure to follow and subscribe so you don't miss a minute of Electrek's high-voltage daily news! Got news? Let us know!Drop us a line at tips@electrek.co. You can also rate us on Apple Podcasts and Spotify, or recommend us in Overcast to help more people discover the show!

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How to Prove 1=0, And Other Maths Illusions - Sarah Hart

Gresham College Lectures

Play Episode Listen Later May 30, 2024 63:37 Transcription Available

In this lecture I will show you some mathematical illusions: “proofs” that 1=0, that fractions don't exist, and more. There are curious and important implications behind what's going on.These “proofs” reveal some very common logical slips that can go unnoticed when we are trying to prove more plausible statements. And the stakes are high. As I'll show you, once you have “proved” one false claim, you can prove absolutely any statement at all.This lecture was recorded by Sarah Hart on 14th May 2024 at Barnard's Inn Hall, LondonThe transcript of the lecture is available from the Gresham College website:https://www.gresham.ac.uk/watch-now/maths-illusionsGresham College has offered free public lectures for over 400 years, thanks to the generosity of our supporters. There are currently over 2,500 lectures free to access. We believe that everyone should have the opportunity to learn from some of the greatest minds. To support Gresham's mission, please consider making a donation: https://gresham.ac.uk/support/Website: https://gresham.ac.ukTwitter: https://twitter.com/greshamcollegeFacebook: https://facebook.com/greshamcollegeInstagram: https://instagram.com/greshamcollegeSupport the Show.

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Fun With Fractions | Dairy Dive 118

MC Podcast

Play Episode Listen Later May 29, 2024 28:18

Join us as we have fun with fractions, no not like Jr. High School. We're looking at protein fractions with Rebecca Kern-Lunbery, of Ward Laboratories. Buckle up, because we're diving deep.

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Egyptian Fractions

The Art of Mathematics

Play Episode Listen Later May 22, 2024 17:21

Neil Epstein, Associate Professor of Mathematics at George Mason University, introduces us to the fractions used by the ancient Egyptians, well before the Greeks and Romans. The Egyptian fractions all had a unit numerator. They could represent any fraction as a sum of unique unit fractions, a fact that was not proved until centuries later. These sums inspired conjectures, one of which was proved only recently, while others remain unsolved to this day. Recent work extends these concepts beyond fractions of integers. Human heritage goes way back, but is still inspiring modern research. --- Send in a voice message: https://podcasters.spotify.com/pod/show/the-art-of-mathematics/message

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Ep.176: Happy 420 Live Stream Held on Saturday, April 20, 2024 [ASMR]

chycho

Play Episode Listen Later May 10, 2024 157:06

- Video on BitChute: https://www.bitchute.com/video/kMamlPd0yJik/ - Video on Rumble: https://rumble.com/v4qhtai-happy-420-asmr-saturday-april-20-300-pm-530-pm-pdt.html - Video on Odysee: https://odysee.com/@chycho:6/Happy420_2024_chycho:5 - Video on CensorTube: https://youtube.com/live/swLjpSjfiTs ▶️ Guilded Server: https://www.guilded.gg/chycho PLAYLISTS: Podcasts: https://soundcloud.com/chycho/sets/chycho ARTICLE: Happy 420! (Almost everything you wanted to know about Cannabis) http://chycho.blogspot.com/2013/04/happy-420.html VIDEOS: Previous Year's 420 Celebration Live Streams at: - 2023: https://www.patreon.com/posts/happy-420-live-82198614 - 2022: https://www.patreon.com/posts/happy-420-2022-65953275 - 2021: https://www.patreon.com/posts/happy-420-live-50729778 - 2020: https://www.patreon.com/posts/youtube-premiere-36424642 - 2019: https://www.bitchute.com/video/lm8FsvOVTf5a/ PLAYLIST: Happy 420 Live Streams (MISSING 2019, see description for link to video on BitChute) https://www.youtube.com/playlist?list=PL9sfzC9bUPxkB5NapnJ2V_PkofEMe40Ij ***SUPPORT*** ▶️ Patreon: https://www.patreon.com/chycho ▶️ Substack: https://chycho.substack.com/ ▶️ Paypal: https://www.paypal.me/chycho ▶️ Buy Me a Coffee: https://www.buymeacoffee.com/chycho ▶️ SubscribeStar: https://www.subscribestar.com/chycho ▶️ ...and crypto, see below. APPROXIMATE SELECT TIMESTAMPS: - Happy 420 Snacks #1 (11:04-14:10) - My Preferred Vaping Device: Arizer Solo I & II (15:11-19:53) - DJ Painting Vaping Session Discussion #1: Happy 420, 2024 (20:01-27:52) - I Never Thought I Would Stick to Vaping the Sex Organs of the Cannabis Plant: Joints Are Dirty (30:03-30:55) - The Dirtiest Joints Are Roach Joints From the Ashtray (32:28-34:04) - How To Turn People Into Pot Heads, Explain to Them the Pleasures of Smoking the Sex Organs of a Plant (38:10-38:54) - Vaping Session #2: Old-school Vape, Arizer Solo I (38:55-39:58) - Edibles - DJ Painting Vaping Session Discussion #2: Happy 420, 2024 (41:34-51:50) - How To Control the Cannabis Munchies (56:04-57:22) - How To Satisfy the Cannabis Munchies (1:04:14-1:06:12) - Why Canada Fell: Math Illiteracy, Americanization of the Education System, Breeding Low IQ Red Rats (1:07:20-1:10:36) - Math - Centralized Power Has Collapsed the Economy To Be Able To Centralize More Power: 2000 Dot-Com Bubble an Example (1:15:26-1:17:24) - Hard Love Trying To Explain to Kids Why They Need Education (1:17:28-1:18:32) - Fractions and Gambling and Maximizing Your Odds on the Craps Table - Cards and Poker - Stopping WEF BS - The Global Majority Is Distancing Itself From the Genocidal Western World: Pending Collapse (1:27:09-1:28:50) - Happy 420 Snacks #2 (1:29:04-1:32:04) - Kitty Cat, Veeya (1:33:04-1:33:42) - DJ Painting Vaping Session Discussion #3: Happy 420, 2024 (1:35:31-1:45:52) - Experiencing Transitional Periods: Weather Fronts, Changing Times, Pending Collapses, Opportunities Plentiful, Lots of Pain (1:42:09-1:45:12) - Live Your Life As You Please, As Long as Your Beliefs Don't Interfere With My Rights To Live As I Please (1:46:20-1:47:09) - The Way We Prevent Tyranny Is by Decentralizing Our Societies (1:47:46-1:50:22) - Some Random Discussion - U2's Bono Is the Equivalent of All of Ireland Taking a Diarrhea Dump (2:03:01-2:04:06) - DJ Painting Vaping Session Discussion #4: Happy 420, 2024 (2:07:26-2:15:44) - DJ Painting Vaping Session Discussion #5: Happy 420, 2024 (2:21:09-2:26:25) - Fallout TV Series, Thumbs-Up With a Smile: 9.5 out of 10 (2:28:09-2:36:37) ***CRYPTO*** ▶️ As well as Cryptocurrencies: Bitcoin (BTC): 1Peam3sbV9EGAHr8mwUvrxrX8kToDz7eTE Ethereum (ETH): 0xCEC12Da3D582166afa8055137831404Ea7753FFd Doge (DOGE): D83vU3XP1SLogT5eC7tNNNVzw4fiRMFhog Peace. chycho http://www.chycho.com

Episode 161 - A candy bar is better than a pie…when it comes to fractions

The Build Math Minds Podcast

Play Episode Listen Later Mar 17, 2024 13:38

I love pies, Pecan Pie especially, but I don't love them during math time. Pies, well circles in general, are an overused visual when it comes to the teaching of fractions. In this video we take a look at visual fraction models that are much better to use and will be helpful to your students as they progress into other mathematical concepts like percentages, ratios, etc. Get any links mentioned in this video at BuildMathMinds.com/161

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Rough Draft Math - Guest: Dr. Amanda Jansen

Faster, Please! â€” The Podcast

Play Episode Listen Later Mar 7, 2024 21:48

Rounding Up Season 2 | Episode 13 – Rough Draft Math Guest: Dr. Amanda Jansen Mike Wallus: What would happen if teachers consistently invited students to think of their ideas in math class as a rough draft? What impact might this have on students' participation, their learning experience, and their math identity? Those are the questions we'll explore today with Dr. Mandy Jansen, the author of “Rough Draft Math,” on this episode of Rounding Up. Mike: Well, welcome to the podcast, Mandy. We are excited to be talking with you. Mandy Jansen: Thanks, Mike. I'm happy to be here. Mike: So, I'd like to start by asking you where the ideas involved in “Rough Draft Math” originated. What drove you and your collaborators to explore these ideas in the first place? Mandy: So, I work in the state of Delaware. And there's an organization called the Delaware Math Coalition, and I was working in a teacher study group where we were all puzzling together—secondary math teachers—thinking about how we could create more productive classroom discussions. And so, by productive, one of the ways we thought about that was creating classrooms where students felt safe to take intellectual risks, to share their thinking when they weren't sure, just to elicit more student participation in the discussions. One way we went about that was, we were reading chapters from a book called “Exploring Talk in School” that was dedicated to the work of Doug Barnes. And one of the ideas in that book was, we could think about fostering classroom talk in a way that was more exploratory. Exploratory talk, where you learn through interaction. Students often experience classroom discussions as an opportunity to perform. "I want to show you what I know.” And that can kind of feel more like a final draft. And the teachers thought, “Well, we want students to share their thinking in ways that they're more open to continue to grow their thinking.” So, in contrast to final draft talk, maybe we want to call this rough draft talk because the idea of exploratory talk felt like, maybe kind of vague, maybe hard for students to understand. And so, the term “rough draft talk” emerged from the teachers trying to think of a way to frame this for students. Mike: You're making me think about the different ways that people perceive a rough draft. So, for example, I can imagine that someone might think about a rough draft as something that needs to be corrected. But based on what you just said, I don't think that's how you and your collaborators thought about it, nor do I think that probably is the way that you framed it for kids. So how did you invite kids to think about a rough draft as you were introducing this idea? Mandy: Yeah, so we thought that the term “rough draft” would be useful for students if they have ever thought about rough drafts in maybe language arts. And so, we thought, “Oh, let's introduce this to kids by asking, ‘Well, what do you know about rough drafts already? Let's think about what a rough draft is.'” And then we could ask them, “Why do you think this might be useful for math?” So, students will brainstorm, “Oh yeah, rough draft, that's like my first version” or “That's something I get the chance to correct and fix.” But also, sometimes kids would say, “Oh, rough drafts … like the bad version. It's the one that needs to be fixed.” And we wanted students to think about rough drafts more like, just your initial thinking, your first ideas; thinking that we think of as in progress that can be adjusted and improved. And we want to share that idea with students because sometimes people have the perception that math is, like, you're either right or you're wrong, as opposed to something that there's gradients of different levels of understanding associated with mathematical thinking. And we want math to be more than correct answers, but about what makes sense to you and why this makes sense. So, we wanted to shift that thinking from rough drafts being the bad version that you have to fix to be more like it's OK just to share your in-progress ideas, your initial thinking. And then you're going to have a chance to keep improving those ideas. Mike: I'm really curious, when you shared that with kids, how did they react? Maybe at first, and then over time? Mandy: So, one thing that teachers have shared that's helpful is that during a class discussion where you might put out an idea for students to think about, and it's kind of silent, you get crickets. If teachers would say, “Well, remember it's OK to just share your rough drafts.” It's kind of like letting the pressure out. And they don't feel like, “Oh wait, I can't share unless I totally know I'm correct. Oh, I can just share my rough drafts?” And then the ideas sort of start popping out onto the floor like popcorn, and it really kind of opens up and frees people up. “I can just share whatever's on my mind.” So that's one thing that starts happening right away, and it's kind of magical that you could just say a few words and students would be like, “Oh, right, it's fine. I can just share whatever I'm thinking about.” Mike: So, when we were preparing for this interview, you said something that has really stuck with me and that I've found myself thinking about ever since. And I'm going to paraphrase a little bit, but I think what you had said at that point in time was that a rough draft is something that you revise. And that leads into a second set of practices that we could take up for the benefit of our students. Can you talk a little bit about the ideas for revising rough drafts in a math classroom? Mandy: Yes. I think when we think about rough drafts in math, it's important to interact with people thinking by first, assuming those initial ideas are going to have some merit, some strength. There's going to be value in those initial ideas. And then once those ideas are elicited, we have that initial thinking out on the floor. And so, then we want to think about, “How can we not only honor the strengths in those ideas, but we want to keep refining and improving?” So inviting revision or structuring revision opportunities is one way that we then can respond to students' thinking when they share their drafts. So, we want to workshop those drafts. We want to work to revise them. Maybe it's peer-to-peer workshops. Maybe it's whole-class situation where you may get out maybe an anonymous solution. Or a solution that you strategically selected. And then work to workshop that idea first on their strengths, what's making sense, what's working about this draft, and then how can we extend it? How can we correct it, sure. But grow it, improve it. Mandy: And promoting this idea that everyone's thinking can be revised. It's not just about your work needs to be corrected, and your work is fine. But if we're always trying to grow in our mathematical thinking, you could even drop the idea of correct and incorrect. But everyone can keep revising. You can develop a new strategy. You can think about connections between representations or connections between strategies. You can develop a new visual representation to represent what makes sense to you. And so, just really promoting this idea that our thinking can always keep growing. That's sort of how we feel when we teach something, right? Maybe we have a task that we've taught multiple times in a row, and every year that we teach it we may be surprised by a new strategy. We know how to solve the problem—but we don't have to necessarily just think about revising our work but revising our thinking about the ideas underlying that problem. So really promoting that sense of wonder, that sense of curiosity, and this idea that we can keep growing our thinking all the time. Mike: Yeah, there's a few things that popped out when you were talking that I want to explore just a little bit. I think when we were initially planning this conversation, what intrigued me was the idea that this is a way to help loosen up that fear that kids sometimes feel when it does feel like there's a right or a wrong answer, and this is a performance. And so, I think I was attracted to the idea of a rough draft as a vehicle to build student participation. I wonder if you could talk a little bit about the impact on their mathematical thinking, not only the way that you've seen participation grow, but also the impact on the depth of kids' mathematical thinking as well. Mandy: Yes, and also I think there's impact on students' identities and sense of self, too. So, if we first start with the mathematical thinking. If we're trying to work on revising—and one of the lenses we bring to revising, some people talk about lenses of revising as accuracy and precision. I think, “Sure.” But I also think about connectedness and building a larger network or web of how ideas relate to one another. So, I think it can change our view of what it means to know and do math, but also extending that thinking over time and seeing relationships. Like relationships between all the different aspects of rational number, right? Fractions, decimals, percents, and how these are all part of one larger set of ideas. So, I think that you can look at revision in a number of different grain sizes. Mandy: You can revise your thinking about a specific problem. You can revise your thinking about a specific concept. You can revise your thinking across a network of concepts. So, there's lots of different dimensions that you could go down with revising. But then this idea that we can see all these relationships with math … then students start to wonder about what other relationships exist that they hadn't thought of and seen before. And I think it can also change the idea of, “What does it mean to be smart in math?” Because I think math is often treated as this right or wrong idea, and the smart people are the ones that get the right idea correct, quickly. But we could reframe smartness to be somebody who is willing to take risk and put their initial thinking out there. Or someone who's really good at seeing connections between people's thinking. Or someone who persists in continuing to try to revise. And just knowing math and being smart in math is so much more than this speed idea, and it can give lots of different ways to show people's competencies and to honor different strengths that students have. Mike: Yeah, there are a few words that you said that keep resonating for me. One is this idea of connections. And the other word that I think popped into my head was “insights.” The idea that what's powerful is that these relationships, connections, patterns, that those are things that can be become clearer or that one could build insights around. And then, I'm really interested in this idea of shifting kids' understanding of what mathematics is away from answer-getting and speed into, “Do I really understand this interconnected bundle of relationships about how numbers work or how patterns play out?” It's really interesting to think about all of the ramifications of a process like rough draft work and how that could have an impact on multiple levels. Mandy: I also think that it changes what the classroom space is in the first place. So, if the classroom space is now always looking for new connections, people are going to be spending more time thinking about, “Well, what do these symbols even mean?” As opposed to pushing the symbols around to get the answer that the book is looking for. Mike: Amen. Mandy: And I think it's more fun. There are all kinds of possible ways to understand things. And then I also think it can improve the social dimension of the classroom, too. So, if there's lots of possible connections to notice or lots of different ways to relationships, then I can try to learn about someone else's thinking. And then I learn more about them. And they might try to learn about my thinking and learn more about me. And then we feel, like, this greater connection to one another by trying to see the world through their eyes. And so, if the classroom environment is a space where we're trying to constantly see through other people's eyes, but also let them try to see through our eyes, we're this community of people that is just constantly in awe of one another. Like, “Oh, I never thought to see things that way.” And so, people feel more appreciated and valued. Mike: So, I'm wondering if we could spend a little bit of time trying to bring these ideas to life for folks who are listening. You already started to unpack what it might look like to initially introduce this idea, and you've led me to see the ways that a teacher might introduce or remind kids about the fact that we're thinking about this in terms of a rough draft. But I'm wondering if you can talk a little bit about, how have you seen educators bring these ideas to life? How have you seen them introduce rough draft thinking or sustain rough draft thinking? Are there any examples that you think might highlight some of the practices teachers could take up? Mandy: Yeah, definitely. So, I think along the lines of, “How do we create that culture where drafting and revising is welcome in addition to asking students about rough drafts and why they might make sense of math?” Another approach that people have found valuable is talking with students about … instead of rules in the classroom, more like their rights. What are your rights as a learner in this space? And drawing from the work of an elementary teacher in Tucson, Arizona, Olga Torres, thinking about students having rights in the classroom, it's a democratic space. You have these rights to be confused, the right to say what makes sense to you, and represent your thinking in ways that make sense to you right now. If you honor these rights and name these rights, it really just changes students' roles in that space. And drafting and revising is just a part of that. Mandy: So different culture-building experiences. And so, with the rights of a learner brainstorming new rights that students want to have, reflecting on how they saw those rights in action today, and setting goals for yourself about what rights you want to claim in that space. So then, in addition to culture building and sustaining that culture, it has to do—right, like Math Learning Center thinks about this all the time—like, rich tasks that students would work on. Where students have the opportunity to express their reasoning and maybe multiple strategies because that richness gives us so much to think about. And drafts would a part of that. But also, there's something to revise if you're working on your reasoning or multiple strategies or multiple representations. So, the tasks that you work on make a difference in that space. And then of course, in that space, often we're inviting peer collaboration. Mandy: So, those are kinds of things that a lot of teachers are trying to do already with productive practices. But I think the piece with rough draft math then, is “How are you going to integrate revising into that space?” So eliciting students' reasoning and strategies—but honoring that as a draft. But then, maybe if you're having a classroom discussion anyway, with the five practices where you're selecting and sequencing student strategies to build up to larger connections, at the end of that conversation, you can add in this moment where, “OK, we've had this discussion. Now write down individually or turn and talk. How did your thinking get revised after this discussion? What's a new idea you didn't have before? Or what is a strategy you want to try to remember?” So, adding in that revision moment after the class discussion you may have already wanted to have, helps students get more out of the discussion, helps them remember and honor how their thinking grew and changed, and giving them that opportunity to reflect on those conversations that maybe you're trying to already have anyway, gives you a little more value added to that discussion. Mandy: It doesn't take that much time, but making sure you take a moment to journal about it or talk to a peer about it, to kind of integrate that more into your thought process. And we see revising happening with routines that teachers often use, like, math language routines such as stronger and clearer each time where you have the opportunity to share your draft with someone and try to understand their draft, and then make that draft stronger or clearer. Or people have talked about routines, like, there's this one called “My Favorite No,” where you get out of student strategy and talk about what's working and then why maybe a mistake is a productive thing to think about, try to make sense out of. But teachers have changed that to be “My Favorite Rough Draft.” So, then you're workshopping reasoning or a strategy, something like that. And so, I think sometimes teachers are doing things already that are in the spirit of this drafting, revising idea. But having the lens of rough drafts and revising can add a degree of intentionality to what you already value. And then making that explicit to students helps them engage in the process and hopefully get more out of it. Mike: It strikes me that that piece that you were talking about where you're already likely doing things like sequencing student work to help tell a story, to help expose a connection. The power of that add-on where you ask the question, “How has your thinking shifted? How have you revised your thinking?” And doing the turn and talk or the reflection. It's kind of like a marking event, right? You're marking that one, it's normal, that your ideas are likely going to be refined or revised. And two, it sets a point in time for kids to say, “Oh yes, they have changed.” And you're helping them capture that moment and notice the changes that have already occurred even if they happened in their head. Mandy: I think it can help you internalize those changes. I think it can also, like you said, kind of normalize and honor the fact that the thinking is continually growing and changing. I think we can also celebrate, “Oh my gosh, I hadn't thought about that before, and I want to kind of celebrate that moment.” And I think in terms of the social dimension of the classroom, you can honor and get excited about, “If I hadn't had the opportunity to hear from my friend in the room, I wouldn't have learned this.” And so, it helps us see how much we need one another, and they need us. We wouldn't understand as much as we're understanding if we weren't all together in this space on this day and this time working on this task. And so, I love experiences that help us both develop our mathematical understandings and also bond us to one another interpersonally. Mike: So, one of the joys for me of doing this podcast is getting to talk about big ideas that I think can really impact students' learning experiences. One of the limitations is, we usually spend about 20 minutes or so talking about it, and we could talk about this for a long time, Mandy. I'm wondering, if I'm a person who's listening, and I'm really interested in continuing to learn about rough draft math, is there a particular resource or a set of resources that you might recommend for someone who wants to keep learning? Mandy: Thank you for asking. So, like you said, we can think about this for a long time, and I've been thinking about it for seven or eight years already, and I still keep growing in my thinking. I have a book called “Rough Draft Math: Revising to Learn” that came out in March 2020, which is not the best time for a book to come out, but that's when it came out. And it's been really enjoyable to connect with people about the ideas. And what I'm trying to do in that book is show that rough draft math is a set of ideas that people have applied in a lot of different ways. And I think of myself kind of as a curator, curating all the brilliant ideas that teachers have had if they think about rough drafts and revising a math class. And the book collects a set of those ideas together. Mandy: But a lot of times, I don't know if you're like me, I end up buying a bunch of books and not necessarily reading them all. So, there are shorter pieces. There's an article in Mathematics Teaching in the Middle School that I co-wrote with three of the teachers in the Delaware Teacher Study Group, and that is at the end of the 2016 volume, and it's called “Rough-Draft Talk.” And that's only 1,800 words. That's a short read that you could read with a PLC or with a friend. And there's an even shorter piece in the NCTM Journal, MTLT, in the “Ear to the Ground” section. And I have a professional website that has a collection of free articles because I know those NCTM articles are behind a paywall. And so, I can share that. Maybe there's show notes where we can put a link and there's some pieces there. Mike: Yes, absolutely. Well, I think that's probably a good place to stop. Thank you again for joining us, Mandy. It really has been a pleasure talking with you. Mandy: Thank you so much, Mike. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2024 The Math Learning Center | www.mathlearningcenter.org

school education arizona students math ground delaware tucson arizona middle school ear jansen plc rough draft exploratory fractions rounding up mike it mike you nctm mike yeah mike well doug barnes mike so mathematics teaching

Play Episode Listen Later Mar 7, 2024 22:10

Education was among the first victims of AI panic. Concerns over cheating quickly made the news. But AI optimists like John Bailey are taking a whole different approach. Today on Faster, Please! — The Podcast, I talk with Bailey about what it would mean to raise kids with a personalized AI coach — one that could elevate the efficacy of teachers, tutors, and career advisors to new heights.John Bailey is a colleague and senior fellow at AEI. He formerly served as special assistant to the president for domestic policy at the White house, as well as deputy policy director to the US secretary of commerce. He has additionally acted as the Director of Educational Technology for the Pennsylvania Department of Education, and subsequently as Director of Educational Technology for the US Department of Education.In This Episode* An opportunity for educators (1:27)* Does AI mean fewer teachers, or better teachers? (5:59)* A solution to COVID learning loss (9:31)* The personalized educational assistant (12:31)* The issue of cheating (17:49)* Adoption by teachers (21:02)Below is a lightly edited transcript of our conversationEducation was among the first victims of AI panic. Concerns over cheating quickly made the news. But AI optimists like John Bailey are taking a whole different approach. Today on Faster, Please! — The Podcast, I talk with Bailey about what it would mean to raise kids with a personalized AI coach — one that could elevate the efficacy of teachers, tutors, and career advisors to new heights.John Bailey is a colleague and senior fellow at AEI. He formerly served as special assistant to the president for domestic policy at the White house, as well as deputy policy director to the US secretary of commerce. He has additionally acted as the Director of Educational Technology for the Pennsylvania Department of Education, and subsequently as Director of Educational Technology for the US Department of Education.An opportunity for educators (1:27)Pethokoukis: John, welcome to the podcast.Bailey: Oh my gosh, it's so great to be with you.We'd actually chatted last summer a bit on a panel about AI and education, and this is a fast moving, evolving technology. People are constantly thinking of new things to do with it. They're gauging its strengths and weaknesses. As you're thinking about any downsides of AI in education, has that changed since last summer? Are you more or less enthusiastic? How would you gauge your evolving views?I think I grow more excited and enthusiastic by the day, and I say that with a little humility because I do think the education space, especially for the last 20 years or so, has been riddled with a lot of promises around personalized learning, how technology was going to change your revolutionize education and teaching and learning, and it rarely did. It was over promise and under-delivered. This, though, feels like it might be one of the first times we're underestimating some of the AI capabilities and I think I'm excited for a couple different reasons.I just see this as it is developing its potential to develop tutoring and, just in time, professional development for teachers, and being an assistant to just make teaching more joyful again and remove some of the drudgery. I think that's untapped area and it seems to be coming alive more and more every day. But then, also, I'm very excited about some of the ways these new tools are analyzing data and you just think about school leaders, you think about principals and superintendents, and state policy makers, and the ability of being able to just have conversations with data, not running pivot tables or Excel formulas and looking for patterns and helping to understand trends. I think the bar for that has just been dramatically lowered and that's great. That's great for decision-making and it's great for having a more informed conversation.You're right. You talked about the promise of technology, and I know that when my kids were in high school, if there were certain classes which were supposedly more tech adept, they would bring out a cart with iPads. And I think as parents we are supposed to be like, “Wow, every kid's going to have an iPad that's going to be absolutely amazing!” And I'm not sure if that made the teachers more productive, I'm not sure, in the end, if the kids learned any better.This technology, as you just said, could be different. And the one area I want to first focus on is, it would be awesome if we had a top-10-percent teacher in every classroom. And I know that, at least some of the early studies, not education studies, but looking at studies of using generative AI in, perhaps, customer service. One effect they notice is kind of raising the lower-performing group and having them do better. And so I immediately think about the ability to raise… boy, if we could just have the lowest-performing teachers do as well as the middle-performing teachers, that would seem to be an amazing improvement.I totally agree with you. Yeah, I think that was the BCG study that found when consultants used gen AI—I think, in that case, it was ChatGPT—everyone improved, but the folks that had the most dramatic improvement were the lowest performers in the consulting world. And here you could imagine something very similar for teachers that are teaching out of field—that happens a lot in science and mathematics. It's with new teachers, and the ability of helping them perform better… also, the ability, I think, of combining what they know with also what science and research is saying is the best practice. That's been very difficult.One of the examples I give is the Department of Ed has these guides called the What Works Clearinghouse Practice Guides, and this is what evaluation of research, and studies, and evaluation has to say, “This is the best way of teaching math, or the best way of teaching reading,” but these are dense documents, they're like 137 PDF pages. If you're asking a new teacher teaching out of field to read 137 pages of a PDF and apply it to their lesson that day, that's incredibly difficult. But it can happen in a matter of seconds now with an AI assistant that can read that practice guide, read your lesson, and make sure that you're getting just-in-time professional development, you're getting an assistant with your worksheets, with your class activities and everything. And so I totally agree with you, I think this is a way of helping to make sure that teachers are able to perform better and to really be an assistant to teachers no matter where they are in terms of their skill level.Does AI mean fewer teachers, or better teachers? (5:59)I recall a story, and I forget which sort of tech CEO was talking to a bunch of teachers, and he said, “The good news: in the future, all teachers will make a million dollars a year… bad news is we're only going to need like 10 percent of you” because each teacher would be so empowered by—this was pre-AI—by technology that they would just be so much more productive.The future you're talking about isn't necessarily a future of fewer teachers, it's just sort of the good part of it, which is more productive teachers, and any field where there's a huge human element is always tough to make more productive. Is the future you're talking about just… it's not necessarily fewer teachers, it's just more productive teachers?I think that's exactly right. I don't think this is about technology replacing teachers, I think it's about complimenting them. We see numerous studies that ask teachers how they spend their time and, on average, teachers are spending less than half of their time on instruction. A lot of it is on planning, a lot of it is on paperwork. I mean, even if we had AI that could take away some of that drudgery and free up teachers' times, so they could be more thoughtful about their planning or spend more time with students, that would be a gift.But also I think the best analog on this is a little bit in the healthcare space. If you think of teachers as a doctor, doctors are your most precious commodity in a healthcare system, you want to maximize their time, and what you're seeing is that now, especially because of technology and because of some tools, you can push a lot of decisions to be more subclinical. And so initially that was with nurses and nurse practitioners so that could free up doctor's time. Now you're seeing a whole new category, too, where AI can help provide some initial feedback or responses, and then if you need more help and assistance, you'd go up to that nurse practitioner, and if you need more help and assistance, then you go and you get the doctor. And I bet we're going to see a bunch of subclinical tools and assistance that come out in education, too. Some cases it's going to be an AI tutor, but then kids are going to need a human tutor. That's great. And in some cases they're going to need more time with their teacher, and that's great, too. I think this is about maximizing time and giving kids exactly what they need when they need it.This just sort of popped in my head when you mentioned the medical example. Might we see a future where you have a real job with a career path called “teacher assistant,” where you might have a teacher in charge, like a doctor, of, maybe, multiple classes, and you have sort of an AI-empowered teaching assistant as sort of a new middle-worker, much like a nurse or a physician's assistant?I think you could, I mean, already we're seeing you have teacher assistants, especially in higher education, but I think we're going to see more of those in K-12. We have some K-12 systems that have master teachers and then teachers that are a little bit less-skilled or newer that are learning on the job. I think you have paraprofessionals, folks that don't necessarily have a certification that are helping. This can make a paraprofessional much more effective. We see this in tutoring that not every single tutor is a licensed teacher, but how do you make sure a tutor is getting just-in-time help and support to make them even more effective?So I agree with you, I think we're going to see a whole category of sort of new professions emerge here. All in service by the way, again, of student learning, but also of trying to really help support that teacher that's gone through their licensure that is years of experience and have gone through some higher education as well. So I think it's a complimentary, I don't think it's replacing,A solution to COVID learning loss (9:31)You know, we're talking about tutoring, and the thing that popped in my head was, with the pandemic and schools being hybrid or shut down and kids having to learn online and maybe they don't have great internet connections and all that, that there's this learning-loss issue, which seems to be reflected in various national testing, and people are wondering, “Well great, maybe we could just catch these kids up through tutoring.” Of course, we don't have a nationwide tutoring plan to make up for that learning loss and I'm wondering, have people talked about this as a solution to try to catch up all these kids who fell behind?I know you and I, I think, share a similar philosophy of where… in DC right now, so much of the philosophy around AI is, it's doomerism. It's that this is a thing to contain and to minimize the harms instead of focusing on how do we maximize the benefits? And if there's been ever a time when we need federal policymakers and state policy makers to call on these AI titans to help tackle a national crisis, the learning crisis coming out of the pandemic is definitely one of those. And I think there's a way to do tutoring differently here than we have in the past. In the past, a lot of tech-based tutoring was rule-based. You would ask a question that was programmed, Siri would give a response, it would give a pre-programed answer in return. It was not very warm. And I think what we're finding is, first of all, there's been two studies, one published in JAMA, another one with Microsoft and Google, that found that in the healthcare space, not only could these AI systems be not just technically accurate, but their answers, when compared to human doctors, were rated as more empathetic. And I think that's amazing to think about when empathy becomes something you can program and maximize, what does it mean to have an empathetic tutor that's available for every kid that can encourage them?And for me, I think the thing that I realized that this is fundamentally different was about a year ago. I wanted to just see: Could ChatGPT create an adaptive tutor? And the prompt was just so simple. You just tell it, “I want you to be an adaptive tutor. I want you to teach a student in any subject at any grade, in any language, and I want you to take that lesson and connect it to any interest a student has, and then I want you to give a short quiz. If they get it right, move on. If they get it wrong, just explain it using simpler language.” That literally is the prompt. If you type in, “John. Sixth grade. Fractions. Star Wars,” every example is based on Star Wars. If you say, “Taylor Swift,” every example is on Taylor Swift. If you say, “football,” every example is on football.There's no product in the market right now, and no human tutor, that can take every lesson and connect it to whatever interest a student has, and that is amazing for engagement. And it also helps take these abstract concepts that so often trip up kids and it connects it to something they're interested in, so you increase engagement, you increase understanding, and that's all with just three paragraphs of human language. And if that's what I can do, I'd love to sort of see our policymakers challenge these AI companies to help build something that's better to help tackle the learning loss.The personalized educational assistant (12:31)And that's three paragraphs that you asked of a AI tutor where that AI is as bad as it's ever going to be. Oftentimes, when people sort of talk about the promise of AI and education, they'll say like, “In the future,” which may be in six months, “kids will have AI companions from a young age with which they will be interacting.” So by the time they get to school, they will have a companion who knows them very well, knows their interests, knows how they learn, all these things. Is that kind of information something that you can see schools using at some point to better teach kids on a more individualized basis? Has there been any thought about that? Because right now, a kid gets to school and all teacher knows is maybe how the kid did it in kindergarten or preschool and their age and their face, but now, theoretically, you could have a tremendous amount of information about that kid's strengths and weaknesses.Oh my gosh, yeah, I think you're right. Some of this we talked about in the future, that was a prompt I constructed, I think for ChatGPT4 last March, which feels like eons ago in AI timing. And I think you're right. I think once these AI systems have memory and can learn more about someone, and in this case a student, that's amazing, to just sort of think that there could be an AI assistant that literally grows up with the child and learns about their interests and how they're struggling in class or what they're thriving in class. It can be encouraging when it needs to be encouraging, it can help explain something when the child needs something explained, it could do a deeper dive on a tutoring session. Again, that sounds like science fiction, but I think that's two, three years away. I don't think that's too far.Speaking of science fiction, because I know you're a science fiction fan, a lot of what we're describing now feels like the 1995 Sci-Fi novel, The Diamond Age and that talked about this, it talked about Nell, who was a young girl who came in a possession of a highly advanced book. It was called the Young Lady's Illustrated Primer, and it would help with tutoring and with social codes and with a lot of different support and encouragement. And at the time when Neil wrote that in '95, that felt like science fiction and it really feels like we've come to the moment now—you have tablet computers, you have phones that can access these super-intelligent AI systems that are empathetic, and if we could get them to be slightly more technically accurate and grounded in science and practice and rigorous research, I don't know, that feels really powerful. It feels like something we should be leaning into more than leaning away fromJohn, that reference made this podcast an early candidate for Top Podcasts of 2024. Wonderful. That was really playing to your host. Again, as you're saying that, it occurs to me that one area that this could be super helpful really is sort of career advice when kids are wondering, “What I should do, should I go to college?” and boy, to have a career counselor's advice supplemented by a lifetime of an AI interacting with this kid… Counselors will always say, “Well, I'm sure your parents know you better than I do.” Well, I'll tell you, a career counselor plus a lifetime AI, you may know that kid pretty well.Let's just take instruction off the table. Let's say we don't want AI to help teach kids, we don't want AI to replace teachers. AI as navigators I think is another untapped area, and that could be navigators as parents are trying to navigate a school choice system or an education savings account. It could be as kids and high school students are navigating what their post-college plan should be, but these systems are really good with that.I remember I played with a prompt a couple months ago, but it was that, I said, “My name is John. I play football. Here's my GPA. I want to go to school in Colorado and here's my SAT score. What college might work well for me?” And it did an amazing job with even that rudimentary prompt of giving me a couple different suggestions in why that might be. And I think if we were more sophisticated there, we might be able to open up more pathways for students or prevent them from going down some dead ends that just might not be the right path for them.There's a medical example of this that was really powerfully illustrative for me, which is, I had a friend who, quite sadly a couple of months ago was diagnosed with breast cancer. And this is an unfolding diagnosis. You get the initial, then there's scans and there's biopsies and reports, and then second and third and fourth opinions, it's very confusing. And what most patients need there isn't a doctor, they need a navigator. They need someone who could just make sense of the reports that can explain this Techno Latin that kind of gets put into the medical jargon, and they need someone to just say, what are the next questions I need to ask as I find my path on this journey?And so I built her a GPT that had her reports and all she could do was ask it questions, and the first question she said is, “Summarize my doctor notes, identify they agree and where they disagree.” Then, the way I constructed the prompt is that after every response, it should give her three questions to ask the doctor, and all of a sudden she felt empowered in a situation where she felt very disempowered with navigating a very complex, and in that case, a life-threatening journey. Here, how can't we use that to take all the student work, and their assessments, their hobbies, and start helping them be empowered with figuring out where they should be pursuing a job or college or some other post-secondary pathway.The issue of cheating (17:49)You know I have a big family, a lot of kids, and I've certainly had conversations with, say, my daughters about career, and I'll get something like, “Ugh, you just don't understand.” And I'll say, “Well, help me, make me understand.” She's like, “Oh, you just don't understand.” Now I'm like, “Hey, AI, help me understand, what does she want to do? Can you give me some insights into her career?”But we've talked about some of the upsides here and we briefly mentioned, immediately this technology attracted criticism. People worried about a whole host of things from bias in the technology to kids using it to cheat. There was this initial wave of concerns. Now that we're 15 months, maybe, or so since people became aware of this technology, which of the concerns do you find to be the persistent ones that you think a lot about? Are you as worried, perhaps, about issues of kids cheating, on having an AI write the paper for them, which was an early concern? What are the concerns that sort of stuck with you that you feel really need to be addressed?The issue of cheating is present with every new technology, and this was true when the internet came out, it was true when Wikipedia came out, it was true when the iPhones came out. You found iPhone bans. If you go back and look at the news cycle in 2009, 2010, schools were banning iPhones; and then they figure out a way to manage it. I think we're going to figure out a way to manage the cheating and the plagiarism.I think what worries me is a couple different things. One is, the education community talks often about bias, and when they usually talk about bias, in this case, they're talking about racial bias in these systems. Very important to address that head on. But also we need to tackle political bias. I think we just saw that recently with Gemini that, often, sometimes these systems can surface a little bit of center-left perspective and thinking on different types of subjects. How do we fine-tune that so you're getting it a little bit more neutral. Then also, in the education setting, it's pedagogical bias. Like when you're asking it to do a lesson plan or tutoring session, what's the pedagogy that's actually informing the output of that? And those are all going to be very important, I think, to solve.The best case scenario, AI gets used to free up teacher time and teachers can spend more time in their judgment working on their lesson plans and their worksheets and more time with kids. There's also a scenario where some teachers may fall asleep at the wheel a little bit. It's like what you're seeing with self-driving cars, that you're supposed to keep your hands on the wheel and supposed to be at least actively supervising it, but it is so tempting to just sort of trust it and to sort of tune out. And I can imagine there's a group of teachers that will just take the first output from these AI systems and just run with it, and so it's not actually developing more intellectual muscle, it's atrophying that a little bit.Then lastly, I think, what I worry about with kids—this is a little bit on the horizon, this is the downside to the empathy—what happens when kids just want to keep talking to their friendly, empathetic, AI companion and assistant and do that at the sacrifice of talking with their friends, and I think we're seeing this with the crisis of loneliness that we're seeing in the country as kids are on their phones and on social media. This could exaggerate that a lot more unless we're very intentional now about how to make sure kids aren't spending all their time with their AI assistant, but also in the real life and the real world with their friends.Adoption by teachers (21:02)Will teachers be excited about this? Are there teachers groups, teachers unions who are… I am sure they've expressed concerns, but will this tool be well accepted into our classrooms?I think that the unions have been cautiously supportive of this right now. I hear a lot of excitement from teachers because I think what teachers see is that this isn't just one more thing, this is something that is a tool that they can use in their job that provides immediate, tangible benefits. And if you're doing something that, again, removes some drudgery of some of the administrative tasks or helps you with figuring out that one worksheet that's going to resonate with that one kid, that's just powerful. And I think the more software and systems that come out that tap that and make that even more accessible for teachers, I think the more excitement there is going to be. So I'm bullish on this. I think teachers are going to find this as a help and not as a threat. I think the initial threat around plagiarism, totally understandable, but I think there's going to be a lot of other tools that make teachers' lives better.Faster, Please! is a reader-supported publication. To receive new posts and support my work, consider becoming a free or paid subscriber. This is a public episode. If you'd like to discuss this with other subscribers or get access to bonus episodes, visit fasterplease.substack.com/subscribe

Fun and Effective Approaches: Engaging Methods to Teach Fractions

The Teaching Made Easy Podcast

Play Episode Listen Later Mar 6, 2024 17:57

Teaching fractions isn't always as easy as pie. But, it doesn't have to be a drag! You can liven up your math lessons with these fun and effective approaches to teaching fractions! Why do that? Well, finding engaging methods to teach fractions can make all the difference in terms of understanding and retention. Let's explore how to teach fractions using creative and interactive strategies that will help your students grasp fractions in a way that is stinkin 'simple! --- Send in a voice message: https://podcasters.spotify.com/pod/show/farrah-henley/message Support this podcast: https://podcasters.spotify.com/pod/show/farrah-henley/support

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27. Teaching Tips and Strategies for Multiplying Fractions

Elementary Math Chat

Play Episode Listen Later Feb 13, 2024 18:20

Multiplying fractions is no problem, top times top and bottom times bottom…it's that simple, right? Well, not always. In this episode, I share teaching tips, strategies, and activities to help build conceptual understanding of multiplying fractions. You'll hear tips for adding engagement to these lessons along with common mistakes you can expect to see. Topics include: Multiples of unit fractions (4th grade) Multiples of non-unit fractions (4th grade) Multiplying fractions by whole numbers (4th and 5th) Multiplying fractions by fractions (5th grade) I also share a unique idea for a small group activity in the Teaching Tip of the Week. Resources Mentioned: Multiples of Fractions Work MatAnimal Puzzles (for small group activity) Test Prep Kit Episodes Mentioned: Episode 20: 5 Kagan Strategies to Boost Engagement and Build Community Connect with me:InstagramJoin my NewsletterJoin the 4th Grade Math Facebook GroupFollow my TPT Store To view the show notes with the full transcript, head to https://krejcicreations.com/episode27. P.S. Has this podcast been helpful for you? If so, screenshot an episode, add it to IG, and tag me @krejci_creations. This helps spread the word to other teachers!

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Episode 1356: The Western Civilization

how did i get here?

Play Episode Listen Later Feb 9, 2024 83:12

Hello friends! Reggie O'Farrell from indie noise-pop band, The Western Civilization is my guest for episode 1356! Their new album, Fractions of a Whole drops next Friday, February 16th and they're celebrating with a release tour around Texas. They'll be playing at The 13th Floor in Austin on the 16th. Go to www.thewestercivilization.com for show dates, music, videos and more. Reggie and I have a great conversation about starting the band with his musical partner, Rachel Hansbro in Houston in 2002, touring, having success with the band, taking a break from the band for a few years, the Houston music scene of the early 2000's, making Fractions of a Whole, studios, producing, dogs and much more. I had a great time getting to know Reggie. I'm sure you will too. Let's get down! Get your act together at Space Rehearsal, Recording and Video Spaceatx.com

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Ep 189: The Operator Meaning of Fractions

thinking meaning scaling operator talking points fractions

Play Episode Listen Later Jan 30, 2024 22:30 Transcription Available

Finding a fraction of something can be a tricky concept. In this episode Pam and Kim run through two strings that help reason about using fractions as operators.Talking Points:How are unit fractions related like halves, fourth and eighths? Scaling and Over Strategy with unit fractionsFractions as operatorsWhy is scaling a unit fractions critical? Knowing that three fourths is the same as three 1/4s? Knowing non-unit fractions from fractions is a precursor.Thinking of fractions only as part-whole is not sufficient. A different problem string using the operator meaning of fractions.Check out our social mediaTwitter: @PWHarrisInstagram: Pam Harris_mathFacebook: Pam Harris, author, mathematics educationLinkedin: Pam Harris Consulting LLC

Ep 188: Fractions - The Whole Matters

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Play Episode Listen Later Jan 23, 2024 32:19 Transcription Available

For students to really reason about fractions they need to be comfortable recontextualizing them as parts of different wholes. In this episode Pam and Kim take a closer look at fraction problems where the unit matters.Talking Points:Kim and the beachReasoning and teaching are different skillsGaining flexibility with different unitsSmoothie preferences

fractions

Fractions

Who's Tom & Dick

Play Episode Listen Later Dec 15, 2023 27:37

Welcome to the second Christmas Episode of "Whos Tom & Dick"We discuss our preparations for Christmas living with Cancer and Heart Disease and how we cope with our illness during this festive period.Martin Explores "Make a Wish", a UK charity making dreams come true for children with critical conditions, but did you know they do a lot more than this, Listen to the Podcast this week to see where some of their money is spent. https://www.make-a-wish.org.uk/Patrick tells us about his recent check with the Cardiologist, and his Ultrasond. This is a real eye-opening discussion into Blood Flow, Ejection Fraction (EF), and Diastolic Dysfunction in Grades 1,2, and 3. Patrick also describes the ECG and his experience of a heart attack whilst connected to an ECG and exactly what happened.Martin comes up with his "Joke of the Week" a regular spot in our Podcasts now.We have some interesting comments and letters from listeners this week and try to help them through.https://www.whostomanddick.com#HeartTransplant#EbsteinsAnomaly#RareCondition#HealthJourney#LifeChangingDiagnosis#MentalHealth#Vulnerability#SelfCompassion#PostTraumaticGrowth#MedicalMiracle#BBCSports#Inspiration#Cardiology#Surgery#Podcast#Healthcare#HeartHealth#MedicalBreakthrough#EmotionalJourney#SupportSystem#HealthcareHeroes#PatientStories#CardiologyCare#MedicalJourney#LifeLessons#MentalWellness#HealthAwareness#InspirationalTalk#LivingWithIllness#RareDiseaseAwareness#SharingIsCaring#MedicalSupport#BBCReporter#HeartDisease#PodcastInterview#HealthTalk#Empowerment#Wellbeing#HealthPodcast#ChronicIllnessCheck out our new website at www.whostomanddick.com

Making Sense of Fractions and Decimals - Guest: Susan Empson, PhD

Play Episode Listen Later Dec 7, 2023 23:11

Rounding Up Season 2 | Episode 7 – Making Fractions More Meaningful Guest: Dr. Susan Empson Mike Wallus: For quite a few adults, fractions were a stumbling block in their education that caused many to lose their footing and begin to doubt their ability to make sense of math. But this doesn't have to be the case for our students. Today on the podcast, we're talking with Dr. Susan Empson about big ideas and fractions and how we can make them more meaningful for our students. Welcome to the podcast. Susan. Thanks for joining us. Susan Empson: Oh, it's so great to be here. Thank you for having me. Mike: So, your book was a real turning point for me as an educator, and one of the things that it did for me at least, it exposed how little that I actually understood about the meaning of fractions. And I say this because I don't think that I'm alone in saying that my own elementary school experience was mostly procedural. So rather than attempting to move kids quickly to procedures, what types of experiences can help children build a more meaningful understanding of fractions? Susan: Great question. Before I get started, I just want to acknowledge my collaborators because I've had many people that I've worked with. There's Linda Levi, co-author of the book, and then my current research partner, Vicki Jacobs. And of course, we wouldn't know anything without many classroom teachers we've worked with in the current and past graduate students. In terms of the types of experiences that can help children build more meaningful experiences of fractions, the main thing we would say is to offer opportunities that allow children to use what they already understand about fractions to solve and discuss story problems. Children's understandings are often informal and early on, for example, may consist mainly partitioning things in half. What I mean by informal is that understandings emerge in situations out of school. So, for example, many children have siblings and have experienced situations where they have had to share, let's say three cookies or slices of pizza between two children. In these kinds of situations, children appreciate the need for equal shares, and they also develop strategies for creating them. So, as children solve and discuss story problems in school, their understandings grow. The important point is that story problems can provide a bridge between children's existing understandings and new understandings of fractions by allowing children to draw on these informal experiences. Generally, we recommend lots of experiences with story problems before moving on to symbolic work to give children plenty of opportunity to develop meaningful fractions. And we also recommend using story problems throughout fraction instruction. Teachers can use different types of story problems and adjust the numbers in those problems to address a range of fraction content. There are also ideas that we think are foundational to understanding fractions, and they're all ideas that can be elicited and developed as children engage in solving and discussing story problems. Susan: So, one idea is that the size of a piece is determined by its relationship to the whole. What I mean is that it's not necessarily the number of pieces into which a whole is partitioned that determines the size of a piece. Instead, it's how many times the piece fits into the whole. So, in their problem-solving, children create these amounts and eventually name them and symbolize them as unit fractions. That's any fraction with 1 in the numerator. Mike: You know, one of the things that stands out for me in that initial description that you offered, is this idea of kids don't just make meaning of fractions at school, that their informal lived experiences are really an asset that we can draw on to help make sense of what a fraction is or how to think about it. Susan: That's a wonderful way to say it. And absolutely, the more teachers get to know the children in their classrooms and the kinds of experiences those children might have outside of school, the more of that can be incorporated into experiences like solving story problems in school. Mike: Well, let's dig into this a little bit. Let's talk a little bit about the kinds of story problems or the structure that actually provides an entry point and can build understanding of fractions for students. Can you talk a bit about that, Susan? Susan: Yes. So, I'll describe a couple types of story problems that we have found especially useful to elicit and develop children's fraction understandings. So first, equal sharing story problems are a powerful type of story problem that can be used at the beginning of and even throughout instruction. These problems involve sharing multiple things among multiple sharers. So, for example, four friends equally sharing 10 oranges. How much orange would each friend get? Problems like this one allow children to create fractional amounts by drawing things, partitioning those things, and then attaching fraction names and symbols. So, let's [talk] a little bit about how a child might solve the oranges problem. A child might begin by drawing four friends and then distributing whole oranges one by one until each friend has two whole oranges. Now, there are two oranges left and not enough to give each friend another whole orange. So, they have to think about how to partition the remaining oranges. Susan: They might partition each orange in half and give one more piece to each friend, or they might partition each of the remaining oranges into fourths and give two pieces to each friend. Finally, they have to think about how to describe how much each friend gets in terms of the wholes and the pieces. They might simply draw the amount, they might shade it in, or they might attach number names to it. I also want to point out that a problem about four friends equally sharing 10 oranges can be solved by children with no formal understanding of fraction names and symbols because there are no fractions in the story problem. The fractions emerge in children's strategies and are represented by the pieces in the answer. The important thing here is that children are engaged in creating pieces and considering how the pieces are related to the wholes or other pieces. The names and symbols can be attached gradually. Mike: So, the question that I wanted to ask is how to deal with this idea of how you name those fractional amounts, because the process that you described to me, what's powerful about it is that I can directly model the situation. I can make sense of partitioning. I think one of the things that I've always wondered about is, do you have a recommendation for how to navigate that naming process? I've got one of something, but it's not really one whole orange. So how do I name that? Susan: That's a great question. Children often know some of the informal names for fractions, and they might understand halves or even fourths. Initially, they may call everything a half or everything a piece or just count everything as one. And so, what teachers can do is have conversations with children about the pieces they've created and how the pieces relate to the whole. A question that we've found to be very helpful is, how many of those pieces fit into the whole? Mike: Got it. Susan: Not a question about how many pieces are there in the whole, but how many of the one piece fit into the whole. Because it then focuses children on thinking about the relationship between the piece and the whole rather than simply counting pieces. Mike: Let's talk about the other problem type that was kind of front and center in your thinking. Susan: Yes. So, another type of story problem that can be used early in fraction instruction involves what we think of as special multiplication and division story problems that have a whole number of groups and a unit fraction amount in each group. So, what do I mean by that? For example, let's say there are six friends and they each will get one-third of a sub sandwich for lunch. So, there's a whole number of groups—that's the six friends—and there's a unit fraction amount in each group that's the one-third of a sandwich that they each get. And then the question is how many sandwiches will be needed for the friends? So, a problem like this one essentially engages children in reasoning about six groups of one-third. And again, as with the equal sharing problem about oranges, they can solve it by drawing out things. They might draw each one-third of a sandwich, and then they have to consider how to combine those to make whole sandwiches. An important idea that children work on with this problem then is that three groups of one-third of a sandwich can be combined to make one whole sandwich. There are other interesting types of story problems, but teachers have found these two types, in particular, effective in developing children's understandings of some of the big ideas and fractions. Mike: I wonder if you have educators who hear you talk about the second type of problem and are a little bit surprised because they perceive it to be multiplication. Susan: Yes, it is surprising. And the key is not that you teach all of multiplying and dividing fractions before adding and subtracting fractions, but that you use these problem types with special number combinations. So, a whole number of groups, for example, the six groups unit fractions in each group—because those are the earliest fractions children understand. And I think maybe one way to think about it is that fractions come out of multiplying and dividing, kind of in the way that whole numbers come out of adding and counting. And the key is to provide situations story problems that have number combinations in them that children are able to work with. Mike: That totally makes sense. Can you say more about the importance of attending to the number combinations? Susan: Yes. Well, I think that the number combinations that you might choose would be the ones that are able to connect with the fraction understandings that children already have. So, for example, if you're working with kindergartners, they might have a sense of what one half is. So, you might choose equal sharing problems that are about sharing things among two children. So, for example, three cookies among two children. You could even, once children are able to name the halves, they create in a problem like that, you can even pose problems that are about five children who each get half of a sandwich, how many sandwiches is that? But those are all numbers that are chosen to allow children to use what they understand about fractions. And then as their understandings grow and their repertoire of fractions also grows, you can increase the difficulty of the numbers. So, at the other end, let's think about fifth grade and posing equal sharing problems. If we take that problem about four friends sharing 10 oranges, we could change the number just a little bit to make it a lot harder to, four friends sharing 10 and a half oranges, and then fifth-graders would be solving a problem that's about finding a fraction of a fraction, sharing the half orange among the four children. Mike: Let me take what you've shared and ask a follow-up question that came to me as you were talking. It strikes me that the design, the number choices that we use in problems matter, but so does the space that the teacher provides for students to develop strategies and also the way that the teacher engages with students around their strategy. Could you talk a little bit about that, Susan? Susan: Yes. We think it's important for children to have space to solve problems, fraction story problems, in ways that make sense to them and also space to share their thinking. So, just as teachers might do with whole number problem-solving in terms of teacher questioning in these spaces, the important thing is for the teacher to be aware of and to appreciate the details of children's thinking. The idea is not to fix children's thinking with questioning, but to understand it or explore it. So, one space that we have found to be rich for this kind of questioning is circulating. So, that's the time when as children solve problems, the teacher circulates and has conversations with individual children about their strategies. So, follow-up questions that focus on the details of children's strategies help children to both articulate their strategies and to reflect on them and help teachers to understand what children's strategies are. We've also found that obvious questions are sometimes underappreciated. So, for example, questions about what this child understands about what's happening in a story problem, what the child has done so far in a partial strategy, even questions about marks on a child's paper; shapes or tallies that you as a teacher may not be quite sure about, asking what they mean to the child. “What are those? Why did you make those? How did they connect with the problem?” So, in some it benefits children to have the time to articulate the details of what they've done, and it benefits the teacher because they learn about children's understandings. Mike: You're making me think about something that I don't know that I had words for before, which is I wonder if, as a field, we have made some progress about giving kids the space that you're talking about with whole number operations, especially with addition and subtraction. And you're also making me wonder if we still have a ways to go about not trying to simply funnel kids to, even if it's not algorithms, answer-getting strategies with rational numbers. I'm wondering if that strikes a chord for you or if that feels off base. Susan: It feels totally on base to me. I think that it is as beneficial, perhaps even more beneficial for children to engage in solving story problems and teachers to have these conversations with them about their strategies. I actually think that fractions provide certain challenges that whole numbers may not, and the kinds of questioning that I'm talking about really depend on the details of what children have done. And so, teachers need to be comfortable with and familiar with children's strategies and how they think about fractions as they solve these problems. And then that understanding, that familiarity, lays the groundwork for teachers to have these conversations. The questions that I'm talking about can't really be planned in advance. Teachers need to be responsive to what the child is doing and saying in the moment. And so that also just adds to the challenge. Mike: I'm wondering if you think that there are ways that educators can draw on the work that students have done composing and decomposing whole numbers to support their understanding of fractions? Susan: Yes. We see lots of parallels just as children's understandings of whole numbers develop. They're able to use these understandings to solve multi-digit operations problems by composing and decomposing numbers. So, for example, to take an easy addition, to add 37 plus eight, a child might say, “I don't know what that is, but I do know how to get from 37 to 40 with three.” So, they take three from the eight, add it to the 37 get to 40, and then once at 40 they might say, “I know that 40 plus five more is 45.” So, in other words, they decompose the eight in a way that helps them use what they understand about decade numbers. Operations with fractions work similarly, but children often do not think about the similarities because they don't understand fractions or numbers to, versus two numbers one on top of the other. Susan: If children understand that fractions can be composed and decomposed just as whole numbers can be composed and decomposed, then they can use these understandings to add, subtract, multiply, and divide fractions. For example, to add one and four-fifths plus three-fifths, a child might say, “I know how to get up to two from one in four-fifths. I need one more fifth, and then I have two more fifths still to add from the three-fifths. So, it's two and two-fifths.” So, in other words, just as they decompose the eight into three and five to add eight to 37, they decompose the three-fifths into one-fifth and two-fifths to add it to one and four-fifths. Mike: I could imagine a problem like one and a half plus five-eighths. I could say, “Well, I know I need to get a half up. Five-eighths is really four-eighths and one-eighths, and four-eighths is a half.” Susan: Yep. Mike: “So, I'm actually going from one and a half plus four-eighths. OK. That gets me to two, and then I've got one more eighth left. So, it's two and an eighth.” Susan: Nice. Yeah, that's exactly the kind of reasoning this approach can encourage. Mike: Well, I have a final question for you, Susan. “Extending Children's Mathematics” came out in 2011, and I'm wondering what you've learned since the book came out. So, are there ideas that you feel like have really been affirmed or refined, and what are some of the questions about the ways that students make meaning of fractions that you're exploring right now? Susan: Well, I think, for one, I have a continued appreciation for the power of equal sharing problems. You can use them to elicit children's informal understandings of fractions early in instruction. You can use them to address a range of fraction understandings, and they can be adapted for a variety of fraction content. So, for example, building meaning for fractions, operating with fractions, concepts of equivalence. Vicki and I are currently writing up results from a big research project focused on teachers' responsiveness to children's fraction thinking during instruction. And right now, we're in the process of analyzing data on third-, fourth-, and fifth-grade children's strategies for equal sharing problems. We specifically focused on over 1,500 drawing-based strategies used by children in a written assessment at the end of the school year. We've been surprised both by the variety of details in these strategies—so, for example, how children represent items, how they decide to distribute pieces to people—and also by the percentages of children using these drawing-based strategies. For each of grades three, four, and five, over 50 percent of children use the drawing-based strategy. There are also, of course, other kinds of strategies that don't depend on drawings that children use, but by far the majority of children were using these strategies. Mike: That's interesting because I think it implies that we perhaps need to recognize that children actually benefit from time using those strategies as a starting point for making sense of the problems that they're solving. Susan: I think it speaks to the length of time and the number of experiences that children need to really build meaning for fractions that they can then use in more symbolic work. I'll mention two other things that we've learned for which we actually have articles in the NCTM publication MTLT, which is “Mathematics Teacher: Learning and Teaching in PK–I2.” So first, we've renewed appreciation for the importance of unit fractions and story problems to elicit and develop big ideas. Another idea is that unit fractions are building blocks of other fractions. So, for example, if children solve the oranges problem by partitioning both of the extra oranges into fourths, then they have to combine the pieces in their answer. One-fourth from each of two oranges makes two-fourths of an orange. Another idea is that one whole can be seen as the same amount as a grouping of same-sized unit fractions. So, those unit fractions can all come from the same hole or different wholes, for example, to solve the problem about six friends who will each get one-third of a sub sandwich. A child has to group the one-third sandwiches to make whole sandwiches. Understanding that the same sandwich can be seen in these two ways, both as three one-third sandwiches or as one whole sandwich, provides a foundation for flexibility and reasoning. For those in the audience who are familiar with CGI, this idea is just like the IDM base ten, that 1 ten is the same amount as ten 1s, or what we describe in shorthand as 10 as a unit. And we also have an article in MTLT. It's about the use of follow-up equations to capture and focus on fraction ideas in children's thinking for their story problems. So basically, teachers listen carefully as children solve problems and explain their thinking to identify ideas that can be represented with the equations. Susan: So, for example, a child solving the sub-sandwiches problem might draw a sandwich partitioned into thirds and say they know that one sandwich can serve three friends because there are three one-thirds in the sandwich. That idea for the child might be drawn, it might be verbally stated. A follow-up equation to capture this idea might be something like one equals one-third plus one-third plus blank, with the question for the child, “Could you finish this equation or make it a true equation?” So, follow-up equation[s] often make ideas about unit fractions explicit and put them into symbolic form for children. And then at the same time, the fractions in the equations are meaningful to children because they are linked to their own meaning-making for a story problem. And so, while follow-up equations are not exactly a question, they are something that teachers can engage children with in the moment as a way to kind of put some symbols onto what they are saying, help children to reflect on what they're saying or what they've drawn, in ways that point towards the use of symbols. Mike: That really makes sense. Susan: So, they could be encouraged to shade in the piece and count the total number of pieces into which an orange is cut. However, we have found that a better question is, how many of this size piece fit into the whole? Because it focuses children on the relationship between the piece and the whole, and not on only counting pieces. Mike: Oh, that was wonderful. Thank you so much for joining us, Susan. It's really been a pleasure talking with you. Susan: Thank you. It's been my pleasure. I've really enjoyed this conversation. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability. © 2023 The Math Learning Center | www.mathlearningcenter.org

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Ep 162: Comparing Fractions Pt 2