Episode 175: Experience First, Formalize Later – An Interview With Sarah Stecher
Today we speak with Sarah Stecher from Grand Rapids, Michigan. Sarah has experience teaching courses from Geometry all the way through to AP Calculus in the most diverse public high school in Michigan. She is now the chief lesson designer at Math Medic, a website that provides free lesson plans for teachers, covering the entire high school math curriculum.
The discussion today will focus around the teaching model developed for Math Medic — Experience First, Formalize Later (or EFFL for short!). In this teaching model, students interact in small groups on an activity before the teachers formalize the learning with a whole group discussion. Sarah has found that this teaching model promotes curiosity and engagement, opens up access to a wider variety of students, and produces confident learners capable of flexible thinking.
- What is the Experience First, Formalize Later (EFFL) way of teaching and learning?
- How to promote curiosity in your classroom, foster positive collaborative discussions, and create learners capable of flexible thinking.
- Free resources to start teaching with this model.
Sarah Stecher: So in our model, after students have worked in their groups, then they're putting their answers up on the board and when students share out their answer and a student is talking about the starting value on a graph, we as the teacher and the students as well, with the red pens, they'll write, "Oh, we call this a Y intercept," and so this formalizing later is really, "Okay, you already have all of the ideas. You've already done the thinking and the reasoning. In math, this thing that you've already thought about, that you've already discovered, it has a name and that's all it is." And so bringing concepts back down to crosstalk.
Kyle Pearce: Today, we speak with Sarah Stecher from Grand Rapids, Michigan. Sarah has experienced teaching courses from geometry all the way through AP calculus in the most diverse public high school in Michigan. She is now the Chief Lesson Designer at Math Medic, a website that provides free lesson plans for teachers covering the entire high school math curriculum.
Jon Orr: The discussion today will focus around the teaching model developed for Math Medic, Experience First, Formalize Later, or EFFL for short. In this teaching model, students interact in small groups on an activity before the teachers formalize the learning with a whole group discussion. Sarah has found that this teaching model promotes curiosity and engagement, opening up access to a wider variety of students and produces competent learners capable of flexible thinking.
Kyle Pearce: All right, my friends let's hit it. Welcome to the Making Math Moments That Matter podcast. I'm Kyle Pearce.
Jon Orr: And I'm Jon Orr. We are from makingmathmoments.com and together-.
Kyle Pearce: With you, the community of math moment makers worldwide, who want to build and deliver problem based math lessons that spark curiosity-.
Jon Orr: Fuel sense making-.
Kyle Pearce: And ignite those teacher moves. All right. My friends, we are about to jump into an awesome conversation here with our friend, our new friend, Sarah, from just over the border in Michigan, Jon and I here over in Ontario, we are only a few hours away. And Jon, we just hung up the call with Sarah and I feel like we've got an awesome math moment maker trying to do a lot of the same things that we're up to with our classes as well. Right?
Jon Orr: Yeah. And I said this in the interview, you'll hear that in just a moment, is that it's amazing to hear people who are doing similar things, calling it different things, but we're all coming together and going, these are the things that matter in your classroom through practice and through trying different things. People are like this works and it's great to hear it from many voices. We've got many voices here on our podcast promoting this idea of, I guess we'll use the EFFL phrase here from Sarah, where it's you experience this math activity first, and then you formalize it, you consolidate it later. We've been talking about that here on this podcast since inception. It's great for Sarah to come on and go, hey, we're doing that over here as well with lots of other folks on the podcast sharing similar stories. So you're about to hear that here. And she's got some great tips and resources for you to use in your classroom all for free. So stick around to get that info. Kyle, here we go.
Kyle Pearce: All right. Here it is. Hey, hey, there Sarah. Thanks for joining us here on the Making Math Moments That Matter podcast. We're really excited to bring you on. I know you and Jon just met for the first time, but you and I have had a chat recently as well. And yeah, we're really geeked to get going. How are things where you are? Let people know where are you coming to us from and how are things going?
Sarah Stecher: Yeah. Thank you, Kyle. I'm very excited to be here. I am from Southwest Michigan. So, just across the border from you guys, in that Grand Rapids area. So yeah, that's where I'm coming from. And yeah, pretty cold outside right now. It is snowing today, even though it is almost April, but-.
Kyle Pearce: I got to say though, today was crosstalk.
Jon Orr: It started warm.
Kyle Pearce: ... better. Yeah, was a lot better than the past week, I felt anyway with spring's supposed to be here. So hopefully it's a sign of good things to come. Right?
Sarah Stecher: Exactly. If we think spring is coming, I think it'll come.
Jon Orr: I don't know, Southwestern Ontario is, and I'm sure it's the same for where you are since you're just across the border, it's like spring is one minute and then it's winter and then it's summer, and then it's back to winter and then it's summer. And then it's no spring, spring doesn't come usually. It's like foam, it's gone.
Sarah Stecher: Yeah. You never really know what you're going to get.
Jon Orr: Anyways, let's keep this going here. We can talk about the weather, us Canadians all the time, but Sarah do us a favor here, let our listeners know about your journey. How did you get into education? But also maybe before that, why don't you let our listeners in on, what's your role in education currently? And then how did you get into that?
Sarah Stecher: Yeah. So I have wanted to be a teacher for a very long time, probably since elementary school. In elementary school, I wanted to teach the elementary grades and then I got to middle school and I wanted to teach middle school, and then high school, same thing. And in those later years, I just really fell in love with math. And so I knew that, that is really the subject that I wanted to focus on. And so when I went to college, I knew that, that's what I wanted to do. So I attended the university of Michigan and was a math major there and was also getting my teaching certificate in the school of education. Yeah. And so then after graduation, I got my first job, actually it's the same school that I have been at here in Southwestern, Michigan. And it's been a really great fit there.
I taught geometry and pre-calc and AP calc. So got to see a lot of the high school courses and meet a lot of students. And it was really wonderful. And then on the side I had been working with colleagues on writing lessons and thinking about how we could make math class feel different for our students and change the experience that students were having learning mathematics, and to make it more student centered instead of just lecture. And so that has been a project that we have been working on for a few years now. And so, yeah, that's keeping me pretty busy these days, but it's also super exciting. I feel like I have learned more math in the last few years than probably all of the years before though. It's been really good.
Kyle Pearce: That is awesome. And I feel like your story and I had an opportunity to speak with you and your team, your collaborating team. And I felt like there was so many crossovers, so many connections to your journey, to this goal that you have to make math class feel different for students, which I think is great. And that's why you're here today. So we're going to dive deeper into that. Now when I go back and I think about, you were saying, you knew really early on that you wanted to be a teacher and you were almost more and more eager as you went through the grades. So this next question, which you know is coming, everyone knows this is coming. I'm really curious about, because I feel like maybe this might be hard for someone who maybe had this passion all the way through. So we're going to just slam this on you here to think about when we say math moment, what's that math moment for you? Could you name one or there are too many for you, what's coming to mind right now?
Sarah Stecher: Yeah, this was actually, as I was saying about this, I knew this question was going to be asked, so I was trying to narrow it down and it was actually hard for me to think about, but I think the moment that I landed on actually happened during college. And so this was in a modern algebra class. It was a class that I was taking in one of my upper level math classes. And I remember this so distinctly because I remember it was three days where I truly hated math, and I was like, I don't think this is for me. I don't think I can do this. I've always super enjoyed math, and now I don't. And so I just remember that as our teacher was teaching us this, there was just all these symbols on the board and there was a packet that was handed out. And not only did it have some terms and vocabulary that I didn't know, but just things that I did not understand.
And yeah, so I was feeling very frustrated that this wasn't something that I could figure out. And I remember blocking off some time in my schedule. I'm like, okay, figure out modular arithmetic, try to figure out what's going on. I was so used to math always making sense and you can just keep asking questions and then eventually you'll get there. And this is the first time that I felt that math, it felt really arbitrary to me. It was random and maybe this was just the part where you just have to memorize it. And so I was really trying to figure out what was going on. And I remember a couple days later, we actually did this problem in class called the chicken nugget problem.
And it was all about if chicken nuggets are sold in packs of six or nine or 20, what kind of orders can you make? What's the largest order that you can't make? And we were just going, thinking about this problem. And by the end of this, by the time we had debriefed that activity, it just dawned on me that modular arithmetic is really just the visibility with remainder and all these symbols about A plus Q times R and module M is really just saying, oh, when you divide this by six, you're still going to have two left over. And so, yeah, I remember that now and I laugh about it, but I just remember the frustration and thinking, okay, maybe this is where my math journey ends. Maybe I've made it here, it's been a good run, but I'm crosstalk.
Jon Orr: Time to hang it up.
Sarah Stecher: Exactly.
Jon Orr: Yeah.
Sarah Stecher: Call it a day. Yeah.
Jon Orr: Oh, that's a great moment. And there's so much to unpack in there, I think, but I wonder, Sarah, we always say these moments and they pop into our minds, and I firmly believe for a reason. And for you, I'm sure there is that reason. How do you think your moment, you describe the impact that I think it had on you then, but maybe what impact did it have on you personally then, or who you were then and how you've grown into the person you are now? And then also, third part, how does that affect crosstalk.
Sarah Stecher: Too long to remember.
Jon Orr: I know. How does that affect your teaching now?
Sarah Stecher: Yeah, so I think for me, it, first of all, gave me a lot of perspective on what math learning can feel like and that it can feel frustrating. And when it does feel frustrating, I was very, very shocked and surprised at how quickly I could turn to not liking math after having liked math for 14 years. And so I was just very surprised at how quickly that turnaround could be. And thinking about crosstalk.
Jon Orr: Stabbed you in the back.
Sarah Stecher: Yes, exactly. Math and I were like crosstalk.
Kyle Pearce: I thought we were friends.
Sarah Stecher: Yeah. I thought we were friends. And so just how quickly that was able to change. I thought also a lot about what would it have looked like if we had maybe started with that chicken nugget problem instead of the notes with all the symbols and all the notation. And I think that has really played a very large part in how I teach now, which is starting with some experience or something that students can understand, something that feels relevant or just something very accessible. And then adding in some of that formality later, and thinking about my students and how I can reach them and how I can make math accessible and so that they can figure it out. Just trying to be very intentional about, yeah, giving them opportunities to make sense of math and giving them that agency and empowering them that they can figure it out, and math is figure-outable if you want to call it that. And math does make sense. So, that's been the influence on my own teaching journey.
Kyle Pearce: I love it. I think I just heard a bit of a Pam Harris line that she uses all the time that math is figure-outable, and it truly isn't. So I have two things. There's a question and a comment, or maybe a reflection, something that stuck out to me. So my question for you first, and then I'd like to follow it up because I don't want to let it slip. My question for you is, when you ran into that modular arithmetic experience and you felt like, Jon said it's like math stabbed you in the back, or your journey was over, maybe you weren't meant, I thought I was in good shape, but here I am. I'm wondering, do you look back on that now? And if you were to look before that experience, did you find that math very really actually challenged you? Were you just one of those students who was able to "get it" and maybe didn't have a ton of experience with real productive struggle to that point? Or was this something different?
The reason I'm asking is because I know that was my problem, is that, anytime things were hard for me until I hit university where no one wanted to help me, it was like, I always got bailed out. I got bailed out by the teacher. I got bailed out... Somehow the struggle didn't last very long for me. And it was like, I was able to overcome it very quickly. And I'm just curious for you, is that similar to your experience, or was it something different? What about that maybe caused you to have such trouble because you did overcome it eventually and your mindset had shifted so, it seems quickly, I'm sure you were struggling for a while. So I'm curious about that.
Sarah Stecher: Yeah. So I would say, I don't think this was the first time that I experienced challenge in a math class. I just remember that in my prior experiences, when something got hard, I just had this, I don't know, I guess belief that, okay, if you just keep working at it, you can figure it out. I was never the student that just didn't study and that got As, I always, I think I just very much enjoy the learning process. So I enjoyed taking time to really try to understand what was going on as well as, I was that student that was always asking questions and trying to understand it from every angle, because to me, until I fully understood it, I couldn't really own it. And so maybe what made this experience in college different is I had actually worked to try to figure out what was going on. I was doing at my house and trying to figure it out. And there was just some kind of block there where that wasn't happening.
And usually, I think in the past, I had just gotten myself out of that or I had asked the right questions to be able to make sense of what was happening or, and the whole class is struggling, so then we go over it again the next day. So, yeah. So I think that was maybe the difference there. And I think in general I do actually enjoy it when it gets challenging. And so yeah, this felt a little bit different because I just remember being very frustrated at that challenge and not excited by that challenge.
Kyle Pearce: Nice. Yeah. That really piqued my curiosity. And then I was curious as to a bit of the reasoning and I like how you described, I wonder if we started with that chicken nugget problem first, and you went on to describe this idea and what I was hearing you say was really starting with something that was contextual, there was a context there. You're creating meaning out of an interesting question, or your curiosity could be sparked and then you could eventually work through the details later. For us, we talk about getting through that consolidation phase. And I know for you and your friends, you call it experience first and formalize later. So I want to give you an opportunity here, a little bit of a platform to take us through that a little bit. And something that really resonated with us, when we read and went to your website, Math Medic. We looked at the approaches that philosophically and pedagogically, the approach is very similar. It's very on par with what we're trying to do and that's what really resonated for us.
So I want to flip it to you now and I want you to describe, what does that mean to you to Experience First, Formalize Later? And let us know, how did you land on that? Was this something that you were doing from day one as a teacher, or did this develop over time? I would really be curious to hear how you landed in this place that you're in right now.
Sarah Stecher: Yeah. So Experience First, Formalize Later is what we call our student centered teaching model. And there's lots of ways that you can do discovery based learning and student centered learning. And this is just something that we found, worked really well for us and our students. And basically the gist of it is, is that we're flipping around the traditional model where instead of the teacher starting by giving the students the notes or, okay, here are the important words and here are the symbols that you're going to need to know, and now let's practice. Instead, the students start in their groups. So they're working in groups of three or four, and they're working through some kind of an activity that guides them to the big idea of the lesson, except for that they're coming up with it for themselves. And so it's all about the students constructing the knowledge instead of the teacher holding that knowledge.
And I think the main idea philosophically is a lot of these topics that we teach in school, yes, there's a lot of formality, especially as you're getting into the upper grades of pre-calc and calc, but oftentimes the underlying idea, there's this main principle that we really want students to understand. And we're trying to demonstrate to our students that what they may have experienced as math class in the past where it's a lot about answer getting and just following the right steps and doing the procedure, that isn't really truly what mathematicians do. It's not really what math is as a subject. And so we came up with this model of, okay, how do we get them to say these things that generally we, as the teacher would say?
And so, yeah, my colleagues developed it and we've been working on it together. And so thinking about, okay, what questions can I ask the student so that they're having those aha moments? And when we're writing the lessons, like that moment that I had with modular arithmetic, when I realize, oh, it's divisibility with remainder. Those are the moments that we're hoping to create where they're like, oh, solving a system by elimination is just like comparing and seeing what's different between the two and trying to get that difference to only be due to one variable. And so thinking about, okay, what do we want students to understand about this lesson? And then creating an experience where they are arriving at that. And then very similar to your model, there's a debrief where students are sharing their solutions and then there's a chance to do this formalizing because there are vocabulary words and there is formal language and notation that students are expected to know, but it's all about when we add that.
So in our model, after students have worked in their groups, then they're putting their answers up on the board. And when students share out their answer and a student is talking about this starting value on a graph, we as the teacher and the students as well, with the red pens, they'll write, oh, we call this a Y intercept. And so this formalizing later is really, okay, you already have all of the ideas. You've already done the thinking and the reasoning. In math, this thing that you've already thought about that you've already discovered it has a name and that's all it is. And so bringing the concepts back down to their core and having students be the ones that own that learning.
So I guess in regards to your other question about, have I always been doing this? No, I definitely started my first year with notes. I wrote the notes. So I was like, oh, I think this is a really good explanation. I think they'll really get it. I'm really trying to parse out everything and make it super clear. And I think what I realized is it was working for very, very few of my students. And there was probably five students that were really with me and they were asking good questions, and they then knew how to solve the problems. But there was a lot of students that I was missing with that approach as well as, teaching felt like a little bit of a drag. I remember crosstalk.
Kyle Pearce: Been there. Yeah.
Jon Orr: We need to shake it up. It's like crosstalk.
Sarah Stecher: Exactly. Yeah.
Jon Orr: ... you need to shake things up for yourself a lot of times. And, Sarah, I love it when we hear all of a sudden, there's this idea that you've had, or you've heard, and all of a sudden you're hearing it everywhere. You use the word flipping the class around and we've often called that we're teaching math through problem-based lessons. We call it the real flip class, not the flipped math class that had a lot of popularity a couple years ago with videos, but flipping it around, you just explained, which was really nice. We were talking with James Tanton a couple months ago and he's a big fan of doing this too. And he's like you grab the really hard question from the textbook and just bring it to the front and teach it like it's common sense. It was, there was a lot of similarities there. And then you hear from Peter Ludalle, who's asking kids to get up at the boards and you give them the task to start. And then you think about how to use the five practices.
All these ideas, they all come to together and it's always rewarding and it gets calming to know that so many other people are coming to the same conclusion around the same time, because in our career's lifetime where all of a sudden it's like the last 10 years, this divergence it's all coming together. And it seemed to not, I'm sure it was there before, but it's maybe gaining in popular a lot more now. I know that we do this too. And I can say what my students are doing in my class or how they respond, but I'm always curious. And also people on the podcast would probably rather hear from you than Kyle and I every week, all the time.
But Sarah, what are the comments you hear from kids when you teach this? They probably come into your classroom having been taught the other way, the way that you described before, but what do they say about, do you do any at the end survey to see what how do kids respond after you've taught using this technique? I'm curious some comments, good, bad, mix of both. Oh, I don't like this. Just tell me how to do it. That kind of thing.
Sarah Stecher: Well, I can definitely start by saying that September is a very rough month. And there crosstalk.
Jon Orr: Just because you're missing summer or why is it? Tell us more.
Sarah Stecher: It is so rough because I feel like the students are resisting everything I'm trying to do. And I think, Jon it's like, what you said is they come in having learned math in a very, very different way. They're so used to the teacher telling them here's step one, here's step two, here's step three. If you just follow these steps, then the question on the homework will be just like it. The question on the quiz will be just like that. And if you can just follow these steps and do this procedure, basically do a lot of mimicking, then you're going to be successful in this class. And there are a lot of students that they have been successful in that model. And they're the students that are now in pre-calc and AP calc. And that method works for them.
And so they come into my class and I try to every year, I think I get slightly better at the beginning of the year of framing about this crazy thing we're about to do and how different it's going to feel. But it's definitely a shock to the system that, oh, wait, we're starting? You haven't taught us yet. And so they come in with this idea that, oh, well, if the teacher hasn't told me, then I can't possibly know what I'm supposed to be doing. And so September is full of parent emails and parent calls and all coming from a good place. They're concerned. They want their child to do well.
Kyle Pearce: Yeah. Advocating for their children and yeah, absolutely.
Sarah Stecher: Exactly. And so it just takes a lot of time to share why we are teaching in this way. And so generally around then, October, November, they're getting into it a little bit more. They're used to the routine of like, okay, yeah. We start class by working in our groups and the teacher's not just going to bail me out. When they do have questions, I often am that annoying teacher that just asks them a question back. And I just like, well, what do you guys think about that? I'm like, I don't know. And then walk away. And so they start to get used to that. They are going to be the ones responsible for figuring things out.
And then the spring is always the best time of the year because students are coming back and they're like, this was really different. But I feel like I understand math so much better. And to me that is always so rewarding. And every fall I have to remind myself of, okay, we're at the beginning of the year, I know where students will be by the end. And so, yeah, it's just super rewarding that, yeah, that students feel that way.
Jon Orr: Yeah. Awesome. Thanks for that. And I was just going to toss in here, my wife teaches kindergarten and she says September till December is brutal because they're trying to show kids how to use routines and all these things about interacting with each other. But she says springtime is just golden, because everything runs itself.
Kyle Pearce: Well, and think of the slippery slope that is for a kindergarten teacher. You've come so far and you're like, wow, feel this great sense of accomplishment. And then you have to start it all over again. It's almost like crosstalk.
Jon Orr: And then you have to do it again.
Kyle Pearce: Yeah, totally. Totally.
Sarah Stecher: Exactly.
Jon Orr: And Kyle, we have to do that every semester. So I was just thinking, by spring I'm like, oh yeah, that makes sense. They had all this time to let it sink in, whereas our semester system, we're starting, you're doing it all over again in February.
Kyle Pearce: Yeah. And what you're sharing resonates with us. It reminds us of when we began this journey, I think too something that it feels even harder and it feels even more difficult to manage when you're trying to make this shift in your pedagogy. So you're like you believe it from a philosophical perspective, but you're like, I've never done it yet. What if I'm wrong? So you have this doubt and, of course, as you do this more and more, and you know like you're saying, you have confidence in where you're going. That obviously gets a little better. And I definitely know, I struggled with this with a lot of students, especially early on, I'm finding now, more so, that I'm trying to be more explicit.
And I don't know if you and your team feel the same way, but being very explicit in the why we're doing what we're doing helps, it still doesn't satisfy all the students, or all the parents sometimes. Because again, they just want to make sure that, okay, am I going to be okay if we go through a whole year of math, or a whole course of math in this way? I'm wondering, what about from teachers? You've got a group of teachers, you all teach in the same school, there's the three of you. I don't know if it expands beyond that. Are there other teachers in the department doing this or open to that, but what about, is there pushback from other educators? What does that look and sound like? And I guess my wonder is the consistency, because one thing I know that Jon and I struggled with was that we felt like we were making these changes. We did them together, but we are in different districts.
So not just different schools, but different districts. And we worked together to try to ensure, hey, when I had a bad lesson, I'd tell Jon about it. Be like, hey, well, did you do this? Or maybe try that. That was great. But when you're the only one in a department doing something, you can spend an entire year, imagine the kindergartners, you spend the entire year on routines. But then the grade one teacher has no routines. It's like, it goes all out the window. And the same is true when we try to convince students that, hey, math is something different than maybe what you thought it was. But if they go into another classroom where it's back to the way things were, you got to stop and wonder sometimes, I know we did some good things during that year, but for that student as a whole, it's more like a blip in time versus like any consistency. So give us a heads up on that. Did you get pushback from teachers, and what does it look like now? How is that going for you folks?
Sarah Stecher: Yeah. So, in our department, it's a pretty big math department. We have a pretty big school. So our math department is about 15 teachers. And so I would say definitely very mixed in terms of teaching styles and teaching approaches. I would say our school is, they do give us a lot of autonomy, which is very nice about how we teach the content that we teach. We have common assessments and we have the standards that we have to meet. So I would say there's a few other teachers in the department that have gone all in on these lessons and are doing that every day with their students. There are other teachers that probably don't use our lessons. They stick with maybe a little bit more of a traditional model, or just things that they're more familiar with. I wouldn't say there has been pushback in terms of why are you doing this? But I wouldn't say that everybody's been like, okay, let's all do this as a department.
But I have been very pleased and very impressed that some teachers have, even after teaching 15 years, some teachers that are willing to say, okay, let's try this new thing. I have binders full of notes, but I'm willing to try something different. I would say most of the teachers are probably trying a lesson here or there instead of the everyday of it. Yeah, my colleague and I, we were just talking that there are now students that have actually, so Experience First, Formalize Later, we call it EFFL, E-F-F-L. But we are just noticing that there are now students in our school that have EFFL-ed for four years because they keep going back and forth between Luke and Lindsay and I. And so seeing that impact has been very, very cool, seeing students that for four years are doing this kind of teaching style. So, that has been very cool to see.
I think that is not the case for every student. We also have students that switch at semester, which can be very hard, and doing very different types of instruction. I remember I took over a teacher's class second semester and I was like, why are these kids not talking? I had such a hard time getting them started. So that is a little bit tricky, but I think students are really resilient. And I think if we provide those environments where they can show us and each other how brilliant they are, I think the students always surprise us in a very positive way about how quick they are to be willing to try new things, even when it feels unfamiliar.
Jon Orr: Right. Yeah. Sparked a question for me, is that when you said there's some kids that get four years of this style of teaching, yeah, FFL. I know I already asked you this about what the kids say, but I'm wondering anecdotally, what's the impact on a student who has gone through this in to say crosstalk.
Kyle Pearce: Yeah, because you said it was cool to crosstalk.
Jon Orr: ... secondary. Right?
Kyle Pearce: ... see, so I'm curious, what, anecdotally would you say, what's cool to see? What do you find is different or varies from maybe a student who might not have that experience? I'm super curious.
Sarah Stecher: Yeah. So I think the biggest comment I hear from students is, one, that they felt like that their teacher believed in them and trusted them, that they were able to figure things out for themselves. And I had never really thought about that until I heard that from a student that it does put a lot of faith in them that I think, hey, I think you are very capable of figuring this thing out. I don't think you need me to tell you step by step, what to do or to spoon feed you the information. And I think students also telling me that they understand something in a much deeper way, as well as that they find math way more rewarding.
A lot of these students, maybe they were successful in other models, maybe they weren't, but when I see that they are enjoying math more and not just being good at math, but also struggling with math, or that productive struggle and realizing that sometimes that struggle is really, really helpful and it doesn't always have to be avoided. I think for me, that's a big win, because I think those are the kinds of skills that they're going to take into everything later on.
Kyle Pearce: I love it. That's super cool. And today I was actually at a school teaching and using, like I say, very, very similar approach. They were coming into circumference and pie and area of a circle. And the teacher, we were having a discussion and this teacher was like, here's what I was going to do, but it was very like, I said, why don't we try to flip that a little bit and let's do that. We had an opportunity to do it today. And it was so great to be in this classroom and to have, I'm describing, I'm like, what would we call this? We were using cylinders to help us with an exploration. We wanted to have a student try to estimate or predict what's longer, the distance around the top of a glass or the height of a glass.
And it's a great question because your intuition wants you to pick height. So we have a vote and all this stuff and as we're talking about these things, one, I kept saying the distance around the top of the glass, the distance around the top of the glass, eventually a student's like, isn't that just the perimeter? And I'm like, yeah, that is. So I'm like, why is that better to say, talking about your formalize later. It's like, why is that better to say than the distance around the top of the glass? And they're like, well, that's a lot of words. And I'm like, yeah. And then I'm like, it's actually a special perimeter that has a special name. There's actually with rectangles, there's some rectangles have a special name. I'm like, can you think a rectangle that has a special name?
And they're like, what's this guy talking about? I'm like, it's called a square. Have you ever heard of a square? And they're like, yeah, I've heard of a square. So we're formalizing, doing all this stuff, which is great. Now I'm wondering though, because you're teaching, so I'm working with, we're getting more abstract when we're dealing with circle measurement, but I'm wondering for those other high school teachers, like you had mentioned like teaching geometry, teaching even AP calc, a lot of people think, okay, that's great. You can do this sort of thing with maybe not so abstract concepts. So concepts that are a little bit more, or maybe easy to create a context, or to introduce concretely or visually. As we get into more abstract ideas, what does that look like and sound like? And I know this is off the cuff here, this question, but do you have an example of what that could maybe look like and sound like for someone who's trying to visualize this as they're driving to work right now, listening in?
Sarah Stecher: Yeah, sure. I think that is definitely one of the challenges in math education that we wanted to address where it seems like a lot of this good work in math ed is happening in the middle school grades. And what does it actually look like when doing this with really, really abstract concepts? And so I think oftentimes we still use a context, not necessarily just to make it interesting in real world, but to make it accessible to students. Earlier I was alluding to this lesson about elimination, and we could talk more about that. But it's not necessarily that has to be a context that makes things curious or interesting for students. So oftentimes it can be, for example, let's say you're doing trigonometry and they've just spent all this time talking about similar triangles.
And so thinking about, okay, so you have this similar triangle and why is a similar triangle good? Well, okay, it helps you figure out, now that you know the scale factor, you can find all these other sides. Well, what if you didn't have the similar triangle, but you still wanted to figure out the missing sides? Well, and then we start talking about the sign and the cosign and the tangent. They give us these ratios that in this one tiny measurement, it gives you all three of the ratios all at once instead of actually having to have that similar triangle next to you. And so it's like every single similar triangle all wrapped up in this one value and think about how valuable that is. And so it's a lot of that tying it back to something that they already know, and it could be something from a previous unit. It could be something from a previous year, could go way back, but relating it to something that is accessible and something that makes sense in some place so that they have a place for it to land instead of just feeling like something arbitrary.
I guess one lesson that jumps out at me right now, point-slope form of a line. I think a lot of students are really familiar with slope intercept form. It's the most common form. And personally I love point-slope form and just, we have this lesson it's talking about Coldstone and it's talking about, okay, if you know the price of I got to have a size ice cream with three toppings, and then you're trying to find the price with all these other number of toppings. Do you actually have to go back and find the base price with no toppings, or if you've already figured out the price per topping, can you just add on to what you already have? So if you have three toppings and you want four toppings, sometimes students' gut instinct is like, okay, I have to first go all the way back, find the Y intercept and then add in four more. I'm like, okay, well, if you already know what three toppings costs, you can add one topping from there and get four toppings.
And so we make this big thing about, okay, well, why didn't you add four toppings? I thought you wanted four. And they were like, oh, we only need to have one more, because we already have the three that are included. And then in the debrief, and the formalization, we're like, okay, why did you only add one? Well, you had three, you wanted four, it's one more than what was already included. And so by doing this and this repeated reasoning over and over, they're realizing, oh, this anchor point, this starting value, doesn't have to be the Y intercept, it can be anything and I can add and subtract from there. And so getting them to think about the reasoning process and the thought process behind why something works and just taking their ideas and their intuition and their logic. And then, okay, this has a name, it's called point-slope form. And they already know why it's so useful instead of me trying to convince them.
Jon Orr: It's always great to think about what is the big picture here? Why does this make this?
Kyle Pearce: Why does this matter?
Jon Orr: Yeah. Why is this super important? How can I put my students in a situation? Doesn't have to be context driven, it can be puzzle driven, or a situation where I'm like, okay, this is awkward. Can I figure out a different way to do that? And then have your students say, I need a different way to do this, or I've figured out this way, there's got to be an efficient way. Sometimes those are the questions you want to ask yourself when things get abstract. So thanks.
Sarah Stecher: Exactly. Yeah. We call that breaking their tools, making them want this other thing. Yeah.
Jon Orr: That's good. I like that phrase, breaking their tools. Sarah, I know that you guys at Math Medic have created some resources for teachers. So we've been chatting about what's happening in your class, but you guys have been busy creating resources for our listeners and for lots of teachers all out there. And I think it's all, it's for free. You're just like, hey, here's what we're doing, guys. You guys want to do it too? So can you tell us more about that? Where folks can go to get these?
Sarah Stecher: Yeah, for sure. So we've been creating these lessons and we wanted to share them with other teachers so they don't have to do all of that same work. I think something that's great about the math community is we can share our resources. And so our website is mathmedic.com. And there you're going to find a whole bunch of lessons for pre-calc, for algebra two, for geometry. And now we are finally getting into our final course, which is algebra one. So the other three courses are complete. Algebra one, we just dropped the first two units and we'll be finishing the rest of those by the end of this year. And so what you'll find when you go on the website, once you go to the course that you want, you can look at the unit that you want and the lesson, and each lesson is going to have the handout for students as a word and a PDF in case you want to make any edits to it.
It's going to have the answer key, so you can see what student solutions are and then what the teacher adds on. You're also going to see just some, like a teacher guide with the experience first part, like what should you be doing while students are doing the activity? And then the, formalize later, what are some of those things that you want to bring up in the debrief? What special things are you looking for? And there are some ideas for monitoring questions. What can you ask as students are working? And so we've created that for every lesson, and then there are some review activities that you can download and use with your students. So we just hope that you find those helpful and just try them out.
Kyle Pearce: I love it. And I apologize, I smiled a little bit and I hope I didn't throw you off. Those who were listening, you can't see us, but if you're on YouTube, you can, but all you may have heard Jon's cat show up on the scene. I didn't see him this time, but my goodness gracious, it's like every single time we're recording that cat likes to show up and I can't help myself. So Jon's panicking to turn his mic off and Marty's just like, whatever, I don't care. So that is awesome, Sarah, and thank you again for sharing for the audience. Hopefully they weren't as distracted as I was by the cat, but that is fantastic that you've spent this time, you're sharing it with the community, and that is such an awesome, awesome thing. And I'm sure folks at home are going, yeah, good on you for doing this work.
I'm wondering here, before we hang up, if there was just one thing, we know people can't remember every little detail. If there was one big idea that you're hoping folks walk away with from this conversation, what would it be? If it was one thing that you could do to influence that math moment maker community, what would it be there, Sarah?
Sarah Stecher: Yeah, I guess I would say that I think my goal for my own students and for all of us is really that we can make our math classrooms ones where students are doing mathematics and not just learning facts about mathematics or memorizing mathematics. And I know Peter Ludalle has that quote that we sometimes plan under the assumption that students can't or won't think. And so I guess my encouragement for the Math Moment Maker community is to just reenvision what it could look like when we understand how brilliant our students are. And we create the kinds of environments where they can flourish and they see that math is something that does make sense and something that they can figure out for themselves.
Jon Orr: Awesome. Great takeaway for our listeners. And Sarah, we want to thank you so much for joining us here on this episode of the Make Math Moments That Matter podcast, we are going to put your links in the show notes. So links to Kelp Medic, Stats Medic, Math Medic, and some of the blog posts you guys have been writing links to those lessons for everybody to get their hands on. So thanks so much. And, hey, take care and do that great job that you've been doing in your classroom.
Sarah Stecher: Thank you so much. Have a great night.
Kyle Pearce: You too. Thanks so much. Cheers, Sarah. We'll chat soon.
Sarah Stecher: Bye.
Kyle Pearce: As always math moment makers, both Jon and I learn so much from these conversations and I'm sure you heard some of the similarities in terms of our philosophical approach. This idea of, I was going to say formalized first, come on now, experience first and formalize later, that is right up our alley and something that I think's really cool, if you do check out their resources, is that, like Jon and I say all the time, everyone is a little different. Your personality is different. Your teaching style is different. Your philosophy can be the same and maybe how you go at it might look and sound a little bit different. So go check it out. Maybe some of the things there are more in line with how you think, or how you would like to deliver a lesson, but ultimately at the end of the day, what we're trying to get away from is rushing to that algorithm. That's what I hear from this.
Jon, it's like, we're not going to rush to the algorithm. We're going to get formalized and we're going to help students with algorithms later. We're going to co-construct and co-develop these algorithms with our students when it makes sense to do so.
Jon Orr: Totally. And I hope everyone pulled that away from this episode. And if you're still grappling with some of these ideas, we'd love to hear about it. You can always reach out to us on any of our social media at Make Math Moments on Twitter, on Facebook, on Instagram, you can join our private Facebook group, ask questions over there and community jumps in way before Kyle and I can even get to it to answer your questions. So do that over there and, hey, you're not alone. We can do this better together.
Kyle Pearce: Absolutely. And if you haven't hit subscribe on your podcast platform, make sure you do so. And for our friends, we have an ever growing audience of people now listening/ watching the podcast on YouTube. Hit the subscribe button over there, hit that notification bell because we also release a separate, shorter video that takes you through a concept like recently, Jon, you were sharing how we do our animations when we're trying to spark curiosity. And he does so with apple keynote software. However, if you're a PowerPoint user, you can take a lot of these ideas and you can do things like that crosstalk.
Jon Orr: Hey, Kyle crosstalk.
Kyle Pearce: ... in that software.
Jon Orr: ... we showed it in the video, how to do it with PowerPoint.
Kyle Pearce: Oh my gosh, Jon, I should probably watch the whole thing, shouldn't I, hey? There you go. Awesome crosstalk.
Jon Orr: We did it.
Kyle Pearce: Amazing. Amazing. I didn't realize you were that multi talented, Jon. I thought you were just like explore.
Jon Orr: I explore for people.
Kyle Pearce: Me, I don't even have PowerPoint on my computer, so I don't even know how to get going, but it is possible. So check out that video over on YouTube and remember all show notes, all resources, all links that you heard in this episode can be found on makemathmoments.com/episode175. That's right. We have 25 more episodes to hit that 200 mark, Jon, can you believe it? That is bonkers. Transcripts are there. Download those transcripts, read them from the web and take them with you, or just go and have a browse around. We've got lots of awesome math moment maker fun over there. So my friends, until next time, I'm Kyle Pearce.
Jon Orr: And I'm Jon Orr.
Kyle Pearce: High fives for us.
Jon Orr: And high five for you.
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