Pam  00:01

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

Kim  00:10

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  00:18

So, let's get at it! We know that algorithms are amazing, historic achievements, but they're not good teaching tools because mimicking step-by-step procedures can actually trap students into using less sophisticated reasoning than the problems are intended to develop.

Kim  00:33

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

Pam  00:40

So, let's get it to it. Kimberly. How are you today? 

Kim  00:44

I'm good. You? You've been traveling. And...

Pam  00:47

Whew, yes.

Kim  00:48

...I mean tons and tons and tons of traveling. And you sent me a text and said, "Hey, I want to tell you about Sandra." And that was the entire text. "I'll tell you about Sandra" and a quote. So, here we are. Me asking.

Pam  01:03

Bam. So, when I was in northern Alberta, had a fantastic time traveling around the North Pole. Again, I'm just chuckling a little bit. Met some amazing people. Had a really, really good time. Spent a lot of time in cars. They drive a lot out there. Sandra said, "Hey, let Kim know. One of the things I really appreciate about the podcast is that you guys don't agree on everything, and that you kind of beat out your disagreements for everybody to hear. And I smiled at her. I said, "Yeah, I really love that... Most of the time."

Kim  01:41

Yeah. I mean, you know, what I find interesting is we agree on the big, important things.

Pam  01:48

Fundamental.

Kim  01:49

Yeah. 

Pam  01:49

Underlying. 

Pam and Kim  01:50

Yeah.

Pam  01:50

Absolutely. 

Kim  01:51

And I think... You know, I think maybe because we kind of like melded and  and that kind of stuff over time. But yeah, there are definitely things that we see differently. And, you know, I don't know if it's because we have different teaching experiences. 

Pam  02:06

I mean, we definitely had different growing up experiences. We were in different places. We kind of went different directions. And I think there's some... I have some holdover sometimes, of some things that I did that I haven't thought through that I'll bring up, and you'll go, "Um... Do you really think that?" And then I'll have to actually think about it. And there's plenty of times where I'm like, "Maybe not. Yeah, let me rethink that."

Kim  02:29

Yeah. And I also taught far younger students. So, you know, I think that just like more elementary. Like you, you helped me consider long term effect. And I think, you know, when we talk really young students, then I can share experiences. But anyway. Well, I'm glad Sandra loves our disagreements.

Pam  02:50

We'll try to keep that going. Yeah.

Kim  02:52

Yeah, yeah.

Pam  02:53

Alright, bam.

Kim  02:54

So, you've been traveling a ton, a ton, a ton, but we've also been super busy. I mean...

Pam  03:00

Getting a lot of stuff done. Yeah, we've been working, working hard. 

Kim  03:03

Our team's great. Yeah. And lots of good content. Books. We just had the launch of the Trap of Algorithms. That's really exciting. 

Pam  03:10

So exciting. So, if you haven't heard of it yet, Developing Mathematical Reasoning - Avoiding the Trap of Algorithms is out and on the Amazon bestseller list! So super exciting. Within not even a whole week of being out for sale, Amazon has us as a best seller. We have a little flag and everything that says, "Best seller". It was best seller in the specific category of Secondary Education at one point, and then another one... Oh, now I can't remember what the other one was. Anyway, so fantastic. We're super excited that that's going well. 

Kim  03:42

Yeah.

Pam  03:42

We've had a blast launching that. If you haven't checked it out yet, it's your new book study book. Yeah, we're very excited. And, like you said, getting lots done.

Kim  03:51

Yeah. And, you know, I really, really love our Journey members. I know you do too. And we actually...

Pam  03:59

That's our coaching group, yeah.

Kim  04:00

Yeah, yeah.

Pam  04:00

Mmhm. 

Kim  04:01

So, we recently, like in the last month or so, did this event with Journey, and we thought that we would bring it to the world. So, we spent some time with Journey talking about sequences. Yeah, sequences of tasks, some ideas. Yeah, sequences are important. 

Pam  04:17

Mmhm.

Kim  04:17

So, we thought we would do a couple of episodes for you, listeners, about sequences of tasks. 

Pam  04:23

Yeah, that went so well, and everybody was really excited about it. We'll bring the parts of it that make sense to a podcast.  So, I also want to circle back to something. A few weeks ago, I was on the podcast Math on the Rocks with the fantastic Duane Habeker and Cole Sampson. And Kyle Atkin was there as well. I think it's just Duane and Cole that actually usually run Math on the Rocks. And I had a great time on that podcast. And after, I did some work with them. One of the things that Duane said has just really been resonating with me and I've been thinking about. He said, "So, you've identified like a small set of major strategies." He's like, "Major! I'm really thinking about that. Like, what is... What do you mean major strategies?" And we sort of talked about the fact that we really believe that there are these major relationships, that if we can build in kids brains, they lead to major strategies, so that kids can solve any problem that's reasonable to solve that a calculator.

Kim  04:30

Yeah.  Yeah.

Pam  05:25

Which then means we don't need a general solution that can solve any problem that's reasonable to solve without a calculator because we have these few major strategies. And the great news is, is that if we build those few major strategies, we've also built the math.

Kim  05:41

Mmhm.

Pam  05:42

They now have these major relationships, the literal mathematical connections in their in their brains, the neural connection. So, super important. Kind of an interesting thing that we just... I'll just say, again. There are major strategies. And they're right that I've spent a lot of time talking about Problem Strings, in part, because we get a big bang for our buck towards getting those major strategies built. We can do a lot of mathing in a short period of time. And if you've been listening to the podcast at all, you know that we often do Problem Strings on the podcast, because that's a fantastic way of really helping build those major relationships that lead to those major strategies. 

Kim  06:24

Yeah.

Pam  06:24

But. But you just mentioned sequencing. 

Kim  06:27

Yeah.

Pam  06:28

There are other tasks that we... So, if you were in my university class, or when Kim was in the classroom, or if we go into classrooms, we don't just do or promote just doing Problem Strings. There are other tasks. 

Kim  06:41

Yeah. I'm glad you said that, because I think it can be misunderstood that  Problem Strings are the thing. That's the only thing. And so, yeah, absolutely. We believe very strongly in really rich investigations. Kind of like a messy, lob it out, get kids involved and talking with each other, and tinkering. Those investigations, to bring out big ideas, are super important.

Pam  07:10

Mmhm. 

Kim  07:10

We also love games. And kids reinforce things that they're learning. Gives us an opportunity as teachers to circulate to see what kids are saying and doing. You just get a different feel when kids are playing games. 

Pam  07:24

It's a brain change up. It's a different kind of brain activity. 

Kim  07:27

Yeah.

Pam  07:28

Yeah. 

Kim  07:28

There are also so many. Man, so many other good routines. Too many to name. But other really good routines that you can do in classrooms to have a little bit of a difference. You know, Problem Strings are our favorite. We've done some episodes on other routines. But just to reinforce and come at relationships in a slightly different way. We also, you know, feel strongly about kids having opportunity to demonstrate their understanding. You know, we're not adverse to paper, pencil type of work.

Pam  08:01

Mmhm, mmhm.

Kim  08:02

So, there's lots of pieces. But if you listen to episodes and you think it's just Problem Strings, this episode is for you because we're going to talk about kind of how to put some of these things together.

Pam  08:13

Yeah. How do we sort of sequence? 

Kim  08:15

Yeah.

Pam  08:15

Parts of that. So, one of the things we do in our coaching group that we call Journey is that we talk about sequencing tasks. You might hear me in a presentation, or work with me in a workshop where I'll talk about a part sequence. How do I say that? Partial? That's what I'm looking for. A partial sequence of tasks. But we've also created entire units sequences. Those are in our Building Powerful Mathematics workshops. So, some of the parts of those that include kind of what Kim just mentioned. More open stuff. And then some smudgy, curated number kinds of tasks where we analyze student strategies with some parts missing in them. And what we also need to really connect operations. And, like you said, we think that there should be some practice involved. But how do you practice? Does that look like 1-29, odd? Or are there other? Are there better ways of practicing? So, let's dive into... I just mentioned kind of a lot of things. Let's sort of dive into some of that. 

Kim  09:15

Yeah. So, can we focus today on kind of some foundational stuff for young learners? 

Pam  09:21

That sounds great. Yes. 

Kim  09:22

Okay, so let's talk foundation for a second because when we talk about the foundations of young addition, a lot of people will think you start by having students learn and memorize facts. Like, that's the starting point, right? You got to know the facts first, and then you can (unclear). 

Pam  09:42

Then you can do real math. 

Kim  09:44

(unclear). Yeah, because... And if you are algorithm focused, it might be true that you need kids to know single-digit facts first before they can do any larger problems. But we... 

Pam  09:54

Well... And sorry, I'm going to interrupt you. If you're algorithm focused, it's not might be true. It is true. Like, if you... Are we going to disagree here?

Kim  10:04

No. No, go ahead.

Pam  10:05

If a teacher's goal just to get kids to correctly repeat steps of an algorithm, then you've got to have kids have those facts rote memorized, so that they can. Or it will be drudgery, right? They'll be counting and doing. 

Kim  10:21

Yeah, that's what I was going to say.

Pam  10:22

Over and over and over, they're going to be doing those single-digit facts. Yeah, sorry. 

Kim  10:25

Yeah. There's some very traditional teachers who think they want kids to memorize facts, but they're still going to progress them through. They just will leave them counting on their fingers. So. So, anyway, in any case, you're going to start with single-digit facts. But we say there are foundational relationships that lead to major strategies. And one of the really important relationships that we think is important is 10 (unclear).

Pam  10:50

So, again, for young learners. Yep, 10. Yeah, absolutely.

Kim  10:54

Ways to mess with 10. And we... I was going to say what we believe about facts, but I'm going to hang on to that for just that for just a second. But we believe that ten-ness is a really important thing.

Pam  11:07

So, that's one of maybe the common threads. That if we think about sequencing tasks, we would make sure that combinations of 10, and how you think about teens compared to 10, and how numbers relate to 5 and 10. That that we would have a common theme all about 10.

Kim  11:26

Yeah.

Pam  11:26

That that could sort of run through a series of tasks. 

Kim  11:29

Yeah, I'm going to add to that, that adding 10 to a single-digit or adding single-digit plus 10. So, there's so much about 10 that we would want. We also think a really important thing to work with is doubles. 

Pam  11:43

Yep, yeah.

Kim  11:44

Because then you could do so much off of those doubles. 

Pam  11:47

Absolutely. Agreed. 

Kim  11:48

We also think that helping students understand deeply the commutative property is really important. Not just, "3 plus 4 equals 4 plus 3, kids."

Pam  11:59

Or call it, "Here. Memorize your turnaround facts."

Kim  12:02

Yeah. Yeah, yeah. So, really understanding why and how the community of property is at work.

Pam  12:07

Feeling it. 

Kim  12:08

Yeah. We also think a really important relationship is the idea of adjusting.

Pam  12:13

Oh, yeah. 

Kim  12:13

So, like a little more than something, a little bit less than something, and why that can be true. So, those are major relationships and understandings we think are important, foundationally, for young learners.

Pam  12:25

Yeah, and there's some major strategies that come out of really messing with these young relations. Or these early. Maybe I'll say "early" not young. Or early relationships that we need. And it's not even like six. It's just three. There's three major relationships that cover all of the addition and subtraction strategies. So, if we're really focusing on major strategies, we would say there's six strategies. There's three for addition, three for subtraction. But there's just really three major relationships that feed into those strategies. So, it's not about memorizing six strategies. It's really developing three major relationships. And bam, we've got kids reasoning toward the end of having single-digit facts at their fingertips and so much more,

Kim  13:11

Yeah. And so, these relationships that you're talking about, if you were to say a list of three addition and three subtraction, it can feel like, "Oh, wow. I... You know like, having kids work with these six, there's so many." But if you can step outside of like the name, or step outside of the details because it's addition or subtraction. If you can see the big picture, the properties at work, then you can see how they're all connected. And I think that's really important for us as teachers to say, "How are these things connected? And what relationships am I building?" so that they can do some of the same work in addition and subtraction. It doesn't feel so disconnected.

Pam  13:52

Yeah, and so maybe I'll just say them again. Like, it's really all about partners of 10, getting to 10 in some way and using that. Or adjusting a little bit. So, doing a little too much and adjusting back. Or doing not quite enough and adjusting up. And then using double.

Kim  14:08

Yeah.

Pam  14:08

Like, being able to really own doubles, and then kind of doing that adjusting, back and forth, off of those.

Kim  14:14

Yeah.

Pam  14:15

Yeah. 

Kim  14:15

So, let's talk for a little bit about these sequences. So, we... There's kind of some categories that we think about. So, in sequencing, we think about bits that are kind of messy. There's some messy work to be done. And then we also think of some bits that are strategy focused. And maybe that's where people hear a lot about Problem Strings. But in general, focusing on strategy. But then there's also this bit about practice in some way.

Pam  14:42

Yeah, those are some nice categories. And to be clear, when we say "messy", we don't mean that kids are... That the classroom's unstructured, and kids don't know what to do, and that it's messy in that we send them off, "Hey, go discover math. Hope you figure it out. Come back. I know I could tell you, but I'm going to be mean and just make you go find it on your own." We don't mean that at all. 

Kim  14:43

Yeah. Right. 

Pam  14:46

We do mean that the problems the kids are messing with are a bit more open, a bit more... Maybe open-ended, but for sure open-middled. And there's some grappling to do for kids. It's going to take them a little bit longer to dive in and like really figure out what the question's... Not be unclear what the question is asking, but really to dive in to understand the relationships that are happening. Go ahead. 

Kim  15:32

And it's messy in the sense that there are ways to go about that work that is not just one way, and there are several possibilities that are available to them. It's not messy like, "Let me send you away for 45 minutes, and hope you figure something out, and then I'm going to call it done." It's so much more relational as the teachers working with kids, and circulating, and seeing what's happening, and nudging. So, it's not at all what sometimes people think. Just like, "I'm going to send these kids away, and they're going to do it, and hopefully they discover it on Yeah. their own."

Pam  16:09

Not that at all. Let me give you... Let's maybe give an example, so we can be a little more clear. So, could we take young learners and say to them, "Hey, I've got 10 ice creams. Some are vanilla. Some are strawberry. How many vanilla? How many strawberry?" And then let these young learners start thinking about that. We could even give them like two colors where I have white on one side and red on the other side. And we could, you know like, "Here, show me. Show me an example of 10 ice creams where some are vanilla and some strawberry. Show me another example." And then I could walk around. I could say, "Can you come up with another one?" And then as I'm walking around and some kids got a bunch of them, then I'm like, "Do you have them all? Do you have all the possible combinations?" So, there's... It's messy in that way. It's open. The kids could say, "Well, I don't like strawberry. I'm going to have all vanilla." And then I could say, "Well, are there other possibilities?" And then really, to keep kids challenged, then I'm... Now, I'm not going to probably ask all kids right off the bat. Some kids are really still fussing with, "How do I record mine? You know, I now have the counters, so I've got 9 red up and 1 white up. But now I need to record that as 9 plus 1 equals 10." And the teachers circulating and helping kids record what they've got. And then if kids got a bunch recorded, now I'm asking that kid, "How do you know if you have them all?"  And the kid starts thinking about being... What's the word I want? Help me. If I'm trying to know I have them all? (unclear) organized.  Yeah. Like, I'm trying to organize and find a pattern, so I can be... There's an S word. Trying to be...

Kim  17:36

Structure?

Pam  17:38

Yeah, keep going.

Kim  17:40

Systematic? 

Pam  17:41

That's the word I wanted! Well done, Kim!

Kim  17:44

Read your mind. 

Pam  17:45

Nicely done. Can I be systematic about how? And we're not going to force kids to be systematic. But, boy, we're nudging that way.  That's good mathematical behavior. So, that's an example of something that we might do. And then we might follow that where, now that we've kind of organized that a little bit, could we organize it in a 10 frame? And then that would help us kind of look at do we have them all? And the 10 frames are great because they build off the 5 and the 10 structure. And so, we can also look at equivalencies. Like, all of these combinations gave us 10 ice creams. We can also look at connecting addition and subtraction. Well, if you had 6 vanillas and 10 total, how many strawberry do you have? And we could connect that with a missing addend problem but also with subtraction. We could also do some messier things, where to build an open number line by looking at some kind of tasks that could help kids think about measuring something. Like, for example, I know one task that we've written is if we have a kid doing a sack race. And we talk about, well if this kid did 4 hops, and their partner then started and did 6 hops, where did the partner end? Well, they ended 8 hops later. And notice that we've got kind of this hop thing where it's a span. I'm sort of spanning this distance. And we could kind of connect that to an open number line. So, those are all examples of things that are a little bit messier. I don't know if I have another word. Where the kids are kind of investigating and we're really building foundational kind of ideas. But let's be clear, we're not really getting... We're not cinching a lot there. And that's okay. There's a place for this messier stuff. But we have to also include more cinching kinds of things. So, that was kind of your second. Or do you want to say something before I go into the second category? 

Kim  17:51

Yeah. No, I was going to say just that in these kind of messier, like kids messing situations, lots of different ideas can come out. So, in that way, you're tinkering with big ideas of mathematics. So, when you were talking about like the sack race type idea, you could also bring out if the first kid did 4 and the second kid did 6, you would be at 10. But what if the partners switched and the first guy went 6 and the second girl went 4? Then you're talking about the commutative property. So, lots of different ideas can come out in the same investigation. And then you get to tinker with what do you want to work with students on? 

Pam  19:28

First.

Kim  19:28

Because then you're going to want to eventually work on all of them, right? Yeah. Yeah. 

Pam  19:33

Nice. Nice, nice, nice. So, one of the other categories you talked about was being more focused because we've just now been messy and got all these ideas percolating and bubbling up, and we got to focus on some of them. So, one of the things that we like to do with younger learners is to use addition charts. Not as a retrieval device to retrieve addition facts.

Kim  20:32

Yeah.

Pam  20:32

But more as a pattern finding device where kids can look at that addition chart and go, "Huh. Like, I'm noticing that there's tens here. Well, there's a pattern where, if I add these numbers together. Like, check it out. In this diagonal, they all add to 10. Huh. In this horizontal, they all add to 10. What was happening? Oh, it's all these numbers plus..." Or, sorry not add to 10. Add to the number. "It's like the numbers repeat. Well, it's because I was adding 0. Ooh, they're all 1 more. Ooh, it's because I was adding 1." Like, there's these patterns that can come out of the addition chart that can be super helpful. We can also analyze strategies. So, if we want to build foundations for strategies, one of the things that we can do is look at student work where we've smudged out part of their work. So, for example, if I've got a student who's adding 7 plus 3 on a number line, and they've got a jump to 7, and then a jump of 3 more to 10. But I smudge out how long the jump was and where they landed. Just enough. I have to be careful. You know, there has to be enough information on there that the kids can fill in the other parts. Then I can say, "How did this student use that strategy?" Let me give you maybe a better one. What if I had a student doing something like 15 minus 5, and then minus 2 more. And again, I've smudged a little bit of that, but when the kid fills everything in, they can go, "Okay, I can see what the student was doing was doing 15 minus 7. But they did minus 5 first, and then 2 more." Then another student, or maybe the same student, did a different problem. 13 minus 3, and then they subtracted 4 more. And then another student did 17 minus 7, and then they subtracted 1 more. And the point that we would want to be pulling out is what did the student do every time? They subtracted to 10. They removed to 10, and then they removed the extra. Oh, that's one of the major strategies. So, by smudging out part of the work and having students fill in the smudges, and then asking them to kind of like take a bird's eye view and what was the student doing every time? Bam, we start to develop the foundations for that student to own the strategy of removing to 10. So, that's an idea that we can do. We can also do things where, once they've kind of started building those relationships towards those major strategies, create anchor charts that that make those relationships visible. So, we might have an anchor chart that says, "What are the subtraction strategies we've been looking at?" Well, one of them would be Remove to Ten, and we could write Remove to Ten and have an example on a number line. And then we've been using doubles, and we could put that on there. And we could say we've removed a bit too much, Like, remove 10 and adjust. An example of that would be 17 minus 9. I could say, "Well, I'm going to do 17 minus 10 because I know what that is. Oh, but I only meant to minus, to subtract 9, so I have to adjust up." So, kids can look. We can anchor those strategies and put them on the wall, and kids can use those in their work when they're doing other tasks. Kim, you mentioned one other category, which is we believe kids need to do some practice. But boy, our practice might look a little different than some of the practice that we maybe experienced as students. One of the things that we can do for practice are things like... We've talked about it in other episodes, so we'll put this in the show notes. Where we can create hint cards for kids. Where you interview kids. Facts you know. Facts you don't know yet. And the facts that you don't know yet, let's just focus on that smaller set of facts. But as we're going to flash card kids with those facts, on that flash card... We're going to call it a hint card. Let's create a hint that would be helpful to you. Just a hint. Not too much. Don't give it away. Don't make it too crazy to read, but don't make it too hard that it doesn't help me either. And write your personalized hint on that card. Bam, we want kids to actually practice those. And there's some other things that we could do to practice, including games. It would be super helpful for kids. And we want to sort of sequence these messy tasks with strategy, focused tasks with practice tasks, and rinse and repeat.

Kim  24:40

Yeah. So, listeners, when you are creating sequences for your students, you might want to consider, what is it something that I can do that's kind of messier with my kids, and then strategy focused, and then practice. And we've listed several ideas here, but you can be thinking about what do my kids need? What are the foundations that we're working with? And what could I string together that would help nudge kids along based on what I know I can do?

Pam  25:08

Nicely said. And if you're interested in a sequence, kind of like we just described, we've actually planned that out for you in an easy to read format in a project that we've been doing with Hand to Mind, called Foundations for Strategies. There's an addition and subtraction for young learners, addition and subtraction for single-digit numbers that you could check out. Hand2mind, Foundations for Strategies. We'll put that link in the show notes where you can just do the sequence that we've sort of suggested with all the helps and teacher notes. But since it Hand2mind, it's super easy to to read. Alright. Kim, thanks for chatting about sequencing younger tasks. Let's sequence some older tasks to come. Yeah? 

Kim  25:49

Yeah, sounds great. 

Pam  25:50

Alright, in our next episode. Ya'll, 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!