Sarah Schaefer: And so you’ll see kids exploring this idea, when they present their ideas as the teacher starts to structure that learning, we require students to come up with two different methods to solve a problem. That’s really a big piece of the methodology is that there’s more than one way to attack this and it’s important that you see different perspectives. It’s important that you can communicate-

Kyle Pearce: Today we speak with Sarah Schaefer who is the go-to person when learning about Singapore methodology. Sarah is the principal at Mathodology in Atlantic Beach, Florida and unleashes a bunch of great tips and ideas on how to use the Singapore methodology in your math classroom.

Jon Orr: You’ll also hear about how the Singapore method falls in line with the five practices and how you can use lesson study to avoid teacher isolation. Let’s get to it.

Kyle Pearce: Hit it. Welcome to The Making Math Moments That Matter podcast. I’m Kyle Pearce from tapintoteenminds.com.

Jon Orr: And I’m Jon Orr from mrorr-isageek.com. We are two math teachers who together-

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

Jon Orr: Fuel learning-

Kyle Pearce: And ignite teacher action. Jon, are you ready to get into some Singapore methodology?

Jon Orr: Yes sir, Kyle, we are super pumped to bring you this episode. Did you know we’re at over 160 ratings and over 65 reviews on Apple Podcast? Help us reach a wider audience by taking a minute right now to hit the five star button inside your podcast app. And if you’re as kind as playing with dice, I love that username, you’ll give us feedback like this.

Kyle Pearce: Dive into the rabbit hole. This podcast introduces listeners to the very best in educational researchers, math education literature, and other math educators. This is the best rabbit hole to dive into, whether you’re a novice to or you’re experienced with ideas like standard space grading or thinking classrooms. After each episode, I walk away with a new education author or blogger to look up or a new idea to try in my classroom. Thank you for this truly amazing professional development podcast.

Jon Orr: And thanks a million to playing with dice for leaving us that five star rating and fantastic review. It’ll only take a minute, so hop into your podcast app now and do us a solid.

Kyle Pearce: Awesome stuff. Please do. And before we get into this talk with Sarah, we are super excited to share another professional learning opportunity for the Math Moment Maker community.

Jon Orr: In our brand new webinar, we will be unpacking how to turn your textbook into a curiosity machine by applying the make math moments three part framework, including sparking curiosity with the curiosity path.

Kyle Pearce: So be sure to get yourself registered for one or even more than one of the webinar dates available by visiting makemathmoments.com/webinar. And while we encourage you to have a look at the webinar sign up page at makemathmoments.com/webinar, to toss those dates into your calendar, never fear if your schedule changes as we will email you the replay link after the last webinar has been delivered.

Jon Orr: We also want to take this moment to give you a quick heads up that our online workshop Making Math Moments That Matter has just opened for registration. We run the online workshop twice a year and registration is open now and closing on Friday, January 31st. You can learn more at makemathmoments.com/onlineworkshop.

Kyle Pearce: If you’re listening after January 31st, 2020, registration has closed, but you can still head to makemathmoments.com/onlineworkshop to join the waiting list in order to be notified for your next opportunity to participate. So head on over to makemathmoments.com/onlineworkshop and get yourself signed up.

Jon Orr: And now let’s get on with our talk with Sarah.

Kyle Pearce: Hi there Sarah. Welcome to the Making Math Moments That Matter podcast. We’re super excited to have you on the show today. How are things? Are you down in Florida right now or are you traveling around doing workshops?

Sarah Schaefer: No, I’ve taken a pit stop. I am home. I’m in Atlantic Beach, Florida and I’m excited to be here, thanks for asking me on the show.

Jon Orr: Wonderful. We are glad to have you here on the show. It’s recording this in the summertime and when I think Florida summer I just start to sweat right away. Can you tell me more about Florida in the summer?

Sarah Schaefer: Yes, it’s hot here. It’s pretty humid in the morning but I lived in Massachusetts and moved down here in 2004, and so I’d rather the hot than the cold. So I’ll take the summer. I’ll take the June, July, because I get it back. That January, February, March is nice place to be.

Kyle Pearce: That’s a good trade off.

Sarah Schaefer: That’s right.

Kyle Pearce: Nice. Sarah, can you help our listeners understand a little bit more about yourself? We just talked a little bit there but maybe elaborate a little bit more on where you are specifically and your role in education, and also how did you get into the education field? Like the background story there.

Sarah Schaefer: I think as a kid I really was intrigued by numbers. I can remember back to being in third grade, I had an older sister that was in high school and I would just try to take her algebra one homework and try to figure it out. So I knew there was something about numbers that was intriguing to me, the challenge of trying to figure things out. So when I went to school, I studied math and computer science. I didn’t really know what I wanted to do with it, but I knew there was, again, an interest there.

Sarah Schaefer: And it wasn’t until my junior year in college that my path kind of crossed the School of Education and I ran into two women in particular, Carol [inaudible 00:06:40] and Carol Findell at Boston University, who shaped more of the education side and integrating that math love with the education world. And so it was really their leadership that got me into education. I became a teacher. I taught then at the middle and high school levels implementing the techniques and the strategies I had learned from Carol and Carol.

Sarah Schaefer: And it wasn’t really until I guess, my youngest child or my oldest child actually was in school that you kind of wear a different lens when you’re a parent. And I was watching the way that she was learning and I realized that I like to say she could do school, she could compute, she could get an answer, but she was really lacking in number sense. And I was seeing the same things in my high school classrooms. And so I thought, gosh, there’s got to be a better way.

Sarah Schaefer: And I had worked at a pretty dynamic school, The Bolles School in Florida and under, Scott Smith was a leader there in curriculum and Dr. Trainer and I spoke my concerns to them and they said, “Well, what do you want to do?” And they gave me the liberty to go investigate different curriculums and how we could start to develop that number sense at those early ages. And it was a blessing. I got to just jump in and try things out and investigate. And that took me on my journey to what people refer to as Singapore math and really learning it through doing it. There were no trainers out there at the time, so I had to learn it by being in the classrooms and it spiraled from there.

Sarah Schaefer: I began working with our teachers at The Bolles School and then another school would call me, and that spiral then to a business where I consult with schools and I helped schools learn about it and implement it. And then I’ve also done now a little bit authoring along that avenue as well.

Kyle Pearce: That’s fantastic. Something I really like is just hearing about how you’ve had a lot of experience in middle school and high school. And I think Jon and I can relate from that secondary teacher perspective where students would come into our grade nine classes and we’d commonly say things like, oh, they don’t know their math facts or they aren’t fluent, they don’t have any number sense.

Kyle Pearce: And it’s really common to hear those things because it is something that so many students struggle with. But to actually do something about it I think is really awesome, because for many years I didn’t do anything about it. I would mention it, I’d say it, but then it was sort of like I was putting it on someone else. So it’s really awesome that not only were you given the opportunity to actually go and explore other curriculum, but to actually do the work because let’s be honest, it is hard work and we’re really excited to dive into the journey of Singapore math for yourself in your own professional learning.

Kyle Pearce: But before we do, can we dig back to your own K through 12 experience and talk a little bit about a math moment that you remember from your math class. Some people talk about something that helped to shape what made them into a teacher. Others talked about some maybe not so positive experiences. So what comes to mind when you think about math class?

Sarah Schaefer: I think there’s probably two moments that stick out for me. One was a sixth grade class where I remember the teacher, [Mrs. Lemay 00:10:17], and she would really harp on the idea of speed and drill. And I’m a competitive person by nature and King Richard, Richard was the one that always got it first and it drove me crazy. So I just wanted to be like King Richard. So that was something that I remember, and looking back now, realizing that that idea of speed and computation doesn’t necessarily mean that you’re good at math.

Sarah Schaefer: A second instance that I can remember is in high school I was taking a course honors geometry trig, and I got a progress report early on that said, you are in danger of failing. And I thought, oh my gosh, I’ve never gotten something like this. We only had one assessment at the time and I thought, I’ve really got to learn math from a different perspective. Again, I was big into computing and what step came next and it kind of made me think about, there was more to math than just the computation. And so from that point on I began to look at the patterns and the beauty of math and how to connect it. And so that was the turning point for me that junior or sophomore year, that class and that particular possible failure notice that reshaped me.

Kyle Pearce: For sure, and I think I relate to both of those moments. And in one way I think so many people do think that math is about being fast and speedy and sometimes ashamed about that. It reminded me of this, I just went and did a presentation at our local rotary club, they wanted a math presentation and we asked that same question about their math moment and a teacher, not a teacher, sorry, these are just community members, but teachers have said the exact same story and maybe you’ve heard of this game, it’s called Around the World.

Kyle Pearce: And somebody will stand up and the teacher will tell them a math fact, like a multiplying fact, like six times seven. And if they get it right, they continue standing and then if they get it wrong, they have to sit down and they’re out. And then the teacher goes around the room kind of like that. And so every time we’ve asked a math moment, somebody will tell us that speed and memorizing kind of thing that’s, they have a lot of embarrassment when they have to sit down and that always comes out. It came out in that community group, it comes out with teachers. And it reminded me when you said about speed sometimes we think a big factor.

Kyle Pearce: And the other one, you know what that reminded me of when you said like you realized that there was more to learning math than just the memorizing and the calculations. And I was reminded of my grade 11 or 12th year in high school we had a class called finite math, which was mostly about combinations and permutations. And those are the questions like, how many ways can Sally stand beside Jamal in a line. And I remember I had a very similar, hey, you’re failing this class notification quarter the way through. And it was because I was that memorizer, I was that procedural guy that could just follow the rules with like function math, but all of a sudden it came to problem solving with like how many ways problems. You have to think about logic and more than just plug numbers into formulas and I struggled big time when it came to that, and it was my first kind of eye-opening experience that there’s more to just plugging numbers into formulas to do some math here. So I related big time too, both of your memories there too.

Kyle Pearce: I’d like to move into your passion here about Singapore math. Many people I think have heard of the term and name, but I think there’s a lot of people that don’t know exactly what that is. Would you mind elaborating on what is Singapore methodology and the approach to teaching?

Sarah Schaefer: I like how you changed to Singapore methodology because there really is no such thing as Singapore math. When I got the opportunity to go to Singapore, I first would spend my mornings watching the teachers teaching in the afternoons. Again, my path crossed Bon Har and he was training teachers at night and they would say to me, “Where are you from? What are you doing?” And I said, “I’m an American teacher and I train teachers in Singapore math.” And they said, “What is that?” And I said, “What you guys are doing.” And they said, “Sarah, we just teach math.” They’re like, “Math is math.”

Kyle Pearce: It’s like new math, right? Common core is new math. And it’s like, no, it’s [crosstalk 00:14:41]. It’s just math.

Sarah Schaefer: It is, it’s just math. So taking this approach, what I’ve learned over the years, I’ve been probably doing this 15, 17 years or so, is that the way that the teachers had their curriculums that had come out of Singapore, they’re really based around what we call five guys, five researchers. And those researchers aren’t specific to Singapore. Actually, none of them are from Singapore. And so the idea is that if we … these five guys, Jerome Bruner. We’ve heard of Jerome Bruner, he’s an American psychologist.

Sarah Schaefer: And his idea is that mathematics in the CPA approach is what you’ll hear people refer to when they refer to Singapore methods. And this is, C standing for the concrete idea, P standing for a pictorial representation, and A standing for that abstract. And what Bruner reminds us is that kids really need to move through those models and also connect those models to really have true understanding. And so you’ll see in a Singapore approach a lot of that. But again, that’s not specific to Singapore, that’s best practices in mathematics. It’s based around a guy named Richard Skemp.

Sarah Schaefer: Richard Skemp reminds us about relational ideas and helping students take it from a childlike way of representing it to more of an adult like. And I think where that might differ from a traditional type of program is we typically would start with that algorithm or abstract method and then we wonder why kids can’t remember how to do it. And what Skemp will remind us is that you haven’t helped to connect that instrumental understanding to that conventional understanding.

Jon Orr: And that’s great, I’m so happy that you broke it down so simply. I mean, because I think so many people think it’s a textbook or it’s something completely different. But then at the end of the day when you talk about CPA, some in the US tend to call it CRA for concrete representational abstract. We here, I’ve used concreteness fading as a bit of a way to reference it. And really at the end of the day, I think you just nailed it.

Jon Orr: And something that I’ve shared a few times when we present is this idea of being more Prince. And in math class we tend to start with symbols and unpack what they mean rather than actually starting with the concrete. And I always joke around that Prince, he started in concert in the concrete’s phase, he was there, you saw him, you could touch him, probably couldn’t touch him, you’re probably really far back, but you get the idea.

Jon Orr: And then all of a sudden, now you’ve got this image in your mind of what it is to go to a Prince concert and you see posters and all of these visuals. And then finally, then and only then did he have the guts to represent everything that is Prince as a symbol. We always say be more Prince in math class. And it sounds like that’s essentially the mantra of this methodology is go ahead and be more Prince. Would you argue that that’s at the root of it all?

Sarah Schaefer: Absolutely. I think one mistake we made early on was seeing that [inaudible 00:17:59] as separate ideas, and wondering why our kids aren’t still getting it. And what we had to, adjustments we had to make, and that really happened after I went over to Singapore seeing how we display things on our board and make that connection between the CPA because we have some kids during a lesson that really need to be in the C, that concrete, but others are in that A. But how do they relate to each other? And that’s the challenge for teachers is making that connection.

Kyle Pearce: For sure.

Jon Orr: And I’m reminded of our talk with Peg Smith, the five practices about connecting the different solutions together. And it reminds me of how hard it is to connect the concrete to the abstract and vice versa if you had to go back. So I definitely think that that’s true for sure.

Kyle Pearce: And something else too there, Jon, I just wanted to toss in. I think when you’re first introducing a concept like very, very early on, obviously concrete is the logical place to start. Like when you think about anything in the real world, it’s like it’s real, it’s there, you can experience it, but then it’s not like it goes away. And I’ve heard some people say that’s why they don’t like the term concreteness fading because it shouldn’t go away. Now it’s like you add to it and as you add to it the concrete should still be there, it should still be apparent. We don’t want to put that stuff away. And it sounds like really what Sarah you’re articulating is that it is about making those connections and kind of going and flowing forwards and backwards along those three different types of representations.

Sarah Schaefer: Absolutely. And I think what you hit on there is that depending on what phase of the concept you’re in, the concrete might be more in the students’ hands where the recording and the abstractness would be in the teacher’s hands. And then that scaffolding naturally occurs where that concrete in that pictorial model might move more towards the teacher and the recording now moves to those students.

Kyle Pearce: Beautiful.

Jon Orr: Awesome. Yeah, true. We’ve just talked about CPA or CRA or concrete fading model. I’m wondering what else is surrounding the Singapore methodology?

Sarah Schaefer: So I think when you walk into a classroom, things that you should notice in a classroom that says they’re using the methods, and again, I want to reiterate, it’s best practices. You’ll see these things in classrooms that don’t necessarily use this approach as well. But this idea that when you give a task, a lesson will start with what we call an anchor task, a high leverage task. You’ve heard it with Jo Boaler’s work, you’ve heard it with Richard DuFour’s work.

Sarah Schaefer: There’s some kind of engaging tasks that requires some exploration, that’s Vygotsky with his zone of proximal development. And so you’ll see kids exploring this idea, when they present their ideas as the teacher starts to structure that learning, we require students to come up with two different methods to solve a problem. That’s really a big piece of the methodology is that there’s more than one way to attack this, and it’s important that you see different perspectives. It’s important that you can communicate your perspective.

Sarah Schaefer: A lot of times with parents they’ll say, why do they have to solve it in two ways? And that’s one of the main philosophies is that my way isn’t the only way and it’s again important for kids to see those connections. So you’ll see that anchor task idea that starts the lesson. You’ll see the multiple methods within those methods. You’ll see that CPA approach and those connections being made.

Sarah Schaefer: I think another big idea is this idea of reflection and journaling. We know that the learning, what we say to our teachers is it’s, Dewey’s idea is that, I quote, that the learning doesn’t take place in the exploration itself. It takes place in reflecting on the experience. And so reflection and journaling, whether it’s reflection verbally or in a journal is an important part of the mathematics classroom. And helping teachers incorporate that journaling in is part of the work that we do.

Kyle Pearce: That’s an awesome summary. I’ve been typing in making sure I’m making some point form notes and I’m sure Jon and I, when we go and add this to our show notes page, we’ll make sure to have something really nice there for people to grab. And it’s so interesting to me because there’s so many different methodologies out there and approaches and names, and sometimes people are talking about the same things. But really what I’m hearing is so much of even what is highlighted in the principles to actions book from NCTM. This is good teaching practice is what I’m hearing, which is great.

Kyle Pearce: And something that really resonated with me and I hope it did with those who listened to the podcast regularly, is the Dewey quote that you just referenced there. Learning doesn’t take place in the exploration itself, it takes place in the reflection. And at the end of every episode, Jon and I tend to throw something at the audience about reflecting and ensuring and we say, are you sketchnoting? Are you journaling? Are you talking with a colleague? What are you doing to reflect on what you just heard in this episode? Because otherwise it’s so true. It’s like you heard it and at the time you think you learn something, but then it just tends to wash away footprints in the sand.

Kyle Pearce: So you really need to build that in for ourselves as educators when we’re learning. But then now also what does that look like in our classroom to ensure that students aren’t just doing it because they have to do it, but because they actually understand that why we’re doing this is to actually do the learning.

Sarah Schaefer: Yeah, and I think the point there is that, that reflection it’s … I always say we use math, we milk math for all it’s worth. In this idea is that life is about reflecting back on your experiences and learning from them. And as teachers, I’ll say, we can do the math, but it’s a lot harder to stand up in front of somebody and teach it. And when you have to verbalize and communicate and kind of think about best ways, that’s where you’re really learning it best. And so I’m glad that you’re emphasizing that reflection piece as professionals as well as what we expect from our students.

Kyle Pearce: Yeah, I think that reflection is super important. I want to go back a little bit to when you said, you require two different methods to solve the problem and then the students are open to solve it in different ways. What is the helping teachers at that part look like in say your trainings? Do you have any tips for teachers? How do teachers handle, like kids are working, what might that look like? And then what does a teacher’s next steps to kind of bring that all together? I’m trying to imagine this as a teacher who’s hearing this for the first time and thinking, okay, I want to try this, but what do I do at that part?

Sarah Schaefer: Yeah. So you hit on it with some of the other practices with NCTM and the five practices. And we look at this idea of anticipated responses, and as a teacher we have to know what I call my target method. So for a particular lesson, you have a method that you’re trying to structure your lesson to get the kids towards. We’ve got to know what that target method is.

Sarah Schaefer: We also have to really think about the number choice and the problem itself and what do we anticipate the kids might naturally do with it during the exploration. Because during this exploration, we need to be quiet and allow the kids to explore it. So we don’t really have a say in the way that they’re going to go about it. But as we structure the learning, the kids that I choose to share methods, it’s not a random choice. I’m really thinking about which child is going to lead me to that method. And a book that I’m working with right now, the think mathematics book, it’s listened to what teachers struggled with and they said, Sarah, I’m not quite sure which methods I should pick. It’s laid out for them. It’s showing them method one, it’s showing them what method two or method three could look like. So as they start to anticipate and decide who they’re going to choose, it’s taking some of the guesswork out of it for a teacher.

Sarah Schaefer: A lot of times teachers will say, I can’t choose every student and I can’t hear all the different methods. And that’s right, we can’t. But again, that journal is a place where those kids could share those methods. But as a teacher we need to be pretty selective on what our target method is, anticipating what the students may say and then structuring the conversation to get to that target method.

Kyle Pearce: Right. So it sounds like obviously, as you mentioned, five practices we just had Peg Smith on. Well, when this episode goes live, it’ll be a little while back now, but it was episode 33 when Peg Smith was on and we were talking about the five practices and we’re hearing a lot of the same ideas here that really, again, planning is so important in anticipating what are we after. And I like how you had mentioned using this term target method and you’re not forcing anyone, so we want to be clear on that. And you were very clear about that, that you are allowing students to do the exploration, allowing them to essentially prove to themselves that they can problem solve, that they have tools in their tool belt.

Kyle Pearce: You’ve obviously selected tasks, that anchor task is being selected and it likely has that low floor and a high ceiling and it’s developmentally appropriate for the students in the room and students are going to solve that problem. But the teacher, we don’t want kids to just solve it any way they want and then walk away because then they haven’t learned anything new. And I’m thinking back to when Cathy Fosnot was on the show and she talks about horizontal mathematizing versus vertical mathematizing. And if we did an anchor task and students did it two different ways and everyone was successful doing it and we called it a day, which I’ll be honest, I’ve done that in my career, it’s nothing to be ashamed of. But now I recognize that really all I was doing was giving students purposeful practice with the tools that they had in their tool belt. But I didn’t help them build any new tools.

Kyle Pearce: It sounds like this idea of this target method is really all about, okay, taking students from where they are now and helping them to see that there’s actually other ways to do this. And I’m guessing that it’s leading to a more, a higher level of sophistication. I’m wondering, what does that look like as you’re planning out a lesson? Is there any go-to methods or strategies that you have? And I know you’ve referenced the think mathematics books, so we’ll definitely include that in the show notes and that helps you with that process. But what does that look like or sound like to you? Is there anything that you can give to teachers out there who are listening and saying, okay, I want to do this, but it sounds a little scary or I’m not sure where to begin.

Sarah Schaefer: I think with the idea of methods, we need to understand what efficient and inefficient methods look like. Obviously we’re going to have kids that do things in an inefficient way and our job is to kind of lead them to more efficient ways of doing that. So what are those ways? We need to go back, depending on the concept. Let’s say we’re looking at addition, subtraction, multiplication, division with whole numbers for instance.

Sarah Schaefer: The big idea surrounding those operations is really decomposing numbers and breaking them apart. And how we break numbers apart depends on the problem that you’re giving me. So if I have 52 minus three, I might think of 52 differently than if I had 52 divided by three. And really giving them the idea that operations are about breaking numbers apart and putting them back together. And at the end of the day there’s a conventional method that these students will be using that vertical algorithm. And how does that breaking the numbers apart in a childlike way connect to that convention away.

Sarah Schaefer: Teachers in their planning phase need to almost work backwards and say, what does the conventional algorithm require students to truly understand, and what does that look like in a concrete, and then maybe a more childlike way of recording it. And then when they’re planning they’re kind of connecting those ideas. So most of the work is in the planning. It’s not in delivering the lesson, which requires for elementary teachers especially, time to go back and look at how a concept was developed and where they fit into what we call that spiral. Am I at the beginning? Am I in the middle? And if I’m at the end and I have a student that still doesn’t understand how to do it, how do I go back to earlier parts of that spiral so that they can connect ideas? So it’s really looking at concept development.

Jon Orr: We’ve heard this a lot lately where teachers are saying the anticipation stage is so important because we need to know where our kids will come from, where they’re going to go, and how do I move them along that kind of learning journey, that landscape of learning that … Oh, Kyle, help me with the name.

Kyle Pearce: Oh, Cathy Fosnot.

Jon Orr: Yeah, Cathy, sorry. Cathy Fosnot mentions like, we were just talking with, I think-

Kyle Pearce: Nicki Newton.

Jon Orr: Yeah, we were talking with Nicki Newton yesterday.

Kyle Pearce: It’s been a long week.

Jon Orr: Yeah, we’ve had a lot of interviews this week. We were talking about Nicki Newton and she was elaborating on landscapes of learning and [crosstalk 00:31:54].

Kyle Pearce: Trajectory, like different continuums.

Jon Orr: Yeah. Those are huge things to think about. And I think sometimes we overlook that, we just go into lessons and go, I’m just going to do this lesson because someone said it was awesome, but don’t actually do the planning part that’s necessary. Even still I think teachers are, they’re not sure about what the planning looks like. They might plan it but like, I’m just going to do this problem, and I’m going to mostly think about classroom management but not the math behind all of the different methods and how that landscape looks.

Sarah Schaefer: I think in planning the question that maybe we start with is, okay, I’m teaching concept A, what did this look like? What were the non-negotiables that students had to have in order to teach this concept? And I really have to understand that because if they don’t have those skills, then how do I incorporate it? Because I can sit there and talk and check a box and say I taught lesson A, but doesn’t necessarily mean those students have learned topic A. And it’s my duty to go back and look at those non-negotiable skills and understand them to deliver a lesson.

Jon Orr: When we’re thinking about planning here, I wanted to think about like what that looks like, are teachers … Do you encourage in your trainings teachers to do this on their own, in groups, or how does that look for you?

Sarah Schaefer: I think early on when I work with teachers we talk about Richard DuFour’s four critical questions. And looking at what is it that we want our students to learn today. And working in a collaborative environment with a group of teachers. If we’ve provided resources for teachers that if you’re alone in your school and you’re the only one teaching third grade math, here’s a group of other third grade teachers from other schools that you could work with. But collaboration is key, both in your classroom with students and in your own planning. So you say, what is it we want them to learn? His second question is, how do we know they’ve learned it? What are we looking for? What is their evidence going to be? What are we going to do if they don’t learn it? Because we’re anticipating there’s going to be students that aren’t going to get it the way we think they are. And what are we going to do if they already know how to do it?

Sarah Schaefer: And so in the planning phases, we’ll open a textbook page and we’ll look at the anchor task and we’ll say, okay, let’s act it out. Let’s think about these four questions and let’s plan according to these questions. And typically the teacher feels much more confident in delivering the lesson and in questioning throughout the lesson when they take that up approach.

Kyle Pearce: Right. That’s so important. And I think we’re hearing that message over and over again with different guests coming on to the podcast. And that’s so important that we work with other people. And actually we just interviewed Robert Kaplinsky yesterday. His episode would have come out just before this one. And he was referencing just this idea that until he actually started doing the math ahead of time with colleagues, he actually never even thought that other people would be solving the problem any other way. You just see the world through your own eyes. And it’s so difficult for us individually to think of what kids might do. And I think over time we get better at this. Because our experience allows us to, I guess it becomes more predictable to us because we’ve seen it before.

Kyle Pearce: But until we sit down with other people and actually see that, wow, there’s other ways that we can approach this. It’s very hard for us to actually adequately anticipate what might happen in the lesson. So I’m picturing this idea of us co-planning as being a really, really critical part of teaching. And a lot of times we think about the time in front of kids, but in reality I think it’s the time ahead and the time after we work with kids where we’re doing all of this heavy thinking so that when we’re in the moment things just happen.

Kyle Pearce: I’m wondering, I know you’re an advocate of lesson study. Did you want to maybe elaborate a little more on like how someone might be able to, let’s say in their own building, start up some sort of little mini lesson study. Are there any tips or tricks that you could give them just to kind of promote people with, hey, listen, let’s do this together. Why are we doing this in silos? Why are we shutting our doors when we could all help each other out, and in turn not only help ourselves, we’re really helping kids in our own classrooms. Do you have any get started moves that they might make in order to make that happen?

Sarah Schaefer: Yeah. Currently we just gathered some teachers and said, what are some bigger ideas in mathematics that you’re interested in? We had teachers comment on the idea of how do we get kids to become more natural at problem solving? How do we build visualization? And is it important? What role does visualization play in a mathematics classroom? Another group wants to look at how to develop that metacognitive thinking. I know you had John Hattie on earlier. And the importance of getting kids to develop that self-talk and how do we do that.

Sarah Schaefer: These groups of teachers have expressed interest in what I call a bigger idea. And then we use the math lessons to say, okay, it doesn’t matter if we’re doing a second grade, a fifth grade, a first grade, whatever grade, it doesn’t matter the content. What are different catalysts that we can incorporate into any classroom? So we try ideas out. And I think in lesson study in other countries which differs from the US is that they actually try these ideas out. When they find things that work, people implement them into textbooks. They implement them into different professional reading.

Sarah Schaefer: And for us in the US what my experiences is, is that we as a group of teachers go try these things and we find out great things and we kind of keep them to ourselves and we keep trying them. And so I wish, or my hope is that when these groups of teachers that they voiced these ideas. For example, how do we get a student that doesn’t want to do anything in a class? We all have had that student that sits there and you’ve done everything you possibly can. And we watched her four days and our big aha moment was we do nothing. When we let them sit there, there’s different maybe body gestures that we could do, but in every classroom when we did nothing, those kids eventually became engaged in the lesson, and it was amazing to us.

Sarah Schaefer: I think that’s really important as teachers gather in groups is to say what is of interest to me. Then find some articles to read about them. We’re reading articles right now on visualization. September we’re going to start to put it in practice in classrooms and we’re going to look for what catalysts, we’re going to watch the kids and see what things that we’re doing in different classrooms are creating change in the way that students learn. And then we need to share that with other people once we see it. Because what the teachers had found is that those same things they’re doing in mathematics also works in language arts. It’s also working in their science classroom. And again, we go back to where we started, it’s best practices. So that’s what we’re finding.

Jon Orr: I really liked that you referenced to that, it’s kind of cross curricular too. Lesson studies don’t have to be math lesson, math lesson, math lesson altogether. Not math lesson, I mean like math teacher, math teacher. Especially in the middle of the high school and you get segmentation on teachers. It can be a math teacher planning with a science teacher. And you’re thinking about that because there are some cross strategies that work really nicely. And I’m reminded of our episode 30 with Jed and Gabby on lesson study.

Jon Orr: And I really liked how Jed, when he looked at going into classrooms to observe teachers on the lesson study portion after the planning portion, he said, “You go in and be selfish.” It’s like look for what you can take out of that lesson for yourself. Because every teacher does something a little different. Even if you plan it together and then you want to take something to help you become better as a teacher. And so then look at going into those lessons to observe and bring something back that you can use in your classroom. Because we don’t get into enough classrooms, we don’t observe enough, and there’s so much that we can learn together. So I’m glad that you brought up and talked about planning together and not in isolation.

Kyle Pearce: I really liked this idea you had mentioned about the sharing aspect, and something that I’ve heard a few others mentioned in particular when I was having some conversations with Phil Daro. And he talks about how in Japan, when you actually measure teacher content knowledge at the beginning of their career versus the end of their career, there’s like a significant gain. Whereas in North America, the gain is not … it’s pretty much flat lined. And a lot of it has come down to this idea of, like you had said that we do things in isolation. We also put a lot of pressure on ourselves, like we’re supposed to already know all of this coming out of our pre-service.

Kyle Pearce: So we come into the classroom and we start to do the whole fake it till you make it thing. And while I think in some ways you should or need to do that, just to make sure that you can get through. In a lot of ways, you have to in your mind realize that you’re not expected to know everything yet. We all have things to learn from each other and that it’s not a competition. And go back to that example Jon gave of Around the World. It’s not a competition of who can do the most math facts. It’s about helping one another and learning from one another. So that really like think about it in year 30 I hope that I have this thing to a point where I feel like pretty good about my lessons.

Kyle Pearce: I don’t want to be just working with the same skills, the same expertise that I had in my year one or year five. I want to continually be developing so that in year 30 I’m feeling like fairly confident about what I’m doing in the classroom. I know I’ll never get there, I’ll never be done, but at least I’ll feel like, wow, I’ve really come a long way and that things are going much more smoothly than they did earlier in my career. So Sarah, I want to shift gears a little bit. This was not planned, but I heard you say the words spiral and you referenced this idea of depending on where you are in the spiral. And my assumption is that you’re describing how the content is shared with students throughout.

Kyle Pearce: And folks who have listened to the podcast know Jon and I are big advocates about interleaving and spiraling. We talk a lot about retention and challenges there. And we even have a mini course that folks can dive into. We’ll put that in the show notes so that folks can dive into that little mini course and learn more about spiraling. Is this the same type of spiraling that you’re referencing in the Singapore methodology? Is it different? Can you just tell us a little bit more about what you were describing when you said, depending on where you are in the spiral?

Sarah Schaefer: Sure. So the textbooks are written in such a way that they use this idea of a spiral approach where there’s a slight variable that’s changing from lesson to lesson. So when a teacher says to me, I want to move chapter four to chapter two where I’m going to combine these two lessons, I’m a little cautious because this idea that when we change too many variables at once, we might overload the students or take it to a place where they’re not able to figure it out on their own. And it has to be a lot more teacher directed. So that’s the way that the textbooks are written in this spiral idea.

Sarah Schaefer: The other idea is that within that, topics come up. So for example, if we look at addition and subtraction, it might be introduced this idea of the vertical algorithm per se at one grade level, and then it might come back later within that grade level or it might come back at the next grade level. We’ve got to know if I’m a third or fourth grade teacher teaching the addition algorithm, where did that spiral start? And then what is the expectation at my grade level? Am I in what I call a learn phase? Am I in a consolidate phase? Or am I in an application phase? Because as you referenced before, that concrete pictorial and abstract representation will look different in those different phases.

Sarah Schaefer: I need to understand both that the textbook is written in a spiral. So from day to day as a teacher, I need to say what’s different about today’s lesson than yesterday. There’s something slightly changing in that idea. And then also within this lesson, where are my students? Are they in that learn consolidate or apply? Because my questioning and the choice of materials that I use will look different.

Kyle Pearce: Fantastic. So it sounds like we are referencing the same type of spiraling and I really love how you’ve articulated that, really a common challenge. And we have a lot of people who reach out to us when they try to spiral their thinking in their minds, like spiraling is, let’s just mix this thing up in a blender and let’s just go and do it. And then what they end up finding is like you had said, either students are overwhelmed or they’re not sure where they’re going or even the teacher doesn’t feel like they know where they are, in reality it’s because they don’t, there’s no logic or rhyme or reason to it. So I really like that. That to me seems like a really good way to think about it, is like what small pieces are changing.

Kyle Pearce: And I think the big part is when we’re anticipating ahead of time, like when I’m going to shift gears from this idea to another idea, why did I just do that? Did I do it just because I’m trying to mix it up or did I do it because there’s some sort of connection that can be made. If we go back to this idea of making connections and how important that is, obviously from day to day we don’t want it to be like, again, siloing now days instead of like, the idea is we don’t want to be siloing big units together so that there is no connection. But we also don’t want to silo things down to the lesson level where now every single day is so disconnected from the next, that now there’s no connections at all. I think that’s a really, really good piece that you shared.

Kyle Pearce: There’s probably people listening right now and they’re going, wow, this makes sense, and they want to learn more. Where might they get started if they wanted to start using that methodology? As you mentioned, it’s not like a magic wand. It’s not like, hey, go and snap your fingers and suddenly you’re doing Singapore math. It’s like, where might they start if they wanted to start scratching the surface on this methodology, maybe learning a little more about it and maybe doing small things in their classroom to start nudging or inching their way towards this methodology.

Sarah Schaefer: The big thing for teachers and what I hear from them is they like to see it, they like to see it and they like to hear it. So they can read about it, there’s plenty of books out there that claim they do Singapore math. Singapore math became a real buzz word as people started to look at what is this idea? So really again, it’s best practices. So I would say go back and look at developing what is that anchor task, what is that exploration you’re going to do? And then focus just on that first part of the lesson, what anticipate those responses. So read about how there’s a lot of NCTM articles in both the magazines, the monthly magazines that come out, but a lot of the literature as well on how do we anticipate student responses.

Sarah Schaefer: And then really focus on how you’re going to choose the students and structure those anticipated responses. I would recommend taking one unit, one chapter and focusing maybe on that first part of the lesson, that delivery of the anchor task where you give it, you step back and you set your timer. Because we want to jump in and save the day too often and we need to set the timer to make us say, [inaudible 00:48:25], let the kids have a little bit of time on their own and then work on your scaffolding and as you said, reflect back day after day, how did it go? Take a picture of your board and look at when you put things on the board, were you helping kids connect ideas. And don’t try to do too much too fast.

Sarah Schaefer: And then whether it’s different websites, I know on our website the methodology one and the thinkmath-us.com. We’re going to this year really be sharing some of those videos and those pictures of this stuff going on. So then now you can say, okay, that’s what I thought it looked like. Now let me go back, videotape yourself. That’s the best piece of professional development and observation you can do. But again, take small bites, don’t try to do it all, it’s too much to do it all. So decide what you’re focusing on and then really look at developing that piece of it.

Sarah Schaefer: And then maybe the next chapter you add a new layer. But for some people it’s a mind shift because it’s not necessarily the way that they learned or were taught. I don’t know about you guys, but I’m sure you had the same teachers as me where it was, they demonstrated one problem and then I went and did it 30 times and it was exactly the same method. And this approach is a little different for people. Less is more, focus on smaller pieces.

Jon Orr: Yeah. Well, we totally agree with you on that because we both taught very much the same way we were taught for many years before we started to make small changes like you’re suggesting. And I really liked that you’ve emphasized the beginning and the planning stages again. And it’s about content knowledge and the more you can learn about the topic that you are going to bring out with your students or the method that you want to bring out with your students, the more you can learn about how it connects with the other ideas around that topic, the better prepared you can be.

Jon Orr: We know that that knowledge can help us and that’s something that Kyle and I have been trying to do in our Academy. We have number of tasks in there and not only those anchor tasks that you’re suggesting, because we built them like that. For each one of those we have kind of full walkthroughs on like what to look for and on how to start planning and how it connects with some of the other math. We are right on board there with you and want to help all those teachers get there. We’re wondering, Sarah, is there anything else you’re working on? We’re getting close to the end here, we’re wrapping up soon. But is there anything else that you … what are you working on right now?

Sarah Schaefer: I think one thing in helping teachers that you talked about in your workshops is we’re creating more of these lesson study opportunities in different regional workshops, to help teachers work together and kind of unpack the math and unpack the concept to look at that spiral. So that’s something this year that I’m pretty passionate about. In my journey, I’ve gone from the classroom and the teacher and then the trainer and the consultant and whatnot. And I’ve now dabbled in the idea of authoring.

Sarah Schaefer: And so I’ve written a few of the lessons within the think mathematics, but this year it’s the first year that we have the full series out. And so distributing that and supporting schools in implementing that is a project I’m working on now. And then in my work in the classrooms as well, what we noticed was the kindergarten level and the kindergarten classroom is different in the US than in Singapore.

Sarah Schaefer: A lot of schools that I was working with would ask me, we want this idea of this play-based in this concrete and pictorial and we’d like what our upper grades are doing, but it’s just, we want more work there. The past three or four years we’ve really focused on that and we created a program that we tested in schools last year, and this year that program is now out in schools, it’s called developing roots. And so that’s something, again, I’m supporting schools that are using that and integrating this idea of this play-based, hands-on idea for kids to build meaning and deep meaning in the mathematics. That’s my plan for this year, it’s a busy plan. I’ve got a lot of schools and teachers that are really excited and passionate about it, which is what drives me. And so I’m excited to continue to support them.

Kyle Pearce: Definitely that is quite a schedule, quite a bit of projects on the go. But as I know from experience having had the opportunity to both meet you as well as watch and hear both you and Dr. [inaudible 00:53:03] present, fantastic. Jon and I, we feel like the messaging that you’re sending is exactly what we’re sending to teachers as well. We really, really feel strongly about how important the concrete experiences are and making those connections across the different representations. So fantastic, we are so happy to have you on the show.

Kyle Pearce: We’re going to include all of those details in the show notes. We have a long list here that we will be including for this episode. But if you had one website that you wanted to throw out here just for someone to remember as they’re driving, where will they connect with you, Sarah, if they want to learn more?

Sarah Schaefer: I would say mathodology.com. We’re currently working on building that even more and then linking it to that thinkmathematics-us.com. So the combination of the two, but mathodology.com is where you could find out more and ask questions and see some of these opportunities and when they’re available.

Kyle Pearce: Awesome. We want to thank you so much for joining us today on the Making Math Moments That Matter podcast. We hope you have a fantastic remainder of your day. Stay cool down in Florida and I’m sure we will bump into you sooner or later be at NCTM or otherwise.

Sarah Schaefer: Absolutely. Thanks for having me and thanks for what you guys are doing. I enjoy listening to your podcast and sharing it. I heard a teacher last week say, “I’m sick of listening to music as I’m training for my marathon. You have any podcasts that I can listen to?” And I said, “I got the one for you.”

Kyle Pearce: Awesome.

Sarah Schaefer: Getting you guys out there is part of my mission as well.

Jon Orr: Awesome. Thanks Sarah. Thanks for joining us.

Kyle Pearce: Thanks so much. We really appreciate it. Awesome. Have a great day. We’ll talk to you soon.

Sarah Schaefer: Thanks for having me.

Kyle Pearce: We want to thank Sarah again for spending some time with us to share her insights about Singapore methodology and her mathodology website and resources for you, the Math Moment Maker community.

Jon Orr: As always, how will you reflect on what you’ve heard from this resource rich episode? Have you written ideas down, drawn a sketch note, sent out a tweet or called a colleague? Be sure to engage in some form of reflection to ensure that [inaudible 00:55:15] learning sticks.

Kyle Pearce: Again, we want to remind you that we are super excited to share another professional learning opportunity for the Math Moment Maker community. In our brand new webinar, we’ll be unpacking how to turn your textbook into a curiosity machine. Who would’ve thought? Everyone’s saying that you can’t use a textbook nowadays. Well, we’re going to show you how you can and how you can apply the make math moments three part framework, including the sparking curiosity pass.

Jon Orr: So be sure to get yourself registered for one or more of the webinar dates available by visiting makemathmoments.com/webinar. While we encourage you to have a look at the webinar signup page at makemathmoments.com/webinar, to toss in the dates in your calendar, never fear if your calendar changes, as we will email you the replay link after the last webinar is delivered.

Kyle Pearce: We also want to take a moment to let you know that our online workshop for Making Math Moments That Matter has just opened for registration. We run this online workshop twice a year. It is self paced. You get to log in when you’re ready to watch the lesson videos. The registration is now open, but it does close on Friday, January 31st. So learn more about this transformative experience at makemathmoments.com/onlineworkshop.

Jon Orr: If you’re listening after the January 31st, 2020 deadline, you can still head to makemathmoments.com/onlineworkshop, to join the waiting list in order to get notified of your next opportunity to participate. That’s makemathmoments.com/onlineworkshop.

Kyle Pearce: In order to ensure you don’t miss out on new episodes as they come out each week, be sure to subscribe on iTunes or what we now call Apple Podcast, or your favorite podcast platform. And if you like what you’re hearing, please share the podcast with a colleague and help us reach a wider audience by leaving us a review on Apple Podcast, and tweet us your biggest takeaway by tagging at make math moments on Twitter.

Jon Orr: Show notes and links to resources from this episode can be found at makemathmoments.com/episode59. Again, that’s makemathmoments.com/episode59.

Kyle Pearce: Well, you know what that means. Until next time, I’m Kyle Pearce.

Jon Orr: And I’m Jon Orr.

Kyle Pearce: High fives for us.

Jon Orr: And high fives for you.