Podcasts about fractions

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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.

Instructional Recipes For Teaching Math

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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.

The Key to Understanding Fractions and Algebraic Thinking

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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.

73. Ellie's Passion for Fractions Taught Right

The Teaching Toolbox - A Podcast for Middle School Teachers

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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Class 3 --Math Ch-6 (Fractions) DPS Nacharam - CBSE

The Delhi Public School Podcast

Play Episode Listen Later Feb 10, 2025 4:28

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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

love discover teach rethinking fractions tara flynn

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!

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!

love discover numbers math mix multiplication fractions hege decimals

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

utah concerns oakland final game fractions

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

concerns oc mlb playoffs sean o fractions scott mitchell rob bradford

FULL SHOW | Sit, shake, fractions!

Nick, Jess & Simon - hit106.9 Newcastle

Play Episode Listen Later Sep 18, 2024 86:39

Can we build a new show theme song with the help of AI? We want to know how nice is your partner? You won't believe where Ducko lost his jacket and Jess grossed out her friends!Subscribe on LiSTNR: https://play.listnr.com/podcast/nick-jess-and-duckoSee omnystudio.com/listener for privacy information.

ai shake fractions

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!

Lobbying For Morning Games

The Restump Podcast

Play Episode Listen Later Jun 14, 2024 43:16

Our post bye game form isn't the Dog's breakfast we perceive and an AM start in WA doesn't hurt! Should we Lobby the AFL for more byes? Time to Restump Podcast the Saturday morning dog fight. For some reason we're all of the belief we're still in holiday mode or we're jet lagged when we turn up after the bye and consequently our form isn't flash. However, we're misleading ourselves because we're 8 wins and 7 losses from our 15 post-bye games.So, we've got a 53% post-bye win strike rate which is about 8 percentage points greater than our overall historical win rate of 45%. Does that suggest we're better coming off the bye?Fractions and odds and percentages aside, every game is its own bag and that includes tomorrow at Marvel against the Dogs.After not putting a foot wrong in his two quarters from his last two games as the sub, Corey Wagner unfortunately makes way for the return of Matt Johnson, who is a really good in.With Aaron Naughton and Sam Darcy out of the Bulldogs, maybe we took the opportunity to strengthen the midfield depth?The Dogs have some quality ins though with Corey Weightman back in since sustaining the arm injury against us 7 games ago. They also have brought back Ed Richards who adds some heat to their midfield in both defensive and offensive manners. And they lose Alex Keith which is a positive for us.Front and centre on the team strategy white board should be, “avoid playing to Liam Jone's strengths”. Whoever is on him at any given stage needs to lead him away from the action.The Dogs were seemingly flying but they got opened up last week by the Lions. Without two of their three tall timber goal kickers, too much pressure fell on to Jamara Ugle-Hagan's shoulders. Corey Weightman will no doubt alleviate some of that pressure but they're not the same side without Naughton and Darcy.This is a very winnable game and it gives us the chance to pt a little mini space between us and those outside the eight if we do grab the four points here.Jojo couldn't let another Rory Lobb moment pass quietly so he has apparently gone gimmicky again. He wouldn't divulge what he's done but we'll evidently find out tomorrow before the game.And if the cost of living is biting, study up and an enter the Mi Casa Property Boutique metres gained competition. We have a jackpot so there is $100 spend at 2 Brothers Foods on offer or a $200 spend if you go one of the two specials, Bailey Banfield or Rory Lobb.Bounce down is not far off so we better get into it. Big second half of the season starts now so feel free to come along for the ride. You'll learn nothing, you'll throw 40 minutes away and you might even be lesser for it but what if that isn't the case? Can you really take that chance? Support the Show.

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

romans greek human associate professor egyptian mathematics george mason university fractions

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

pies candy bars fractions pecan pie

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

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

teaching teach engaging methods approaches fractions

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

Unlocking the Joy of Math: Innovative Approaches for Beginners

Unschooled Unleashed: Unlock Your Child's Genius

Play Episode Listen Later Jan 29, 2024 33:04

Ep 188: Fractions - The Whole Matters

christmas uk podcasts cancer joke grades heart disease cardiologists ecg christmas episode fractions blood flow

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

Let me bore you to sleep - Jason Newland

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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Red Journal Podcast November 15, 2023: Breast Cancer: Fields and Fractions

ASTRO Journals

Play Episode Listen Later Oct 12, 2023 47:28

Editor-in-Chief Sue Yom co-hosts with Dr. Jean Wright, Breast Section Editor and Associate Professor of Radiation Oncology and Director of the radiation oncology Breast Cancer Program at Johns Hopkins University. Joining is Dr. Juliane Hörner-Rieber, Managing Senior Physician and Associate Professor at the Department of Radiation Oncology of Heidelberg University Hospital, who was supervising author of an article published this month, "Non-inferiority of Local Control and Comparable Toxicity of Intensity-modulated Radiotherapy With Simultaneous Integrated Boost in Breast Cancer: 5-year Results of the IMRT-MC2 Phase III Trial." Also joining is is Dr. Danielle Rodin, Associate Section Editor, radiation oncologist at Princess Margaret Cancer Centre, and Assistant Professor at the University of Toronto, who first-authored our Oncology Scan this month, "The Internal Mammary Node Irradiation Debate in Node-Positive Breast Cancer: Case Closed."

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#856 Let Me Bore You To Sleep “NEVER NEEDED FRACTIONS” Jason Newland (27th JUNE 2022)

Play Episode Listen Later Oct 6, 2023 63:51

#856 Let Me Bore You To Sleep “NEVER NEEDED FRACTIONS” Jason Newland (27th JUNE 2022) by Jason Newland

sleep bore new land fractions jason newland

#92 - Who Are Your Favorite Television Neighbors?

Last Call Trivia Podcast

Play Episode Listen Later Sep 12, 2023 47:13

Episode #92 of the Last Call Trivia Podcast begins with a round of general knowledge questions. Then, we're bringing things full circle with a round of Fractions Trivia!Round OneThe game starts off with an Alcohol Trivia question about the French liqueur that's used in both a Cadillac margarita and a B-52 shot.Next, we have a Children's Book Trivia question about the titular book series characters that have a younger sister named Sally and pets named Spot and Puff.The first round concludes with a Sports Trivia question about the basketball star who changed his name in 1971.Bonus QuestionToday's Bonus Question is a follow-up to the Sports Trivia question from the first round.Round TwoOur Team wouldn't be complete without you, so join us for our theme round of Fractions Trivia!The second round begins with an Anatomy Trivia question that asks the Team to identify the body part that roughly half a percent of the population has an extra of.Next, we have a Buildings Trivia question about an iconic skyscraper on Fifth Avenue in New York City.Round Two concludes with an Astronomy Trivia question about the planet that was the third to have a ring system discovered.Final QuestionFor the Final Question of today's game, the category of choice is Television. Won't you be our neighbor?The Trivia Team asked to name four TV shows based on the names of the main characters' neighbors.Hey Trivia fans, we'd love to hear what you think about the Last Call Trivia Podcast! Share your thoughts with us in this short survey: https://forms.gle/9f5HqDV5CLPWkjoZ9To learn more about how Last Call Trivia can level up your events, visit lastcalltrivia.com/shop today!

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Practical Ways to Build Strength-Based Math Classroom - Guest: Beth Kobett, EdD

Kid Friendly Joke Of The Day

Play Episode Listen Later Sep 7, 2023 24:28

Rounding Up Season 2 | Episode 1 – Practical Ways to Build Strengths-based Math Classrooms Guest: Beth Kobett Mike Wallus: What if it were possible to capture all of the words teachers said or thought about students and put them in word clouds that hovered over each student throughout the day? What impact might the words in the clouds have on students' learning experience? This is the question that Beth Kobett and Karen Karp pose to start their book about strengths-based teaching and learning. Today on the podcast, we're talking about practices that support strengths-based teaching and learning and ways educators can implement them in their classrooms. Mike: Hey, Beth, welcome to the podcast. Beth Kobett: Thank you so much. I'm so excited to be here, Mike. Mike: So, there's a paragraph at the start of the book that you wrote with Karen Karp. You said: ‘As teachers of mathematics, we've been taught that our role is to diagnose, eradicate, and erase students' misconceptions. We've been taught to focus on the challenges in students' work rather than recognizing the knowledge and expertise that exist within the learner.' This really stopped me in my tracks, and it had me thinking about how I viewed my role as a classroom teacher and how I saw my students' work. I think I just want to start with the question, ‘Why start there, Beth?' Beth: Well, I think it has a lot to do with our identity as teachers, that we are fixers and changers and that students come to us, and we have to do something. And we have to change them and make sure that they learn a body of knowledge, which is absolutely important. But within that, if we dig a little bit deeper, is this notion of fixing this idea that, ‘Oh my goodness, they don't know this.' And we have to really attend to the ways in which we talk about it, right? For example, ‘My students aren't ready. My students don't know this.' And what we began noticing was all this deficit language for what was really very normal. When you show up in second grade, guess what? There's lots of things you know, and lots of things you're going to learn. And that's absolutely the job of a teacher and a student to navigate. So, that really helped us think about the ways in which we were entering into conversations with all kinds of people; teachers, families, leadership, and so on, so that we could attend to that. And it would help us think about our teaching in different ways. Mike: So, let's help listeners build a counter-narrative. How would you describe what it means to take a strengths-based approach to teaching and learning? And what might that mean in someone's daily practice? Beth: So, we can look at it globally or instructionally. Like, I'm getting ready to teach this particular lesson in this class. And the counter-narrative is, ‘What do they know? What have they been showing me?' So, for example, I'm getting ready to teach place value to second-graders, and I want to think about all the things that they've already done that I know that they've done. They've been grouping and counting and probably making lots of collections of 10 and so on. And so, I want to think about drawing on their experiences, A. Or B, going in and providing an experience that will reactivate all those prior experiences that they've had and enable students to say, ‘Oh yeah, I've done this before. I've made sets or groups of 10 before.' So, let's talk about what that is, what the names of it, why it's so important, and let's identify tasks that will just really engage them in ways that help them understand that they do bring a lot of knowledge into it. And sometimes we say things so well intentioned, like, ‘This is going to be hard, and you probably haven't thought about this yet.' And so, we sort of set everybody on edge in ways that set it's going to be hard, which means, ‘That's bad.' It's going to be hard, which means, ‘You don't know this yet.' Well, why don't we turn that on its edge and say, ‘You've done lots of things that are going to help you understand this and make sense of this. And that's what our job is right now, is to make sense of what we're doing.' Mike: There's a lot there. One of the things that I think is jumping out for me is this idea is multifaceted. And part of what we're asking ourselves is, ‘What do kids know?' But the other piece that I want to just kind of shine a flashlight on, is there's also this idea of what experiences have they had—either in their home life or in their learning life at school—that can connect to this content or these ideas that you're trying to pull out? That, to me, actually feels like another way to think about this. Like, ‘Oh my gosh, we've done partitioning, we've done grouping,' and all of those experiences. If we can connect back to them, it can actually build up a kid's sense of, like, ‘Oh, OK.' Beth: I love that. And I love the way that you just described that. It's almost like positioning the student to make those connections, to be ready to do that, to be thinking about that and providing a task or a lesson that allows them to say, ‘Oh!' You know, fractions are a perfect example. I mean, we all love to use food, but do we talk about sharing? Do we talk about when we've divided something up? Have we talked about, ‘Hey, you both have to use the same piece of paper, and I need to make sure that you each have an equal space.' I've seen that many times in a classroom. Just tweak that a little bit. Talk about when you did that, you actually were thinking about equal parts. So, helping students … we don't need to make all those connections all the time because they're there for students and children naturally make connections. That's their job ( chuckles ). It really is their job, and they want to do that. Mike: So, the other bit that I want to pick up on is the subtle way that language plays into this. And one example that really stood out for me was when you examined the word ‘misconception.' So, talk about this particular bit of language and how you might tweak it or reframe it when it comes to student learning. Beth: Well, thank you for bringing this up. This is a conversation that I am having consistently right now. Because this idea of misconception positions the student. ‘You're wrong, you don't understand something.' And again, let's go back to that again, ‘I've got to fix it.' But what if learning is pretty natural and normal to, for example, think about Piaget's conservation ideas, the idea that a young child can or can't conserve based on how the arrangement. So, you put in a, you know, five counters out, they count them and then you move them, spread them out and say, ‘Are they the same, more or less?' We wouldn't say that that's a misconception of a child because it's developmental. It's where they are in their trajectory of learning. And so, we are using the word misconception for lots of things that are just natural, the natural part of learning. And we're assuming that the student has created a misunderstanding along the way when that misunderstanding or that that idea of that learning is very, very normal. Beth: Place value is a perfect example of it. Fractions are, too. Let's say they're trying to order fractions on a number line, and they're just looking at the largest value wherever it falls, numerator, denominator, I'm just throwing it down. You know, those are big numbers. So, those are going to go at the end of a number line. But what if we said, ‘Just get some fraction pieces out'? That's not a misconception 'cause that's normal. I'm using what I've already learned about value of number, and I'm throwing it down on a number line ( chuckles ). Um, so it changes the way we think about how we're going to design our instruction when we think about what's the natural way that students do that. So, we also call it fragile understanding. So, fragile understanding is when it's a little bit tentative. Like, ‘I have it, but I don't have it.' That's another part, a natural part of learning. When you're first learning something new, you kind of have it, then you've got to try it again, and it takes a while for it to become something you're comfortable doing or knowing. Mike: So, this is fascinating because you're making me think about this, kind of, challenge that we sometimes find ourselves facing in the field where, at the end of a lesson or a unit, there's this idea that if kids don't have what we would consider mastery, then there's a deficit that exists. And I think what you're making me think is that framing this as either developing understanding or fragile understanding is a lot more productive in that it helps us imagine what pieces have students started to understand and where might we go next? Or like, what might we build on that they've started to understand as opposed to just seeing partial understanding or fragile understanding from a deficit perspective. Beth: Right. I love this point because I think when we think about mastery, it's all or nothing. But that's not learning either. Maybe on an exam or on a test or on assessment, yes, you have it or you don't have it. You've mastered or you haven't. But again, if we looked at it developmentally that ‘I have some partial understanding or I have it and … I'm inconsistent in that,' that's OK. I could also think, ‘Well, should I have a task that will keep bringing this up for students so that they can continue to build that rich understanding and move along the trajectory toward what we think of as mastery, which means that I know it now, and I'm never going to have to learn it again?' I don't know that all things we call mastery are actually mastered at that time. We say they are. Mike: So, I want to pick up on what you said here because in the book there's something about the role of tasks in strengths-based teaching and learning. And specifically, you talk about ‘the cumulative impact that day-to-day tasks have on what students think mathematics is and how hard and how long they should have to work on ideas so that they make sense.' That kind of blows me away. Beth: Well, I want to know more about why it blows you away. Mike: It blows me away because there's two pieces of the language. One is that the cumulative impact has an effect on what students actually think mathematics is. And I think there's a lot there that I would love to hear you talk about. And then also this second part, it has a cumulative impact on how hard and how long kids believe that they should have to work on ideas in order to have them be sensible. Beth: OK, thank you so much for talking about that a little bit more. So, there's two ways to think about that. One is, and I've done this with teams of teachers, and that's bring in a week's worth of tasks that you designed and taught for two weeks. And I call this a ‘task autopsy.' It's a really good way because you've done it. So, bring it in and then let's talk about, do you have mostly conceptual ideas? How much time do students get to think about it? Or are students mimicking a procedure or even a solution strategy that you want them to use or a model? Because if most of the time students are mimicking or repeating or modeling in the way that you've asked them, then they're not necessarily reasoning. And they're building this idea that math means that ‘You tell me what I'm supposed to do, I do it, yay, I did it.' And then we move on to the next thing. Beth: And I think that sometimes we have to really do some self-talk about this. I show what I value and what I believe in those decisions that I'm making on a daily basis. And even if I say, ‘It's so important for you to reason, it's so important for you to make sense of it.' If all the tasks are, ‘You do this and repeat what I've shown you,' then students are going to take away from that, that's what math is. And we know this because we ask students, ‘What is math?' Math is, ‘When the teacher shows me what to do, and I do it, and I make my teacher happy.' And they say lots of things about teacher pleasing because they want to do what they've been asked to d,o and they want to repeat it and they want to do well, right? Or do they say, ‘Yeah, it's problem-solving. It's solving a problem, it's thinking hard. Sometimes my brain hurts. I talk to other students about what I'm solving. We share our ideas.' We know that students come away with big impressions about what math means based on the daily work of the math class. Mike: So, I want to take the second part up now because you also talk about what I would call ‘normalizing productive struggle' for kids when they're engaged in problems. What does that mean and what might it sound like for an educator on a day-to-day basis? Beth: So, I happened to be in a classroom yesterday. It was a fifth-grade classroom, and the teacher has been really working on normalizing productive struggle. And it was fabulous. I just happened to stop in, and she stopped everything to say, ‘We want to have this conversation in front of you.' And I said, ‘All right, go for it.' And the question was, ‘What does productive struggle feel like to you and why is it important?' That's what she asked her fifth-graders. And they said, ‘It feels hard at first. And uh, amazing at the end of it. Like, you can't feel amazing unless you've had productive struggle.' We're taking away that opportunity to feel so joyous about the mathematics that we're learning because we got to the other side. And some of the students said, ‘It doesn't feel so good in the beginning, but I know I have to remember what it's going to feel like if I keep going.' I was blown away. I mean, they were like little adults in there having this really thoughtful conversation. And I asked her what … she said, ‘We have to stop and have this conversation a lot. We need to acknowledge what it feels like because we're kind of conditioned when we don't feel good that somebody needs to fix it.' Mike: Yeah, I think what hits me is there's kind of multiple layers we consider as a practitioner. One layer is, do I actually believe in productive struggle? And then part two is, what does that look like, sound like? And I think what I heard from you is, part of it is asking kids to engage with you in thinking about productive struggle, that giving them the opportunity to voice it and think about it is part of normalizing it. Beth: It's also saying, ‘You might be feeling this way right now. If you're feeling like this,' like for example, teaching a task and students are working on a task trying to figure out how to solve it and, and it's starting to get a little noisy and hands start coming up, stopping the class for a second and saying, ‘If you're feeling this way, that's an OK way to feel,' right? ‘And here's some things we might be thinking about. What are some strategies'—like re-sort-of focusing them on how to get out of that instead of me fixing it—like, ‘What are some strategies you could think about? Let's talk about that and then go back to this.' So, it's the teacher acknowledging. It's allowing the students to talk about it. It's allowing everybody … it's not just making students be in productive struggle, or another piece of that is ‘just try harder.' That's not real helpful. Like, OK, ‘I just need you to try harder because I'm making you productively struggle.' I don't know if anyone has had someone tell them that, but I used to run races and when someone said, ‘Try harder' to me, I'm like, ‘I'm trying as hard as I can.' That isn't that helpful. So, it's really about being very explicit about why it's important. Getting students to the other side of it should be the No. 1 goal. And then addressing it. ‘OK, you experienced productive struggle, now you did it. How do you feel now? Why is it worth it?' Mike: I think what you're talking about feels like things that educators can put into practice really clearly, right? So, there's the fron- end conversation maybe about normalizing. But there's the backend conversation where you come back to kids and say, ‘How do you feel once this has happened? It feels amazing.' This is why productive struggle is so important because you can't get to this amazingness unless you're actually engaged in this challenge, unless it feels hard on the front end. And helping them kind of recalibrate what the experience is going to feel like. Beth: Exactly. And another example of this is this idea of … so I had a pre-service teacher teaching a task. She got to teach it twice. She taught it in the morning. Students experienced struggle and were puffed up and running around, so engaged when they solved it. Beyond proud. ‘Can we get the principal in here? Who needs to see this, that we did this?' And then she got some feedback to reduce the level of productive struggle for the second class based on expectations about the students. And she said the engagement, everything went down. Everything went down, including the level of productive struggle went way down. And so, the excitement and joy went way down, too. And so, she did her little mini-research experiment there. Mike: So, I want to stay on this topic of what it looks like to enact these practices. And there are a couple practices in the book that really jumped out at me that I'd like to just take one at a time. So, I want to start with this idea of giving kids what you would call a ‘walk-back option.' What's a walk-back option? Beth: So, a walk-back option is this opportunity once you've had this conversation—or maybe one-on-one, or it could be class conversation—and a walk-back option is to go look at your work. Is there something else that you'd like to change about it? One of the things that we want to be thinking about in mathematics is that solutions and pathways and models and strategies are all sort of in flux. They're there, but they're not all finished all the time. And after having some conversation or time to reason, is there something that you'd like to think about changing? And really building in some of that mathematical reflection. Mike: I love that. I want to shift and talk about this next piece, too, which is ‘rough-draft thinking.' So, the language feels really powerful, but I want to get your take on, what does that mean and how might a teacher use the idea of rough-draft thinking in a classroom? Beth: So rough-draft thinking is really Mandy Jansen's work that we brought into the strengths work because we saw it as an opportunity to help lift up the strengths that students are exhibiting during rough-draft thinking. So, rough-draft thinking is this idea that most of the time ( chuckles ), our conversations in math as we're thinking through a process is rough, right? We're not sure. We might be making a conjecture here and there. We want to test an idea. So, it's rough, it's not finished and complete. And we want to be able to give students an opportunity to do that talking, that thinking and that reasoning while it is rough, because it builds reasoning, it builds opportunities for students to make those amazing connections. You know, just imagine you're thinking through something, and it clicks for you. That's what we want students to be able to do. So, that's rough-draft thinking and that's what it looks like in the math classroom. It's just lots of student talk and lots of students acknowledging that ‘I don't know if I have this right yet, but here's what I'm thinking. Or I have an idea, can I share this idea?' I watched a pre-service teacher do a number talk and a student said, ‘I don't know if this is going to work all the time, but can I share my idea?' Yes, that's rough-draft thinking. ‘Let's hear it. And wow, how brave of you and your strength and risk-taking. Uh, come over here and share it with us.' Mike: Part of what I'm attracted to is even using that language in a classroom with kids, to some degree it reduces the stakes that we traditionally associate with sharing your thinking in mathematics. And it normalizes this idea that you just described, which is, like, reasoning is in flux, and this is my reasoning at this point in time. That just feels like it really changes the game for kids. Beth: What you hear is very authentic thinking and very real thinking. And it's amazing because even very young children—young children are very at doing this. But then as you move, students start to feel like their thinking has to be polished before it's shared. And then that gives other students who may be on some other developmental trajectory in their understanding, so much more afraid to share their rough-draft thinking or their thoughts or their ideas because they think it has to be at the polished stage. It's very interesting how this sort of idea has developed that you can't share something that you think in math because it's got to be right and completed. And everything's got to be perfect. And before it gets shared, because, ‘Wait, we might confuse other people.' But students respond really beautifully to this. Mike: So, the last strategy that I want to highlight is this one of a ‘math amendment.' I love the language again. So same question, how does this work? What does it look like? Beth: OK, so how it works is that you have done some sharing in the class. So, for example, you may have already shared some solutions to a task. Students have been given a task they're sharing, they may be sharing a pair-to-pair share or a group-to-group share, something like that. It could be whole class sharing. And then you say, ‘Hmm, you've heard lots of good ideas today, lots of interesting thinking and different strategies. If you'd like to provide a math amendment, which is a change to your solution in addition, something else that you'd like to do to strengthen it, you can go ahead and do that and you can do it in that lesson right there.' Or what's really, what we're finding is really powerful, is to bring it back the next day or even a few days later, which connects us back to this idea of what you were saying, which is, ‘Is this mastered? Where am I on the developmental trajectory?' So, I'm just strengthening my understanding, and I'm also hearing … I'm understanding the point of hearing other people's ideas is to go and try them out and use them. And we're really allowing that. So, this is take, this has been amazing, the math amendments that we're seeing students do, taking someone else's idea or a strategy and then just expanding on their own work. And it's very similar to, like, a writing piece, right? Writing. You get a writing piece and you polish and you polish. You don't do this with every math task that you solve or problem that you solve, but you choose and select to do that. Mike: Totally makes sense. So, before we go, I have the question for you. You know, for me this was a new idea. And I have to confess that it has caused me to do a lot of reflection on language that I used when I was in the classroom. I can look back now and say there are some things that I think really aligned well with thinking about kids' assets. And I can also say there are points where, gosh, I wish I could wind the clock back because there are some practices that I would do differently. I suspect there's probably a lot of people where this is a new idea that we're talking about today. What are some of the resources that you'd recommend to folks who want to keep learning about strengths-based or asset-based teaching and learning? Beth: So, if they're interested, there's several … so strengths-based or asset-based is really the first step in building equity. And TODOS, they use the asset-based thinking, which is mathematics for all organization. And it's a wonderful organization that does have an equity tool that would be really helpful. Mike: Beth, it has been such a pleasure talking to you. Thank you for joining us. Beth: Thank you so much. I appreciate it. It was a good time. 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

education talk writing students math classroom practical ways fractions piaget build strength mike it mike yeah mike so mike there beth it karen karp beth so

Fractions

Play Episode Listen Later Aug 29, 2023 1:11

Fractions --- Support this podcast: https://podcasters.spotify.com/pod/show/kid-friendly-joke-of-the-day/support

fractions

#11: De Interieur Club Zomergasten: Ontwerper Osiris Hertman

De Interieur Club Podcast

Play Episode Listen Later Aug 21, 2023 20:17

Osiris Hertman studeerde af aan de Design Academy in Eindhoven en ging daarna aan de slag bij Marcel Wanders. Al snel had hij door dat hij zijn eigen designstudio wilde. In De Interieur Live Talks dit jaar tijdens Design District in Rotterdam zei Osiris: "Ik wil die bakker zijn voor wie je speciaal 3 kilometer omfietst. Omdat hij de lekkerste croissants van de stad heeft." Osiris ontwerpt wereldwijd het interieur voor particuliere woningen alsook zakelijke en commerciële projecten zoals showrooms, restaurants, hotels en wellness resort. Door zijn ervaring met highend interieurs maakt Osiris daarnaast designs voor woonmerken zoals Enzo Pellini acoustic walls, Fractions shutters, JEE-O douches & kranen. De Interieur Club is hét inspiratieplatform voor de interieurprofessional. Ben jij interieurstylist, interieurontwerper of architect? Of werkzaam bij een interieurbedrijf? De Interieur Club zorgt voor verdieping en connecties zodat jij stappen kan maken als interieurondernemer. Wij bieden de wekelijkse interieurpodcast, De Interieur Club Academie met cursussen, netwerkborrels en inspiratieblogs zodat jij jezelf kan ontwikkelen. Iedere week een inspirerend en informatief gesprek voor de interieurprofessional. Kijk voor meer informatie en de agenda onze website: ⁠www.deinterieurclub.com

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Ep 162: Comparing Fractions Pt 2

Play Episode Play 34 sec Highlight Listen Later Jul 25, 2023 17:03 Transcription Available

What are some important relationships that students can use to do Real Math with fractions? In this episode Pam and Kim discuss some important relationships that can help develop deeper understanding about fractions for fractional equivalence and comparison.Talking Points:A listener's reviewUsing benchmark fractions to compareUsing a number line to compare fractionsHow does understanding unit fractions help compare ugly fractions?Why we use symbols to represent complex ideasCheck out our social mediaTwitter: @PWHarrisInstagram: Pam Harris_mathFacebook: Pam Harris, author, mathematics educationLinkedin: Pam Harris Consulting LLC

comparing fractions

Ep 161: Comparing Fractions Part 1