The Stenhouse Blog

Sneak Peek at Building Fact Fluency: A Toolkit for Multiplication & Division

Posted by admin on Jan 22, 2021 11:41:15 AM

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In this episode of Teacher's Corner, Graham Fletcher and Tracy Zager share a sneak peek of the upcoming Building Fact Fluency: A Toolkit for Multiplication & Division and discuss the purpose and challenge of creating intriguing and accessible contexts for students. 

 

 

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Read the transcript

Tracy: Hello everyone. It's January 11th, Monday morning. Graham Fletcher and I, I'm Tracy Zager and I'm talking to my buddy Graham Fletcher. We've been hard at work on the Building Fact Fluency multiplication division toolkit. While everyone else was having family fun over break, we were texting each other about lemons and paints and we're having a ball and we thought we would take a few minutes today to talk to you about what we've been up to. Good morning, Graham.
Graham: Good morning, Tracy. Nice to come out of the weeds here. We've been tackling this now for some time. Super excited to start field testing it here in the next week or so. I'm super excited to just dive in and talk about what we've been doing, what we've learned from the addition and subtraction kit and how we're continuing to build that fact fluency, not only in a K2 classroom but now extending it three, five, and then also into the middle grades for sure. So yeah, thanks for having me.
Tracy: It's always awesome to talk to you. The truth is, I will tell you all. The truth is we were on another call before we even started this, because we talk so many times a day about this project. We're both living it and we just love it. Graham, for somebody who's new to the whole idea of the Building Fact Fluency toolkits, how would you describe them? What are they and what's... Let's start with what are they, both of them?
Graham: I guess in order to the toolkit, one of the things that really got me to the toolkit was I loved number talks. I love number talks, the number string. I've been using them as a classroom teacher. I've been using them as a coach and supporting just that work at a district level for numerous years.
Graham: But one of the things where I would become super frustrated is I realized that students would know their strategies. Whether we're talking about just simple addition and subtraction, like that make-a-10 strategy, when we were doing number talks, kids would know that make-a-10 strategy, but then the second that I would turn and take that say nine plus seven that they're able to use with a number talk, whenever we would contextualize that, and what that means is by putting it into like a word problem that they'd have to solve, students would no longer use that make-a-10 strategy and they revert right back to counting strategies.
Graham: Which isn't a problem, but I knew that students knew the strategy. They knew how to think flexibly and accurately. They had that efficient strategy of making 10, but they would never apply it in context. At the ground level, that is what Building Fact Fluency, the toolkit is all about. It's about building a fact fluency through problem solving.
Graham: Kassia, one of our colleagues that we've knocked heads with a couple of times here throughout this project is she says that fact fluency should be an outcome of rich problem solving. A lot of the times what I realized is before we started tackling this toolkit is throw myself right under the bus here, is I would put fact fluency as a granular idea. Where I take it, put it on a shelf and then I would just tap into it whenever we'd needed it.
Graham: But it would never be that fact fluency was interwoven through all the rich mathematical things that we were doing. At the heart of it, context, Building Fact Fluency through a context is what separates this kit from a lot of things that are out there.
Tracy: You're making me think about how fact fluency has historically been a prerequisite to interesting math, like a gatekeeper. It's an equity issue because so many kids are not allowed access to interesting and worthy math problems until they know their facts. They're over at the round table doing worksheets on facts, instead of doing the rich problem solving they can be doing. Rather than viewing fact fluency as some prerequisite to doing all the work that follows, we're getting at fact fluency through that interesting problem solving and rich tasks.
Graham: Absolutely. I start thinking about, especially when we start looking at third grade with multiplication. It's like, let's get our facts out of the way and then we'll start problem solving. Where the problem solving and the facts, they need to be interconnected. You're right when you talk about it being a gatekeeper. Absolutely. We have so many students who feel as if they're not mathematically competent because they don't know their facts.
Graham: When we've talked about this numerous times is it gets pretty crazy how in third grade, students are expected to know all of their facts in one year, like let's just cram everything on students. And how many students leave third grade not knowing their facts and now they've developed this negative disposition about themselves and how they relate to math.
Graham: That's what we're trying to compete with here. Is how do we allow students the accessibility to facts and not see that just because you don't know your facts based off of speed, we're now focusing on that interconnectedness and relationships of number, which is what fact fluency is really all about, for sure.
Tracy: Totally. I have feelings about that placement in third grade. I have feelings about that. One thing when we were writing the addition subtraction kit, we were in the standards, the facts aren't nailed down till second grade. But you've got kindergarten and first grade and second grade to get there, and there are partial benchmarks in kindergarten, first and second grade.
Tracy: But in the addition, I'm sorry, in the multiplication division progression, they're front-loaded in the standards and we're not in the business of writing standards but we are in schools regularly, or we would be in COVID. We're still working with teachers and coaches all the time, and we're just not seeing kids get those facts all the way done in third grade. That's just doesn't seem to be reality.
Tracy: Do you want to talk a little bit about how we built a multi-year progression into the kit, just based on... It's certainly standards aligned, but based on the reality that most kids are not going to get them all in just one year.
Graham: Yeah, I think that's a great starting point. One of the things that we leaned on a lot, there's years and years of research. But one of the big resources that we leaned on was Gina Kling and Jenny Bay-Williams' book of fact fluency. They talked about two sets of facts, foundational facts and then derived facts.
Graham: When you and I were probably students growing up and we were in third grade, it was a know your facts, but used to be zeros, one, two, three, four, and it would be in that sequence, in that chronological order. What research has shown is that there's ways that we can group facts together, but that helps build more sense.
Graham: For instance, if we know that students in a K2 class, in a kindergarten through second grade class are coming to us with an understanding of tens, fives, twos, skip counting by tens, skip counting by five, skip counting by twos, then that seems like the natural place where we'd want to start in third grade.
Graham: Then from understanding those foundational facts, zeros, ones, twos, fives, tens, and also squares was a really beautiful thing that I was introduced at during this journey that we're on right now, is just like students know their doubles facts in addition and subtraction, students also naturally gravitate towards squares, like four times four or six times six.
Graham: Those square facts also become a foundational fact, and then from there, we can develop those derived facts, which would be your threes, your sixes, your fours. You can get to your fours from doubling your twos. Then those other facts come after we've really established those foundational facts. But then when we start looking at multi-year, those first foundational facts, what we've done is we're building those off of an understanding that happens in a K2 classroom.
Graham: But then also as we begin looking at extending for third grade and fourth grade, is we don't want to look at just fact fluency as isolated single digit facts. Students should also be getting to really leverage those properties. For instance, if we start multiplying three times 13, we maybe decompose that 13 into a 10 and a three.
Graham: Even as much as we're talking about fact fluency in this kit, we've been really intentional about layering in scaffolding and extensions wherever needed. So that we're showing that fact fluency isn't an isolated skill that we just put in a box and you know it and stuff it on a shelf. We show how this is really, this understanding of number and number relationship, how it really sets up for success, not only with fact fluency, but also with multi-digit.
Graham: That's where we begin to layer in fact fluency with third grade, but then it also supports fourth grade, fifth grade, and as you can see with, as we're writing the facilitator's guide right now, it really dives into that multiplicative and proportional reasoning where we really fall short in middle school. That's how it's layered in throughout the different grade levels.
Tracy: You're making me think about some of the folks that we've leaned on, the tons of work that's gone on for decades in how children's thinking about the multiplication and division in particular develops. Two of my favorite books that are just powerhouses together are from the CGI crew from Carpenter and Frankie and Linda Levy is thinking mathematically, I think it's called Connecting Arithmetic and Algebra.
Tracy: Then from Virginia Bastable and Debra Shifter and Susan Jo Russell, their book which I think is called Connecting Algebra to Arithmetic. I might have those, their titles are interrelated, but through that work and others, we keep looking at how the properties get uncovered right from the beginning with the single-digit facts. It's not that the kids learn their facts and then later on in algebra, they should learn their properties.
Tracy: It's that a kid who thinks of sevens as fives and twos has just discovered the distributed property. And uncovering it in the single-digit facts is so important because then they extend that work into the multi-digit facts and it ends up being the foundation of all the algorithms, certainly partial products and things like that, but also all the proportional reasoning and rates and ratios and all sorts of things that come later. Finding an equivalent fraction that it's all connected.
Tracy: We've been talking a lot about how those single digit work. It's just one long progression up through much later mathematics of that connection between arithmetic and algebra. It's really exciting to have that algebra work right from the beginning. But that might sound scary to folks if they don't remember these words. You know what I mean? If people think, wait, distributed property, isn't that what you do in middle school? Why would you be talking about that in third grade?
Graham: Right. Absolutely. I think I would say for me personally, this has been one of the biggest shifts between the addition and subtraction kit to the multiplication and division kit. I like the term that you've used just when we've been talking back and forth, as you talk about how we're actually baking the properties right into the context.
Graham: I'll throw myself right under the bus here. As a third grade teacher of eight years, whenever I would start tackling multiplication in fact fluency, I would never really try to leverage the properties and I would leave those as a standalone skill, and if we could get to the properties, we'll get to the properties. But don't really worry about the fancy names.
Graham: What we've really tried to do here is really showed that those properties are the gatekeepers to building a rich, robust understanding of fact fluency. When that happens, students can now leverage this for years to come. It might sound scary. If you start hearing about properties as a third and fourth grade teacher, that does get a little scary.
Graham: It was scary for me, which is probably one of the reasons why I avoided it in the early years of teaching but the more that I've understood the properties, those are our super powers. When we understand those, we can apply them just as you'd shared before, across all different numbers, not just with fact fluency but we start talking about fractions, absolutely.
Graham: So yeah, these properties and the resources that you mentioned as well, like everything that they're talking about in these resources that we've leaned on and countless teachers and educators have leaned on talks about the importance of properties. But too often, I think properties only get talked about in fifth grade and definitely in middle school, but let's start sooner. But I think the multiplication and division kit is definitely, that's something that we're really trying to highlight as much as possible.
Tracy: We have these schematic things where Graham and I look across the kit and think about, where do we see the associated property emerge? Where do we see the... We got to get commutative up early because when a child figures out the commutative property, the next thing is they realize they only have to learn half the facts because they can just flip them around.
Tracy: But figuring that out is not obvious because it depends on the problem type. Graham has been working on creating contexts for students to discover those properties through the tasks. Maybe we should talk a little bit about that. What would be a series of tests, we call it a lesson string and the kit, where we can start introducing some of these properties in accessible, everyday ways, using everyday objects. Your house is full of stuff right now. What kinds of stuff did you get to record these photos and videos?
Graham: It's great. On this journey, both my girls, those of you who are familiar with me and my work, my girls, my wife and I, we have two daughters. One who's now currently in fifth and one in eighth, and we tear the house up. We've been taking pictures all over the place but one of the things that we've tried to do is use these contexts.
Graham: A context is a lot of the times when we talk about properties, those contexts are a void of properties. It's just naked numbers. One of the things Tracy, where you just mentioned a lesson string. There's a lot of really good resources out there, like student facing resources, but they lack coherence. What I mean by that is, you might have a number string over here, and then you might have a game over here, and then you have word problems over here and you have all these great resources, but there's nothing that really connects all of them together. They're on individual islands.
Graham: When you're talking about a lesson string, what a lesson string is is it's a series of individual student-facing activities. But what happens is we'll use one context, so that ties all of those components together. A funny story is about two, three weeks ago, I'm walking out of the Sams. I live here in Georgia. You might have a Costco where you are, and I'm walking out of Sam's with 119 lemons. I look like a nutter just walking out.
Graham: What we're doing then is we're using this context of lemons and we're using the context of lemons for the factor of seven. What I end up doing is I end up taking those 119 lemons and putting them into bags of seven lemons per bag. That's the context that we're talking about so here we'll just talk about lemons.
Graham: One of the nice things, so here's how a lesson string will work out. To start off, there's a three-act task. This is different. We're using a three-act, then the addition and subtraction kit, for those of you are familiar with that. One of the things that we realized is that we want to jolt student's thinking right at the beginning of when we're building understanding. We want to leave a mathematical residue.
Graham: What we'll do is at the beginning of each lesson string, there's a three-act task. If those of you are familiar with Dan Meyer and his work, big shout out to Dan for creating three-act tasks, that we have a lemon context with a three-act task. That launches this idea of, hey, we're going to start playing around with lemons as our context and putting them into bags of seven lemons. Then we build from there.
Graham: Then we have pictures of the lemons, which instead of going with a number string, it's now like a number talk image string. And just a series of images that are now connected that pull out the math and the properties that we were talking about right there. So you'd have the three-act task, then you would have an image talk.
Graham: Then what we gradually do is we start to strip away that context of lemons, and we begin to replace it with a tool. It might be colored counters, and we gradually strip away the context and we decontextualize that lemon piece of it. We'd have a three-act task, an image talk, a tool talk, and then we would get right down to naked numbers.
Graham: Those are the components that are the same in the lesson string for both addition and subtraction and multiplication and division. But there's also some extra things that we've included in the lesson talks for multiplication and division. I'm a huge fan of Open Middle, Robert Kaplinsky, Nanette Johnson, they've gone ahead and created a beautiful crowdsource website. They're probably one of my favorite activities to get that intentional, repetitive practice that students need with fact fluency.
Graham: We've designed a Open Middle problems for each one of these factors. Now you're going to have a lot of opportunity to practice these facts with open middle type problems. We've also included in these lessons strings for each factor, true-false statements. Tracy, I know you're a big fan of true-false statements. Can you talk a little bit about the true-false statements that we've included?
Tracy: I love them. If you think about this context, this is a grouping context. Some of the contexts involve things that come in groups and some of the contexts involve things that come in arrays. Then some of the contexts will involve area, measurement and comparison. But we have a lots and lots of groups and arrays because those are the most powerful models for learning about multiplication. They're the reason multiplication happens.
Tracy: This context, we have things that come in groups, and so this is a really nice place for us in the true-false, to push at some of those properties in a group in context. For example, we mentioned the community property before. We could put up on the board, we could put three times seven equals seven times three.
Tracy: Kids could be thinking about that lemon context, so they've got three bags, each with seven lemons in them. Is that the same as seven bags each with three lemons in them? The thing is, it's not exactly the same from a grouping point of view. If you put those two pictures up side by side and you said, are these the same or different, well, they have the same total but it's a really different story if you're making lemonade.
Tracy: In the true-false, we could put those side-by-side and say, when we're working with the numbers, three times seven equals seven times three, is that true or false? Can kids figure out if that's true or false without calculating each side? Can they reason through why that might be true or false? With each tour, first there will be some number sentence where sometimes it's true and sometimes it's false and we're pushing at properties. That's just a commutative example in that case.
Graham: I know that's one of your favorite, because what it does is it gets kids to generalize. I think that's a big piece when we start talking about properties, understandable, does this always work? Does a times b, is that always going to be the same as b times a. And getting kids to not only look with those individual numbers, but we want them to generalize those patterns as much as possible.
Graham: So yeah, the true-false is definitely something that I'm glad that we're adding in here because it really gets students to step back and say, is this true? Is this always the case? Is this not the case? And we want them asking those, internalizing those numbers in those relationships as much as possible. Another thing that we've included along with the true false is same, different.
Graham: Here what we'll do is, Tracy gave an example of the seven bags with three lemons in each bag, and then three lemons, sorry, three bags was seven lemons. But showing two images side-by-side and just asking students, what's the same, what's different? When students are doing that comparison analysis of the two images, well, what they're doing is they're probably going to use informal language for the properties.
Graham: That's a beautiful place because now students are bringing their understanding to the table, and then we can begin to formalize their informal language and get them to generalize and see those properties. Properties are everywhere and I'm super excited that students like eight and nine and 10-year-old boys and girls are going to have much more access to the properties, which are seldom ever really explored, especially in a third and fourth grade class.
Graham: That's it. Oh, and the last thing with lesson strings is there's also a game as well. I think we all like games. One of the big things with the games that we've included is, Tracy, you and I have talked at length about this, we are not big fans of roll and record games. Like just roll two dice and multiply it. Cover-Up that number. One of the things that we've intentionally done is designed brand new games that are all strategy-based games.
Graham: It's like you roll the dice and you have to do this, or you could do this. You now have choice and in that choice, you're now probably multiplying and doing way more practice than you would have if you just rolled and record. Is there anything else that I'm thinking that we might be missing here at the lessons string? That's it. We've got the... Go ahead.
Tracy: The card talks, this is a new thing as well, or the teacher size decks of cards. You want to talk about that a little bit?
Graham: Yeah. In the addition and subtraction toolkit, one of the things that we loved about that was the opportunity to put five frames, 10 frames and double 10 frames in students' hands, decks of cards to where they can play with that. We initially wanted to try and carry that. In the past I've created what I call multiplication subitizing cards. We were trying to figure out a way that we could fold those into the kit and get decks.
Graham: What we realized is that would push more of speed on students and that's not something what we're after. What we want students to do is be super thoughtful when they're looking at subitizing. If you're not familiar with subitizing, subitizing is ability to recognize a quantity and a set without counting it. Well, we've used that same understanding with addition and subtraction. Now we've applied it to multiplication and division. But with these cards-
Tracy: Give an example of what a subitizing image would look like for some fact. Like three times seven, what does subitizing look like there?
Graham: What that would look like is a three times seven, so if we were seeing that, we'd say three groups of seven, so you'd see three circles on a card and you'd see seven dots inside each of those three circles. Now when kids are doing that and what they might look at that three times seven, they might say, oh, I know that two sevens is 14 and I need one more group of seven. Which right there, that's the distributive property for getting at three times seven, two times seven plus one times seven.
Graham: By not just putting those cards in students' hands but by allowing a teacher to have one big display card that they can show to the class, we now allow students thinking time to breathe. It's not about, show the card, tell the answer. It's about, show the card, how did you find out how many dots are on those cards?
Graham: I really like the fact that we're now slowing down and telling those students that normally to just blast out the answer, whoa, we know that you have the answer but we don't want to hear from you yet. We actually provide those students, some of our brightest students in the classroom who maybe just need a little bit of think time, we now invite them to take their time, turn and talk with their partner, what do you see, how are you seeing this, how are you putting it together? And we now that's where the conversations about properties will come out. The card talks, I'm glad you brought that up.
Tracy: In the kit, there'll be a big deck for grouping cards. Just to be clear, kids probably aren't going to subitize as many as seven dots. The idea there that I think Graham is talking about is like, look at the picture and be able to see those groups. I think of those cards as the grouping cards where we're showing, there might be three circles with seven dots in them and somewhere else in the deck is seven circles with three dots in those, so kids can see the difference when you're grouping.
Tracy: Then the second deck of cards are array, cards where it's blank arrays for all the facts. Those are especially nice for the computer to property because you don't need two different cards to show three times seven and seven times three. You just hold up your card and then you rotate it 90 degrees, and it's the same thing.
Tracy: This is a really nice device for helping kids start to make the idea about how multiplication commutes, and then they can transfer that thinking to grouping. They won't do that by accident. We built that all into the kid but... The card talks, teachers might... Like some of the ones that Graham's building, you might see two cards side-by-side. It might be a grouping card in an array card or two grouping cards, or it might be for grouping cards and you say, find all the ones that make 20 or find all the ones that make 24.
Tracy: We talked a lot about how to use these cards in a small group. It's a really nice tool for a small group where because they're big enough for all the kids to see. It's a really nice assessment tool. Also Graham, you want to talk about that a little bit? You saw it right away as something teachers could use to assess facts.
Graham: Absolutely. Just sitting down with students. A lot of the times, I think about assessment and I think that's a bone of contention for both of us. We think about time tests and how many times students would again, build that negative math disposition because of time tests. One of the things that we're trying to be really intentional about is taking away that speed element and really trying to get students to uncover those properties.
Graham: By just showing two cards to students and just listening to them reflect, I can be jotting down annotative notes to say, oh, they're familiar with these ones. Or one of the nicest things is you could give the kids a whole deck of cards and just have them go through it and say, pull out the ones that you don't know, maybe you're a little bit shaky on and you had to maybe... like you don't know them right away.
Graham: One of the things that we've really liked about this journey is kids will tell you what they don't know. Give the kids the cards and say, which ones do you have a hard time with? They'll tell you which ones they don't know. A lot of the time we don't provide students an opportunity to be driving that learning bus that they're on.
Graham: A lot of times we tell them how to drive the bus. Giving them the cards or giving them an image and saying, what's the same, what's different, which ones are hard, which ones do you just know just like that? A lot of times, kids will do the assessing and the grading for you simultaneously, which is nice.
Tracy: That's awesome. Assessment is baked all the way through the toolkit. Let's see, I just said baked again. Maybe I got to stop overusing it. But we're constantly talking about assessment as sitting beside kids and listening to them as they work. That's the root of the word. Rather than the traditional ways that facts have been assessed, which has been really problematic for so many reasons, it doesn't give you valid information and it's not really a teaching tool and it's super stressful to kids to do those time tests.
Tracy: We have thoughtfully threaded formative assessment all the way through, so that you'll have a really good handle on where your kids are with their understanding of the properties and if they have those foundational facts and then if they're extending them to the derived facts and if they're extending all of that into multi-digit, because there are so many chances to listen to kids. While they're playing games, while they're solving problems, you're going to get to use your five practices strategies. While they're engaged in the three-act task or the practice problems are all really rich.
Tracy: One thing that is harder about this kit Graham and I have talked about a lot is we don't like contrived stuff. Addition, subtraction contexts are a little bit easier because the problems can make sense, but like Graham likes to say things become a gong show, right?
Graham: Yeah, absolutely.
Tracy: What's an example where you thought it would... I don't mean to put you on the spot, but if you thought of a context and then you're like, wait a minute, I can't make a story out of that. That sounds ridiculous. Who would go into Sam's and buy 119 lemons. That's not going to be a story that makes sense to kids.
Graham: Absolutely. I think that's a great example. The lemon, so when we're making the three-act task and when we're... Lemons themselves, if I was just to get a bunch of lemons and then just put them on a table or put them on a four in a classroom and take a picture, that just seems so contrived. Why on earth does this nut job have 119 lemons on a school floor.
Graham: In order to make it more relatable for students and more accessible for students, is I went over to my neighbor's house, he has a white pickup truck, and I build the whole context in the back of my neighbor's pickup truck. In doing so, kids can now say, oh, they're in a lemon farm. Those students who are out in California, they're probably familiar with lemons and the types of fruit that they get out there, but there's a lemon from, so now we're bringing it to life.
Graham: He's a lemon farmer. Okay, so now he's got to take them. He's got to put them in the bag before he goes. Trying to make them as real as possible. I think a lot of the times kids can't relate to what's happening in their class and that's one of the things that we've really tried to do. Is make it so kids can look at it and they have access to it because it's relatable to them. One of the things that we've tried to with in choosing the context is try to make them to where they're as gender neutral as possible so all students can relate to them, but also as great neutral as possible.
Graham: We're not going to put Comic Sans, Fontan and Chevron borders on every single thing. And we've talked about that, but you know there's kids in middle school. There's kids in middle school that need access to this. It's almost like we don't want it, it's geared towards third grade but unfortunately, we have so many students in middle school, in high school who haven't been provided the opportunities to play and explore.
Graham: When building these contexts, when kids look at it, we don't want kids to say, Oh, that's baby math. We want them to look at it and we want them to be genuinely intrigued and curious. What do you notice? What do you wonder? Big shout out to any veteran and Max Ray there. But yeah, so trying to make it as authentic as possible for students, no matter gender or no matter age, I think that's something we've really been super intentional about throughout this whole process with both addition, subtraction and now with multiplication and division.
Tracy: 100%. Then the other thing we really care about is accessibility for everybody. Part of why Graham was the perfect person to do these kits is that he has such a beautiful foundation in multimedia math tests. Those of you who go to his website and do the three-act math tasks know how good he is at this. He's just brilliant at this part.
Tracy: What he talks about, I wondered whose white pickup truck that was when I was looking at the photos. He always finds amazing surroundings for the photographs that hint at a story. There's an element of storytelling in all the work here that you don't need to be able to read English to access. All of the contexts start with either images or a short video and there's no reading to be done, and the images and the video grab you right away. Sometimes they're really funny.
Tracy: The other day he texted me the first act of the bobbers. It's Graham going fishing and he's walking down the dock and he's whistling, and I'm cracking up over here because it's just funny. There's an element of humor in some of them. Like when he went to visit his in-laws, he took the stuff for the bobbers context so he could take great pictures on the dock. Like the paints tasks, it's an array of little paint pots of all these different colors. It's like a beautiful symmetric array. But next to it is some water with brushes in it and a canvas and the edge of a painting, and so students can see...
Tracy: I just love the way you do that, Graham. Often it's on the edge of the photo, there's just a little glimpse that tells you, this is a real context, a real story, or part of something that I can actually imagine. Not one of these pseudo context, math workbook problems. Then kids right away can start to make sense of it. We were looking for contexts that are accessible to kids with a wide variety of cultural backgrounds and lived experiences so that they can mathematize the things that they see in their everyday lives. That's part of what this kit is about.
Graham: Right. I'm glad that you brought up the bobbers because when I first sent the picture of the bag of bobbers and you're like, I don't... At first it sounded like you might've been like, "I don't know if a lot of kids are going to be able to relate to that." If you just go show a bobber to some students who are probably living in some of our bigger cities, they don't have access to fishing, so that can be really scary to just pull out a bunch of bobbers.
Graham: Instead, by using that video of me walking down a pier with a fishing pool whistling, and then dropping it and then now the water brings the bobbers, so now the bobbers now have purpose and by painting that picture, and a lot of times, I think the word you just used there, mathematize, we try and put things in a book and kids can never internalize what's actually happening in print.
Graham: How is it that we can present information and tasks and activities to students where all students have access to it? Because a lot of the times when students look at a word problem, they just shut down and say, "I can't do that." But every single student can notice or wonder something from an image. Trying to make it, I think that accessibility is definitely at the forefront of every decision we really made throughout the kit, for sure.
Tracy: It's no afterthought. Inviting students in and making this accessible to all kids is something that... That's the first step for each of the contexts. We're thinking about, we need things that naturally come in groups and we need things that naturally come arrays and we need to be able to have comparison problems, but they've got to be contexts that kids can relate to. And if they don't have the vocabulary for it, the vocabulary and the understanding of the meaning of the story will come out naturally in informal language in that conversation about what do you notice, what do you wonder?
Graham: Right. Trying to make it as language rich as possible. I think a lot of times, if kids can't talk the language, they're never going to be able to write about the language. Whenever possible, we need to provide students that opportunity to talk about things. One of the differences that I've seen with both kits, and before I started working on this project, whenever I'd create a three-act task, it wasn't so much about storytelling, it was more about just designing a task.
Graham: But I really think what's happened now is working with the addition and subtraction and the multiplication division. I've now becoming a mathematical storyteller. The importance of having that white pickup truck, the importance of walking down the pier, all of these smaller things that can be really overlooked from us as teachers are now a truly important piece to bring and invite the mathematics into the classroom.
Graham: Trying to become a better storyteller is something that this whole project has really allowed me to work on is interweaving the context through the entire lesson string. But it's now a story because they're contextualizing, decontextualizing all through this context, but it's always about the story of the lemons or the bobbers. It's definitely been an area that I've grown in terms of my task development as a task writer, which is something I love doing.
Tracy: That's awesome. It's become a family affair. The first one you did was the peaches. You went peach picking with your kids and shot the three-act right there on the peach farm
Graham: Yeah, the girls think they're movie stars, which is great. It's been awesome. I'm super thankful to my wife and my daughters for jumping in and joining me on this journey. It's a selfish journey but it's been great because they've made it a family selfish journey. It's been great, the whole process. Super excited to get this out in teachers' hands as soon as possible, for sure.
Tracy: People might be curious about timing on that. We are working super hard. Both of us are working as hard as we can on this. This kit is harder than the addition subtraction for many reasons. I don't even know if I can nail down why it's harder. Do you have a handle on that, like why is this one harder?
Graham: Because I think it comes down to... For me, it's about, and we floated around there today, is we start talking about not making it forced. There's so many things that we put in front of students that are forced so we have to find natural context for students, and we missed the boat. I start thinking about coin collecting and putting coins inside a acquaint album.
Graham: Well, that's a natural context where you see arrays. Thinking about cartons of eggs. I start thinking about putting little matchbox cards on shelves, on a display case. We've gone a little bit slower but it's because I think we want to really find those contexts that are authentic for students.
Tracy: Well, and I needed to find the right hot wheels. Because I'm so happy. I haven't even sent them to you, but we have our first cover drafts. We'll get them to you, I think this week. There's a hot wheel that's, it's an orange hertz with flames on the side, and I've already told everyone at Stenhouse, that's how I want to go out. I want you to take me on the car like that.
Tracy: We've needed to find and develop really great context, ad also there's the complexity of, when you're working on your twos, you're also working on... When are you working on your twos and when are you working on your sixes? If you're doing times six, are you working on your twos, are you working on your sixes? And which strategies might kids bring into play?
Tracy: There's a lot more overlap. There's a lot more flexibility and strategies in multiplication division, because kids might break the numbers up by addition, they might break it into chunks that way, or they might break it up by multiplication, or they might use doubling and go from there. It became like, how do we organize this? Graham and I spent a lot of time creating the frameworks and the structures and the sequencing around the factors. Then he's been creating tasks, like you can't believe they're awesome. Where we are now is you're about two thirds done with creating the tasks?
Graham: Yeah, they're all done. Starting to get them. Basically, we want to throw all of the games in front of students as well. We want to break everything, see what works, see what doesn't work. What's nice is we've had such amazing feedback from the addition and subtraction kit with just students who normally wouldn't have an opportunity to advocate for their own learning in class, that are speaking up and whether it's remote or in-person having all of these being digital.
Graham: We're testing it out in a couple of different districts, and as we do that, what's nice is teachers will then be able to provide feedback and students will be able to provide feedback. That number doesn't work on the game. Was that intentional that we don't have that number there.
Tracy: That's right.
Graham: And then we should get that back in about two months and then we just start rolling everything into the press and hopefully get it out sometime in the fall. That's the timeframe for the multiplication and division kit.
Tracy: We're going to work as hard as we can if we can get it into teachers' hands before the school starts this year. That is our goal. Everyone at Stenhouse is United behind that goal and we're just going to have to do the best we can. It's just, there are so many, I can't even describe how many files. Literally thousands of files goes into creating this because we're also building a huge companion website where all of these pieces are housed so that teachers can both project images for kids to talk about in a really easy way.
Tracy: There's also a whole bunch of professional learning videos, which Graham will record this spring. Once the resources are created, he'll be recording professional learning videos, where he takes you into classrooms and he teaches these routines himself and then reflects on what went well, what did we notice? What might I do next with that kid? I loved the videos of the addition, subtraction kit and it ends up being really like an online course.
Tracy: It was hours and hours worth of Graham on video, and so we'll do that in the spring and get that on the website as well. We'll be working all winter, spring, summer to get it to you as fast as we can. So far it's beautiful, the things that I've seen the design team on this is just amazing. We're gunning. We swear we're working hard as we can.
Graham: What's awesome is where you and I take like really, really messy ideas and then we can throw them to the team, and they'll just make them beautiful. The whole Stenhouse team has been phenomenal. They take ideas and they only make them so much better. When you're looking at those images, when you're looking at images, there's a lot of work and intentionality that went behind those. So yeah, super pumped, absolutely.
Tracy: Everyone who's working on this products just loves working on it because it's creative and it's different and it's fun and it's going to make a huge difference in kids' lives. When I think about the role that fact fluency, what has been done in the name of fact fluency for kids over many years, it's often been one of the least favorite parts of math class.
Tracy: I think when teachers are using the building fact fluency toolkits, it can become one of the most favorite parts of math class and that alone just makes me so happy. That we can make that kind of difference for people. Because there's no resource like this that exists. If you know, I don't want to be doing time tests. Well, what do I do instead? The answer to that question has not been obvious.
Tracy: Number talks alone, number talks are wonderful, but they're not going to get you all the way to fact fluency. What Graham has been building is a resource to answer that question. Well, how do we teach back fluency if we care about kids understanding? How do we teach fact fluency if we care about how kids feel in math class, and their confidence, and their belief in their ability to solve problems, because they've solved lots of problems already.
Tracy: This is now a resource that aligns with what we know about how we could be teaching fact fluency and I'm just so excited to put it out into the world and see what people are doing with it. Thanks Graham for all the work because it's awesome.
Graham: That's it. I'm a firm believer, all of us are smarter than one of us. We've been working hard to continue to support and build and build that capacity with teachers so that they can then in turn go and support students for years to come. It's not just fact fluency in third grade, it's fact fluency for success for life really.

Topics: Math