Pam  00:01

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

 Kim  00:08

And I'm Kim Montague.

Pam  00:09

And you have found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But it is about sense making, and noticing patterns, and reasoning using mathematical relationships. We can mentor students to think and reason like mathematicians do. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keep students from being the mathematicians they can be.

Kim  00:39

Hey, Pam. 

Pam  00:41

Hey, Kim. 

Kim  00:42

It's our anniversary. 

Pam  00:45

Happy anniversary! Woah!

Kim  00:47

Three years! Can you believe we've been doing this for three years? I mean, I know we talked for a million hours for 20 years. But three years of doing the podcast. Thank you, thank you, thank you, listeners. It's been super fun to do. And we are so grateful for you listening. So, Pam, you dragged me along this journey, and I dug my heels in for quite some time, but I am so glad that you did. And I'm super happy to be doing this with you.

Pam  01:18

Well, and I'm going to back at you. There's no way that I could have done it without your organization. And shall we say prodding? Because once you committed, then you were like, "Alright, then we're doing it." And yeah, so it's been a fantastic partnership. 

Kim  01:32

It's been fun. Yep. So, listeners, if you have learned something by listening to the podcast, we would absolutely love to hear from you. You can tag Pam on social media, leave a rating, or comment on whatever platform that you listen to. It's not only super meaningful to us, but it helps others as well. So, listen, Pam, I have an idea for today, and I am going to give you some topics. People sometimes say to us like, "Where do I start? You have 150 something episodes." Like, "Where do I get people started." And so, that's what we're going to do today. So, everyone is telling you that it's about learning loss. I'm telling you that these are the things that if you get a handle on, they will transform your math class. So, I picked some topics that I thought would be the quintessential Pam Harris, Math is Figure-Out-Able, you got to know this, this is the thing. And okay, so I've set a timer on my phone, and I'm going to give you some... I'm not playing.

Pam  02:37

It's like a speed round?

Kim  02:39

Yeah. This is speed round, and you get like a minute and 45 seconds. And I'm going to give you (unclear). You don't have long. So, I'm going to give you a thing, a topic.

Pam  02:50

Okay.

Kim  02:50

And you get a little time, and then we're going to point listeners to what's another place that they can go for more information. But if you are a brand new podcast listener, or even if you've been around a while, this is your episode of "What do you need to know?" Okay?

Pam  03:06

Okay, I got to tell you. I'm like, I've assumed the stance. Like, I feel like I'm about to, like serve a volleyball or shoot my free throw, or like I'm bouncing a little. I got my weight evenly distributed. I'm shaking it off. You know, I have stretched my neck a little bit. Okay, I'm good. I'm ready. I'm ready.

Kim  03:21

Alright, your first topic is the Development of Mathematical Reasoning. Go.

Pam  03:28

Bam! So, it is so important that we realize the trap of the algorithms are that they don't help us develop reasoning. They keep us mimicking steps. And so, if we realize that mathematicians think in this very sophisticated way, and there are there is a hierarchy of reasonings that build toward that. It's not about memorizing a bunch of skills and rote memorizing sort of the basics, so then you can magically one day do what mathematicians do. It's actually about creating relationships and connections and students brains first, so that they are solving problems using counting strategies. But as soon as they can count and solve problems using Counting Strategies, we've got to build their Additive Reasoning and we have to help them think in terms of bigger jumps than one at a time. And then, we have to go from Additive Reasoning to Multiplicative Reasoning and help students, help all of us think in terms of bigger chunks than one group at a time. That's thinking multiplicatively. And then, we have to build their Proportional rReasoning, where we have a quantities varying in tandem. They're related and varying in tandem when we have to be able to scale two or more quantities that are related and they're scaling multiplicatively. And then, if we want to build from there, then we want to build Functional Reasoning, which is... I don't have time to describe Functional Reasoning. Where we really want students thinking and reasoning about relations, and functions, and graphs, and transformations. We have to build on that. And so, if we don't build those kinds of reasonings, and all we get our students getting answers to particular questions. If students or just getting an answer to a multiplication or division question-- [Timer dings, Pam laughs] There's my timer.

Kim  05:10

That's what you get. No, I'm not playing. You get like a short amount of time. But here's the thing. We've talked about these topics, right? So, if you, listener, want to know what the end of that sentence is, if you want to know more about Pam's Development of Mathematical Reasoning, I urge you to check out Episodes 5 and 6, and 67 and 68. Because it's a huge topic and super important. Alright, Pam. Deep breath. You ready for the next one?

Pam  05:36

Yeah, I'm shaking my head. Okay, shake it off. 

Kim  05:38

Okay. Alright, a huge, huge topic. Strategies versus models.

Pam  05:45

Go. 

Kim  05:46

Go.

Pam  05:46

So, strategies are the way that you mess with the relationships and the problem, the way you tinker, the intuition about how to attack the problem first, the way that you're going to use the relationships to reason through the problem. The model is the way you make that thinking visible. It's the tool you use to help you keep track of what you're doing or to see the relationship, so you kind of know how to attack next. We need to be clear about the differences between these two because when we are muddy as teachers, then students are muddy. Students don't actually need to be so clear on this. Teachers need to be clear. Where do you see this arise? When students say something like, "I did a number line." 

Kim  06:32

Yeah. 

Pam  06:32

That's a model, and what we need to then follow with is, "How did you use a number line? Where did you jump?" Just the other day, I worked with teachers from Australia, and I said, "How did you solve it?" And the teacher said, "I did the jump strategy." And I said, "So you jumped on a number line?" "Yeah, I did the jump strategy." And I said, "So, that's a model. What did you do on the number line? What relation...?" And she's all, "Oh, well, I got to the 30." And I'm like, "Oh, so you got to a friendly number." That's a relationship that is the strategy of Getting to a Friendly Number, and then adding the rest. The student says, "I did an area model," then your job as a teacher is to be clear, that's a model. "What did you do in the area model?" We see this confusion come up in things like the area model, where teachers have said, "Oh, this is a new strategy. Alright, everybody ready? Draw a square." What's the word? Divided into four sections, and then always split the numbers by place value, always [Timer dings] do Partial Products by place value.

Kim  07:25

That's it. That's all you got. Sorry.

Pam  07:29

We want to chunk the area in more efficient ways, so that their reasoning multiplicatively. Okay, whew.

Kim  07:35

The end. This is hard, right? This is hard. These are big topics. So, for more about strategies vs models, check out Episode number 9. It was kind of an early one because it was super important to get out there. Alright.

Pam  07:50

Absolutely. 

Kim  07:51

Take a deep breath. 

Pam  07:51

Alright. Whew! Okay.

Kim  07:54

This one's going to be tough because it's massive as well. Algorithms are not the end goal. Go. 

Pam  08:07

We have traditionally seen mathematics education as, "We want kids to get answers." And so, what do we do? How do we get an addition answer? We do this algorithm. How do we solve a quadratic equation? We use this formula. We maybe divide it into two linear binomials. We set them equal to 0. We use a 0 product property. But we tell kids, "This is what you do." 

Kim  08:28

Yep.

Pam  08:29

When we do that, we get kids who can perform steps and get answers, but they're not building their reasoning. My question or my mandate, then is to say, "If it's not about algorithms, then what is it?" And I think this is what I add to the work of mathematics education. There's many people out there that are saying, "Yeah, help the kids get conceptual understanding because then the end is the algorithm." And I'm saying, "The end is not the algorithm." But then, we have to answer the question, "What is the end?" And so, through my study, I have suggested to the world that there are these major strategies that kids need to develop, based on these major relationships. And those are the end goal. Just this morning, I was speaking with a teacher in Germany who said, "You know, they've been doing this great work in second grade and third grade. Were kind of concerned about what the fourth grade teacher is going to say." And I said, "Well, honestly, I got to be clear with you. I'm kind of concerned a little bit about what you're doing in second, third grade." Because I've talked to these two teachers well enough to know they're doing good work getting kids to think, but they also don't own yet the major strategies that they need to own for second and third grade. What does that mean? It means their kids are muddy. You might have a kid... I shouldn't say "muddy". Let me clarify that. You might have a student [Timer dings] who's doing decent-- Ugh! You might have a student who is doing this work, but they're only using one of the major strategies, and a different student is using a different major strategy. I need that one kid to own them, all four of the major strategies, at least well enough that next year that teacher can build from there. Or they're going to hit problems where that one strategy they own won't be good enough, and then it will fail. So, we need to own those major ones. Okay, I'm done.

Kim  10:11

I feel like I need a second timer. Oh my gosh. Okay.

Pam  10:16

I only went like seven seconds over.

Kim  10:18

Okay. Alright. So, algorithms are not your goal, and we would suggest they shouldn't be anybody's. You can hear more about that in Episode 2. And then, we actually revisited this topic because we wanted to give people a picture of, "If not algorithms, then what?" And you can check out Episodes 94 through 99, where we talk about the major strategies for each operation. That's all I'll say about that.

Pam  10:45

Maybe I'll pipe in there. Mimicking algorithms is not the end goal. Okay.

Kim  10:49

Right. Right. Alright. 

Pam  10:50

Okay, alright. 

Kim  10:51

Okay, something also very quintessential, "Pam Harris" is the topic of perspectives and what that brings to the table. Go.

Pam  11:03

Bam! I'm going to start talking before you say "Go". So, I think one of the reasons that the conversation that we're having in math education today is so confusing is because we have different perspectives, and when we communicate not understanding each other's perspectives, we talk past each other, and we miss communicate, and we're having arguments using terms that we're defining differently. And so, I think it's super important to realize that there are people out there who believe, who bought into the myth that math is a disconnected set of facts to memorize and rules and procedures to mimic, and the answer is all important. And therefore, they teach that way. And so, when we have conversations about conceptual understanding, they say, "I mean, sure, we could do some work. Do we have to waste time? Like, our goal is to get them to mimic this stuff and rote memorize this stuff. Let's just sing some songs, get them to rote memorize the facts, and then I'm going to tell some stories, so they can mimic the procedures. They're getting answers. Move them on. I don't want to waste time doing this other stuff." And I think that is completely understandable if their perspective is, that's the definition of math. I think we have a different group of people who was always sort of clear that they could use relationships in their mind to solve problems. They were always kind of mathing. But the way they were taught, it seems they were taught the way those other teachers were thinking about math, and somehow, they were able to see through that rote memorization and create relationships. Therefore, they think that is the way to teach and teach the way they were taught. And if you have the math gene, you'll do what they did. And if not, well, stinks to be you, and you're just going to have to follow the rules. To that perspective, [Timer dings] I say, "No, we can all do that." We just have to know it's a thing. So, if we can understand those different perspectives, we can all teach Real Math. Woah, that's hard. This is hard, Kim.

Kim  12:57

I'm impressed, though, because you're totally on the fly right now.

Pam  13:00

I'm going to have to go like to a spa afterwards and get a massage and take like a hot bath, like relax after. Whoa! Okay, I can do this.

Kim  13:07

So, listeners, this is not... There is no script for this. Pam is like, go for 1 minute. 40. So, I'm impressed. Okay, Pam (unclear).

Pam  13:18

Wait, tell them the episodes. (unclear).

Kim  13:19

Oh, sorry. Oh, for more about the perspectives, 24 through 28. Episodes 24 through 28. We also have (unclear).

Pam  13:28

You can hear me talk about those slower there. I'll talk slower about those. 

Kim  13:32

And actually, each perspective got their own episode. And if you are curious about which perspective that you formerly had, but you're moving into Real Math perspective, you can take our quiz that's in the show notes. Okay.

Pam  13:44

Go. 

Kim  13:45

Okay, the next one, Pam, if you can get it before the timer goes off, I'll owe you...

Pam  13:50

I can do it! I can do it!

Kim  13:51

...some ice cream because it's not going to happen. The next one is Problem Strings. Go.

Pam  13:59

Problem Strings are an instructional routine. So, it is a way of teaching. It's not the only way of teaching, but it's a super helpful, important way of teaching that we get from the Netherlands. The Freudenthal Institute did a lot of work with it. There's a lot of research that came out of schools of thought in Great Britain about variation tasks, where basically we start with a problem, and then we vary the next problems in very certain ways to put patterns up in front of students, and allow them to grapple with the relationships, and use the patterns and connections that they're seeing and making, for the next problem. And then, refine their thinking for the next problem. And so, it's it's not about... It feels maybe a little bit like a number talk or what I would call a problem talk. But it's a sequence of problems, where the sequence matters, where the conversation in between each problem matters. You can't give a Problem String to a kid and say, "Go, and learn it on your own" most of the time. Most of the time, you have to say, "Here's the first problem. Everybody solve it. Let's have conversation." Make thinking visible. Maybe compare strategies. Next problem. "Do it however you think about it." Elicit those strategies out of students. Make thinking visible. Make connections between the problems. And by the end of the Problem String, students are noticing and using a pattern that they might not have noticed and used before. So, it's a very systematic, intentional, purposeful sequence of problems with very intentional conversation between the problems to build [Timer dings] relationships and connections in students heads, so strategies become natural outcomes. That's a Problem String.

Kim  15:44

Well done. Alright, more about Problem Strings can be found in Episodes 33, and 71, and 72. Check that out. Alright, we have three more.

Pam  15:56

Okay, okay. I can do it! I can do it! 

Kim  15:58

The next really important topic.

Pam  16:00

I'm pulling up my hair.

Kim  16:01

Okay. The next important topic is everybody learns, everyone grows. Go.

Pam  16:09

So, this is my way of saying that all students can do more Real Math than fake math. Meaning, that all students can reason at wherever they are. We can help them reason more sophisticated-ly with the math from where they are. We can ask them questions and determine kind of what they own, and then give them questions that are on the edge of their Zone of Proximal Development. And being the more knowledgeable other, we can give them questions that are just on that edge, make their thinking visible, and they can then grow from there. So, sometimes people will say, "Give students low floor, high ceiling tasks." And I say, "But that doesn't mean that you give them a task that they can already do, and that therefore they don't learn through." You want to give them a task just on the edge of their Zone of Proximal Development. And the task needs to be open enough, it needs to be rich enough, that it'll be on the edge of everyone's Zone of Proximal Development in some way. So, it's a rich enough task that students have to reach, and therefore by that reaching, by that grappling with relationships, they actually create more connections, mental connections, so that their brains grow in sophistication. And so, everybody learns, and everybody grows. So, I'm not the one that's looking for one outcome from a task. I'm not, "Okay, by the end of today, all students will blank..." because I don't think that's reasonable. Now, I can say, "By the end of this task, all students will have grown. All students will now be thinking more sophisticatedly. And in these ways." [Timer dings] But I'm not looking for a specific outcome. I'm looking for growth of all students in that task. I did not do that one justice at all, Kim.

Kim  17:57

Okay, but here's the thing. The beautiful thing is, they can listen to another episode. That's Episode 39 for a full meal deal about that topic. Totally fine.

Pam  18:10

Alright cool. 39. Go hear it better (unclear).

Kim  18:11

Alright, here we go. 

Pam  18:12

Alright.

Kim  18:13

Our mantra, Pamela. Know your content, know your kids Go.

Pam  18:18

Ah, well, it's very sweet that you say "our mantra" because it has become our mantra, but it started as yours. One day, Kim said to me, "Pam, it's like everything we talk about can kind of be capsulated into these two ideas. You got to know your content. And you got to know your kids. And if you know content, and you know your students well, then everything else follows from that." But we have to start with those two main pillars that you know the content. Meaning, you don't just sort of have a bunch of procedures and an isolated facts at your grade level, but you know how to develop to get there because you're going to have students that aren't quite there yet. And you know, how to develop from there the things that are coming on the horizon, so that you can help students. You can start to open that horizon for them. But you also...I'll use a Cathy Fosnot term here...you're not just looking vertically below you and above you content wise, but you're also thinking deeply, horizontally on that landscape of learning. You're also thinking about deepening connections and encouraging kids, challenging students to make multiple connections to strengthen the way that they're thinking about certain things. So, you've got to know your content. But then, in that same vein, you've got to know your kids, so that as you work with a student, you can give them questions right at the edge of their zone of proximal development. You can choose the Problem String that's just right for your class because it allows all students access, everyone will be able to dig in and grapple with something, but it will also [Timer dings] stretch all students in some way. So, you got to know your content, got to know your kids.

Kim  20:02

Nice. I'm feeling for you.

Pam  20:06

I'm sweating. I'm sweating right now. I'm sweating.

Kim  20:09

Oh, my gosh. Okay. To hear more about "know your content, know your kids", check out Episode 74. Alright, the last one that I have prepared is Do, Say, Represent. Go.

Pam  20:22

Oh, golly. Ha! Okay. So, "modeling". Super tricky word. It has so many meanings, and it's been co-opted by people to mean different things when I don't think it means certain. I think it has like eight legitimate meanings in mathematics, and I think it has at least one non-legitimate meaning that some people are using. So, what we would suggest. I could tell you about all the "not's". I'm going to spend my minute and a half I have left talking about the "do's", the "what to do". We recommend that you give students problems on the edge of their Zone of Proximal Development that they can grapple with. And then, as they are doing that work, you pull out what they are doing in their heads. You elicit their thinking. You help them say what they're doing. So, they're doing, they're grappling in their heads. You help them say, put words to what they're doing. And as you pull those words out, and they get more clarity on the relationships they're using because they're having to put words to it and you're helping with that, then you represent their thinking and you make their thinking visible, so that then it is point-at-able, and discussable, and comparable. So, the way to help students learn more and more math is to give them something worth chewing on, elicit that thinking, help them sharpen that thinking by putting words to it, and then help them sharpen that thinking by making that thinking visible. Ya'll, it's the thinking they're doing, [Timer dings] make that thinking visible, and then you can compare, and grow, and point at, and discuss, and get even more sophisticated about that. Thank you. 

Kim  22:12

Alright.

Pam  22:13

Well, I got to end it with, that was also Kim. Kim is the one that said, "It's almost like there's things you can do. But you can say more than you can do. And you can represent more than you can say." And we've now taken that sort of model of what you can Do, Say, and Represent into that's actually the way we then want to help kids learn.

Kim  22:36

You're tired.

Pam  22:38

Now, I'm done.

Kim  22:38

We can do more than we say.

Pam  22:39

I'm tired? Do I sound tired? Did I just say that wrong?

Kim  22:42

You said it backwards. We can definitely do more than we can say. We can say more than we can represent. You have just unloaded your brain. 

Pam  22:48

You're serious? I said it backwards?

Kim  22:49

Yeah, it's okay.

Pam  22:50

Holy cow. Okay (unclear). 

Kim  22:52

They're going to listen to Episode 128, and hear more about it.

Pam  22:57

Yowza! What did I say? You can say more than you can do?

Kim  22:59

That's how I heard it. Maybe I'm wrong.

Pam  23:04

We do stuff... We can do more than we can clearly say, and we can say more than we can, then it's easy to represent it. But it's also the... Okay, wow.

Kim  23:13

Yeah.

Pam  23:14

Yes, I am tired. 

Kim  23:15

This is your opportunity. You get like one more, 1 minute to tell us anything that you think the listeners need to know about the Math is Figure-Out-Able movement, or something that I didn't grab, or you can say something about what's next.

Pam  23:32

Ah, okay, nice. I'll just finish by saying, we need to do more with this modeling thing that we kind of the last couple things that we ended with.

Kim  23:40

Yeah.

Pam  23:40

The idea of making thinking visible, I'm clear that we need more on that. I've kind of spent my time spent helping teachers rethink the way they think. So, giving teachers the "Aha" of "Oh, my gosh, I actually can think and reason about math. We actually can teach kids to..." I've spent my time doing that and not as much time putting work into helping teachers then know what to do next, and so that's the work to come. Especially the idea of modeling and how we can use making thinking visible to help kids learn more and more math. That's what's coming.

Kim  24:14

Yeah. So good. Listeners, we thank you. We are so happy to have a place where we can grapple, and word vomit, and just like wrestle with some ideas. We are super grateful for you for listening, for sharing, for giving us your ideas, and for pushing back, so that we also grow.

Pam  24:33

Absolutely. Thank you so much! Thank you 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. Let's keep spreading the word that Math is Figure-Out-Able!