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Episode 150: Rethinking Fractions – An Interview with Shelley Yearley

Oct 11, 2021 | Podcast | 0 comments

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Today we speak with Shelley Yearley all about how we can rethink fractions in the mathematics classroom! Shelley is a fellow Ontario educator, researcher and author who has held many roles over her career which you will hear all about today including her most recent role as she has returned to the classroom!

Stick with us and you learn why you should focus on unit fractions before rushing to the algorithm for operating with fractions; what the Fraction Learning Pathway is and how it can help you in your classroom; and, why students & many adults struggle with fractions and what we can do about it.

You’ll Learn

  • How we can understand fractions at a deeper level (as educators)
  • Why students & many adults struggle with fractions and what we can do about it.
  • Why you should ask students what they know
  • Why you should focus on unit fractions before any operations with fractions
  • What is the Fraction Learning Pathway and how you can use it in your classroom.
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FULL TRANSCRIPT

Shelley Yearley: I think it might be fair to say a recurrent theme has been that it's very important to make connections both for teachers and students. And I think one of the things that we do a lot is we focus on making connections to, I don't know, something we call daily life or real world connections. And that is quite ambiguous because none of us have a common daily life, but I think crosstalk

Kyle Pearce: Hey, Hey, Math Moment Makers, today, we speak with Shelley Yearly all about how we can rethink fractions in the mathematics classroom. Shelley is a fellow Ontario educator, research, and author, and she's held many roles over her career in education which you're going to hear about in today's episode, including her most recent role as she has returned to the classroom.

Jon Orr: Yeah, stick with us, and you're going to learn about why you should focus on unit fractions before rushing to algorithms for operating with fractions, what the fraction learning pathway is and how it can help you in your classroom. And finally, why students and many adults struggle with fractions and what we can do about it. Kyle, let's go.

Kyle Pearce: All right, let's do this. Welcome to the Making Math Moments that Matter Podcast, I'm Kyle Pearce.

Jon Orr: And I'm Jon Orr, and we are from makemathmmoments.com, and together-

Kyle Pearce: With you the community of Math Moment Makers worldwide who want to build and deliver math lessons that spark curiosity-

Jon Orr: Fuel sense-making-

Kyle Pearce: And ignite those teacher moves. Welcome my Math Moment Maker friends to an episode where we get to bring on another fellow Ontario educator, but not just educator, a researcher, also an author and just an all round awesome person. You are going to do a lot of learning today right there John.

Jon Orr: For sure, for sure, and you might be looking at the length of the episode and you're like, "Oh, sometimes those 30 minute episodes hit the right spot, those 40 minute episodes are perfect." And you're like, "This one's a little longer guys." And it's because we've had such a great conversation and that you are going to get a lot out of this. So stick with us and hey, let's just go right now. Let's get right into it.

Kyle Pearce: Hey, hey there Shelley, thanks for joining us here on the Making Math Moments that Matter Podcast. We're really excited to have an Ontario friend. We were just discussing. I don't think we're dealing with the same thunderstorm that's going through. I think you might have a different one in your part of Ontario. However-

Jon Orr: It's a big one.

Kyle Pearce: As soon as I said thunderstorm, maybe it's the same one, who knows? But we are super excited to have you on the show. And we really want to dive into a bit of your backstory and also dive into some of the work around a concept that is near and dear to both Jon and my heart. So how are you doing tonight and how are things on your end of Ontario?

Shelley Yearley: I'm well, I live in a beautiful part of Ontario. I don't know if there a non-beautiful part, but I think where I live is particularly lovely. But tonight, yes, we are facing severe thunderstorm warnings, watches and tornado warnings. It's very interesting.

Jon Orr: It always confuses me the warnings of the watches.

Shelley Yearley: Yeah. I have to read carefully.

Kyle Pearce: Yeah, I'm sure it has something to do with probabilities which could be a whole other episode here.

Shelley Yearley: Yeah, actually I was helping a student today with some math and we were talking about statistics and just how we use statistics without even realizing that we're using statistics. So there you go, the warnings and the watches are definitely a great example. So things are good here.

Kyle Pearce: Good to hear, good to hear. Tell us a little bit more about yourself, tell our listeners a little bit where specifically in Ontario are you, and then let's get into a little bit of backstory along Shelley. How long have you been teaching and what that teaching journey looked like for you that led you to where you are now?

Shelley Yearley: Sure. So I am in Muskoka, I live in Huntsville. I was actually born here and was raised here and thought I was never coming back when I was a teenager and I've ended up living here my life. So this is all good. I think in terms of my personal journey, I really have so much gratitude for the opportunities that I've been allowed to pursue through my career. So I started out as a high school student.
I wasn't really sure what I wanted to do. I was okay in math. I wouldn't say I saw myself as a particularly strong math student, and I know from listening to each of you that that's a bit of a common thread amongst us perhaps. I also knew though that I really didn't want to do a lot of writing. So my father thought I should be an accountant and I did not want to do that. And so teaching was an option that was suggested to me and I thought, "Okay, why not?"
And then when I went to university, I did a joint degree in math and computers thinking in both cases that would get me out of my writing career. I have the irony, I'm sure it's not lost on you.

Kyle Pearce: Yeah. True, true.

Shelley Yearley: So I graduated, I actually trained for primary junior education and I did a concurrent program. So I had lots of experience in primary junior and a little bit in intermediate. And then couldn't get a job after I graduated in primary junior, but could get a job at the local high school. So I went back, got my intermediate and senior qualifications and then was successful in that endeavor to get a job.
But when I got my job, it was ... Destreaming was taking a hold at that time. And there was a lot of turnover in the math department. And so very quickly I became the department head very early in my career. And then somehow, and I'm still not sure how got invited to a meeting at the ministry when they were starting to work on tips and went to the meeting and then ended up being involved very heavily in the tips project which is one thing led to another.

Jon Orr: Now just, I want to jump in here. A lot of our listeners, Shelley wouldn't know what the tips are. Not because everybody in Ontario and especially in the middle school ages know in Ontario, or may have seen that resource. I know that I have, and it was a great resource when I first started teaching, but we have majority of listeners here listening to us right now are from United States or from Australia. Would you mind, we'll digress here a moment because I think that's a great resource.

Kyle Pearce: Yeah. Yeah. And I've got to say I get so many educators like when the elementary curriculum changed last year and now with destreaming back. So there's a lot we can dig into here based on what you just said, but I get emails so often educators ask me, do we have any new resources that align with the curriculum like, and most often the blank that's filled in there, that is like tips. And in particular, our grade nine teachers who have taught maybe the grade nine applied course.
So tell everybody what that is. I think it's still a really valuable resource even though maybe it might not be fully connected to the Ontario curriculum in that order, but where did that start? And I'm actually curious for my own sake because I used that resource for so many years. How did that come to be and what is?

Shelley Yearley: So it's an acronym, and if my memory serves me correctly, I think it's Targeted Implementation and Programming Supports, but I might be slightly wrong with the P there. Basically Myrna Engels was working at the ministry and she was a secondary teacher from the Markham area which is now York board. But at that time in Ontario, we had a curriculum which in Ontario states the learning goals for the students, I guess is one way to describe it.
Unlike in the states where the textbook is the curriculum sometimes. So our curriculum is a list of things that students will know and do by the end of a grade with no teaching support. So the tips was one of the earlier that we had profiles before that which were also a similar project that were designed to help teachers bridge the gap from this list of expectations to actually programming and assessing student understanding in the classroom.
So as many of the projects I've been involved in seem to do, this project started relatively small and then ballooned out to this mass of undertaking. But our first task was merely to look at the curriculum and basically, we didn't call it that then, but spiral or sequence that curriculum in a way that would make sense and not necessarily be restricted to the strand. So that was task one.
And then task two was to get a group of writers, and we've been so fortunate in Ontario for so many years to have writers come from the classroom to do the writing and not be the ministry resource. So it's actually by teachers for teachers. So we had lessons for a lot of the programs. Some courses we had all, some we had some, and we did grade 7, 8, 9 to start, and then eventually, they did nine, 10 and 11. And when the curriculum changed, they had tips and they called it For RM.
So it was tips for the revised mathematics. And then it evolved into research. And I think I wouldn't be telling a lie if I said that all of the math fees learning tools probably evolved out of that research and programming work that was done where we said, "Oh, we need a virtual manipulative for this." And so then we ended up with people writing and creating virtual manipulatives.

Kyle Pearce: Love it. I love it. Yeah, and there's so many things we're just jotting down notes of links to throw in there. So you've got two segments of listeners or viewers if you're watching on YouTube, whoever's with us on YouTube right now, our Ontario friends who are probably like, "Yes, I've used these tools." And then everybody else outside of Ontario, folks, you definitely want to check out the show notes.
We'll link them up. The beautiful part about tips as well is that it is wide open. So it was a resource that was available. Of course, people could have stumbled upon it using Google or whatnot. But if you're from the US for example, it's unlikely that a random math resource would show up. And then the maths inaudible tools I hear are undergoing a little bit of a revamp right now.
So that HTML 5, I know that Flash was depreciated. So those tools are slowly reemerging. So some good links coming out really early here. And I would love to know Shelley Yearley when we say the math moment that pops into your mind. You had already referenced that you didn't really see yourself as being necessarily comfortable ... No, I shouldn't say comfortable, but-

Shelley Yearley: No, you can say comfortable.

Kyle Pearce: Yeah, yeah. With mathematics, I would argue I was that kid. I just meandered through and I'm going to call it a bit of luck and maybe a bit of opportunity for my family. I ended up doing okay. What's the math moment that pops into your mind when you think back to your experience in K to 12 or K to beyond 12 mathematics?

Shelley Yearley: Well, it's a very interesting question for me, and it's not one that I easily answer because I feel like I had really good teachers in school. So that was one thing that I felt I could have picked, but I really wanted to just go by back to that destreaming, the first iteration of destreaming in Ontario. I was a young teacher, newly out of teachers college essentially. And I had this opportunity to go whitewater canoeing in south of Barry's Bay with the school, students from the school would go and then teachers would go to supervise and I had my qualifications, so I could go unsupervised.
And two of my math teaching colleagues also wanted to go on this trip. And so we made a plan that we were going to go on the trip, supervise the students around the campfire at night, have a talk about what destreamed grade nine math might look like next September. And so work together. So it's a moment in my career when you look at the timeline, but it's many moments compiled within that year what we decided to do was ask the kids on the first day what they were interested in, list all the topics on the board and then find the math in their topics and map it back to the curriculum and teach by topic.
So we did building construction, we did astronomy and we did music and then there were others. And so we had nothing, we had old textbooks which we cut up and repurposed parts of. We literally cut them up. This was the old fashioned days where you cut stuff and you glued it on to card stock for kids to work on. So they're so dynamic. And we grouped the kids different ways, and we obviously had a range.
When that destreaming happened, they didn't retain what had been basic pathways so they merged all three pathways for people who aren't in Ontario that would be workplace college and university altogether in one class. And I think that's my math moment, because what it did was math had to all always been to me somewhat theoretical and I could get it, but it was a little bit a guessing game lots of times.
And when we started by looking at the topics of interest of the students and then connecting the math into that topic, not only did it make it really accessible and understandable to me, but what I saw was students starting to value math as a tool that they could use and actually do use in their life instead of a game that you play in school to get a grade. And so for me, that was astounding.
I wouldn't say we did it perfectly. I wouldn't say that everybody loved it, and there were certainly times where we were exhausted and slightly burnt out, but it was a lot of work. But I do think that seeing kids have those moments where they could go home and talk to their parents about what did in class and their parents could say, "Yeah, actually I did the same sort of thing at work today, but on a bigger scale." It was really powerful.

Jon Orr: Yeah, that's such a great memory and such a cool idea to change what may have been in our classrooms and the way math class may have looked before that moment and with the switching into destreaming way back then and back in the 90's and it's come full circle again, we're doing that this year. We're going back to destreaming for our grade nine program.
And I'm curious, you tried that so many years ago, and I'm wondering how much of that stuck with you from year to year to year and where you are now? How much of an influence was that one year carried forward about how getting the students hooked into what they want to learn about and then tying it back to something practical? So I'm just curious what that looked like stretching forward to where your role is in education now.

Shelley Yearley: I would say it's being immense on many levels. One of the biggest things it did for me was I just became a real risk taker because that's what we did every single day. And I didn't get hung up in a perfect lesson and I could stand in front of the class and say, "Okay, well, we did that yesterday. It worked for some of you, but not for others. So we're going to come back and do something different today."
And I really think that it helped me to understand the variety of learning needs within the classroom, really young in my career. I think it was my first or second or maybe third year. I think I carried all that. The spirit of all of that forward, it's been ... I was out of the classroom for 19 years. So I worked at the board for just over a decade. And then I went and did the research at the ministry and then I returned to the classroom. Well, it's awesome now because this is my third September back. It wasn't such a great experience.

Jon Orr: You come back at the right time.

Shelley Yearley: Yeah, on the first year crosstalk.

Jon Orr: I came back last three school years have been quite the school years.

Shelley Yearley: That's great. Yeah, I came back and there was a strike and then a pandemic and there were some pretty severe personal issues. Not all of which have been resolved and everybody's fine and healthy, but there were some pretty severe personal issues at home that all erupted the minute I walked into classroom and when I left the classroom 19 years ago, I was taking attendance on a Scantron sheet. And when I came back, everything was digital.
So it was a big upheaval, but I do think that that idea about asking kids and giving kids choice, I haven't been able to do it to that extent because I'm just trying to figure out it's like being a brand new teacher again in many ways, but giving kids choice and not being so uptight about everybody doing the same thing and letting them find the math where they find the interest has definitely stayed with me.

Kyle Pearce: I love it. I love it. There's something interesting that you had said too. I know it was early in your career when that first attempt at destreaming happened. We'd probably argued maybe it was for different reasons and purposes. And I'd like to believe that we've done so much learning to bring us to where we are today where we're attempting this destreaming again and what I believe is for the right reasons, but something that really resonated with me is what you learned from that experience because I'm going to probably argue that educators today feel like they didn't receive a lot of support because we're coming out of a pandemic, curriculum has changed.
Lots of changes are going on and it just happened. But the part that really resonated with me and I believe wholeheartedly that a lot of educators are going to look back on this year, this first year of destreaming for them. And they're going to see how much they grew from that experience. So you get put into this spot where you feel like, "Oh my gosh, I have no idea." I have a colleague who said he was losing sleep in the summer because we care so much and has done things so well and this particular educator has actually run a really effective grade nine academic and a grade nine applied program.
Students loved going to the classroom and it was just, it was really heavy trying to imagine what that's going to look like and sound like because it's just brand new. But I believe that educators are going to look back on this and they're going to have grown quite a bit and hearing your story really. I hope others who are listening or are teaching these streaming early, and they're wondering like, "How am I going to do this? How am I going to figure this out?" That they will figure it out and it's not going to be perfect like you had said, but we're going to learn a ton and it's going to help impact and influence our career and how we teach moving forward.

Shelley Yearley: Absolutely. And John Ross was a professor at Trent many moons ago, and he had done some re search around consultants and how long it took a consultant to become effective in their role. And it wasn't until the third year that the consultant actually had their feet under them and knew how to maneuver within the system and had a good understanding of the work that they were doing and the strategies to do it. And that has been another thing that I've really held onto because I feel like as a teacher, we're all doing the best we can possibly do for our kids at any given minute.
And we're all hopefully learning a little bit more every day. And so to be graceful with ourselves too, we would be graceful with students if they couldn't do it perfectly the first time. And so we should be graceful with ourselves too.

Jon Orr: Mm-hmm (affirmative). Very true.

Kyle Pearce: Absolutely.

Jon Orr: Yup, very true. We could talk probably all day about destreaming math than we have in the past.

Kyle Pearce: We didn't even have that on the agenda crosstalk.

Shelley Yearley: I know.

Kyle Pearce: Just, but yeah, I think it's super valuable.

Jon Orr: To keep things moving along here. We are eager to dive and focus some of your recent work in math. And in particular, when we hear your name and in collaboration with your colleagues, Kathy Bruce, and Tara Flynn what we always hear or what we always think is fractions. What we love to do is dive into that with you and love to hear, how did you get down this rabbit hole of focusing you've got a book coming out on fractions, you've got some resources.
I think on the ministry websites on fractions. And so I guess before we get into the meat and the potatoes here, we're wondering what led you down this rabbit hole of this is where I should focus my time and effort.

Kyle Pearce: Arguably one of the concepts that most human beings find scary and, but you all, something led you down in. So we're really curious to kind of figure out how did it come to be.

Shelley Yearley: Sure. So again, Myrna Engels was at the ministry and there was an opportunity for some funding and it was called KNEAR I think, and it was a knowledge network funding opportunity. And Kathy Bruce and Myrna Engels wrote a proposal to do some research and they were granted some money to do this work and it was collaborative action research. So they had money to release teachers from classes and meet with them and investigate the area of fractions because it's a known area for students and teachers and adults in general to struggle with, they got them money. And there it is.

Kyle Pearce: Yeah, for our YouTubers, I figured I'd give them something to have a peek at while we're chatting.

Shelley Yearley: So anyways, they got the money and then they needed somebody to work at the ministry arms length to coordinate the research and be the liaison, if you will. And they asked me to do it. Again, just one of those opportunities that I thought I could do, and it was fractions and I was a secondary teacher. So that was one thing I thought I must have some skill in.
Three weeks into the first month, I'm quite open about the fact that I was wondering how I was going to get out of this so that people didn't figure out that I didn't know anything about fractions because once I started digging into it, I realized, and my understanding was really so superficial.

Jon Orr: Would you elaborate on that point just a moment? Like as a high school teacher, why did you all of a sudden think you had this superficial knowledge of fractions?

Shelley Yearley: Well, one of the biggest things was I was thinking about, and I can still see this moment where a student didn't understand. I forget the exact question, but let's say it was three times one-third, and they didn't understand why the answer was one. So that's easy, I just drew three circles on the board and I partitioned them each into thirds. And I just told them, "See? I've got one, two, three, three-thirds."
And within the first week I was reading about how challenging circles are as a representation for fractions. And then also re-doing a lot of reading and really reflecting on the ways that we use fractions, the constructs of fractions that we use. And we just slide between them and we don't really unpack them. So we often talk about fractions as part hole and that's clear, two-thirds of the people on the screen identify as male I would say. And then we do use fractions as ratios.
So especially in high school, we write ratios as a fraction for slope, for example, or for a tangent ratio. But we don't actually tell kids that's a ... Well, okay. I didn't actually tell kids. I know some teachers did, but-

Kyle Pearce: Or we didn't explicitly.

Shelley Yearley: that that's a ratio or part whole.

Jon Orr: Yeah.

Kyle Pearce: I totally can resonate with that. I didn't even realize what I was saying when I said trig ratios, but then I wrote it as one quantity over another quantity. And I just was like, "Well, that's a ratio." But then when I defined what a ratio was, it didn't really match what I just did. So then it's like, "But don't worry about that, just copy it down and move on." That's the typical approach.

Shelley Yearley: Yeah, yeah. And we had elementary teachers in our group who were using the language, I think it's actually from the Ontario curriculum previously, part of a whole or part of a set and part of a set meant more like a ratio than, and not part of a hole. So we kept surfacing these messy things in our curriculum and in our understanding.

Kyle Pearce: It's so interesting because more and more as I was going down the proportional reasoning rabbit hole which obviously you bump into fractions all along that path. And just as you're mentioning, if you go from curriculum to curriculum, I was blessed to be a part of a symposium of different folks from around north America that came together to essentially discuss different aspects of proportional relationships and proportional reasoning.
And that was one of the things we noted right away was that every curriculum uses its own language which oftentimes contradict each other. And then you go down and you realize that even some of the research suggests this researcher is using that language and then this researcher may not, or maybe doing something that ... It's like but then these don't really make sense.
And it really caused for ... My eyes were wide open at that point. It's very humbling, and I'm sensing that from what you're saying where you go on and you're teaching in a classroom and you believe you know something and you realize, "Oh my gosh, I actually don't have clarity around so many of the ideas that start poking their head out in the junior grades and into those middle grades." And we just gloss over them and move along.

Shelley Yearley: That's right. And even young students we found in our research that when we gave student sets and they were in younger grades, primary, junior, that they would often do a part, part fraction. But then at the very same time, they might use part whole reasoning to describe that fraction. So we had students in one class, it took us forever to figure out what they were doing, but they had taken pattern blocks.
So let's say they had four yellow hexagons and six red trapezoids. So they wrote the fraction four-six. And then on the far side of the page, they had this definition of fraction and that enumerator tells you the count and the denominator tells you how many you have in the set in total. And so we said, "Well, but you have four six here. So can you connect between this part, whole idea and what was part?" And they didn't see the discrepancy in that definition.

Kyle Pearce: Yeah, that is so fascinating. But at the same time, it's like when I look at it from the other angle, you go, "I guess it makes sense that if we are just doing things and we're hoping, I feel like we get caught doing this often is that we sort of hope everyone caught the drift, especially when we do an investigation, you go tada."

Shelley Yearley: We were hoping.

Kyle Pearce: Yeah, and at the end of the day, it's like, "Well, actually they didn't make that connection." Even though I thought, or you think it's maybe so obvious that, "Hey, you're going to pick up on that." But some do, and some don't and just think of how often that happens in a math class from early years, all the way into high school where enough of those I hope they picked up on the concept without being explicit at the end of a lesson can really be difficult.

Shelley Yearley: Well, and I think that's where part of our research is so powerful is that we not only, and really, the credit on this goes to Dr. Kathy Bruce, and Tara Flynn because they pay as much attention if not more to the teacher content knowledge as they do to the student responses. So when we were doing our research, it was always about working with teachers and building their content knowledge.
And then at the same time, trying to figure out what interventions we could do to support student's learning. And so that when a teacher saw that in a classroom, they wouldn't just say it was wrong and that they would say it's actually mathematically correct, but it's not the answer that I was looking for here because I was hoping you would treat this like a whole set or whatever the case may be.
And we saw that in other examples. We all use a fraction as a quotient to figure out what our score was on a test, but we don't actually ever identify the students or when I say we, I mean in north America, we traditionally don't identify that to students as one of the constructs of a fraction. And so when I was in a very young class and the teacher wrote a fraction on the board and asked the students what they thought that might be about, and one of the student said division and because the teacher didn't have that knowledge, they said, "Oh no, that's not quite right."
And so we really also wanted teachers to understand that we use these fractions on different ways. And we would often say when we were working with teachers, "This is an activity for you. This is not an activity for you to go and do with your students. This is a teacher activity so that you understand." We use fractions in multiple ways and very fluently and not realizing the difference between those.

Jon Orr: Right. And in your research, let's say I'm a teacher and I'm starting to work with, I'm going to getting into the adding of fractions and eventually we'll be getting into the multiplying of fractions, but I'm a teacher at that level and why I'm not throwing at a grade level just because some of our listeners are different. They're at different places in the curriculum or the standards of when they were going to do that. But I'm wondering in your research, if I'm a teacher who has traditionally taught fractions that very procedural way in, in the way I imagine it the way sometimes I see my kids bring home stuff and this is the way fractions are taught about adding fractions and multiplying fractions.
Would you recommend to that listener right now saying like, "This is where you'd want to start." You've mentioned that it definitely has to ... Part of this has to start with educating the teacher and getting their content knowledge up. So there's that piece, but also where would the teacher, what lessons would the teacher start with or what tip or starting point could you get a teacher who wants to make this transition to be less procedural, but start to build this content knowledge and help their kids build that content knowledge.

Shelley Yearley: I would start by asking them what they know and you'll probably be amazed at what they do know. And we've done that in very young classes. We've asked grade four students to just say what they know and kids will describe different things. They say there's an infinite number of fractions between any whole numbers and the teacher's jaw hits the table because the teacher didn't talk about that yet.
So I would start by asking students what they know, and then I think the other thing would be to pick up on the idea of the unit fraction. So a unit fraction is any fraction that a one in the numerator. So one-half, one-third, one-fourth, those are very familiar ones, but it could be one thousandth or one-2,000,562, it doesn't matter, as long as it has a one in enumerator, it's a unit fraction and the power of unit fractions is that every other fraction is sum and a product of those unit fractions.
So if you think of any fraction, three-sevenths, it's one seventh plus one seventh plus one seventh. And if you want to write that as a multiple, it would be three times one-seventh. So when we first started our research, actually we would have said, if I was talking to you 10 years ago, I would've said, "Well ,that in Ontario is like a grade seven, eight concept." But now I actually say that in any location is a primary concept because students right away, as soon as they start representing see that they add fractions by counting up the numerators and retaining that unit of sevenths in the example I did just like they add measurements together.
One centimeter plus two centimeters is three centimeters. I don't change my unit of centimeters. And when they connect that addition and multiplication, then using the inverse operations, connecting subtraction to addition and division to multiplication, make that so much easier.

Kyle Pearce: I love it. I love it. And you took the words out of my mouth. I wanted to talk about the unit fraction. I'm so happy you gave the out example. And I was thinking as Jon was asking that question, I was almost going to jump in and say like, "I'm wondering if you're a teacher, especially if you're let's say middle years or higher and you're going, how do I do this?" A lot of times I think teachers have this pressure.
They feel this pressure that they're so far down the line that they're like, "How do I go all the way back to help make something conceptual?" And I think the unit fraction is such an easy way to start, especially since you gave the example of measurement and centimeters or inches or feet. The examples go on and on and on. I could talk about thousands or I can talk about Xs or all of these ideas.

Shelley Yearley: Pies.

Kyle Pearce: Yeah.

Shelley Yearley: When you get to radian measure.

Jon Orr: As soon as you said this counting, I was seeing a little radian measure in your unit circle too.

Kyle Pearce: Absolutely. To me, that was such an epiphany for us thinking about the unit fraction that it's been right there, it's been there the whole time, but we didn't really know it was there. We didn't really know how helpful it could be, and it also brings me back as well, working in the primary classroom and talking with teachers about why it could be helpful before we use, let's say standard notation to use the actual unit fraction as a unit, writing it like in words.
So one-fifth, then two-fifths. If you ask young children, "How many fifths do I have in total? It's how many cookies do I have in total? How many socks do I have? How many centimeters? How many fifths do I have in total?" And it becomes very intuitive, much like how mathematics really could be if we do things conceptually. So to me that's a mind blowing piece and dare I mentioned the fractions learning pathway is such a great place as well to when I'm thinking, if someone's thinking like, "Well, how do I learn more about this?"
I'm going to steal this right from you and say, "The fractions learning pathway is such a great tool because it's like PD, but also provides activities and resources for students." I'm going to bring it up on the screen. Do you mind giving us maybe a general idea what is the fraction learning pathway and how might an educator dive into this tool and how do you see it being used?

Shelley Yearley: Sure. So as I mentioned, we started the research with Kathy. We had a one year grant. It quickly became an eight year project, well it morphed into an eight project. And then over that time, we just were accumulating so much learning that we wanted to provide a really easy to understand resource for teachers. But actually, can you go not to EduGAINS? Can you go to fractions learning pathways with an S on the end?

Kyle Pearce: .CA right? Yeah, there we go. Perfect.

Shelley Yearley: There that's the one. Okay. So this one. This site's a little bit more updated than the EduGAINS. So we wanted to create a visual reference for teachers so they could just hold this in their head. And when they're instructing, they could use it as a bit of a tool to help see where their students are at. So you'll see there's three colors, the greenest unit fractions, and really emphasizing unit fractions.
And also, just to say looking at the fact that as soon as a student starts to talk about any fraction, they automatically have to think about the compliment which just goes back to your idea about addition. So if I think about having one fourth of something, I'm automatically realizing I don't have the other three-fourths to make the hole. And so that's that implicit addition for students that we can build on.
So you'll see here for unit fractions, I won't read them all, but there are six cells and they're not necessarily linear. They're definitely connected, and we notice that as students move through, they can sometimes leapfrog. We've noticed by highlighting some things over others, comparing fractions then gets into comparing an equivalence. And also that idea of density which is very, was previously very underplayed in our Ontario curriculum. So students didn't actually get a lot of experience with looking for fractions between fractions or between numbers.
And then lastly on this version is the addition and subtraction. And you'll notice just at the left side, it's focusing on counting and composing and decomposing much in the same way that we would work with students around their whole number operation sense. So we really try to build that in so that it is a more natural and transition for students. And then if you scroll down, you'll see the purple which is-

Kyle Pearce: Oh, this is new since I've been here.

Shelley Yearley: Yeah, you got to come to this site, not the EduGAINS. So this is our parallel for multiplication and division. One thing I will just say as often, we think multiplication and division is easier as if we just think about instructing using the algorithm, we might think that's the easier of the two, but you can see the size of that pathway is almost the same as all the other put together.
Now, if you click on any one of these cells, they should be hyperlinked. I'm really hoping this works, good. Perfect. And so each one of those cells is hyperlinked to a page that gives you some more information on that concept. And then you'll see on the right-hand side, most of them have tasks. They also have additional prompts. And then for Ontario folk, they have the cricket connections, but I have to tell you that's to the old curriculum. So you may have to do a little bit of work on that, but we did put some information in there to help teachers with those instructional challenges.

Kyle Pearce: I love it. I love it. This is fantastic. And we've used this in my district quite a bit, and again, has really aided in my own understanding of fractions and what is really happening there. And I couldn't agree more about the operating with multiplication and division. That again, an algorithm is not very helpful. You have to sometimes stop and wonder how often are you multiplying fractions in real life with an algorithm, right? You're probably going to use a calculator.
So there's so much opportunity for reasoning when we actually do the work, when we actually dive in and actually look at what it really means. And all of this work here is so key for us to be gaining a real understanding. And I'm just sitting here thinking if there's an educator listening right now and you're going, "I think we can all raise our hands." Lots of students struggle with fractions.
There is so much even just right here that could get you thinking maybe a little bit differently in helping you so that you stop spinning your tires because I know I've done it in many, many concept areas, but in particular in fractions, you spin your wheels where you just keep doing the same thing over and over again, right? Louder and slower.

Shelley Yearley: Absolutely. With circles.

Kyle Pearce: Yes, yes, exactly. So there's so much to dive into here and I also want to give you an opportunity. I know you have a book coming out with your colleagues here. We'll call them partners in crime. We have to think of a fractional name.

Shelley Yearley: I know, we need something.

Kyle Pearce: The trio, the thirds. I don't know, but tell us a little bit more about the upcoming book and where, and when people might be able to find that.

Shelley Yearley: Sure. I will. I'm going to ask that you actually just, I just want to talk a little bit about this website for one second. So if you go to the top, so right now you're on the pathways and then you can see there's classroom resources. So all of the tasks are organized there and people can go just find different tasks. But then the piece that people might appreciate is the next two links at the top.
Sorry, the last one, the research link, but there are three links in that ... Oh, sorry. So I just noticed, I forgot. I've been writing the book and not working on the website. So I noticed that also in classroom resources, there are maps to show people how they might ... Those planning supports, show people how they might use this resource across the span of a grade. Yeah, so those are all hyperlinked by month concept. We had our teachers track what they were doing and then they collated that information into a sample for people.
And then under the research, we have three really strong lit reviews that we've written and they go into depth about the cultural practices on instruction of fraction. So people may find that interesting to read through. And then we have shorter summaries for people who don't care to read the 50 whatever page, however many pages, those things are. So we have some short or two pages generally speaking, some maybe four.
And for leaders, there are two more tabs there. So if you're running professional learning, there's a couple of tabs or if you're interested in watching some videos, there's some tabs there that you can find more information. So it's really packed full. There's much more here than on EduGAINS. Just we decided to transition to a format that was more AODA compliant, and so there's lots housed here that's not on EduGAINS but you can see there that the focus of that is mostly classroom practice with a bit of professional learning and support for board personnel.
And so then what we did when we did our research was we had a diagnostic assessment that we used with students to do a pre-impose to figure out is what we're doing making any difference at all. And so for the book, what we've done is we've taken those assessments and we've identified eight core concepts for fractions that all evolve around the idea of the unit fraction and the importance of the unit fraction. And so we've taken those questions, those concepts, and we've got two or three tasks for each concept as well.
So the idea is that as a teacher, you have these grade nine destream students in front of you and you don't know where they're at with fractions, you can pick some questions, administer them. We have a nice, straightforward scoring or coding guide that we unpack for you with the questions. And you can figure out where your students are at, where their needs are and then pick the task or tasks that best will build their knowledge the way that we need it to be built. So it's the missing link out of this piece is that assessment and the tasks.

Kyle Pearce: Beautiful. This is great. There is so much more here than the last I had visited, but yeah, this is fantastic. This was when we were really diving in and doing a lot of our learning, we were diving in here. There's a lot there, and now you have this additional piece. There is so much. So we will definitely be linking that up in the show notes. I'm so happy we have the conversation so that now I'm getting re-exposed here to some of the resources.
So thank you so much for that work, and I'm wondering, I know this one's turned into a long one Jon, we got talking again. We've been trying to get better at this, but I'm wondering there Shelley, if we have some educators from all around the world, different grade levels, if there's one thing that you would like them to take away from this conversation today. There's so many different pieces that I'm sure they could walk away with, but what would be the big piece that you'd like them to walk away with today when they think back to this particular episode?

Shelley Yearley: I think it might be fair to say a recurrent theme as being that it's very important to make connections, both for teachers and students. And I think one of the things that we do a lot is we focus on making connections to, I don't know, something we call daily life or real world connections. And that is quite ambiguous because none of us have a common daily life. But I think more than that, it's important to make connections to other content areas for students so that no matter where they end up, they understand if they're counting radian measure or if they're counting pieces of pizza at their counting unit fractions, and they can do that in the same way.
So I feel like in the work that we do, whether we're in the classroom or a district level, or even at a provincial or state level, that helping people to see the deep connections that exists between mathematical concepts is probably one of the most important things that we can do. And I think when we do that with students and we say to a student, "You know what? You are great at counting by centimeters. So let's just take that and apply that to counting by unit fractions, and then let's build on that." We help build student efficacy and we, we know that that is the piece that we need to have in place to have students be successful, so.

Jon Orr: That's a great takeaway, and thank you for sharing that for sure. As Kyle said, we are going to put all the links that you shared here in the show notes page. So if you want to dive into what we've been talking about, head on over to our show notes page, stay tuned for the link to that. Shelley, what can our listeners learn a little bit more about you? You've got that book coming out, maybe give them a heads up of where they can learn about that, but also learn about how they could learn more from you.

Shelley Yearley: Sure. So I think the fractionslearningpathways.ca definitely has a lot of resources on it. The book is called Rethinking Fractions, Eight Core Concepts to Support Assessment and Learning. And it's being published by Pearson education Canada. And I believe it's a January release date, January, 2022, and so they can get more information there.
I am on Twitter, but I've actually taken a little bit of a break from Twitter. For people not in Canada, we are having a federal election, and I just haven't been able to bring myself to spend much time on Twitter these days, but people can reach me through Twitter. I do still monitor the email and stuff, so.

Jon Orr: We'll be putting those links to the books and your Twitter all on the show notes page as well. Shelley, thanks so much for joining us here on the podcast. We always learn so much from our guests, and especially from you tonight and looking forward to putting these things into practice. And I hope our listeners are as well. So thanks so much for joining us.

Shelley Yearley: Thank you. And thank you for all of the time and effort that you put into this. I've been enjoying so many of the podcasts and I find it's a great way to refresh and reenergize myself. So I'm delighted to be a part of it. Thank you.

Kyle Pearce: Thank you so much for your work and I'm sure we will bump into you soon. Hopefully, I feel like Jon, we've been saying it for two years now hopefully at a face to face conference sometime soon we will bump into you.

Jon Orr: Keep saying that.

Shelley Yearley: Yeah, for sure.

Kyle Pearce: Awesome. Have a great night. And we look forward to chatting with you again. As we is my friends, both Jon and I learned so much. And actually, we were chatting with Shelley after the call just about how we learned so much from recording these podcasts and these episodes, we get some benefit out of like doing the thinking ahead of time. What questions are we going to ask?
We get to do the learning during the episode and then find finally we get to do the learning as we write the show notes page. So the question we have for you is how are you going to ensure that the learning sticks so they don't wash away like footprints in the sand?

Jon Orr: Yeah, so a great way to hold yourself accountable is to write it down or, hey, even better call a colleague or talk about it with your partner. These are all great methods to help you formalize what you've learned here in this episode. You could also hit us up on our social media channels and connect with us and share the learning with us. Head on over to app Make Math Moments on social media, or you can head to our private Facebook group, Math Moment Makers K to 12.

Kyle Pearce: Yeah, and make sure too this episode, and many of the other episodes are being recorded with video on YouTube. So they're coming out each week plus all the other YouTube content. So make sure to go in there, subscribe, hit that notification bell and leave us a comment there. Remember, show notes and links to resources, including full transcripts as well as today.
Remember, that Fraction Learning Pathway, the link and the resources to even tips and all of those great resources can be found over on the show notes page at makemathmoments.com/episode150. Again, that is makemathmoments.com/episode150. Well until next time Math Moment Maker friends, I'm Kyle Pearce.

Jon Orr: And I'm Jon Orr.

Kyle Pearce: High fives for us.

Jon Orr: And a high five for you.

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