Pam  0:01  
Hey, fellow math-ers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam, a former mimicker turned math-er. 

Kim  0:10  
And I'm Kim, 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  0:18  
Y'all, we know that algorithms are super cool human achievements, but they are not good teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop.

Kim  0:31  
In this podcast, we help you teach math-ing, building relationships with your students, and grappling with mathematical relationships. 

Pam  0:38  
We invite you to join us to make math more figure-out-able! Okay, that was probably the least interesting intro that we've done for awhile.

Kim  0:48  
My neck is bothering me today. Is that weird?

Pam  0:50  
Oh, no! I hate that. That's the worst!

Kim  0:53  
It's like a little bit of a crick in my neck.

Pam  0:56  
Ooh, sorry about that. 

Kim  0:57  
Eh, it's alright. I'll live. 

Pam  0:58  
Yeah, that's no fun. And now... Okay, and I was just going to say, and I can't get my screen to scroll, but now it's scrolling. So good I can look at my notes.

Kim  1:05  
That might be problematic.

Pam  1:07  
All the things. Can we just stop having system updates? Can I just request that right now?

Kim  1:12  
Oh, man! Well, hey, can we tell the listeners that recently we've discovered that we'll put our stuff on "Do Not Disturb".

Pam  1:18  
Oh, yeah.

Kim  1:19  
But pings still come through. So...

Pam  1:22  
Yeah, yeah. Like, I'm doing a webinar the other day, and all of a sudden my mom's FaceTime request comes through. I was like, "What is happening?!" So, yeah, we definitely need to get into this new system update to see what is happening. Come on, Apple! Anyway, hey, Kim. We were chatting about a webinar that we're about to do. 

Kim  1:40  
Yeah.

Pam  1:40  
With hand2mind. 

Kim  1:41  
Yeah.

Pam  1:42  
We're going to be focusing on our Foundations for Strategies that we've created with hand2mind. We're super excited about it. And we thought we'd share a little bit. We started talking, not necessarily about the webinar, but part of what we did in our Foundations for Strategies, and we thought, "Hey, that's something that we haven't shared on the podcast." And so, yeah, let's dive into that a little bit today. Okay. So, we have four different kits with hand2mind. I'm going to talk about a sequence of tasks that we do in our multi-digit, multiplication/division kit. So, first. And we'll give Kourtney credit for coming up with kind of a Willy Wonka setting. 

Kim  2:19  
Yeah. 

Pam  2:19  
So if you think about Willy Wonka and the Chocolate Factory. Kind of. It's loosely based on that. Where you could think about having a contest where the kids can come in, and they can design their own chocolate bar. So, picture a Ritter Sport. Now, if you don't know Ritter Sport because it's like expensive German chocolate, but my mom's from Switzerland, so I'm all about expensive European chocolate. It's the square bars. And in the square bars, each of the little pieces are cubes. They're square. Square pieces that are three dimensional cubes. Yeah? And so, we kind of based it on that because we're heading towards area, and area is all about square units. And so, we didn't want rectangular units. We wanted these square units. So, we said, "Hey, design your own chocolate bar." And so, one of the things that we do is we give kids base-10 materials where we have like the unit cube, and then the 10 of those in a 10 sort of rod. And then 100 of those. And 100 kind of flat, you know, arrangement as a 10 by 10 array. And then we also gave them these linear pieces, which we'll give Bridges in Mathematics credit for kind of giving us that idea. We want to do a little bit of work with these. Not a ton of work. A little bit of work where we say to them, "Okay, you want a chocolate bar that's..." Now, pick one, Kim. I don't know. 30 by 30 pieces. Maybe we could even make it smaller for this. But like 15 by 8. Maybe that's going to be your chocolate bar, because you don't have to make a square chocolate bar, so you can have these square pieces in it.

Kim  3:48  
Okay.

Pam  3:49  
And so, a kid might say, "Yeah, 15 by 8." And then we give them these materials on a magnetic board, so that the magnetic, linear pieces can kind of outline. What would a 15 by 8 look like? We could kind of outline that rectangle to be a 15 by 8. And then we're like how many chocolate squares are going to be in there? And we actually have the kids, one or two times, slap in all those base-10 materials to be able to count quickly. Now, we don't slap in, you know, a bunch of unit cubes through the whole thing. They could, right? Because that's what the candy bar is going to be able to be a bunch of unit cubes. But we can stick some together, and they can put the 10 by 10 in there, and then the 1 by 10s, and kind of fill it in. And we kind of nudge them a little bit to kind of get it in a place value kind of looking way. But we really want kids to feel area. We want them to feel that it really is this many of those 1 by 1 cubes. Even though we let them do it, you know, with those chunks. Does that make sense so far? 

Kim  4:46  
Wait, hang on a second. Why are you saying 1 by 1 cubes?

Pam  4:49  
The little... As we're putting the base-10 materials on (unclear).

Kim  4:53  
Oh, like the little 1 by 1 by 1. But.

Pam  4:55  
1 by 1 by 1.

Kim  4:56  
Yeah, yeah. Okay, okay. 

Pam  4:57  
Yeah, yeah, yeah.

Kim  4:57  
Alright.

Pam  4:58  
And so, they're sticking them on that tray. 

Kim  5:01  
Yeah.

Pam  5:02  
Yeah. So, we feel like it's really important to feel area. 

Kim  5:08  
Yeah.

Pam  5:09  
Feel all the 1 by 1 by 1s in there. It's kind of... We kind of end up with smart partial products in that way. Well, we kind of end up, sorry, with partial products, where you can sort of look at the 10 by 10, and then the 1s by 1s. And when we call it partial products, it's usually if you had something like if you had 15 times 18, then kids will often break that up, and they'll say, "Well, that's like a 10 plus 5 by a 10 by 8. And so, they'll actually put in, you know, the 10 linear piece by a 10 linear piece. And then they stick in the 10 by 10 area piece that fills in that space. And then they'll put 5 more unit linear pieces down the bottom, and 8 more linear pieces across the top. And then they're putting in 1 by 10s next to those. And then there's that little square in the corner. And then they're, you know, putting in a bunch of 1 by 1 in that corner. Again, so the kids really get a sense and feel for area. Well, we call that partial products. Then, we want kids to get better at smart partial products. So, then we say, "Well, for 15 times 18. Do you really have to cut it up into all those pieces? Could we do bigger chunks? Now, we go to open arrays. Now, we're going to start drawing on... Well, maybe closed arrays on grid paper first. So, where they're drawing stuff on grid paper. But then we're going to kind of look at could we chunk bigger chunks? So, if I have a 15 by 18, could I think about a 10 by 5, a 10 and 5 but just keep the 18 whole? Like, don't cut up the 18 into 10 and 8. So, do you know 10 times 18? Sure enough. That's 180. So, now I'm either drawing that in grid paper, or eventually we go to just drawing open arrays. Then do you know 5 times 18? Now, you're going to snicker, and you're going to go, "Of course, kids don't know 5 times 18." But we just had 10 times 18. So, if you have 10 groups of 18, can you cut that in half to get 5 groups of 18. And now, you can add that 180 plus that 90 to get the total area. So, we move them to more of a smart partial product.

Kim  7:11  
Okay.

Pam  7:12  
Yeah, interrupt.

Kim  7:13  
Can I interrupt you? Yeah, just because I want to... I know you're going to now move forward where we went with this, and I want to kind of back up for a second and name some of the things that happened so far. So.

Pam  7:23  
Please. 

Kim  7:23  
This Willy Wonka-ish chocolate thing created a situation where it was context based for the students. Which I think is important. You know, you've said it, but it's important to name that that was purposeful for us, that we gave them a context that they could mess with. And then...

Pam  7:38  
Hey, Kim? Hey, Kim? Was the context a fake story, just to get kids kind of vaguely interested, so that we could suck them into being willing to do boring math?

Kim  7:47  
I mean, I hope not. I hope that it was... You know, it's obviously a made-up story. There is some engagement in it. But they needed to use the mathematics that they had at that moment to solve the problem. And then in the work that they're doing, we're going to move them. We're going to nudge them along. And so, I think that's part of what we're going to talk about here is that they... You know, their interest is clearly peaked, but that wasn't the whole point. The point was to engage them in a scenario that involved mathematics, that they were forced to work with the mathematics in such a way that we could nudge.

Pam  8:29  
Nice. And I'll quote Cathy Fosnot here to say it makes the mathematics realizable. 

Kim  8:33  
Yeah. 

Pam  8:34  
So, when kids start talking about naked numbers and they just start throwing stuff around, we can say, "Whoa, whoa, whoa. Where are the chocolate squares? Like, what's happening? Like, show me the chocolate bar. Like, show me what's going on," to keep them grounded in what's actually happening.

Kim  8:47  
Right.

Pam  8:47  
Yeah.

Kim  8:47  
And so, when students are just throwing out numbers, we can pull it back to the context to help them make sense of what they're thinking about. 

Pam  8:55  
Nice. Yep, yep. 

Kim  8:56  
And so, context was important for us. Then we gave students manipulatives that they could build with. So, you know, I think side note, right? Sometimes people might hear you talk about manipulatives and think, you know, if they're not listening carefully or if they just hear snippets, they might think that, you know, we're against manipulatives. It's not at all true. We are just for the right ones at the right time. And so, for this context, its important...

Pam  9:21  
The right way.

Kim  9:22  
Yeah, that's true.

Pam  9:23  
Yeah. 

Kim  9:24  
So, for this context, for this situation, we felt it was important that students get to build the chocolate bars for just a moment. Like you said, just a wee bit of time, and then they move to a closed array, a way to represent with sketches, the work that could then replace those manipulatives.

Pam  9:47  
And now I'm going to interrupt you, if you don't mind.

Kim  9:48  
Yeah. 

Pam  9:49  
So, listeners, you might be hearing this as, "Sure. Pam. You're going to have them build the steps of the algorithm, and then you're going to have them draw the steps of the algorithm, and then we're going to get to the abstract steps of the algorithm." So, hear me carefully that we're going to use these manipulatives to give kids a sense of area.

Kim  10:05  
Mmhm.

Pam  10:05  
Like, we really want them to understand area is made up of unit squares. Like, this is an important ... I get why you were pushing on me for cubes earlier. Because we're talking about area.

Kim  10:14  
Yeah.

Pam  10:15  
Because I was thinking about a chocolate bar. So, it's going to have the cubes. But you're right. Area should have unit squares. Thank you for that. Made me think. If I may just tell a brief, brief story. When I was working as a college intern at Hewlett Packard, one of the days... So, I don't know. They still round? So, they made printers back in the day, forever ago. Anyway, so it was one thing I did to make money for university during semesters. And so, one summer, I was working there, and our line chief said, "Hey, we're going to move, and I need you to go figure out how much area we take up right now, so that when we move to the new building, we tell them how much area, you know, space in the new building we need." And he said, "You can grab what's her name, and she can help you." I don't remember her name. Sorry. But a nice gal that I worked with now. So, I was a college intern, and she was a lifer. Like, that was her like real job. And I was like, "Yeah, hey, come on. Let's figure it out." She goes, "Oh, no. I don't do math." And I said, "What?" She's like, "No, I don't do area." And I said, "Oh, no. This is not like school math. Like, this is... all we have to do, literally, is count." Y'all there were squares on the floor. "That's all we have to do is literally count the squares on the floor." And she panicked, white, looked like she's going to faint, and walked away. Never did help me like figure out. Now, I didn't count the squares on the floor because I actually did the rectangular thing. And like it was a little bit... You know, I did it more efficiently. But people don't. You know, we have made area so much like, "Hey, is that where you multiply the sides or you add them together?" You know like, area and perimeter become the so abstract thing. This was our attempt to go, "What is area?" Like, really feel it. Not steps of the algorithm, concretely, so that we can then get them abstractly later. That was not our goal. Sorry, let me just... Sorry, for putting a fine point on that.

Kim  11:54  
That's okay. Well, and so you mentioned that the goal there was to help kids build smart partial products. So, in none of that were we building... I think one time we had... Gosh, maybe I'm remembering wrong.

Pam  12:11  
Place value partial products?

Kim  12:12  
Yeah. And then quickly we moved into smart partial products. Yep, yep. As a more efficient strategy.

Pam  12:17  
And smart partial products. What are bigger chunks? 

Kim  12:19  
Yeah. Bigger, fewer chunks. Yeah. So, then we moved kids off of the manipulatives into a closed array to represent those smart partial products. And then into an open array. So, all throughout, we're keeping with the context, and we were moving students towards models that were representing thinking, but they became more efficient models.

Pam  12:41  
Mmhm.

Kim  12:41  
So, I just want to name those things before you move on to what happened next in the lesson.

Pam  12:45  
Can I also add that those models became more abstract? 

Kim  12:49  
Mmhm.

Pam  12:49  
So, in a way, we sort of with the, you know, 1 by 1, 10 by 1, 10 by 10. Kind of these manipulatives. And then we move to grid paper, where we're sort of outlining those same things on grid paper. And then we move to open arrays, where we're just drawing the chunks of the area that we need. So, those models are getting more abstract. But again, not because we're leading to some steps of an algorithm, but because we're actually helping kids conceptualize area and being more efficient as we go. 

Kim  13:15  
Yeah.

Pam  13:16  
Cool. Okay, so when we got them messing around with smart partials, then we did a lot of fun things. We call them smudge problems, where we gave them a bunch of problems for days, and we would exemplify certain sets of relationships. So, we would have a problem, and we would show what a kid did to solve the problem with an open array, but we would leave parts of them smudged out, and we would say, "Go fill in those parts that this kid obviously did, but we smudged them out. And then let's talk about what were the relationships this particular kid was using today?" And we'd get better at one of the... Or we'd introduce at least one of the major multiplication strategies. And then on another day, we would say, "Well, here's a different kid, and they were solving the same kind of problems but this way. What were those relationships?" So, it's a way to kind of introduce the major efficient multiplication strategies that if we build relationships in kids' heads, then these strategies become natural outcomes. One of the ways that we did that, which I just particularly think is super cool. And I don't know if this was you, or Kourtney, or the two of you combined, but I take no credit for it. Up till now, kids have been designing these chocolate bars, but mostly we've been giving them... I should have said. When we gave them problems to solve, they were really, "Here's a design of a chocolate bar. How many individual squares will this chocolate bar have in the plan?" And so, that's where we were smudging things out. So, it was kind of like an order form. Like, we would give them, "Here's what the kid ordered," and the kid had to figure out the number of chocolate bars, but we smudged out part of the work. Well, now we begin to give them these order forms with this stamp on them that says, "Adjust." And so, we kind of made up this, you know like, "Hey, the Chocolate Factory said, "Oh, we see that so and so wanted to create this chocolate bar. But instead of creating that one, we're going to make you adjust it." Maybe have not enough chocolate that day. Maybe we don't have molds that big. Like, whatever. But we adjust it. And then we had the kids figure out from there what it would be. So, could we... Do you want to talk more, Kim, or do you want to do one together?

Kim  15:16  
Let me mention one thing. 

Pam  15:18  
Okay.

Kim  15:18  
So, in this sequence, you have mentioned the context is like the thread that kind of is throughout. And we've moved students from manipulatives to other models that becomes more abstract. We also did this work where we went from a place value partial product to a smart partial product to now this new strategy that you're talking about.

Pam  15:41  
Mmhm.

Kim  15:42  
So, in my mind, it feels like these little levers that we're pulling where we're not adjusting a whole lot at once, but we're making these small shifts, these adjustments to get kids. They're still grounded in a context, but they're becoming more sophisticated, both in models and in strategies all throughout this thread, these sequences of lessons. And I think it's noteworthy to say because I think it's something that, you know, whatever teachers are building, whatever they're working with, that is something to keep in mind is that, you know, it doesn't... It's okay to make these shifts, these moves in bits and pieces. But messing with the models and the strategies naturally grow over time with those shifts that you're making.

Pam  16:27  
Mmhm. Yeah, nicely said. Okay, so here's an example. Let's say you see this order form, and a kid said, "I'm going to create a 30 by 32 chocolate bar. But then there's a stamp on there that says, "Actually, today, it's..." And they crossed out the 30, and they put a 29 over it. So, now it says, "Adjust, 29 by 32." Kim, what are we suggesting right now that the job of the kid is to find the number of squares in the plan that's going to create this plan for a 29 by 32 chocolate bar?

Kim  17:01  
I mean, when they find the original plan, then if they see the adjustment is just 1 group of 32 less, then we're suggesting the Over strategy.

Pam  17:11  
Yeah. So, often, kids will... And well, and maybe we'll even show the work and smudge part of it out. But they'll be like, "Well, can I find the area of a 30 by 32?"

Kim  17:19  
Yeah. 

Pam  17:20  
So, Kim, what's the area of the 30 by 32?

Kim  17:22  
Oh, you are going to ask me that. That's 960. 

Pam  17:26  
How'd you do it? 

Kim  17:29  
What did I do? I think I thought about three 32s times 10.

Pam  17:34  
Nice, nice. Okay, so a 30 by 32 has 960 of those unit squares. But we want to now... Adjust. It came back. Adjust. It can only be a 29 by 32. So, how are you going to use the 960 to help you think about 29 by 32?

Kim  17:50  
Then I'm just going to subtract 32 from 960.

Pam  17:55  
Nice. Which is?

Kim  17:57  
928

Pam  17:58  
And we could talk about subtraction, but we're not going to today. Cool. So, in a huge way, we are nudging kids. We're introducing this idea that when they hit a problem like 29 times 32, they can go, "Huh. What do I know that I could use to solve this problem?" And they might think about 20 times 32 and 9 times 32. They could. But we also, in this problem, because we gave him the 30 by 32, and then "Eh, adjust," we're suggesting, "Do you know 30 times 32? Can you figure that out? And bam, you just got to get rid of the 32." So, we're introducing the Over strategy.

Kim  18:31  
And even just the language of "Adjust", I think is just a bit of a nudge. It's a bit of a nudge to say you don't have to start from scratch. You just adjust what you already know.

Pam  18:41  
Oh, that is nice. I'm glad you brought that up. Okay, let's do one more quick one. Okay. What if I were to say the order was for a 21 by 40? What would the area of that be? Well, maybe I'll say the order was for 21 by 40, but then "eh, adjust," it's actually 21 by 39.

Kim  18:58  
So, 21 by 40 is 840. But instead of twenty-one 40s, I only want twenty-one 39s. And I'm actually going to think about that the other way, if I can. So, instead of forty 21s, I'm going to think about thirty-nine 21s. Nice. So, I'm going to subtract 21 and get 819.

Pam  19:19  
Sweet, square units. And that is how we're kind of introducing the Over. Now, we're not going to suggest that kids walk away from a couple of these "adjust" or even four or five of these "adjust" problems owning the Over strategy. We're suggesting this as a way to introduce the Over strategy. Then use Problem Strings with lots of problems with the Over strategy to get kids kids good at multiplying by something they know, and then adjusting a little Under, a little Over. And for the Over strategy going a little too big, and then cutting off the extra. Cool.

Kim  19:51  
Yeah. So, two last things I want to mention is that in this work, in this sequence, students started with some collaborative work, and then moved into more independent work. So, not only have we adjusted the models they're using, we've adjusted the strategies they're using, we've adjusted that they're becoming more independent. And then, like you said, with Problem Strings, we don't just do this, you know, this sequence with Rich Tasks. We know that we're going to come back and revisit. You know, it's not a one and done. We're going to come back with Problem Strings to give them more experience.

Pam  20:21  
Nice. And eventually, when we just did that 21 by 39, eventually we're going to write that as 21 times 40 minus 1, times a quantity 40 minus 1. And talk about the distributive property. Y'all, that's the abstract that we're heading for.

Kim  20:38  
Mmhm.

Pam  20:38  
Not some algorithm and series of steps to memorize and mimic. We are using concrete things, and we do make them more sort of representational or pictorial like we did with grid paper.

Kim  20:53  
Mmhm.

Pam  20:54  
And even open arrays. And then we do get more abstract and write down that 21 times the quantity 40 minus 1 equals 21 times 40 minus 21 times 1. We do get that abstract equation format. But just like Tondevold and Flynn would suggest that CRA or the modeling process is not linear, they're all connected. We would also agree that these are all connected. We want kids to really understand area, so that when they compress that knowledge, then they can uncompress it, that they can just use it to solve problems, but they can also get inside it, and really... This is when I don't get them as high school saying, "Miss, area is that the one where I multiply or the one where I add?" Like, they know because they own area. They've really had this experience where the concrete, the representational, and the abstract are all part of a cohesive whole that means mathematics. And in this case, that means multiplicative reasoning. So, what are the abstract things that we're building? We're suggesting we're building strategies. Some of the basics that are needed for that, we need anything times 10. We're suggesting that representing doesn't mean drawing the manipulative, so that then you can... Drawing the steps of the... Sorry, how do I say this? Do the steps of the algorithm with the manipulative, then draw the steps of the algorithm with showing the drawing of the manipulative, and then do the steps of the algorithm. So, if I could cross out what I just said. Eh, not that. Rather, we are talking about representing doesn't mean drawing the manipulative, it means open arrays, and ratio tables, and equations. So, we do want to move towards more abstraction and more sophistication. But yeah, that. 

Kim  22:44  
Yeah. The concrete experience. We're helping kids in Rich Tasks and using context, and then they're representing their thinking rather than the steps somebody else told them. And then we're helping them abstractly record the strategy that they used.

Pam  23:01  
From that narrow set of strategies that we're trying to build, right? It's not just like anything that they're doing. But yeah. Yeah, nicely said. Alright, y'all, there is a better way to conceive of concrete, representational, abstract. It's not a linear path. It is a cohesive, coherent understanding and getting more sophisticated with strategies. Cool. 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!