Episode #120: How Do I Engage My Students When The Math Is Abstract? – A Math Mentoring Moment
In this mentoring moment episode we speak with Adam Love; a first year teacher from Myrtle Beach South Carolina. Adam shares what his first few weeks have looked like in his classroom. Like many teachers he wonders how to generate curiosity and engagement around abstract math concepts in senior classes.
This is another Math Mentoring Moment episode where we talk with a member of the Math Moment Maker Community who is working through struggles and together we brainstorm possible next steps and strategies to overcome them.
Adam Love: I guess trying to create curiosity through some upper level classes, where it's like the conceptual understanding often is a complicated proof, in which if I started to go through with them, they would get lost. And trying to make almost proofs a little bit more fun and engaging and contextual, I guess that's the hardest problem. crosstalk-
Kyle Pearce: In this Math Mentoring Moment episode we speak with an Adam Love, a first year teacher from Myrtle Beach, South Carolina. Adam shares what his first few weeks have looked like in his classroom. Like many teachers, he wonders how to generate curiosity and engagement around abstract math concepts in senior classes.
Jon Orr: This is another Math Mentoring moment episode where we talk with a member of the Math Moment Maker community, someone just like you, who is working through struggles. And together, we brainstorm possible next steps and strategies to overcome them.
Kyle Pearce: Let's do it! Welcome to the Making Math Moments That Matter podcast. I'm Kyle Pierce.
Jon Orr: And I'm Jon Orr. And we're from makemathmoments.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 math lessons that spark curiosity, fuels sense making and ignite your teacher moves. Welcome to another Math Mentoring Moment episode. Jon, I've got to say, these episodes, I think they might be my favorite.
Jon Orr: Oh, I think you're right. I think you're right. Let's get ready for another jam-packed episode. But first, we want to let you know, in addition to producing the podcast each week, Kyle and I have been hard at work preparing our classroom tasks, so you could use them with your students.
Kyle Pearce: Yes, we get up early on the weekends, and we stay up late during the week, just kind of building these passion projects of ours. And over the past year or so, we've been really trying to create full units that are all problem-based. And actually, Jon, we just shared recently a brand new unit called Piggy Banks, where students will explore adding, subtracting and even a little bit of multiplying with fractions and decimals. And we use coins dropped into a piggy bank in order to introduce students to this pretty abstract concept. And also, to ensure that we hit on some of those money concepts, those financial literacy big ideas.
Jon Orr: Yeah, I'm excited for this one. Our tasks are free to use with your class and academy members of the Make Math Moments Academy have access to the full teacher guides that show the teacher moves, as well as student exemplars. So they can run the lesson without a hitch.
Kyle Pearce: Head on over to makemathmoments.com/tasks to see the full catalog of tasks and full units that range from about the third grade through the 10th grade.
Jon Orr: Super exciting stuff. See you over at makemathmoments.com/tasks to run one of these awesome problem-based units in your class today.
Kyle Pearce: All right, now let's jump into our Math Mentoring Moment episode conversation with the Math Moment Maker himself, Adam. Hey, hey there, Adam. Thanks for joining us here on the Making Math Moments That Matter podcast. We love having Math Mentoring Moment episodes. And there's nothing more enjoyable than getting to chat with someone who's a little bit newer to the teaching profession. So I'm sure you'll tell us a little bit more about that. But before we do, how are you doing today? And where are you chatting from today?
Adam Love: I'm doing pretty well. I am coming from Myrtle Beach, South Carolina.
Kyle Pearce: Yeah.
Jon Orr: Nice. Myrtle Beach is always like when I ask my parents where they want to go visit in inaudible they're just like, Myrtle Beach. This is their prime destination, has always been Myrtle Beach since I was always a kid. So I think they went there, not obviously this year, no one went anywhere. But they went there last year. They always enjoy it. Adam, do us a favor, introduce yourself a little bit more. Let our listeners know what grade you teach, what's your school look like? And maybe also kind of give us a little backstory on what your teaching journey has looked like so far.
Adam Love: Yeah, sure. So I'm similar to, I think you, Jon, I think your parents or your dad was a teacher, if I'm correct. So my mom was a teacher. My dad was a pastor, actually. But I've loved teaching ever since, I think first or second grade. So I knew I always wanted to teach. When I was in second grade, I wanted to teach second grade, and when I was in third grade, I wanted to teach third grade. And then that eventually developed to I want to teach history for a while. And I don't know if this is a good advice or not, but someone told me like, not every school needs a history teacher. If you become math teacher, that's in high demand.
And I found out in junior high that, what do you know, I actually really enjoyed math. So I always knew I always wanted to teach. It was just kind of like, the teaching and the kids was what I always loved. It was just, I just need to find something that I enjoy to teach as well. So I went to college for math education, and got my bachelor's. I actually heard about you guys during my student teaching. So I was cleaning buildings to get money, because you have to pay to student teach. And I was just looking for something to listen to while I was cleaning, in the midst of student teaching and planning and stuff. And I found you guys, I think, looking for some project that I was trying to plan. So anyway, found your podcast, I think you guys were on Episode 10, or something like that.
Kyle Pearce: It was early.
Adam Love: Yeah, it was very early. It almost two years ago. I was very thankful, so I felt like you completely opened up resources to me, that I didn't really know were there. I had good experience in high school and college with teachers. And I felt like I learned the concept well, but as far as unlocking just the connections and the way of teaching, and getting away from procedures. Obviously not completely away from procedures, but just understanding what it meant to teach for conceptual understanding was really big for me. So going into student teaching, I tried a lot of different things. And my inaudible teacher was great, she just let me try whatever I wanted. And I did.
I tried making some my own three-act tasks, they did not go well. And even tried some of yours, guys and Dan Meyer's, also didn't go very well. But yeah, I finished up student teaching and then I had an accelerated grad program that I was already on. So finished that up. And then I'm currently in my first year of teaching. So I teach at a small Christian school here. I'm teaching algebra through calculus. So algebra I through calculus. Heavy load, but it's been fun.
Kyle Pearce: Fantastic, what a cool story. And I'm sure Jon, you must be feeling it as well, nothing makes us feel more energized than hearing about how the podcast actually has influenced teachers, and especially our youngest teachers or newest. Sometimes new teachers are doing the second career and things like that. It makes us feel so good that we're able to help with that journey. And something that really resonated with me, anyway, is just how you had mentioned, and I don't think Jon and I mentioned it enough. You had mentioned about how you had a really good experience with really good teachers in math, but maybe some of the connection piece was missing.
And you kind of mentioned a lot of that procedures first approach to teaching. And I think we could relate. I would hate if any of my former math teachers listened and thought that we didn't have a lot of respect for how they influenced us into wanting to get into mathematics. We had great teachers, and they built great relationships. I think we're just blessed that we, very similar to your experience, Adam, at some point we managed to kind of bump into, for me, it was Dan Meyer was kind of my turning point. I know Jon mentions Marian Small was a big turning point for him, as well as Dan Meyer and some others. And it's just awesome to hear that the math community keeps helping push one another. And that's just awesome for us.
I want to turn this conversation and kind of build on it. You had mentioned you had some good experiences with some teachers in high school. I'm really curious to hear about your math moment. So when somebody says, math class to Adam, what pops into your mind? Is it some of those experiences with your teachers you just referenced? Or is it a different experience that you sort of haven't mentioned here yet?
Adam Love: I guess I got two that kind of compare to each other a little bit. So I had a great teacher, a high school math teacher. He's actually one of the reasons I wanted to be a math teacher, just the way that he built relationships and discipleship and all that. But he was very good at teaching procedures procedurally, like, obviously, I don't know the rest of the kids in the class. But as you guys have mentioned, it's easy to catch on to procedures pretty well with textbooks. And he was very good at just decoding textbooks in that way. He didn't discourage conceptual understanding at all, but it was never at the forefront.
An example that I always think of is, I was playing around with the difference of squares on my calculator, and I didn't know the difference in squares at the time. And I just remember, keep doing these on my calculator. I was like, man, there's a pattern going on between these. There's something, I can almost predict it, but I can't quite put my finger on it. And I told him about it. And he's like, hey, just see if you can work it out, and see what you come up with. So I think it was like a week later, I came to him with this epiphany. And it was just the formula for difference of squares. And so I found this and then I realized that this is actually in our book. And he was like, yeah, I tried that once in high school too, and I found out it was already invented.
But I think that moment for me was just, he encouraged that kind of thinking. And maybe if it wasn't always on the forefront. And then I had a college teacher, and she taught all the upper level. But I think she really expanded my conceptual understanding of what was going on. And even now, as I talk to her now, as I'm asking your questions and stuff, just like the lights going on to what she was trying to tell to me, and I'm finally understanding it.
Jon Orr: My guess, Adam, is that that has to be rare to have moments like that. And maybe I'm just speaking from my perspective, because I don't think I had moments like that where things were taught and my teachers enforced or tried to encourage that discovery, that play, that seeing the connections. For you to have multiple teachers to do that, I think that's pretty awesome for you to see that pathway to become a teacher. Because I think, I've said it here on the podcast before. When a student tells me they're going to be a teacher, I think of one of two things. They really like what I've done, or they really hate what I've done, and they're going to do the opposite.
I think when I became a teacher, I knew that I wanted to do something different. But it wasn't because I hated my teachers, like Kyle said, we had a lot of respect for... My father is a very traditional math teacher. And we've had many talks about different ways of teaching things and how kids learn. But I think that's got to be rare, because we have so many people on the podcast of what their moment is, and it's rarely that kind of thing. So that's awesome that you've had that moment, and it's influenced you. I guess I'd like to ask this follow-up question, and I think I probably can see where this is going for you. But being a first year teacher, how do you think those moments are going to influence your decisions, as a math teacher?
Adam Love: I definitely think one thing that I've really tried to plan for and tried to figure out how I can do, is to let kids discover those connections by themselves, but intentionally. I guess that's good teaching, right? Is if you can make them figure something out on their own, when they actually weren't completely doing it on their own. But you were kind of leading there without you knowing it. And I guess that's kind of been the struggle, because I feel like that takes a lot of time and a lot of planning. And you guys have both been first year teachers before, so I think you understand just the lack of time to prep new stuff for four or five preps.
Kyle Pearce: Yeah, absolutely. And something too, as I was listening to you share that about the difference of squares and what's interesting, because I had a recent experience with a teacher. It was a grade five teacher and a student was doing, he described it as, this educator described it as this trick. Where he kept saying, there's this trick. This kid's doing some pretty complex multiplying, and he was doing it very quickly. And he knew his perfect squares. And he would be essentially like derive, he would work his way back and compensate. And he couldn't quite figure, and at first, I couldn't figure it out either. And it was really cool when it was kind of the same thing, like a week later, I was just driving in the car. And then it just sort of came to me. I'm like, oh, my gosh, I wonder if it's that.
So we had a fun time with that. But the part that I loved about your teacher was, I feel like as an educator, and I've done this for so many years, and even with my own children, I struggle to hold myself back from essentially ruining it. What I mean by that is, it would be so easy for your teacher to be like, yeah, that's this thing called different of squares. This thing that you're doing right there, or like, here's how to do it. And we kind of take away, we rob the thinking from the student. And I think that means so much, when an educator can kind of bite their tongue and let the student ride that out. That's so awesome. It's so hard to do, it's hard to perfect, if you can ever perfect anything in this world.
But you try to do it as often as you can. And sometimes that can be really, really challenging. So super cool stuff. Now I want to go, and I want to bring this back to you, Adam, and talk about your current experiences. So you're like fresh, you're in this first year teaching. I can already tell from what you're saying about time and how time is limited. Time is limited for a 30-year teacher, a first year teacher-
Jon Orr: During COVID-19 also.
Kyle Pearce: Yes, exactly. With all these other things going on that we've never done before. Time is non-existent for a first year teacher. So what I want to do is, let's start with a positive moment here from this year. Do you have any recent successes or what you would deem a recent success, from your very, very new teaching journey, that you wouldn't mind sharing with the community?
Adam Love: Yeah, actually, one of the more recent ones was I was working with the sum of polygon. So actually, I think I used Jon, your polygon pile worksheet...
Jon Orr: Right, the worksheet.
Adam Love: Yeah, so I have a geometry honors class, and so I just gave them that as their honors assignment or whatever. And I had one of the kids in study hall, and just, we had gone over, we had done the investigation with your circles to polygons.
Jon Orr: Oh, yes. I just did that this week too. Sorry to cut you off.
Adam Love: No, you're good-
Kyle Pearce: Jon's copping your scope and sequence here.
Adam Love: Yeah. So I felt like I didn't really give that investigation go over well. But one of the students, as they were kind of going through the shape, they were in study hall asking for it. And I remember, we were just kind of trying to look at it, one of the kids said, he was just like, "I wasted all that time on the left side of this shape, and I didn't even need to." It's like both the students that were working inaudible just the light bulb that went on when they saw the parallel line and the transversals, inside of the polygon. And I felt like it finally clicked for them, that these polygons do create parallel lines. So I guess that's been the most recent one with students.
And then actually, I think the biggest one of the year was to start off the year doing Jo Boaler's squares task. I think that got me super pumped for teaching, because it was like my second day of teaching. And it's got so many connections to algebra, to geometry. And getting the kids talking about math was really cool to see.
Jon Orr: Awesome, awesome. We're going to throw all those links that you just referenced in the show notes. So if anyone else is looking or thinking, that might be a good idea too, so you can grab those over in the show notes. But those little successes that we'll build, and I think that will come over time, too. Those are some great shares from you, Adam. So thanks so much for that. And that kind of steers us in a different direction. What we wanted to focus this discussion mostly on today is thinking about any current struggle, maybe it's like a struggle that came from some of those activities that you did have successes on, but you were wondering about. Or what's on your mind lately, that we can help you here with in the next little bit.
Adam Love: Yeah, I have two big questions. So hopefully, we have time to cover them. Having listened to the podcast for a while, I feel like there's so much good content, but I also feel like it's kind of aimed towards, I don't know, grade six to 10. It feels like a lot of the strategies and a lot of the things that pop out of that are there. And so teaching calculus and pre-cal, I know you're in the advanced functions class, Jon. But I guess trying to create curiosity through some upper level classes, where it's like the conceptual understanding often is a complicated proof, in which if I started to go through with them, they would get lost.
And trying to make almost proofs a little bit more fun and engaging and contextual. I guess that's the hardest problem, bringing the upper level stuff into context, and then also at the time constraints.
Kyle Pearce: Now, can we ask you to go a little bit deeper, like so mentioning context, so trying to maybe find context. Why do you suppose context is what's missing? Are you seeing any indicators in your classroom of maybe it just not working the way in your mind you'd like it to? Because obviously, you're feeling like maybe there's something missing there. I'm wondering if we can dive a little deeper to kind of get to the root of what exactly is missing. If it is context, let's say it is context, what do you think the context might bring? Or what else might be just not working as well as you'd like it to?
Adam Love: I guess it would be number one, curiosity, I know that's a big thing, just getting them engaged in the math. And at the same time, I feel like it's really easy to get just very procedural, gradual released response kind of form of teaching. That's just the easiest thing I could do. It's like, yeah, I can just pop up a few examples and do that for going through how to teach limits. But to give some context to it, so that it's engaging in some way. Does that kind of answer your question.
Jon Orr: Yeah, I think you're kind of thinking that there's some abstract concepts, but we want to make that abstract engaging. So I'm wondering now, what right now about your lessons think that this is not engaging? Maybe that's what Kyle was kind of steering at. What are the look-fors right now that you're like, well, what's not working? How do I know it's not working?
Adam Love: Yawns, is that an indicator? I guess that's part of it, is, I just realized that one, my students aren't moving. They're just sitting there. And I don't know, I just use that as an indicator. If they're not writing or moving or thinking or talking in a good way, then I'm just talking. It's not really going in. And it's showed up on assessments and stuff like that. Just seeing, okay, they didn't quite get this as much as I thought they did. And they need to kind of get a little bit more engaged into it, I guess.
Kyle Pearce: You're painting a picture of what you'd like to see. So you had mentioned, moving and talking, and the good talking, I like how you say that too. Because early in my career, and I've got to tip my cap to you. In first year teaching, I was like, I hope nobody talks. Because I was worried about discipline. I would have dreams sometimes in my first year teaching of like, being teaching at the board and asking a student to come in and grab a seat, and them saying no. And then I'm like, oh, now what do I do? So hats off to you for that, for sure, for wanting to kind of draw out this and get students talking, which is great.
I'm wondering if, could you take us through maybe a recent lesson? And maybe it's even more like your routine, of what might a typical lesson look like, sound like? Take us kind of through a bit of that, so we can paint that picture and get a sense as to what that routine might look like, sound like for your students.
Adam Love: Yeah. So I would say with my lower level classes, look a lot different than my upper level classes. So like algebra I, geometry, it's a lot easier to find visual prompts, and three-act tasks and just things to kind of generate conversation, estimation, things to just teaching through task in general. So I guess specifically, looking at calculus, sometimes if I can find some pop culture reference to kind of connect it. So I'd put something up on the screen, I'm trying to pull up a lesson here that I can just refer to real quick. But I guess going through, I guess limits is what we were just on. So just going through a limit laws, and just trying to show, here's the table, here's the function and here's the graph. And trying to do a notice and wonder from that.
So looking at just putting something up and asking them to notice a wonder. And that I don't really get much of response, because second hour in the morning, so they're not really talking much. And so not getting really much of a response, and then just saying, oh, well. And then just go teach it, and then send them to practice it. And just not seeing that really work super well.
Jon Orr: Yeah, and I think it's a common issue where there's a lot of resources for middle school and early high school algebra, to use a three-act math task or to use this resource. Because there's more of that going around. And it makes sense that, mostly, or my grade nine math class tends to be proportions and linear relations, which is very easy to model in the real world. And so the context is there. And when you move into higher level algebra courses and calculus courses, the context becomes, it's pulling at teeth. And it's like, those are the examples where like, really, we're going to look at this context about ice shavings? And it has to do with polynomials? And we're like, what do you mean, come on. That's really drawn at straws. So it's like this abstract math, textbooks try to put that in there, but no one's really believing it.
That context is like, there's no way that I'm going to model that polynomial for that ice sculpture, no one's really doing that. It's not the way that you would solve that problem. Whereas in, say, grade eight, grade nine, you could easily solve these problems with proportions and drawing number lines and tables and ratio tables. So I get where you're coming from. So maybe I'm wondering, let's think about the tasks that you know are getting some engagement. And what do you think, is it about those tasks? Because if you think about it, if we can find parallels in what's really going on with these easy context problems, and how they're generating engagement there, maybe we can take that from those, and these abstract ways of engaging, and use them in these abstract concepts in higher levels.
I'm wondering, what do you think it really is about the say, grade nine, or the grade eight, or the early algebra courses, those three-act math tasks? What do you think is going on there that generates that curiosity with kids, and engagement with kids?
Adam Love: I guess, the first thing that pops in my mind is just ownership, or some clue or guess. I know when I'll have, in the lower grades, just kids immediately... like I had one pop up and try to solve it on the board using the PowerPoint, using the picture itself. And just being able to immediately try to solve the problem visually, because they could make sense of it. So I guess there's the ownership factor, that they can, I guess you're too high, too low, they can kind of get in a ballpark, they can own some part of it. Where they have some skin in the game, I guess, I think it was a phrase that's been used before. So something like that, where there's something on the line in some way. Yeah, that's kind of why I'm asking is, because I got no clue.
Kyle Pearce: Yeah, no, and I love it. And we've mentioned it on some episodes in the past as well, that I know sometimes it feels guests come on, and I hope they don't feel like we're grinding them. But we're asking these questions because we truly want to help you unlock some of those struggles, to kind of bring to the forefront what might be missing. And I think, especially with these middle school tasks, like Jon's mentioned, because they are easier to model, they're more relevant and in the real world, we can bring these contexts in pretty easily. But I think sometimes we can confuse the context itself with what's actually drawing them in. And actually, yeah, your example earlier today about using the polygon pileup. That isn't a contextual problem at all. Actually, it's very abstract. But yet, there's something about it that seems to draw students in.
And we would argue that if we can ask good questions, and I like quoting Dan Finkel, he always says, start with a question. I think one of the hardest parts for us when we get into the older grades is because the concepts are more abstract. Sometimes it's hard for us, as the educator, to even fully understand why we're teaching the concept in the first place. It's great, like Jon's talking about the polynomials and the ice sculpture and so forth. Like it's super contrived, when we try to do it with those types of problems. But sometimes we can ask very abstract problems, but we can do it in maybe in mysterious ways. And I'm sure, when you think about the Curiosity Path, and how we've mentioned it in previous episodes, one of the biggest pieces that we try to get people thinking about is this idea of withholding information.
Sometimes it's not contextual at all, sometimes it can just be about kind of making you look at something and make your head kind of turn sideways and go huh, what's going on there? There's something really interesting happening. And I'm wondering if you're thinking about maybe some of the lessons that did go well this year, and maybe compare them to maybe some of the lessons that didn't go so well this year. If maybe there's anything early in the lesson about those questions we're posing, or those wonders, the mystery that we're trying to provide for our students.
So I'm going to pause there and let you kind of share your thoughts on that. I know that context is really safe for us, but it is hard to do, or harder to do in the senior grades. So I'm wondering, and I think difference of squares is a great idea as well, because it's relatively abstract, or at least it's introduced that way in many cases, yet, there was something about it that sort of drew you in as a student. So I'm wondering if you're kind of thinking ahead to let's say, this week or next week, and you're thinking about some of the concepts that might be coming up. Maybe we can riff on some ideas together here on like, how we might figure out how do we introduce this idea, so that it's a little more curious for students to think about?
Adam Love: Okay, so here's something, derivative rules. When I learned them and went through them, it became very, very procedural very, very fast. It's like, oh, well, you can take the exponent and multiply it by the coefficient, and subtract one. I have no clue what I'm doing, but I can do that. And the chain rule, all those very procedural things. And then I think about it, and this is where the existential question comes from. It's like, why do they even need to know this anyway? Then I start asking that question. And that's just a bad spot to be, if I'm the one asking that question.
Jon Orr: I think those are great questions, because that actually goes to the heart of what we need to help our students understand. Because the first thing they're going to say is, why do I need to know this? And that's always what happens. If a kid has said that to you, most times it's we went too fast to the abstract, and they didn't catch why this was important. What was the big idea here? What is this purpose of me following this procedure, or even solving a problem like this? So a context can help with that, because you put the kids in a context and they're like, okay, well, I see that context, and how I can solve that.
But when you get to the senior grades, those contexts are out the window now, because it's like, we're not going to be in that context. So now, how do I accomplish this? And what I like to think about is thinking about, what's the real reason that this math needs to happen this way, at this time? And so think of your learning goal for the lesson, and you're thinking, okay, I would teach the power rule. But why are we teaching the power rule? Well, the power rule is basically a shortcut procedure that was proven from first principles. So it's like we're doing first principles, but why are we doing first principles all the time, when we can see that after first, we get this other kind of manipulation. And if we do it this way, it's all of a sudden, super short, super quick.
So you're teaching first principles and kids are like, oh, man, do we have to do this every time we have to do a complex problem? Like yes, that's the way I used to teach calculus. It's like, you're going to do it like that every time. And then we're going to introduce this other shortcut, and we're going to prove how it works. And then it's like, oh, this is so much easier, I just love this. Situations like that go a long way of why these procedures were created in the first place. Because most procedures were created in the first place to shorten up a long procedure, that someone went through and were like, you know what, we want to repeat that, but we want to speed it up. And I wonder if I can do it in a more thoughtful, more clever way.
And so procedures morph over time because of that, and then we just hammer procedures to kids. But there's a reason we did that that way, a long time ago. And then there's a reason we sped it up. So, especially in your senior courses, it's like thinking about, how can I put my students, and this is often what I like to do, is how can I put my students... I guess, usually I do it two ways. One is, how can I put my students in terrible situations where they don't want to do this math, but then all of a sudden, there's a shortcut or there's a nice elegant solution that will come out of this. And then they're like, oh, that was so much easier. Just because I can draw a line here, or I can think about how the connection here, or these two things will divide out nicely.
And those things exist. And then that can open the eyes of students, but they had to struggle first. And that struggle of struggling with this complex problem, or brute forcing solutions can create value for the kids learning. Because they're like, ugh, I solved it, yeah, but there's got to be a better way. Oh, I so much more value that solution over there, than the one I was working on, or the one I saw over here. So that value will sink in and then create that math moment. The other situation I tried to use with abstract math, is trying to put students in, like what Kyle has said about the Curiosity Path. It strips a whole bunch of stuff away, and see what we really need, so that we can solve these kinds of problems. And it's not real world, what I'm referring to.
So if you think about like a problem where, how we teach finding the distance between two points on a plane. We'd show, hey, you can use the distance formula. And then you could say, you know what? Well, the distance formula, it comes from Pythagoras theorem, I wonder if I can try to have my students try that strategy out with me not telling them. So one way I did this with my students when we were doing that was, we stripped everything away, we put two dots on the screen and said, find the distance between them. But I stripped the graph away. There was just a blank screen up on the thing, that said two dots.
And then they're like, what do you mean? Do I get a ruler out and measure? And then it's like, well, what would you like? And then now it's like, okay, well, what about a graph? So I put a graph up with no numbers, does that help? Well, it helps a little, but then we don't know what the units are. And then, okay, let me get the units up. Does that help? And then they're like, I think it does. And then students are starting to play with the graph and the units. And oh, now I can see that it's this way, and this way up and this way over. I can draw a triangle. And all of a sudden, it kind of unfolds from there. But it started with mystery. And there's no context, it's just abstract. But those students kind of step their way through it, and then also will feel that value, because they walk through that solution. So I often think of those two ways of trying to create mystery in my class, in a certain way.
And then also, what's the big problem? Like, what's the big idea, that I can put my students in to think there's got to be a better way? So those are two definitely things I think about. They're very big ideas, because they apply to lots of different situations. But if you're thinking about your lessons of like, why the grade nine stuff works better than the grade 12 stuff, it's because of those two things, I think. And those three-act math problems, those problem-based problems, do that well, easily. Whereas we got to work a little harder when we're going more abstract. I'm going to throw it back at you now, Adam, what do you think about some of those suggestions?
Adam Love: So if I can narrow it down, basically, make your students beg for a procedure? Because the math is so painful, is what I'm getting. No, but I think I understand what you're saying, though. It's just, in essence, showing them the reason why we have a procedure is to make things easier. Why we have computers and calculators.
Jon Orr: I think it's more like putting them in a situation so that they are wanting to know why, that they need to do that way. And so it brings the conversation out. And so how can we design the lessons that allow for that? Like my lesson would be great if it had ownership, right? This would be a great way to start designing, if I could strip a whole bunch of stuff away and put them in these situations, they're now starting to have ownership. And discussions are going to be built into something like that. So how can you do that? And it takes time to think about these types of lessons, but that's where we go when we try to think about how to create a new lesson for an abstract idea.
Adam Love: Can I throw a follow-up question off of that? You gave an example with the distance formula. And I tried something similar, but I ran into a problem with that kind of approach, where you just kind of put two dots on a screen. I'm just using that example at this point. And I feel like my students, once I send them into their groups, and they start working on boards or something like that, and I start walking around, I'll just find them standing there, inaudible.
Kyle Pearce: Yeah, this is something that is so common, especially early on and I think again, when you're a first year teacher, you're feeling this, but also when you're a 30-year teacher and let's say you've been teaching a certain way, let's say you've been teaching very procedurally, and you try to shift things. Especially when the shift is happening after the first few weeks of school, like if it's a month in, or maybe it's even three months into the school year, and you're trying to make these changes slowly over time. Like students, they know when something is different. And as humans, we would rather kind of go through the motions. I talked a lot about it during the spring, when COVID had us all working from home, and I had the kids at home. And because I was out of my routine, I felt exhausted all the time, because I had to think all the time. There was nothing that was automatic.
And as humans, and students know this as well, that, to them, it's like, oh, this actually, in the long run, would be easier if he or she, or whoever my teacher is up there, would just go ahead and just give me some procedures to memorize. And that worked for a lot of people. It worked for me, it worked for Jon, it sounds like it worked for you as well. So some students might be like, I can do a whole lot less thinking, if I just boycott this idea of having to actually engage. And I'll be honest, even when I go to conferences, and I'm in a workshop, initially when they say, okay, get into a group and solve a problem. Initially, I'm like, I kind of just want to sit here. But then once I'm in there, I'm like, it's actually a lot of fun. And there's a lot of learning. And I know I'm learning way more when I'm in that group.
And sometimes it does take a little bit of time for that to happen. And there's also like, I would say, this level of awkwardness there. Where students may not know what to do, they don't know how to be cool in this context, where it's like math class is looking maybe a little different than what it has looked like for them in the past. And as students get into the senior years, grade 11, grade 12, if they've gone through, essentially 10 years of education, where maybe math class looked a lot different. They might be just kind of not wanting to stand out, they don't want to do something wrong, they don't want to stand out as that student who's maybe too into the lesson. Maybe that's going to make them feel like they're not popular or not cool, or whatever it might be.
So this does take a lot of time to build that culture. But it is something that if you stick with it, and it is going to be awkward. Like anytime you are in class, and you try to give wait time, I used to try to count in my mind to give students enough wait time. And those few seconds that you'd wait, if it was five seconds, or even 10 seconds, it feels like an eternity. Where you don't hear anything, you're just kind of sitting there. And if you go back to even Episode 21 with Peter Liljedahl. He mentions when he started trying, it sounds like maybe you've listened to that episode, where he tried to get his kids to actually think. And it took two full classes where he lied and said he was going to go and do some photocopying, and he would come back, and the kids hadn't done anything yet. So you're not alone in that journey.
But I do think as students actually do take those first steps, they will start to feel good about it. When they get those mini wins, they start to see the logic there. But it does take a lot of time. And I know for Jon and I, it was something like, luckily at the time, Jon and I were going through a similar change in our practice. So we did a lot of collaborating online over Twitter, just kind of those lessons that blew up in our face. You had mentioned earlier, you tried your own three-act math, and it didn't go so well. You tried some of ours or Dan Meyer's, and they didn't go so well. That's very common early on. And it's really easy to just kind of pack it in. A lot of people, they try it once or twice and they go, no, this doesn't work.
I promise you, over time as that culture builds, and as you start to reflect and kind of weigh those good lessons versus those ones that didn't work so well. There's those little elements that we like to call in the Curiosity Path, and in our framework, that we like to work through. And it does take time, but I promise you, you will get there.
Adam Love: Yeah. And it's interesting, the times that I've seen it work, it's energizing. You see kids discussing and you see them coming up with strategies, you see them almost like forget that they're in math class, I feel like for a second. And it's like, those are the moments, just like, now, this is the way I want to teach. I can see it's like, this is the way to teach. It's just knowing that it's going to take a while to get there across the board. It's like, well, we just need to capitalize on those things, I guess. But yeah, I guess just seeing there's just a lot of room to grow in the, I guess you would say, the fuel sense making part. Knowing how to kind of prod them and push them down the learning goal, without just giving them the answer and teaching them.
Kyle Pearce: That's, I think, a good next step or where you go next step. And that's kind of the question that I want to throw at you, is thinking about this conversation and thinking about some of the learning that you've done in the past and some of the discussions we've had here. I think you would be, because you're a beginning teacher, you're asking these good questions about what's this look like in this class, what's this look like in this class? Our academy has a great community of teachers who are sharing questions like that in the community and the messaging board. And people respond, and there's a lot of sharing that's going back. People are posting what their lessons look like, all inside the community.
Plus, there's the courses that we've built in there about the Curiosity Path and some of the other things. So Adam, I think you would be a great fit for the academy. And we're going to send you a free year access into the academy, so that you can get in there and keep discussing these ideas with us. And also with the community. So I'm going to let you say what you think about that, but also think about, or we'll ask you, what do you think also is also a big takeaway from this conversation? We'll get that from you. And then we'll start to wrap up.
Adam Love: Well, first of all, thank you. That's awesome, very generous. I appreciate that. I guess, as far as a takeaway, would be, I kind of think, just realizing that context maybe is not necessary to still generate good discussion and good mathematical thinking. But just almost kind of hijacking any procedure to figure out, okay, why did we come up with that procedure in the first place? And then withholding the procedure long enough for them to want it, if that makes sense.
Kyle Pearce: I love it, I love it. Those are big takeaways. And I'm definitely like you, were I'm going to always look for a context first, because I feel like it makes it easier. But then when the context isn't there, it's not easy, or it's going to be just too contrived. I think you're right, you start to move to plan B, what else can we do? And tonight at the dinner table, with my own children in grade one and three, we did kind of a an ad lib number talk. And it was just dealing with addition, with three digit numbers. We started with 300 plus 300, then it was 320 plus 300. And basically, I just kept building and changing the quantities of the add-ins just slightly. So they can see patterns that, hey, wait a second, if he had adjusts one of the add-ins by five, then my answer is going to adjust by five. So it was like, they could see this pattern.
And there was no context whatsoever. But they wanted me to keep going, because it was like they could see it and they felt empowered. That my little grade one son is there keeping up with his older sister, adding three digit numbers together. Whereas in class, they're probably dealing with adding to 20 right now. And he's there rocking and rolling, because he's noticing this pattern, he's feeling enabled, he's feeling empowered. And I promise those calculus students are going to feel the same way when we start with things like, for example, you said derivatives earlier in the conversation. And going with like, okay, well, what do they know? And what can I do? Okay, well, if a derivative is like an average slope, it's an average slope. And it's between two points that are really close together, really, really, really close together.
Well, maybe we start with two points that are really far apart. We start getting them to just estimate. Like, what would you say the slope of this line, you got this quadratic here, and you toss two points on it, and you have them sort of predict, what would the slope of the line between those two points be? And getting them estimate and then you reveal what it is. And they're like, yay! Like small wins. No context whatsoever. And then all of a sudden, these points get really close, really close. And then all of a sudden, they're like, basically on top of each other. And they're like, whoa, now what's going to happen? So thinking of these ways, sometimes you have to be a little bit creative, talk with your colleagues, flip us an email, whatever it might be. But there's always a way, I believe that truly.
Sometimes it takes a couple times through a course for you to come up with a great idea to try, or maybe that idea didn't work the way you wanted, you have to modify it for next time. But I truly believe you've got what it takes based on just this conversation, how dedicated you are to try to not make it a procedures first process to learn math class. But more concepts first and build to the procedures. So I love that and I'm so excited that you'll be in the academy with us to do some learning. Before we let you go, we want to know, how are you feeling now after this call, for the Math Moment Maker community to kind of hear from you before we sign off?
Adam Love: Encouraged, I guess would be a one-word answer. Just yeah, encouraged and I guess, saw a next step that hopefully unlocks a lot more observations. Just looking at some things that are super complex and just breaking them down and just trying to make them as simple as possible.
Kyle Pearce: Awesome, awesome stuff.
Jon Orr: Adam, what do you think, since you're going to be getting in the academy, it's your first year starting off, we would love to check in with you on another voice call. We'll be checking in with you on the academy for sure. But another voice call in say next year, or maybe the end of this year, to see how the school year went. Would you be up for another chat with us, coming up in about a year?
Adam Love: Absolutely.
Jon Orr: Awesome stuff. So we'll definitely keep in touch that way, and look for an email from us about getting in the academy. We definitely have enjoyed this conversation and glad that you are a teacher and you are helping kids every step of the way. I think students will be better off just because you're in the profession. So thanks so much, and enjoy the rest of your evening.
Adam Love: Thanks so much, I really appreciate this.
Kyle Pearce: As always, as we said in the introduction, both Jon and I learn so much from these Math Mentoring Moment episodes, but in order to ensure we hang on to this new learning so it doesn't wash away like footprints in the sand, make sure that you're doing something to reflect on what you've learned here. An excellent way to ensure this learning sticks, is to reflect and create a plan for yourself, to take action on something. Even if it's something really small, right now.
Jon Orr: A great way to hold yourself accountable is to write it down, or even better, share it with someone. Your partner, colleagues or a member of the Math Moment Maker community, by commenting over on the show notes page or tagging us at Make Math Moments on social media. Or hey, head over to our free private Facebook group Math Moment Makers K to 12.
Kyle Pearce: And keep in mind, as we mentioned, we have all kinds of new problem-based units and tasks just like the brand new Piggy Bank unit, that explores adding, subtracting, even a little bit of multiplying with fractions and decimals. And we use the money model, that's coins, in order to do so. So head on over to makemathmoments.com/tasks and go check out the Piggy Bank unit or one of the other problem-based units.
Jon Orr: Yeah. Are you interested in joining us for an upcoming Math Mentoring Moment episode, just like Adam did here in this episode, where you can share a math class struggle? You can apply over at makemathmoments.com/mentor. That's makemathmoments.com/mentor.
Kyle Pearce: In order to ensure you don't miss out on any new episodes as they come out each week, be sure to subscribe on your favorite podcast platform.
Jon Orr: Show notes and links to resources from this episode, plus complete transcripts are over on makemathmoments.com/episode120. Again, that's makemathmoments.com/episode120.
Kyle Pearce: Well, until next time, Math Moment Makers. I'm Kyle Pierce...
Jon Orr: And I'm John Orr.
Kyle Pearce: High fives for us.
Jon Orr: And a big high five for you, too.
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