Pam  00:00
Hey fellow mathematicians! Welcome to the podcast where Math is Figure-Out-Able~ I'm Pam.

Kim  00:06
And I'm Kim.

Pam  00:07
And you've found a place where math is not about memorizing and mimicking, waiting to be told or showing what to do. But it's about making sense of problems, noticing patterns, and reasoning using mathematical relationships. We can actually mentor students to think and reason like mathematicians do. Not only are algorithms really not helpful in teaching mathematics, but rotely repeating steps actually keeps students from being the mathematicians they can be. What is...

Kim  00:34
I don't know. Sometimes I feel like I'm not ready to start these things. I started to yawn while you were talking, (unclear)

Pam  00:40
Oh.

Kim  00:41
(unclear). Oh, gosh. In my head, I was like, I hope I'm done yawning (unclear).

Pam  00:43
Because I'm that boring? Is that what's happening? 

Kim  00:46
No, no, no.

Pam  00:47
Man. I don't know that you and I are ever bored talking to each other? 

Kim  00:51
Oh, I mean, no.

Pam  00:52
(unclear).

Kim  00:53
No, boring is not the word. Running out of time. That's always (unclear).

Pam  00:57
That is always. Remember when we just took that trip not too long ago? And we were like, "We had this list we've created for months at a time. We're like, "We're going to talk about this. Hey, and on the plane, we'll talk about that. We'll talk about this. We'll talk about that?" How much of that stuff did we actually?

Kim  01:11
Zero. We were busy doing the things the whole time, and we came home with the whole list. And the rest of our people that we work with we're like, "Wait, I thought you were going to decide (unclear)." We barely got done what we were supposed to get done while we were there.

Pam  01:24
Ah, it was fun, though. We had a good time. 

Kim  01:27
Well, and that's funny that we say that because we don't do that as much anymore. You know like, we each kind of have our own thing that we work on. 

Pam  01:36
Well, you had kids, right? Like, I mean, a big thing that happened is you had little kids. 

Kim  01:40
Yeah.

Pam  01:40
We used to travel more together before you had kids and when your kids were super tiny. 

Kim  01:45
But then, I traveled a bunch when they were little just doing different stuff than you. 

Pam  01:51
Oh, that's true.

Kim  01:51
It was like, "Hey, I need you to go here, and I'm going to go here." And so, it was less of...

Pam  01:57
Together.

Kim  01:57
...we go to places together. Yeah. So, it's funny that you mentioned that because I feel like maybe even the... No, it wasn't the last time we were together. It's been a long time. But I remember going to a particular place. 

Pam  02:11
Together. 

Kim  02:12
Yeah, together. And I think I drove. And we got there, you know, maybe a few minutes early, before we even need to go in and set up.

Pam  02:18
I'm pretty sure I drove.

Kim  02:21
No, I was definitely in the driver's seat.

Pam  02:24
Okay.

Kim  02:25
Just because I remember this so vividly. It's burned into my mind. Because we laughed. We laugh so hard. So, listeners, we went to a place to present together, and it was one of those things where there were a ton of people, and so Pam was like, "Hey, we'll split up. You take a group here to do the..." I don't know "...K-5 group. And then, I'll take the older group. And we'll we'll do our thing at the same time, the same hours, just in two different rooms."

Pam  02:51
Mmhm. Mmhm.

Kim  02:53
So, we got there a few minutes early.

Pam  02:55
We got there early because we always leave early enough. 

Kim  02:57
Oh, sure. 

Pam  02:58
It was long enough drive that if something goes wrong, (unclear).

Kim  03:00
Yes.

Pam  03:01
We got there early. We had some time. I think the school wasn't even open yet. 

Kim  03:04
Yeah, yeah, yeah. So, I guess we were, you know, chatting, chatting, chatting. And at one point one of us, probably me, was like, "Hey, we should probably look at, you know, what we're doing today." And so, I pulled my bag and put it in my lap. And Pam's got her computer bag in her lap. And we're both like kind of going through our stuff to find, you know, our notes or whatever. And we both, you know, put our bags back, and we're holding our stuff in our hands. And we look at each other. And I still laugh every time I think of this. We laughed so hard because it was very classic, classic, Pam and Kim. Because in Pam's hand... No, I'll say in my hands were these beautiful color coded...

Pam  03:48
Typed.

Kim  03:49
...typed notes of kind of times (unclear). 

Pam  03:53
Multiple pages.

Kim  03:54
Multiple pages of like what I wanted to do, and like the examples and the times that we're kind of like, "I'm going to hit this mark." And I look over at Pam, and Pam has an index card in her hand with like maybe four things written on it. And I remember us... I don't think we ever looked at our stuff because I was like, "What are you doing?" And you were like, "What are you doing? That's insane." And it was just we laughed so hard because A, that's kind of our personality in general. But now that I think back on it, it was classic for where we both were at the time. And, you know, I'm a like... I process by writing, and so that's just kind of who I am anyway. But also, it's indicative of kind of where we were, right? That like I really like to think out like all the details, and like what I was gonna say, and, you know, anticipate like questions people were going to ask, and like how I would respond. And that was old hat for you by then, right? You had been. I was still in the classroom for later in our careers. And...

Pam  05:03
I've been presenting to teachers. I've been doing this particular presentation a lot. 

Kim  05:07
Yeah, yeah. 

Pam  05:08
Yeah. So, I had thought through that stuff over time.

Kim  05:12
Yeah.

Pam  05:12
I had made decisions, and notes, and things. And I had... So, if I could just make. Now, "four things". I didn't have four words. I probably had four problem strings.

Kim  05:20
(unclear) like the string that we're going to do. But where I would have written like what I wanted to... Like, I maybe had bolded a problem that I wanted to linger on, or like made a point off the side that I wanted to say. You were like, "I know exactly what I'm going to do with that particular string," and so there was just less.

Pam  05:38
Yeah, I know when to shuck and jive. I know when to backup, I know when to slow down if I need to, I know when... 

Kim  05:43
Well, and... 

Pam  05:44
Yeah,

Kim  05:44
And you just live with index cards in your travel, and you like have like 175 strings, and you'll just like rifle through your stuff, and you're like, "That's the string I'm going to do." Where I would like rewrite it each time. You know, (unclear). But I also later, like now, I'm able to think about, you know, as I help other teachers like in a coaching role, or like when we're helping to prepare people to go like do some presentations for you. You have said like, "Hey, Kim, I spent my time with you." And so...

Pam  06:25
I paid my dues. Your turn.

Kim  06:27
So, yeah. And, anyway, so you and I have talked before about like other travel situations where we go...

Pam  06:34
Well, wait. Yeah, let me... Are you're going to finish? Let me say (unclear).

Kim  06:37
No, I was just going to say there's hotel time.

Pam  06:41
Oh, gosh. But let me say a little more about what you just meant. Like, in other words, when Kim... Or maybe you were going there, sorry, with hotel time. When Kim and I very first started presenting together, I would have the outline. I would know what it was kind of going to be. And we would plan it. And then, we would, you know, go to the place, and the night before in the... The other one we were talking about was close enough that we went there that morning. But, often, you know, we would get in the hotel the night before, and we would stay up and we would go through the presentation. And Kim would ask all these really, really good questions. And I would yawn, and I would answer the questions. Because it was always super late. Like, we had little kids. You know, we left as late as possible. And then, we'd go through it all, and then you would go, "Let's do it again." And I would be like, "Okay. Fine. Here we go. (unclear).

Kim  07:29
You were so patient. So patient. 

Pam  07:31
So, that's what you meant when you were like, "I put in the time. I walked you through that. Then, we did it together, and then we debriefed how it went. And so, then later when other people, I would like, "Kim, will you please go train them? Please don't make me do that again."

Kim  07:46
I don't think you have to train anybody more than me. I like to know all the things. So, yeah. And maybe even at that time...because that was way early days...you might have even had more than an index card at that point. Like, I feel like you might have had some notes. So, anyway.

Pam  08:04
Well, and let me say one more thing. Every time I had a new Problem String. And that doesn't mean the first time I did it. That means the first 10 times I did it. I would have more notes on that card for that Problem String. 

Kim  08:15
Yeah.

Pam  08:15
Until I did it enough that I could kind of then. Alright, now it's a part of me. 

Both Pam and Kim  08:20
Yeah. 

Pam  08:21
So, that's important to note as well.

Kim  08:22
Yeah. So, today, we want to talk to teachers specifically. So, you know, anybody can listen to the podcast. And that's fantastic. We love everyone to listen. But today, we want to talk a little bit to teachers about what does it mean to plan. And so, that was kind of our evolution. Now, we both will have an index card going somewhere. But there is an evolution. And so, what does it look like to get there, to get to the place where it feels natural and you know what you want to do? And that means more than just gathering materials. So, sometimes when people are talking about, "Oh, I got a plan," they might like read the lesson in their textbook and make sure they have their copies or like their example book. But when we say "planning", it means a whole lot more than that. So, we want to share a little bit about that today.

Pam  09:11
Yeah, and we're going to share what we have found to be a helpful resource. You might have noticed that we don't share a ton of things on the podcast. We're super picky about what we share. So, if we're going to share this, we actually like it. We don't think it's perfect, but we think it's pretty thoughtful and well done. Publicly available. We'll give you the link in the shownotes where you can go there. And, hey, I'll just give a little shout out. If you have a resource like that something that's helpful, thoughtful, well done, publicly available that we could share with listeners. We'd invite you to share that with us. So, feel free to send it to Kim@mathisfigureoutable.com. Yep,

Kim  09:47
thanks for that. I'll happily take it. 

Pam  09:50
Send it to me, it will get lost. 

Kim  09:52
Yeah, true.

Pam  09:53
(unclear). Anyway, so this particular resource is a Look For document that was created by New York City public schools. So, there's a group there, super thoughtful. I've worked with several people at New York City public schools over the last few years. I had a fantastic job working with the leadership and also working with teachers in the city. And part of what they've created is a set of three documents called "Look For's". And today, we're going to talk about the Look For's for Student Learning in Mathematics. And so, this particular documents two pages long, and it's broken up into 6 kinds of areas. And it intends to help teachers think about the kinds of things to create a shared vision, kinds of things that they would look for as students are learning mathematics. And it's a way of them kind of, the leadership, of kind of saying, "Hey. Like, this is what we look for when students are learning mathematics. How's that going in your class?" And you can use these to help you plan. So, like one of them is... First of all, there's an overarching... Actually, let me read their very first sentence at the very top of the page. "Teachers, colleagues, and coaches can use this tool to observe evidence of student learning of mathematics. It is not intended to take the place of a formal observation tool." So that is a good sort of description straight from them. As an overarching question, "To what extent are students..." So, they're really like thinking about to what extent are these things happening in your classroom with students? And then here are the 6 areas. One of them is engaging... So, "To what extent are students engaging in reading, writing, and oral communication in mathematics?" Another big area. "To what extent are students reasoning through problems to find solutions?" Another one. "To what extent are students using technology, manipulatives, representations, models, and other tools?" Another one is, "To what extent are students working together to solve problems?" And then, "To what extent are students explaining their thinking as part of completing tasks?" And the last one, "To what extent are students discussing patterns and structures, making conjectures, and developing strategies?"

Kim  12:11
Hey, you know, I'm really glad that you, even though it was repetitive, didn't leave off the "To what extent are students..." because I think there's a lot of weight with that. Because we might say, "Well, my kids talk." "To what extent?"

Pam  12:23
Nice. "My kids use tools." "To what extent?"

Kim  12:26
Right.

Pam  12:26
"My kids make conjectures." "To what extent?" Yeah, that's nice. So, one of the things that I'll mention is under each of those are two columns. And they're the same two columns everywhere. And the first one kind of breaks down each of the big areas, and again asks, "To what extent are students doing more specific things." And then, in the right column, it's super helpful. It actually helps illuminate, I think, this category because it says, "Here's some focus questions," and then it asks questions for you to consider to help you think about what's happening with students in that particular section. So, Kim, to get kind of at how those questions work with the things, we thought that we would ask each other a thing that sticks out to us. And as we talk about what... In other words, listeners, we're not going to go through all of it. We're not going to read it to you. We highly recommend that you download it and look at it. But we did think we would pick out a thing that stuck out to us individually, and we'll kind of talk about that one thing. And that can kind of give you an idea of the structure of the whole thing. 

Kim  13:30
Yeah. 

Pam  13:31
So, Kim. Will you start with what's one of them that stuck out to you? 

Kim  13:35
Yeah, so you told me to pick one. And I... There's a lot of good stuff here. But I picked one maybe because I felt a little convicted. 

Pam  13:46
Oh.

Kim  13:48
Under the Discussing Patterns and Structures section, there was one that said, "To what extent are students connecting current work with mathematics previously studied and seeing where the mathematics is going?" I think maybe why I felt a little bit... Mmm.

Pam  14:07
Pinged?

Kim  14:08
Pinged is a good word. Was because I feel like I do this. Like, I work really hard to make connections to what we've previously done and kind of where it's going. But I wondered to myself do I make the space for students to do this? Like, do I create opportunities for students to connect and imagine how this might come up again? So, when I read the focus questions that kind of go to the side, it says, "Who expresses connections to prior knowledge and experiences?" So, am I raising the idea that there might be a connection, and wondering, and then given the space for them to do the connection? And then, the other question was, "Which students connect the problem that they are working on to the big mathematical ideas?" So, if I say, "Okay, yeah. Yeah, they are." Which students are doing that? Is it the same students every time? Are there like a cluster of students? So, yeah. I mean, I felt like among all of the pages, there were so many good ones. But that one, I felt like I could spend some time thinking about that.

Pam  15:11
Mmhm. So can I. We just did a Live Math with Pam event with our Journey group. And one of the things I'm aware of is I walked away from that hour. It was fantastic. People that showed up, were so thoughtful. We worked really hard on what I think is a difficult thing that teachers grapple with about how to teach Order of Operations in the best way possible. But I'm aware that because it was virtual... Because. It was virtual. I'm aware I didn't do as good of a job of thinking ahead. Hey, planning. About how I could have provided a space for them to have conversation about exactly what you just said. To connect what we were doing. And it was because I didn't think about the fact that it was virtual. So, I remember in the moment saying to myself, "Okay, turn and talk to the person next to you.... Oh, crud. We're going to have to have breakout rooms to do that. We don't have time. I didn't think about that." Like, I literally had this go through my head. 

Kim  16:09
Yeah. 

Pam  16:09
And so, I was like, "Put in the chat what you're thinking about." Which is an okay thing. It's better than not at all. It's not the same, right? To put what you're thinking about the chat is not the same as beating it out with somebody next to you. So, I'm just aware that because I didn't think about that ahead of time. That's a thing that I didn't do as well as in the moment I knew I wished I could have.

Kim  16:29
But I love that you consider that that was the thing to do. Like, you could have walked away and said, "That was fantastic, and there was nothing I needed to improve." And, you know, to be in a space where you go, "I'm reflecting. And I know that's a thing that I want to strive for. And I maybe missed the mark for my own standard for this." And now, it will be in your head for future things.

Pam  16:49
Yes, absolutely.

Kim  16:50
Yeah. 

Pam  16:50
Yeah.

Kim  16:50
For sure. 

Pam  16:50
Yeah. 

Kim  16:51
Okay. Tell me about one that you? 

Pam  16:53
Yeah.

Kim  16:54
You don't have to feel... 

Pam  16:55
That it was convicting?

Kim  16:56
Pinged. Yeah. Just tell me something you're thinking about.

Pam  16:58
So, in the section, "To what extent are students explaining their thinking as part of completing tasks?" It says, "To what extent are students explaining the mathematical basis for what they are doing, rather than simply recalling the steps of their procedure?" 

Kim  17:14
Oh, dang.

Pam  17:15
So, I found that... That's not about convicting me. That was a bit more of me going, "Ooh, I like this document."

Kim  17:22
Yeah. Yeah, for sure.

Pam  17:24
But what I liked about it, one thing I liked about it, is how they sort of stated it. Like, I like the way they phrased that students are explaining the mathematical basis for what they're doing, in part, because I think sometimes teachers hear me say "Mathematizing, mathing, is not about algorithms," and they push back on that. They're like, "What?! No, of course, we need algorithms." And they like instantly shut off. So, I wonder if having part of this is it's not that algorithms are evil... And to be completely clear, I'm not saying that. Algorithms are amazing historic achievements. They're just not very good teaching tools. So, to what extent are students explaining the mathematical basis for what they're doing, I think can help everybody. Even if your goal today is for them to study an algorithm. Is it about recalling steps of a procedure? Mmm, see this one really is. And the focus question says, "What are students doing to explain their thinking? Do students explain why their process makes sense to them and how their thinking may have changed?" If you're only teaching kids one and only one algorithm, that question, you're like, "Wait, how could it have changed?" Also, "How do students express disagreement or skepticism with classmates when they think differently?" That sets up this perspective that students are thinking differently, that there is going to be times where they need to be able to express disagreement or skepticism. And how can we do that? And so, it just sort of sets a really nice tone, and background, and sort of setup for how teachers can go, "Wait, like that's suppose... How do I even get that to happen if all I'm doing is teaching one procedure? What would it look like to have the major strategies be a focus of class? Okay." So, that's one of the reasons why I liked that particular section. 

Kim  19:15
That is really good. And as I hadn't paid a ton of attention to that particular one. But it's funny, because as I was looking at the focus questions, and you were talking about them, my brain went to a different place. Because when it says "...explaining the mathematical basis for what they're doing..." And I think you can focus on the procedure part, but sometimes it makes me wonder like, "Oh, do I spend as much time as I'd like to talking about the mathematical properties that are at work when we're doing a particular strategy?" So, like, am I saying, "Oh, yay! So, you added 10, and then you backed up 1." Am I connecting that with students, or helping them connect, or allowing the space for them to connect to the associative, and commutative, distributive properties that are overarching everything that we are doing? And it's fresh in my brain because I'm really checking for that stuff in the Problem String books a lot right now. But, you know, I know I didn't do as good of a job then in the classroom as I would now. And, you know, I don't know if that's something we talk about as much as we believe in. 

Pam  20:22
Yeah.

Kim  20:23
You know, people like shy away from properties because they're like, "Oh, my kids heard the name, and they have the definition, and they have an example." But that's kind of not what the point is. So, I wonder if that could also go underneath discussing patterns and structures? Anyway.

Pam  20:38
Yeah, absolutely. I would definitely say that an evolution of my work is that I used to really just focus on kind of using the relationships and being able to represent. Well, first, it was using them, and then it was being able to represent them. Always kind of knowing that the properties were behind and at use. But it was later in my work where then I got more explicit about, "Oh, yeah. Like, yes. Let's, let's actually here spend some time discussing thinking about constructing what properties are at use." Kim, do you remember when we created that flowchart? We created a flowchart of what does a mathematician think when they see a multiplication problem? 

Kim  21:18
Yeah.

Pam  21:18
And part of the way... That was so much fun to create.

Kim  21:22
Sure.

Pam  21:22
Because part of what we did was color code it, where there were certain sections of you get to hear and you think to yourself, "Ooh, this relationship pings for me. I'm going to use this strategy." And, "This relationship pings from me. I'm going to use this strategy." And we were able to color code it based on the properties at work. 

Kim  21:37
Yeah.

Pam  21:38
That was definitely an evolution in my work. And in my defense, I kind of had to build my own numeracy... 

Kim  21:44
Yeah.

Pam  21:44
...before I could represent my and other people's thinking. Or at least kind of at the same time. Before then, I would kind of focus on the properties. So, we're all on a journey. We're all making more sense of all the things. Let's just keep on the journey. Yeah?

Kim  21:57
Yeah, yeah.

Pam  21:58
Cool.

Kim  21:59
So, you know, all of this work of doing these, you know, spending time with these Look For's and really asking yourself to what extent are students doing that and asking yourself these focus questions takes time.

Pam  22:12
Mmhm.

Kim  22:12
And, you know, when I spend some time working with teachers, sometimes what I hear is like, "I don't have time to dive into a single lesson. I've got 6 preps. And, you know, I've got to do all the things. And there's no way. Like, I don't have the time for that." And so, what I would say is, "If you want to build the behavior of really thinking deeply about the lesson, you're going to have to start like me maybe with... Maybe not typed. But really thinking through. You know, as we talked about in the beginning of the episode. Really thinking through very deeply about A lesson, about A presentation, and really starting the behavior of analyzing. And over time, you will get to the place where metaphorically you have an index card, and some of these things are ingrained in you, and the questions you ask yourself, and the reflection where Pam, you know, had, you know, at the end of our Journey event where she was like, "Ugh, there's this thing that I wanted to do, and I don't think I did it well. I'm going to bring it back up again when I have another one." You might pick A lesson. Ask somebody to sit with you. Or, you know, even if you take these Look For's and you sit down by yourself and you think, "Okay, I'm going to pick an area. And how deeply have I thought about this particular thing?" Maybe you have a new focus each semester. You know, you could work on this work over a couple of years. Yeah, and I think that's... (unclear). 

Pam  23:36
That's kind of the big point of something like Lesson Study. 

Kim  23:38
Yeah. 

Pam  23:39
Where they really encourage groups of teachers. Or even, like you said, grab a partner, grab a coach. Somebody to help you think deeply about as much as you can, you know? Like, take a lesson like Kim said. Think deeply about that lesson. Make it really good. Survive the rest of the time that week. And then, carve out some more time to think deeply about the next one. And the more that you do that, the more you gain a repertoire of sort of the lessons that built really well. Kim, when I was a young teacher, I did a lot of innovation, innovating, and I would innovate, innovate, innovate, and then I would take a deep breath and turn the page in the textbook for a few days. 

Kim  24:16
Sure. 

Pam  24:17
And then, I would like, "Okay. Alright, I have some bandwidth now. Let me innovate about the next thing." I would innovate, innovate, innovate, and then I would... You know like, you can't change everything all the time. Even crazy people like me. So, we would encourage you don't get overwhelmed, but that this document could be a way to help you create a shared vision.

Kim  24:37
Mmhm. 

Pam  24:38
If you can get some colleagues, your department chair. It would be great if you had, you know, a shoulder teaching partner to create, and it would... Honestly the best thing to do is not to use this Look For but co-create your own. Define what you think is what you want to have happening if students are really mathing, then what would you expect students to be doing? And then, refine that together. That's the, you're going to gain the most bang for the buck. Now, I don't know that you have to start from scratch. It's one of the reasons why we're telling you about this resource. I think you could pull up the one from New York City, and you could say to yourself, "Huh, that's not too bad." I mean, they give some references on the bottom of who they used as their inspiration. You could check those out in all your spare time. Oh, maybe you could just use this as a basis to sort of jump off. You could say, "I don't even know what that means. Let's talk about it." Or, "Let's even focus on this bit, and what does that mean for us here." So, here's a tool, listeners, that you could use to help you co-create, create, define, and refine your vision for what it could look like to math with students. Yeah,

Kim  25:46
and I just want to mention that we hear from a lot of people who say, "You know, I'm kind of flying solo on my campus, and I don't have a coach." Or, "I'm kind of... I love what you guys are talking about. But I'm all alone." I think these documents are a really good place for people to start to say, "I'm going to sit down. I'm going to think about my classroom. What are my students doing?" And so, if you are, you know, solo, and you're trying to share with your teammates, but it's not quite happening yet, this is a good place for you to have somebody outside of you who's asked some really good questions just to get you thinking.

Pam  26:25
Yeah, and pop in our teacher Facebook group and share there, or even better join our Journey group with like-minded teachers who are definitely working on the same kind of thing. Hey, Kim? 

Kim  26:35
Yeah.

Pam  26:35
Let's tell everybody where they can find it. The link is crazy, so the best place to do is go to the show notes. You can also try New York City Look For's. That didn't work really well for me. I didn't find a good Google search for it to come up. So, check out the link in the show notes... 

Kim  26:49
Yep. 

Pam  26:49
...to be able to download. Like I said publicly available. Look For's for Students from New York City. We appreciate all the hard work, thoughtful work that wonderful people they're put into creating it. Hey, everybody, thanks for tuning in and teaching more and more real math. To find out more about the math is Figure-Out-Able movement, visit mathisfigureoutable.com. And keep spreading the word that Math is Figure-Out-Able!