Carmen Sinatra: In terms of the sparking curiosity, it's different from my 12s I find than from my nines. For my nines, I feel like I can grab your jumping frogs or one with the water balloon. I can think of lots of things like that and I can find little videos. But for my 12s, I just don't have that kind of resource or that kind of... Yeah, I don't know if it's just me or if it's my natural way of getting them into the lesson.

Jon Orr: Today, we bring back Carmen Sinatra from Richmond Hill, Ontario. Carmen is back in the classroom with COVID protocols firmly in place, and shares what it has been like teaching from a distance online, and preparing for the part of her schedule, where she'll be teaching from a distance face-to-face.

Kyle Pearce: By sticking around, you'll hear us chat about two benefits of our new all-day math blocks, and what we can do with senior students, who typically learn abstract math concepts, and how to spark their curiosity. This is another math mentoring moment episode where we talk to 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. All right, cue up the music. Welcome to the Making Math Moments That Matter podcast. I'm Kyle Pearce from tapintoteenminds.com.

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

Kyle Pearce: With you, the community of math moment makers worldwide who want to build and deliver math lessons that spark curiosity-

Jon Orr: Fuel sensemaking.

Kyle Pearce: And ignite your teacher moves. Jon, are you ready to dive into, I think this is episode 102 of the podcast?

Jon Orr: Yes, we are pumped for that. Over 100 episodes under our belt. I can't believe it. We want to take a few minutes to say right now that the second annual Make Math Moments virtual summit was a huge success. We had over 20,000 teachers registered, and a huge portion of those people attended live.

Kyle Pearce: If you attended, then you know what powerful messages have been shared, great resources for your classroom that were shared and given out. And I'm sure you walked away rejuvenated to keep making math moments that matter for your students. If you didn't attend, then something big must have come up this past weekend, or you know about the replays happening this week, and you've already scheduled them into your busy schedule.

Jon Orr: That's right. If you're listening to this episode the week, it went live, which is Monday, November 9, 2020. And then, you'll have until this Friday, November 13, 2020 to watch any and all of the 36 sessions that aired over this past weekend.

Kyle Pearce: Holy smokes, 36 sessions. If you visit makemathmoments.com/summit right now, you can watch those sessions like Kaneka Turner's session on making math class safe again, or Chris Lesniak's up for debate talk. Or you could dive into Dan Meyer's session on connected and creative classrooms in a time of crisis. Plus, 33 other hour-long sessions hitting different big ideas from kindergarten through grade 12 and beyond.

Jon Orr: After November 13, 2020, all the sessions will then go inside our Make Math Moments Academy for our members to bench watch all year long. If you're not a member of the academy yet, then you can still join over at makemathmoments.com/academy.

Kyle Pearce: That's right, my friends. And you get your 30 days free, your first 30 days free to gobble up those sessions. So, maybe you want to go a session a day for 30 days, or maybe you want to stick around for the whole year long to dive into that summit, plus the nine other courses, and all of our teacher guides, and units for your classroom. Head on over to makemathmoments.co/academy to dive in, and become a member today.

Jon Orr: All right. Now, let's get on with our discussion with Carmen.

Kyle Pearce: Hey there, Carmen. Welcome back to the Making Math Moments That Matter podcast. This is another math mentoring moment episode, and we were just chatting before we went live, and restating that typically, we don't bring people back so quickly. But back on episode 92, we were chatting about getting ready for this very uncertain school year start during COVID-19.
And we had a great chat, and we thought you know what, let's bring you back on a lot sooner, not a couple months sooner than we normally do. But more or less a couple weeks later. So, we are recording this about a month to the day from when we recorded our original episode. And we're dying to know, how are you doing and what's going on in your world right now, now that we are just into the new school year here in Ontario?

Carmen Sinatra: Well, so I'm at Richmond Hill High School in Richmond Hill. And our time has been very interesting the first few weeks, very uncertain, like you said, at the very beginning. I feel like the last week or so, in particular, this week, I finally started to feel like I'm settling in a bit into, "Okay, this is how it's going to go." But our structure for dealing with COVID-19 is, we have one live class in the morning, and it's two and a half hours.
And that class is a maximum of 15 students. And then, in the afternoon, the rest of the classes are online for approximately 50 minutes. I think I said 40 the last time, but the reality is, it's scheduled for 50. But by the time we actually get everybody on, and doing what we need to do, it's basically crosstalk, right, exactly.

Kyle Pearce: And all that wonderful stuff.

Carmen Sinatra: Yeah. And then, we switch cohorts every day. So, there's two cohorts, and the following day, it's the same class, but different kids, the other half of the kids. And we do that for about two weeks. And then, we switch to a different face-to-face class.

Kyle Pearce: Right. Like a different course?

Carmen Sinatra: Yes, yes. Yeah.

Kyle Pearce: Yeah. Interesting, yeah, go ahead.

Carmen Sinatra: Yeah. It's interesting because I don't have a period one class. So, the last time we did a lot of talk about how we're going to deal with the face-to-face class, and then I haven't had one yet.

Jon Orr: Oh, wow.

Carmen Sinatra: Yeah. I know, it just happened that way. I didn't realize what the schedule that that was going to happen.

Kyle Pearce: So, you're having your prep period, your preparation period first, does that mean that first two weeks? Is this your second week you're in right now?

Carmen Sinatra: Second week of the kids being there, other than we did start on September, 9, 10, 11. Yeah.

Kyle Pearce: So, those two weeks would have been your period one, but you don't have a period one. So, are you just prepping, and then helping the online kids in the afternoon?

Carmen Sinatra: Yeah. So, then my three classes, I have them in the afternoon. So, I have them like 2:00, 3:00, 4:00 right straight. So, 12:55 straight through to 3:30 about, I'm online with my classes.

Kyle Pearce: I'm curious, just because I'm trying to compare to my system, what we're doing, because we're also in week two into the school year with the students. I also had period one prep. What I did the first week, because we have a different system than you. But I was moving around covering teachers from class-to-class. So, I was a substitute teacher for all the first week, going into a class to give that teacher a break, to give them their prep period.
So, I guess I should give why that needs to be the case. So, I think the last time we chatted, I had shared that, I was going to be in a somewhat similar situation to you with my students that we would have this 2.5-hour block of math in the mornings, then we'd have a different 2.5-hour block with a different course in the afternoons. And then, we were doing that every day for our quadmester.
And they changed that about a week before we were back to school. And now, what we're doing is we're doing a 300 math blocks, basically, all day with your class. So, you'd teach your period one class, or your first week of school every day that week is with the same group of students for the whole day. So, from 8:00 until about 2:20, 2:30, where they're in that math class doing math that whole time.
And so, we were just saying before we hit record that there's some good sides to that and bad sides to that. But the kids are like, their brains are just ready to explode doing math for five hours straight, and I could imagine what that might look like for different teachers. But because I had first period prep, I didn't have my class that week.
And so, all the teachers since, they're on duty all day long with their students, they aren't actually given a prep period during that time. So, we organized a schedule as math teachers, where I was the one on prep. I would go into a math class and give that teacher a 76-minute break, where they would go, and have a preparation time to plan their next day, or later that day.
And then, I would take that class for 76 minutes, and do whatever the teacher wanted me to do. It could have been, last week was the first week we did it. So, it looked like here's a lesson on linear relations. Good luck. And I got it the moment I walked in the door.

Carmen Sinatra: Oh, my gosh.

Kyle Pearce: We just did some lessons all day. And now, it's just practice time. And so, it was being given one-on-one help. Sometimes it was, can you fill this time?

Carmen Sinatra: Why don't you just do whatever?

Kyle Pearce: They got hand over the chalk crosstalk. So, we had some discussions on some math ideas that aren't same curriculum based related to math games, but yeah, that's what I was going through, but I was curious-

Carmen Sinatra: And how often do you rotate?

Kyle Pearce: Yeah. So, we every week we would then, so that's week one would be your period one class all day. And then, the next week, we're in week two is now period two.

Carmen Sinatra: Oh, my gosh.

Kyle Pearce: And we would have that class all day. And then, so you wouldn't see once your week was over with that class. You don't see them for a week, and then because you have a different group. And then, you'll flip flop week one to week two for the quadmester until November.

Carmen Sinatra: Wow, it's so strange to me how all across this province, there're so many different models. It's amazing how we all have to figure out how to work our own model. And just, that's what we're given. So, there you go.

Kyle Pearce: Well, I guess that was a long-winded explanation there. But it came from the ideas I just wanted to ask you about your prep period. So, you don't have to do any cover another teacher, and give them a break during their 2.5 hours? I'm guessing no, because they have the afternoon to be online.

Carmen Sinatra: No, we haven't. We've had, I'm going to say regular on-calls if a teacher is away, and it has happened. If the teachers away, then we get an on-call like we normally would. But they're only a quarter of that period. So, kids can have four different teachers come in for their on-call.

Kyle Pearce: That's true. Yeah, 35 minutes here, and every 35 minutes or something.

Carmen Sinatra: Yeah, yeah. And hall duty, things like that. But no, the two-and-a-half hours, I have to admit has been pretty nice. And I'm concerned, because next week, I will not have that. Because next week, I'm starting the period to class for two weeks. I've heard from other people in the department who have just other people in general who don't have that period one off, and they just seem to be a wreck almost.
Because you do the two-and-a-half hours, then we have about, well, from 11:30 until the first class in the afternoon starts around 1:00, 12:55. And then, somewhere in there, if you taught period one in the morning live, then you would have one of the 15-minute blocks you would have free.

Kyle Pearce: Right. Got it. It's so interesting to me how all of these different models across the province. In some ways, I find the ministry here in Ontario, the Ministry of Education, they do give a lot of autonomy to districts. I think we chatted a little bit in the last episode about how they rejected a lot of the initial plans.
And it made you wonder if why they didn't maybe hone in on, "Hey, listen, it's got to be within these particular parameters or whatnot." But it's just so interesting, the different models, and trying to think of the pros and the cons. And I don't know, even if I was sitting there, and your district said to you, you can pick any of these different models. I've heard many of them out there.
My district is in a slightly different model than Jon's where the afternoon is online, but it's with a different cohort. So, it's not as many classes as you're dealing with Carmen, but it's not one class like Jon is dealing with. So, it's like somewhere in between. I have no idea which one I would rather.

Carmen Sinatra: I don't know, either. I know that there's ones that I would not want. I have to say, Jon, I don't think I would want the one that you've got.

Jon Orr: When we're in the third day, there's some good and bad. And actually, I was just having this discussion with teacher at lunchtime about the good that comes out of it. Now, don't get me wrong. Obviously, I would love to go back to our original system. But having the kids all day, and knowing that they're going to be with me, it gives me some freedom to stretch certain lessons out when I always was crunched to get, say, a consolidation or a connect stage in, just to wrap it up before the bell.
Because you've always got those lessons where this lesson runs long, but this lesson runs short. And you're like, because you have all day you can play with that lesson. So, it's like, okay, we're going to stretch this past where I would normally go, and then I'll shorten that one up over there. That part has been really nice. And then, that's one of the two things I was talking about today with a teacher.
And the other one was about students who generally might fall through cracks. This happens. So, it's like, you'd have a 76-minute block. And then, for those of you who have taught 60-minute blocks, or even shorter, the act of having that hour and having say, a class of 30, or somewhere between 25 and 30, whatever class number you have, it's possible that there's a student there that just say, doesn't answer a question that day, or says I'm just not going to participate that much for this hour.
I'm just going to go through. You know there's kids that hide a little bit, and it seems like they can get away with that because it's only 76 minutes. They might go, I'm not going to participate. And they make that choice, and say tomorrow, I will. But then you know, the next day comes and they don't participate fully that day too. And it snowballs. But when you have them all day, they can't hide.

Carmen Sinatra: They can't hide. There's no hiding.

Kyle Pearce: I'm coming for you.

Jon Orr: Yeah. Well, they can't hide. And they know that they can't just opt out of math just for the one hour that they normally would have it. So, what I'm finding is those kids that have been shy and might have opted out before are participating, and they're doing their work. And they're actively involved in discussions, and I have a feeling they're going to benefit from this situation.

Carmen Sinatra: Always good to look at the positives, right?

Kyle Pearce: Yeah. You know what, and especially during challenging times, you really do have to. And the one thing I'll say, as well, just knowing from the teachers I've worked with, my district, my wife is a teacher, she's in elementary, so all the kids are back, and they're doing everything but at a distance. And so, she's having her own struggles. And when you just see, educators are figuring it out.
We knew everyone would, but initially, there was a lot of stress, a lot of anxiety, probably still a lot of stress and anxiety. But everyone is coming together, they're figuring it out. And when I say figuring it out, like Jon said, it's not like it's just like it was, or we would like it to be, but making it work and doing the best with what they can. And I think that's beneficial.
And that's the one thing too, that Jon was saying about, being able to stretch your lessons to play with the clock a little bit during the day a little more. It's like a lot of elementary teachers have that benefit. The downside is you're with the same group all day. So, if you have a group of students or a student who is really driving you bonkers for whatever reason, you don't have that 75 minutes, where you get to hit the reset button, and try to take a deep breath, and go again.
So, there's some of those negatives there as well. But on the positive side, yeah, you really do get to manipulate the day a little bit, and make sure that you have a conversation with every student at some point that day, where it's really easy to miss that in shorter periods.

Carmen Sinatra: Yeah. That's what I'm looking forward to next week. I get to officially meet my nine applied students, and they've been great online, for the most part. There's a few who, honestly, I have had one student who's come once. And I don't know exactly what the issues are, I think it's a lot of tech issues. And some of them, they just don't know how to get on a Google Meet.
They just don't know how to set up the Google Classroom. I have a couple of other students who they just arrived in Canada not that long ago. So, it's completely foreign to them how this is all working. And I'm looking forward to having that opportunity to have them in my class. Because I suspect, there's more there, when we're just talking online, and they're not talking because they don't know how to speak English.
Then, it'll be really good to have them in class, and I'll be able to interact with them. And my class is very small to begin with, it's 15. And they're still splitting them up into two cohorts. So, I'm going to have seven on one day and eight on the other. And I was actually given the option to not split them up, and to have them all together, and just come every other day.
They're not allowed to come every day, because then they would be mixing with the other group, and the whole point is to keep groups apart. And I said, "No, let's split them this time." And part of my reasoning is simply because I want to catch those kids who I haven't talked to much. And I'll have that opportunity, two-and-a-half hours is a lot of time, and only seven kids or eight kids. I'll be able to hopefully get them on the ball, and caught up, and see what they're really thinking about the math.

Jon Orr: Right. Yeah, for sure. That's a wise move to separate them. Actually, I did luck out. So, my grade nines that I have right now this week. Normally, my class size did shrink, because I think some opted to go online, and then they did not reshuffle my class to lump with another class, thankfully. So, I end up only having 11 on my class list right now. And I've had a few not show so far.
So, basically, over this week, we've had eight regular kids every day in that class, which has been awesome to work with eight every day all day. Like I said, they can't hide anywhere, especially because there's eight, and then they're on the ball. They're working hard all day, even though by the end, we could see it in their eyes that their math filled.

Kyle Pearce: Well, Carmen, the last time we chatted, we discussed all kinds of our wonders, and even our worries, and are just trying to get ourselves prepared for this COVID situation. And because you have that first prep, and you haven't actually interacted with some students face-to-face, we may not be able to dive into some of the things that we chatted about back then, but maybe we can chat about some of those same challenges, or maybe different challenges online.
Last time, we were chatting about some of these ideas, for example, having double the time compared to normal class time, and how do we safely use vertical non-permanent surfaces? And actually, Jon and I were chatting before we hit record tonight. And Peter Liljedahl actually messaged us after your episode went live, and said he had listened. And then, he wanted to have a chat.
So, we just recorded another episode with Peter. So, that'll be going live soon as well. I don't know if it's before or after this episode goes live. So, it struck a chord with him as well. And how do we make groups safely, and then we talked a little bit about consolidation prompts. And I'm wondering if we think about what you've experienced over this past week, and a half or so, with your online classes.
I'm wondering what's on your mind now or currently? And maybe first, we can even start with what successes, do you have a success you can share from your experience in the online space? And it maybe, it was something that surprised you, and you didn't anticipate would work as well as it did.

Carmen Sinatra: The online has been interesting. I've had my ups and downs. Definitely have had some moments of success, though, for sure. My nines just seemed to be like such a friendly bunch. My surprise, my pleasant surprise, I guess is that sometimes they just have no inhibitions, the grade nines. They get online, they don't care. They're talking, and you get to the grade 12s. And it's like, I'm talking to a void, nobody wants to say anything.
They're all afraid to speak. But my nines don't seem to be, overall. They're nice in the chat. And they get on, they say hello. One time, it was so nice. There was a student who didn't want to talk. I said you can unmute yourself and just explain what you did. And he wrote, he goes, "No, I'm too scared or something, too shy." And then, another girl just unmuted herself and said, "Oh, don't worry, we're all nice." And I was just like, "Oh, that's so nice."
And since then, he doesn't talk still, but he definitely types. So, that's nice. I've had some really great successes with breakout groups, and doing the breakout extension. It was pretty shaky at first, but I did a breakout group with my twelves. And I have two sections of MHF. And that was really good. I enjoyed that a lot. I got a jam board out for everybody, actually, one jam board.
And then, I had, how did I structure it? I created the teams beforehand. And then, you put links up to other Google Meets that they go to. And so, they have their own little group that they can talk with. And on one jam board, I had 12 different slides, one for each group. And I was able to broadcast out to all of the teams and say, "Okay, this is what I want you to try and do. And this is what I want you to discuss."
And yeah, by the end of it, all of their boards were full. I could see that they were talking to each other. It was really good. My only problem with that was that the time went so fast, and I was so tight at the end. So, that's been a great success. And I did jumping frogs with my grade nines. That was fun. And there was another one that I did. It was jumping frogs, and I forgot, but there was another one that I did with my nines that was really good, too.
But yeah, we've had a lot of fun with doing some of those engagement activities. And that's been the highlight when I do my online, because if it's just me going through a lesson, sure, it gets the job done, I suppose, and be boring.

Jon Orr: Right. We chat about that often. You could technically do any lesson faster. And to the point by just telling them what it all means, and giving them what they need to know. But we know that that's not going to make a math moment that matters. That's not going to fuel any sense making. They can't make those connections if you're just going to do that. That's when kids say why do I do math?
And I don't see the purpose of this because they didn't actually have any meaning or value attached to any of that learning. You just gave it to them, instead of them earning it, and feeling like there was a success there on their own end. And so, we often talk about that of trying to make that happen in your class so that students go, "Yeah, I got something, and it was me." And that's important aspect of learning.
So, those are some awesome successes. I think that we can all pull success from our day. And I think that's important to continue to think about. Carmen, I'm wondering if we dig down a little bit in the sense of what are you struggling with, or what's a problem of practice right now in this current situation that you'd like us to just brainstorm and hash out here for the next 15 or 20 minutes?
And then, we can maybe bring you back on a little later, and keep having these conversations as a regular thing. But what would you like to chat about right now to get the ball rolling to address something in both your current classes/

Carmen Sinatra: There're two things. I think I'm going to talk more about one, because I think I have a handle on one of them a little bit better. So, I've mentioned with my grade 12s that I did a great breakout group. And that was fun, it was a lot of fun. I always feel like I have trouble in general with the grade 12, this is as advanced functions. In terms of the sparking curiosity, it's different for my 12s I find that for my nines.
For my nines, I feel like I can grab your jumping frogs, or one with the water balloon. And I can think of lots of things like that, and I can find little videos, but for my 12s, they just don't have the resource, and I don't know if it's just me, or if it's my natural way of getting them into the lesson is where I used to be. "Okay, here you go, I'm going to tell you everything about remainder theorem, factor theorem, whatever."
I tend to do a lot of, "Okay, you guys try this, you do it." And that works nicely when I was in the classroom, when I could have them up on whiteboards, and I could see them. But it is tougher to manage online. And so, that's how I usually go about it. But are there sparking curiosity ideas you guys have for grade 12s?

Kyle Pearce: This is definitely a common struggle that we hear from educators, especially our senior division. So, that'd be your grade 11 and 12 teachers are saying a lot of those same things that what about, and if we were to maybe compare and contrast. And I know it's nice when you have, for example, some of the Make Math Moments units or studies like jumping frogs.
We didn't really elaborate earlier, but jumping frogs is one of our Make Math Moments tasks that we design with that sparking curiosity in mind in order to fuel that sense making. And so, I'm wondering if you were to look on, and way out when you do your grade nine task, whether it's using jumping frogs, or whether it's using any other task that seems to draw students in.
And when you're doing this with your great 12s, and it sounds like again, more of this online space is where you're feeling this-

Carmen Sinatra: The hardest.

Kyle Pearce: Yeah, exactly. It's just stinging a little bit more. What would you say is different, maybe fundamentally about how the lesson unfolds? If you could paint us maybe a picture. And then, maybe we might be able to land on maybe some elements that maybe are happening on one that aren't in another or vice versa.

Carmen Sinatra: One of the fundamental differences, obviously, between nine and 12 is the level of difficulty of math. And the other video I got from you guys was fast clapper, that was another one. And they love that one. It's so simple. It's fun. And I guess maybe that's something I'm thinking about is that it's a very simple task. And also, its ratios and proportions. I feel like you can find so many things, talk about so many things with that.
And then, when I get to advanced functions, it's just a different kettle of fish. I don't know how to explain it. There's that time intensity too, the kids are, "Okay, we have to cover all these topics." And I know we want to uncover curriculum, and I get that too. But I feel like that's the big difference. And I don't know how to get around it.

Jon Orr: Yeah. And I'm super glad you brought up the fast clapper, which is actually is not our task. It's a task by Nathan Kraft, which we adapted a little bit to generate a little bit more curiosity, a little bit more kind of following the curiosity path. But great task for Nathan. He's definitely provided lots of 3 Acts tasks over on his websites, too. So, awesome stuff there, the fast clapper task.
We'll put that one in the show notes to Nathan's site. But let me ask you this Carmen, because I get where you're going with that, "Hey, these tasks in grade nine are very practical." There're so many real-world situations in grade nine, because the grade nine curriculum here in Ontario is heavily weighted on linear relations, and proportional reasoning.
And so much of what we see in our world is proportional, or has a linear relation built into it. So, it makes a lot of sense that we can find stuff there. But then, when we move to grade 12, all of a sudden, we're dealing with, like you said-

Carmen Sinatra: Polynomial functions.

Jon Orr: polynomials, and we're doing trigonometry with radians, and then all of a sudden, we're now into logarithmic functions, and doing all the algebra around all of that stuff, too. And so-

Kyle Pearce: How could you not be drawn in?

Jon Orr: Yeah. And so, I get what you're saying. So, that's context, but Kyle and I, sometimes we'll say context matters sometimes, but then sometimes it doesn't. So, I'm thinking if you can think about the jumping frog activity, or the fast clapper activity, if you think about the beginnings of those, and if I took context out, which is hard.
If you have to think about, "Okay, let's take context out, what is it about say, those tasks that draw kids in if we say it's not about the thing, or the actual real-worldness, what's going on behind the scenes that makes kids want to keep going with the math?"

Carmen Sinatra: I think for both of those, what I liked about was the element of estimating, and guessing, and saying, what do you think and no worries about are you going to get it right or not? And is he going to beat the record? I don't know. It was something about that being able to estimate that they really liked. And with fast clapper, they just were in awe. They thought, "Wow, that's so cool. Look at this guy, he's going really fast."

Kyle Pearce: Yeah.

Jon Orr: I think I got to agree. And Kyle and I chatted about that many times that I think we get glassy eyed when we see the videos. And we think, "Okay, that's a cool situation." But a lot of times, you can boil it down into thinking that these things withhold information. They don't give everything upfront, just like a scary movie, or mysteries novel that they don't give it all upfront.
And so, there's an element of surprise a little later on. And these elements, I think, even though they're not context, or that context doesn't transfer from grade nine to grade 12. But I think the elements of withholding information, and surprise transfer. So, when you think about lessons in grade 12, when you're thinking about factor theorem, and remainder theorem, and sketching functions, and using the transformations to do all that.
Yeah. It's very abstract, but we want to think, how can I withhold a little bit of the information, and the surprise, or the actual big idea here, and have it slowly come out? And how can I put my students in situations where they don't have that, but they're like, "How can I get more of that information?" Or maybe it's like, how can I get a better solution than what I've done, because what I've done is just so overwhelming, and so tedious?
But there's got to be a better way, there's these moments where it's like, kids just help me, give me something more so that I can keep going. So, that's what I try to think about when I'm grade 12 in starting those lessons. But I'm going to throw it back to you about what you think about that. And maybe does that spark any ideas right now about the topics that you're working on factor theorem, remainder theorem?
Yeah. That makes total sense. And this is why it's great to talk to you guys. Because much like when I've seen Peter, it's like, you're putting together the ideas that I have flown around, the things that I'm thinking of, and making a little more solid. And when I said that I had a great success the other day doing a breakout group. When I think about it, when I reflect on it, that's exactly what I did.
I did a very vague introduction. So, I was withholding information. And we were talking about slopes of secants and tangents. And I just gave them a scenario of, you're driving along, and you set your cruise control, and then you hit traffic. And just before you get to the place you want to get to you get pulled over. And the cop says that you're speeding, and then we talk about, I say, could you be speeding?
Could you not be speeding? And then, I started to give them more little bits of information. And it really worked. I was really pleased with it. Because you could see them on the jam board. They were putting little sticky notes, "Well, what if this, well, what if that?" And that's exactly what I wanted. So, then I was able to give them little bits and little bits of information.
So, I like that idea, and I'm thinking more about that plan I need to implement in other lessons, and do that more consciously. This was almost like it was conscious. But I did it last year this lesson, and I was searching through my stuff. "Do I have it written down? Oh, thank goodness, I did." But yet to be more conscious about that, I think that's what I'm going to aim for.

Kyle Pearce: I love it. I love it. And as you're mentioning, you're using some of these tasks. And there's these elements that are happening. And Jon referenced this idea that oftentimes, we get sucked into the idea that showing a video is what makes kids curious about it. And really, at the end of the day, it's about getting kids wondering, like what's going on here.
And I find that when we get into more complex material, as we move up the grade levels, we tend to move away from that. And I don't know if it's just simply because even conceptually, us as educators, oftentimes we learned much of this material very procedurally, and we might still be working through in our mind how things worked, or how it develops.
But really thinking about why are we teaching this concept? Not because we're going to apply it in our everyday lives, or when is a student ever going to use it? Not like that. But what about this concept are we actually trying to elicit here? And what's the question that I can introduce that's going to get kids wondering, "Huh, what's going on?" And it might not be contextual.
And the problem that pops into my mind, we'll put it in the show notes is magic rectangle. And while it doesn't necessarily apply directly to, let's say, grade 12, it is something that does come up as we get into more abstract algebra. And this idea of completing the square is one of these ideas, it seems like when we get into more complex situations that are more abstract.
We tend to resort back to this gradual release of responsibility approach, where we sort of pre teach it, we state the learning goal upfront, and then we say, "Okay, here's example one, it's an easy one." And we turn to that more traditional sense of I'm going to teach you, and then you're going to try some examples afterwards.
Whereas, if we actually think about completing the square, for example, and we go to one of those problems in the textbook, that usually, we don't get to until kids have done enough of these naked problems, where it's just like, "Hey, just complete the square, complete the square, complete the square." And then, finally, we get to the word problems.
When you look at the word problems, and you go, how can I take this problem? And how can I give it or give enough of the information for kids to start estimating about it. And then, also to help them uncover, or emerge the strategy, or the method that they would need to use in order to really uncover this new topic, it's really mind blowing.
And this magic rectangle task is really just an animation of a rectangle going really thin and long. And then, it goes, it almost looks like a perfect square. It's not. It's actually a rectangle, it's a little longer than it is wide. And it's mesmerizing. And basically, the question we're asking kids is like, when does it have the most area? And we just get them to yellow, when is it?
And then, from there, we build this, now, they're curious, and we start giving them a little bit of information. And almost bread crumbing them down this path, where all of a sudden, oh, my gosh, next thing you know, we're using algebra tiles, and they're doing this thing that will later name is completing the square.

Jon Orr: Kyle, I just wanted to jump in here, just to remind, I know you remember, but we pulled, and created that abstract math problem, because remember, we were driving home from a math conference. And we wanted to get over this idea that everything had to be contextual, we made this problem, a problem we actually pulled from a grade 12 book, whether it was a calculus book, or whether it was the advanced functions book, I can't remember.
But it was that common problem you see in those classes, but also in the Grade 10 academic book, which is like that common is a revenue problem. Every time you increase the price by X $2 for every increase of price $2, you decrease the amount of people who are going to buy by five people, and it was that common. It's a revenue problem, but every increase here, there's a decrease here.
And kids were always getting killed on that question. That was the one that was always asked, coming back from homework. And we're like, "Okay, I guess we just have to show them how it works." But we wanted to redesign that problem that didn't have any context. Because if we can do this rectangle problem, because actually the rectangle, every time the side length increases by two, the width decreases by five.
It was the exact same numbers that we use from the textbook. And so, we ended up, like what Kyle just shared, is we ended up creating an equation for area, which ends up, it's the same as the revenue function, when you do it with the other question. And then, you have to complete the square, or find the maximum if you're going to say, use it for calculus, when you try to figure out when that square, or when that profit is going to be the greatest.
So, we redesigned that problem just to actually take the context out of it, just so to say, "Hey, we don't need context to make these things curious." So, couple of the other things, Carmen, I wanted to just share with you about sparking curiosity, or keeping kids more engaged in say, grade 12 class than say just solve these problems.
Yeah. I definitely would give full credit for Peter, and also, for other teachers who talk about problems strings, where it's like, if you're trying to eventually factor different types of polynomials, you might start with some easy ones that get harder incrementally. And then, they're doing the math boards, and you can be walking around, guiding them along the way, and stopping the class every now and again, talking about a strategy.
That's always great in an advanced functions class and a calculus class, because kids will work during that time, mostly independently, and you can circulate really nicely. The other things that I like to do on a regular basis on those class is I've used two truths and a lie quite regularly. And then, I morphed it with them to call it just truth and lies, where I say we pull up say, a polynomial.
And then, I've written two truths, or I've written three statements, some are true, and some are lie, and they have to figure it out. And so, they have to play with the equation, put stuff in. This point is on this function, or this has a root of this, or this has a multiplicity root of this. And they have to figure those things out. And then, their job is to then create their own.
So, then, they have to create a polynomial, and then three statements that go with it, and then they post it around the room. So, that the other class, the other people in the class have to now figure out their statements, whether their truths or lies. And that's always this nice engaging thing that keeps flow going. And students are actively engaged to do that math.
And the other thing is a great starter with abstract math is creating which one doesn't belong. They're hard to create. But if you went to Mary Bourassa's site, wodb.ca, she's got some advanced functions stuff in there. But definitely, you can create your own, which one doesn't belong, and then that starts the conversation. And then, the fact that the students are talking about those ideas can get things flowing.
So, that all of a sudden, it doesn't feel like you just jumped into showing them how to do stuff, because there are lessons in that course that there's probably no way about it. You're going to have to show them how to solve this problem, or how to use this technique.

Carmen Sinatra: Like we did division of polynomials today.

Jon Orr: Yeah. The division of polynomials, and then how does that relate to the area model, and are we using area model? And then, we always end up having a big... every year, I've taught that it's a big class debate of who loves the area model better, who loves dividing polynomials, long division better? It's an ongoing war, every class I have. It's like you're in this side, or you're in this side, you can't be in both.
I use Desmos a lot just for generating curiosity. You can do a lot with hiding parts of functions, showing the whole function, not whole showing it, asking questions like that. I've got a lot of Desmos activities, where if you're going to do some sketching, or your drawing, half that course is drawing functions and transformations. And one activity that I've morphed into many activities is the matching function.
So, it's like, here's a function, you know what the base function is, but by trial and error, what can you put into this function so that it moves this one on top of that one? And because Desmos is so slick, and it's so real time when you type an equation in, kids can try to map one function on top of the other by trying to see which transformations do what.
And then, you can have a discussion, and a consolidation of all the different transformations, and what that means for the function. So, I think there's lots there to do in advanced functions to round it out, and generate discussion, and generate some curiosity. But I think the biggest thing is thinking about those elements that actually generate curiosity, and seeing how you can fit them in.

Carmen Sinatra: Yeah, excellent. Those are some fantastic ideas. I love Desmos. I have the which one doesn't belong actually bookmarked. So, I just haven't looked at it very much. Sometimes you have so many resources, and you need to actually find the time to really look at things. So, yeah, those are problem strings. I love it. These are all ideas that I can see being just a little more free, I guess.
I'm thinking, the which one doesn't belong, you could easily have a bunch of functions, and then asking which one doesn't belong, and they can have different opinions. And that's not necessarily that there's one right answer. So, that's the many entry points, all those things, I really like about those ideas. And that'll allow some of those students who I know are silent, they haven't said much. And it'll hopefully allow those ones to come out a little bit, and not feel so scared.

Kyle Pearce: I love it. I love it. It sounds like you've got a handful of takeaways today, which is awesome.

Carmen Sinatra: Yeah, definitely.

Kyle Pearce: Yeah, for sure. And something that did pop into my mind, I was going to end it there. But when Jon mentioned polynomial division, I had to throw in one more link. And that's James Tanton and the Global Math Project. He does exploding dots. If you go to explodingdots.org, there's this fascinating journey from place value all the way to basically, this idea of exploding dots of super conceptual, super, it just feels right as students work through it.
And the next thing you know, you are into polynomial division, and doing it with these dots that explode. So, definitely something to check out if you haven't. If I had to say and summarize everything that we've talked about today, before we sign off for the day, and then try to connect with you again in a few weeks' time once you get those face-to-face students in the classroom so we can check in with you, and see how things are working.
I think one of the biggest pieces, if I could only take one bit of learning over these 15 years that I've been in this math game, trying to figure this thing out, is starting with a good, interesting question, is the best way to start any lesson. If I can do that, it doesn't matter if it's visual, or contextual, or if it has a video, or what it is, if I can get kids to chew on an interesting question, it just changes everything.
And I think that's one of the big pieces here. I would be super curious after you go back in with those grade 12s. And you try thinking about the intentionality behind some of these pretty abstract concepts and go, "Okay, what's the question that I'm going to just toss to them, and try to get them estimating, and thinking, and essentially, just kind of playing with?"
And then almost, they'll be wanting to know, so how do we do this? Now, that you've got us drawn in, what do we do now? And that's where we get to dust off the microphone and say, "Okay, now, we can do a little bit of that teacher-led instruction as well. And hopefully, they're a little more drawn in."

Carmen Sinatra: Yeah, I think it's so important to realize that to draw students in, there're so many different ways. And there's been that misconception, I think that's out there for, I don't know how long it's been out there, that we have to have these real-world problems. And it just doesn't always work that way. And sparking curiosity can be done in many different ways.
The gradual release of information, I'm now looking at some other fantastic ideas that you guys have thrown out at me, the two truths and a lie. Lots of ideas where it doesn't have to be, here's a real-world problem. And quite frankly, some students just think that those are boring. They don't get drawn in by those at all. So, defined something that'll be a hook, doesn't have to be so complicated.

Kyle Pearce: Well, Carmen, it has been nothing but a pleasure. Now, we're two episodes deep here with this conversation. And I really do want to continue this conversation on. I think it's so relevant right now. And if it's okay with you, hopefully, we'll be able to get another date in the calendar coming up.

Carmen Sinatra: Fantastic.

Kyle Pearce: Yeah. And we'll see how things work here. And hopefully, just these conversations, we can just chew on some ideas together. I love your big takeaway that there's just so many different ways to spark curiosity. And I would love nothing more than to chat with you next time. And for you to feel like hey, your grade 12 are... it's almost like trickery, right? It's like, I've tricked them into wanting to talk about math.
Because they're definitely at a place now where they want to be cool. And they might want to get good grades. But maybe they don't want to get too into it. It would be really cool if as a team here, as a unit, we can draw out some good conversations, despite the online situation, or a partial online situation that you find yourself in.

Carmen Sinatra: Yeah, exactly. Sounds great.

Kyle Pearce: Awesome.

Carmen Sinatra: I'm looking forward to it.

Jon Orr: Yeah, we are too. And thanks so much, Carmen, for joining us on this episode, and look forward to talking to you again.

Carmen Sinatra: Fantastic.

Kyle Pearce: As always, both Jon and I learned so much from these math mentoring moment episodes, and we are sure you do too. But in order to hang on to this new learning, so it doesn't wash away like footprints in the sand, we must reflect on what we've learned. So, how are you going to go about doing that? One excellent way is to ensure this learning sticks by reflecting and creating a plan for yourself to take action on something that you've heard here today.

Jon Orr: A great way to hold yourself accountable is to write it down, or even better, share it with someone, your partner, a colleague, or a member of the Math Moment Maker community by getting over to the show notes page, or into our free private Facebook group, Math moment Makers K-12.

Kyle Pearce: And just as a reminder, the 2020 Make Math Moments virtual summit just ended this past weekend. But the replays are still up until November 13, 2020. If you head on over to makemathmoments.com/summit, you can join in on the learning while there's still time.

Jon Orr: You could even watch the replay of Marian Small's session, think the math, not do the math, or past NCTM President Robert Q. Berry III, and his session, what does it mean to be an anti-racist teacher of mathematics? After November 13, 2020, all the sessions are going to be then put into the Make Math Moments Academy for our members to gobble up all year long. If you're not a member of the academy, you can join over at makemathmoments.com/academy.

Kyle Pearce: And make sure once you're in that academy, you get 30 days free, so really, nothing to lose here. Stretch out that replay time by heading over to makemathmoments.com/academy to sign up, and gobble up those sessions.

Jon Orr: If you're interested in joining us for an upcoming math mentoring moment episode just like this one here was with Carmen, where you too can share a big math class struggle, if you can apply over at makemathmoments.com/mentor, that's makemathmoments.com/mentor.

Kyle Pearce: Show notes and links to resources from this episode, including downloadable transcripts can be found at makemathmoments.com/episode102, that's makemathmoments.com/episode102. Well, my friends until next time, I'm Kyle Pierce.

Jon Orr: And I'm Jon Orr.

Kyle Pearce: High fives for us, and high five for you.