Pam  0:02  
Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris, a former mimicker turned mather.

Kim  0:11  
And I'm Kim Montague, a reasoner who now knows how to share her thinking with others. At Math is Figure-Out-Able, we are on a mission to improve math teaching. 

Pam  0:19  
Y'all, we know that algorithms are amazing human achievements, but they're terrible teaching tools because mimicking step-by-step procedures actually traps students into using less... Less something. Oh, my gosh. I can't even read my own writing. Less sophisticated reasoning than the problems are intended to develop.

Kim  0:34  
In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships. 

Pam  0:41  
We invite you to join us to make math more figure-out-able in Indiana at the NCTM Spring Conference.

Kim  0:50  
Such a great time.

Pam  0:51  
So, Kim, I have to ask.

Kim  0:52  
Yeah?

Pam  0:52  
Why do you call it spring in February?

Kim  0:55  
We're heading towards spring. 

Pam  0:57  
I mean, is it spring in February anywhere in the world? 

Kim  0:59  
It's actually pleasant here. I thought it was going to be way worse, weather wise.

Pam  1:03  
Pleasant? I mean it's been right around freezing.

Kim  1:05  
Yeah.

Pam  1:05  
There's snow on the ground. 

Kim  1:06  
Oh, well. 

Pam  1:06  
Which it should be in February. Like, it's delightful. Yeah. And we've had a great time. So, we are sitting where there's lots of teachers hanging out. We've just been in sessions. We're actually just about to eat lunch, and we're going to record a podcast. 

Kim  1:19  
Yeah. That sounds great. 

Pam  1:20  
Kim, what are some things that we are thinking about based on some of what we've seen? So, I'm going to start if that's alright.

Kim  1:25  
Yep.

Pam  1:25  
I've been thinking about this for a minute, and I've just... Maybe because I've been thinking about it, it has struck me as, I think we should do a whole podcast on it, so I'm going to tease it just a little bit today. What is the difference, and is it important to differentiate, between curriculum... Oh, look. There's the guy I sat next to in the session where we talked about this very thing. 

Kim  1:48  
Okay, good. 

Pam  1:48  
So, what's the difference between curriculum, standards, textbooks, scope and sequence?

Kim  1:52  
Yeah.

Pam  1:52  
And then, Wade, who I was sitting next to in the session. Hey, Wade! Was talking about content. 

Kim  1:57  
What about resources? 

Pam  1:59  
Okay, and where do resources fit in there? Because, and I was sort of thinking resources were textbooks. But you're right. There could be other resources. There could be supplemental resources.

Kim  2:08  
Yeah. 

Pam  2:08  
Yeah, all sorts of things. And I think that it might be helpful if we all were a little bit more purposeful about the words we use because when we say, "Well, we just adopted a new curriculum," I worry that that is a muddy, messy. I think most of us think... Clearly, we're adopting a new textbook, but it sends messages about what that means and what it comes with, and we get into phrases like implementing with integrity versus fidelity, and blah, blah, blah. Anyway, So, Kim, may we dive in in an episode soon? And kind of dive into all those words?

Kim  2:46  
Yeah. 

Pam  2:46  
Hey, listeners, if you are interested, we would love to hear your definitions, your clarity. Do you think there's a difference between scope and sequence? Content? Curriculum? Standards? Textbooks? Resources? Are there other words that you would throw in there that get mixed up and mashed up that then maybe keeps us from some clarity. 

Kim  3:07  
Yeah, I think that's an important thing that you're saying because we can walk into a situation and have a conversation where we are nodding and agreeing, but could be thinking about two totally different things. And so part of... I think you've talked about this before that part of the really important thing that needs to happen in conversations is that we know what each other is talking about. 

Pam  3:27  
Yeah, define terms.

Kim  3:28  
Yeah, I was just in a session with Susan Loveless, and she was talking about lesson study, and one of the things that she said that I wrote down was "clarity before competence." And so, when she was...

Pam  3:37  
Say more about that. 

Kim  3:38  
Well, she was talking with their teachers, and she said, "We can't expect them to be competent in the thing that we're doing with them unless they are clear about what we even mean." And so, they do a whole... You know, there's a whole lesson study protocol. And then they did work with their teachers. They have a massive district. And she said, "We had to make sure that teachers were even clear what we meant and what we were asking for." And so, she had some really like detailed out things that they did because she said, "We expect teachers to be competent in a lot of things, but we aren't clear with them about what that looks like, and sounds like, and what we're actually expecting." 

Pam  3:38  
Well, especially if we're using all these different words that mean different things to different people.

Kim  4:16  
Right, right. 

Pam  4:16  
That could be super tricky. So.

Kim  4:18  
Yeah.

Pam  4:18  
Maybe that's one of the reasons. Well, it is one of the reasons why we feel so strongly about having shared experiences.

Kim  4:24  
Yeah. 

Pam  4:24  
And say, and then tag the term. Okay with this experience, this is what we mean by these terms. 

Kim  4:29  
Both with students and with teachers. 

Pam  4:31  
Yes.

Kim  4:31  
Yeah.

Pam  4:32  
Yes, yes. Nicely said.

Pam and Kim  4:33  
Yeah.

Pam  4:34  
Cool. Okay. So, another thing that I'm thinking about. We went to the opening session last night. Which I think we both appreciated. Well, maybe I should ask you. Did you appreciate the format? 

Kim  4:42  
Yeah.

Pam  4:43  
It was less one keynote, and it was four leaders in math education who spoke.

Kim  4:51  
Mmhm.

Pam  4:51  
They each had a little bit of time to do their own kind of idea, and then they had...

Kim  4:54  
It's like a panel type situation. Yeah.

Pam  4:56  
Yeah, and then Robert Berry did a great job of asking them questions.

Kim  5:24  
Yeah.

Pam  4:59  
Yeah, and then they sort of separated out into four corners of the room and people could then ask their own questions.

Kim  5:31  
Yeah, that was nice.

Pam  5:03  
That was kind of cool. So, one of the things that I'm thinking about that happened last night was there were some questions around tensions. Robert Berry brought up that we have some tensions that we need to hold, hold that tension. We don't necessarily have to resolve it, because there is this tension between different things. And Peter brought up, Peter Liljedhal, brought up that in mathematics, mathematics is both a noun and a verb. And, Peter, I hope I do right by you, that mathematics is the noun, the content that we need to learn, but it's also the verb, the mental actions that you do. And maybe I could say the process standards or the practice standards. And I really appreciate the fact that he's suggesting that there are both of these things that we need to pay attention to, to work on, to work together to refine. And I think that's one of our main messages at Math is FigureOutAble, that these practice standards, or process standards, or the verb, the... I think he said... Y'all, let's see if I can do it. I would say mathing. He said generalizing, thinking, reasoning. That if we don't have shared experiences, those words sound very fuzzy. They sound very undefinable. And I think that we are going to get pushback, probably rightly so, from people who say, "We can't just create better thinkers." And that's not what we're suggesting here at Math is FigureOutAble. We are suggesting, we are creating, proposing to help develop mathematical reasoning. Not just some fuzzy think better reasoning, but mathematical reasoning. 

Kim  6:20  
Yeah.

Pam  6:20  
I think it's very important that we dive into both the noun and the verb of mathematics.

Kim  5:50  
Yeah, I think Robert Berry's question was actually something like, "What is something that you are in tension with that you're not rushing to get out of tension of?" I can't remember exactly what it was, but it was something about like what are you holding space to stay in and have in tension? And, you know, I've been thinking about that ever since. It was just last night, but like what is the thing that I'm wrestling with? Grappling with is the word that you use.

Pam  7:06  
Yeah.

Kim  7:06  
That I feel like it's worth grappling over? That, you know, that in that grappling and in the learning, that maybe there's so much learning to happen within the grappling that even as, you know, educational leaders or as coaches we're wrestling with, so that, in that moment, we're holding space for our learning. And it's not just rushing to solve the problem. It's not rushing to have an answer at the end.  We are growing and developing through that grappling as well. So, I'm kind of been thinking throughout the morning here about like what am I in tension with that I want to hold space for that maybe even a little bit longer, even though it's uncomfortable?

Pam  7:42  
So, tell me if you think this fits. I'm thinking a little bit about some slides a friend of mine just showed me some slides from a state presentation who shall not be named, where... I'll maybe just raise this caution that if somebody says, "Research has clearly answered this very difficult pedagogical conversation, this very difficult question that has risen in pedagogy and teaching for a long time. And clearly, we have this easy answer," I invite you to consider to be dubious, to be... What's a better word for that? To hold that in tension. 

Kim  8:18  
Yeah. To wonder.

Pam  8:20  
To wonder, to dive in, to read that research, to not take it just as, "Oh well, I guess if someone's telling me research says." Who was the research done with? Was it done in a laboratory setting versus in a classroom? Peter brought up last night, was it things that we can do with one individual student? That might be super effective, but is that necessarily efficient for a whole classroom? And those are two things that he was also holding in tension that we need to seek for efficiency, but we also want effectiveness, and that those two things aren't always... Well, that there's a tension between them, and that we have to work through that. 

Kim  8:55  
Yeah.

Pam  8:55  
Yeah, if anybody just tries to give you an easy answer, I would invite you to consider that easy answer is probably too easy, too cute, too... 

Kim  9:04  
You know, it has me thinking, as you're talking right now, about how when we are talking about developing students, the work that we do is about developing teachers, so that they can make these decisions for themselves. We want to grow them to be judicious problem solvers and be judicious in the research they... Yeah, so that they take it in and they filter it through the lens with their experiences and what they've learned. And, you know, it would be easy maybe to just lockstep follow what somebody said, but that's how we, heads up, how that heads down a path sometimes where we veer off for what we believe and what we veer off into that maybe isn't best because we are just like following a system maybe that isn't right for the classrooms that we're in. Maybe it's not right for everybody. 

Pam  9:50  
And if somebody says, "The Science of Math..." Caution because science is multi-faceted, and it depends on lots of variables. And yeah, that would be one of the  things that I would invite everybody to keep some tension around.

Kim  10:11  
Yeah.

Pam  10:11  
Hold space for the fact that there needs to be a preponderance of evidence before we make rash decisions about moving forward. Alright, let's move on to a third thing. You went to a routine. Excuse me.

Kim  10:25  
Yeah.

Pam  10:25  
You went to a session on routines. Can you tell us? 

Kim  10:27  
Yeah, I went to a session that Amy Lucinta and Grace Kelemanik did with micro-routines. And they, you know, they talked for a while just about two of the micro routines. One of them was called... I'm going to butcher this because I was just so absorbed, and I took some notes. But one of them is called "Interpret With Intention". And what I loved about their routine is, you know, they were talking about how whatever resource that you have, you might put up a graph. Or it would have some sort of image. And putting it up. And it was a very specific like 45 second to 60 second protocol where you put it up, and you give some sentence stems to students, and they are supposed to share their interpretation of that image. And so, they think, they look, they share their interpretation, they turn and talk with a friend. But what they're saying is what might be important to notice about this thing. So, the overall question was what might be important to notice? And then they tell students to turn to their partner and say, "It might be important to notice..." blank. And so...

Pam  11:28  
Are you saying that what might need to be... "What's important to notice?" that's a teacher question beforehand, not to students?

Kim  11:35  
The purpose of this protocol is to say to students, "What might be important to notice about this thing?" And then they share the thing. So, kids are framed with, "My goal in looking at this is to interpret what might be important to notice."

Pam  11:47  
Thank you. Okay, yep. 

Kim  11:48  
And so, then you and I are partners. You say something. I say something. But there's specific sentence starters. And what they said was, you know, "It creates a regular, repeated situation where students are expected to pause and think before they jump in." So, you might start it with like a graph, or you might start it with something that, you know, that you're about to talk about. But it gives kids a chance to look at it, to think about what they think is important, to hear from a partner about what they think is important, and then you can start talking about the thing. And so, you know, I think what struck me is that often in classrooms, we throw something up, we dive in right away, we start talking about it, and the kids are just passive listeners, rather than being forced. "Forced". Encouraged.

Pam  12:34  
Invited.

Kim  12:34  
To look at it themselves and take in that piece of information. And I got to tell you, one of the things I thought about was, you know, my own son sometimes wanders mentally, and then he gets drawn back into it as he's hearing people talk. But as he's engaging in conversation with a friend, he's had several opportunities to hear, to see it himself, to hear what a partner is saying, and then engage in the learning with a teacher. So, I just really liked this particular micro routine. And I got to tell you, one of the things that I heard happen in that session, and I commented to Grace afterwards is in the session, somebody was asking a question, and they said something about, you know, having a low student, and then continue on to say comparatively to a high student. And Grace, just like stopped that conversation in the midst of it. You know, the person was well meaning, and they wanted to describe a situation in their classroom but used the words "high" and "low" several times. And Grace just said, "Listen, I'm going to invite you to come to our later session because we're really working to not use those words anymore to describe students," and then invited them to use a particular different phrase to use. 

Pam and Kim  13:47  
(unclear).

Pam  13:47  
asset perspective.

Kim  13:48  
Yeah. And I said to her later. Like, "You know what? I really appreciated that in that moment, in a room full of people, it can be very uncomfortable to say something to another adult that maybe will make them uncomfortable or maybe defensive." And, you know, I just really appreciated that if that's something that we don't believe in saying about kids, it's going to take educational leaders to hear that, and in the moment, say, "Hey, I need to stop you," and give them other words to use. And when I said something to her afterwards, and I said, "I just appreciated your bold leadership to just stop it right there," she said, "I was looking at the crowd, and I saw a lot of really uncomfortable faces, and so I knew I needed to say something." And she went on to tell me that she went and talked to the teacher afterwards, and the teacher was very reflective, and appreciated being stopped and said that she was working on that language. 

Pam  14:40  
When you said, Grace did it in a very respectful...

Kim  14:40  
It was so respectful. It was... Yes. And, you know.

Pam  14:45  
not ornery or rude.

Kim  14:45  
No, not at all. And she just said, "Listen, I'd like to invite you to our session, so that you can hear why, and... "

Pam  14:52  
Have some alternative.

Kim  14:53  
Yeah.

Pam  14:54  
Yeah.

Kim  14:54  
Yeah.

Pam  14:55  
Yeah. 

Kim  14:55  
It was nice.

Pam  14:55  
Grace and Amy are very reflective, thoughtful math teacher educators, and we really appreciate their work. Yeah, they've done a nice job. Hey, just quickly. One of the other things I think we could do a whole podcast episode on. So, maybe we'll just mention it briefly. 

Kim  15:07  
Yep. 

Pam  15:08  
Another thing that I'm sort of holding tension between is the word "strategy" in math education. So, may I introduce... Or introduce. What's the word I want? May I suggest? Invite? That I think there's at least three. And then I'm going to ask listeners what are the other strategies that we might be unintentionally kind of mixing and melding together, and so then we're not able to be as refined in our conversation and get as much work done as we could? So, here's a couple of examples. I think there are teaching strategies.

Kim  15:44  
Yep.

Pam  15:44  
I think there are mathematical strategies. And I think that maybe there's what we've called "problem solving strategies".

Kim  15:51  
Yep.

Pam  15:52  
Those are three that we kind of came up with right before when we were sort of brainstorming what we're going to talk about today. So, Kim, I'm just going to give you an example of a thing, and I'm going to ask you. 

Kim  16:00  
Pop quiz, okay.

Pam  16:01  
Yeah, pop quiz, pop quiz. If you think it's more of a teaching strategy, more of a math strategy, or more of a problem solving strategy.

Kim  16:09  
Okay.

Pam  16:09  
Okay, so if I say, "I've just solved the problem 64 minus 18 by saying 64 minus 20, and I'm going to add 2."

Kim  16:16  
Mmhm.

Pam  16:16  
Is that a teaching strategy, a math strategy, or problem solving strategy? 

Kim  16:18  
I think that's a math strategy you're talking about.

Pam  16:23  
Because?

Kim  16:23  
Because you're thinking about the numbers and what happens with the numbers in the problem. And you're using Get a Friendly... Get a... Remove a Friendly Number, Over strategy. We're talking about the numbers in the problem. 

Pam  16:40  
Yeah, it's the relationships you're using. 

Kim  16:40  
Yeah. 

Pam  16:40  
to attack. The way you attack the problem.  What if I said,  "Somebody picked 64 blueberries. That somebody picked 18 more blueberries." So, let me give some names. 

Kim  16:50  
Yeah.

Pam  16:51  
"Kim picked 64 blueberries. Kim picked 18 more blueberries than Pam. How many blueberries did Pam pick?" You just solved that problem using a mathematical strategy. What if you were to try to get a handle on that problem and you started drawing a model to represent what's happening, so you can figure out who even had picked more, who picked less? 

Kim  17:11  
I think you're probably using a problem solving strategy, where you might read the problem. You might even circle the numbers in the problem, and you've decided to draw a picture to think about how to solve it. 

Pam  17:25  
And you're not circling because somebody commanded you, "Thou shalt circle the numbers in the problem," but you're sort of helping yourself focus. 

Kim  17:26  
Yeah, one of the problem solving strategies might be identify key information, and so you might choose to circle it, so that the numbers jump out to you. 

Pam  17:36  
And if they were mean and stuck something irrelevant in there, you might cross it out.

Kim  17:40  
Cross it out or underline the question. Yeah. 

Pam  17:42  
What if I said, "Hey, Kim, when you're doing that with kids, you really should have good wait time." In fact, gol, I could use Grace and Amy's. "You should give that problem and let them sit with intention for a minute and say what's important to notice here?" What kind of strategy is that?

Kim  18:00  
Yeah, that's a teaching strategy. 

Pam  18:02  
A teaching strategy. 

Kim  18:04  
Yeah, mmhm.

Pam  18:04  
What if I were to say, "Kim, you better have more..." Let me just pick a better one here. What if you are grouping students?

Kim  18:10  
Oh. Like in the classroom?

Pam  18:11  
Yeah, they're about to work on something, and you're like, "Hey, we're going to work in groups."

Kim  18:19  
Purposeful pairing.

Pam  18:19  
Purposeful pairing, purposeful grouping. 

Kim  18:19  
Yeah, that's... a teaching strategy. 

Pam  18:20  
It's a teaching strategy. Okay. What if I said, "Solve a simpler problem."

Kim  18:25  
That's a problem solving strategy. 

Pam  18:25  
Okay, what if...

Kim  18:25  
How am I doing? Am I getting them right? 

Pam  18:25  
I like your explanations. It's very nice, very nice. Because if you're solving a simpler problem, you're looking at a problem, and you're saying to yourself, "How am I going to solve this problem? Ooh, I wonder if I can make a simpler problem," that therefore is a problem solving strategy. What if you said, "Mmm, with these numbers..." Now, not the blueberry problem, different problem. You said, "Mmm, I think I could double this factor and halve that factor and create an equivalent problem that's easier to solve. What would that be?

Kim  18:51  
Yeah, that's definitely a mathematical strategy. 

Pam  18:53  
Mathematical strategy. So, y'all, I'm just going to invite us to consider that strategy, if we say the word strategy... I don't know, Kim, maybe I'm going to commit. Ooh, can I commit? That I'm going to try to not just say "strategy", and I'm going to try to say it's a "teaching strategy", a "mathematical strategy". I don't use "problem solving strategies" as often.

Kim  19:11  
Yeah.

Pam  19:11  
But to help us just gain some clarity before we can expect... What was that? Confidence...

Kim  19:17  
Yeah.

Pam  19:17  
...in teachers really learning the content if we never talk about mathematical strategies. Which can I just say? We do less of. We talk a lot. We. We go to conferences like this. We talk. We hear from really good people.

Kim  19:30  
I was going to say. Wait a second. We talk about mathematical strategies all the time.

Pam  19:31  
Yeah, we do it all the time. We hear from really good mathematics teacher educators, and mathematicians, and people that are teaching math. We hear a lot of teaching strategies, and we hear a lot of problem solving strategies, and we don't hear as much about mathematical strategies. And I shouted out today in our session Jenny Bay-Williams and John SanGiovanni. Because they do that work of really identifying mathematical strategies. 
 
Pam and Kim  19:57  
Yeah.

Pam  19:57  
Nice. 

Kim  19:57  
Yeah, it's good work. 

Pam  19:58  
Alright, Kim. Having fun here. We got a whole other day at NCTM Spring Conference in Indianapolis. We're excited to learn more. 

Kim  20:04  
You have another session.

Pam  20:06  
Yeah, we've got one more session today with Lisa McConchie on Teaching Mathematics with Brain Science. Or I think we called it "Reaching More Students in Less Time With Brain Science." 

Kim  20:16  
No idea. It's going to be great no matter what you called it. 

Pam  20:25  
Always good to work with Liesl. Yeah. So, here we go. Let's go learn some more.

Kim  20:25  
Alright, sounds great. 

Pam  20:25  
I have no idea how this podcast is supposed to end. 

Kim  20:25  
Oh.

Pam  20:25  
Let's see. How does it usually end? Thank you for joining us to make math more figure-out-able. To find out more about Math is Figure-Out-Able, visit mathisfigureoutable.com. Keep spreading the word that Math is 

Pam and Kim  20:37  
Figure-Out-Able!