Developing & nurturing data literacy using slow reveal graphs to empower students in grades 6-12

I'm so excited to introduce Jenna Lee and I'm gonna let her tell you about her wonderful work. Um, but let's uh welcome her with a round of applause. Hello. I'm excited to be here. So, my name is Jenna. I'm coming to you from Boston um where I am a K through8 math specialist. Um and so today we're going to be talking about developing and nurturing data literacy using a technique called slowreveal graphs. But there's lots of techniques you can use, of course, to work on data literacy with students. We're specifically going to focus on this one and what it might look like in the secondary classroom. Whenever you see like an article or like a session that's about data literacy, I feel like the first sentence that people write is always data is everywhere. Data is the future. And in fact, when you look it up, um, you will not believe how many articles there are about how data literacy is important and how it is our future always. And they always look a little bit dystopian. Um, there's just so like there's so many articles about this. Um, and all of them are like a little bit vague, just basically like you're going to need to know this because it's everywhere. Um, I was kind of struck by this last one, which was the role that data will play in our future. This was the very punchy opening line. Data is not the new oil. Data is the new air. And I was struck by that because oil for me conjures up all these feelings of like consumerism and capitalism and it feels very like it's literally fueling the climate crisis. Um whereas air is like refreshing and immersive and a life source and like something we need and we need it to be clean, which is another issue that's important in data. Um, and I thought this was just so beautiful that like we should not think of data as like something we're fighting against to try and claw our way to the top for the future. That we should like immerse ourselves in it and breathe it in and allow it to help us make good decisions. But I should also warn you that I um cropped that quote a little bit. Here was the quote and then it continued like this. We breathe, generate, and consume data through every step we take and every interaction we participate in. So, it started out for me as like let's breathe in and make all of these wonderful database decisions to kind of like this really fast like surveillance capitalism. Um, and I think that that is something that can happen with data. And also what I just did to you, which was like leaving out a piece of important information, happens in every single data visualization that we deal with. Someone has decided how they're going to collect the data. Someone's decided how they're going to interpret what's important about that data, how they're going to represent that data, and that is all through their own lens. Everything is a story that's being told and we need to help students wrestle with that while also dealing with the mathematical nuances and the very important mathematical skills necessary to even read the graph. So with that, what future do we envision for today's students? So we have this idea of like we can think holistically and breathe in the data and then use that to be good citizens or does anyone recognize this picture? First of all, it is from an older movie. I must have seen it like Yeah, Minority Report. Very insidious dystopian thing. Um, so they use this data they have about what people what crimes people will commit in the future to then um arrest them before they actually commit them. So this is I think one of the worst examples you could have about collecting data and using data. So what do we want students to be able to do? Do we want them to use it in more dangerous and dystopian ways? Do we want them to use it to become good citizens? There's often a very fine line between those. So today we're going to talk about what does it mean to be data literate and how do we support students in developing this data literacy. I know that you're here probably because you saw it was listed as the keynote, but you did have a choice. There are other sessions you chose to be here. So you're at the right presentation if you believe that data literacy is a critical part of global citizenship and that in turn being a teacher is in part creating a network of global citizens that you want to engage students in sensemaking and mathematical discourse. The routine we're going to be exploring does involve sometimes a paper component but is really discourse driven. It's a lot of student talking and you love to celebrate your students brilliant ideas. They're coming in with all these mathematical ideas already before you've taught anything. But also, even with that celebration, you still want to nudge and develop their thinking further. We have so much more math learning we can do. So, with that, here's an example of a graph that I found in an article recently. I want you to take a look for 10 seconds. and think about what do you think this graph is? What's it telling us? All right, let's share a couple quick takeouts that you have. You've had a couple seconds to look at it. What do you think this graph is trying to tell us? Do we have a microphone? Thanks. All right. Do you want to go first? Yeah, I can go first. You stop talking to the microphone for people on Zoom. Okay. It's not going to project much. Um, I'm think if you're asking me the question what the graph's telling me, I think the graph is telling me that the United States spends a lot on health but doesn't have a good life expectancy comparable to how much they spend. How do you know that? Um, well, I first realized USA is what I'm going to guess is representing the United States because I see C. So, I think that's Canada. So, I see two different countries. So, I'm Where is C? Oh, there it is. Yeah, in the middle of that cluster. So, I'm like, okay, these are probably countries. USA. I know that. And then it's red. So, I'm thinking they're probably making me look at that for some reason. And then I see life expect life expectancy as like going down. Yeah, I see that. So, I'm guessing Okay. So, one of these axis is life expectancy. And I see spend going up and down up there to the left and to the right. And spend is up arrow on right. Spend is down arrow on left. Makes me think left, less spend, right, more spend. And I see USA all the way on the right but still below what looks like a median or mean line uh for the overall life expectancy when a lot of countries are below above it. How many of you also focus on that USA dot when you were looking at it? It is red. It does kind of stand out, right? First of all, you have a beautiful data literacy skills. So weren't even a plant in the front row. But yeah, so we see this thing labeled USA. I like that you had like noticed the c for Canada. I think when I was first looking at it, I had like seen as my eye kind of traveled this way, traveled towards n which was like oh it's probably Norway. Um and then I think like deu is Germany like Deutseland. Um which kind of was like I wonder where this article is from that they're like using like here I think this is Turkey. I think this is Estonia. I think this is Spain Espa. Um, but you can see that this is off and you're noticing the labels. And then you had this question of is this a mean line or a median line because it doesn't seem like the quadrants that it's forming are not along the grid lines. So what is that line? We are missing information. So we have some quantitative information here. We have some numbers. We have some numbers here. They don't exactly say what they are. If I had to hazard a guess, I would guess that this is like maybe a mean spending because this is the um spending is down, spending is down, spending is up, spending is up. Oh, so this would be Yeah, this would be the spending, right? So then this would be like spending twice as much as a typical country, spending two and a half times. We spend a lot of money on our healthare system for maybe not great outcomes all of the time. Um, I noticed that these ones were green. I was thinking originally the ones we hear about all the time are like Canada and like the UK and France with their socialized healthare systems. I was like those have to have great outcomes. They're all kind of here though in the like they spend like pretty close to an average amount. They have like a slightly better than average expectancy. Um, but it's not Swain is apparently crushing it. Uh there's a lot of skills that go into us interpreting this. So as I was listening to you talk, you started to reason about what are the axes, right? You noticed instantly the color coordination. You were like, "What is that color? I feel drawn to this." Also, red often feels like bad. Green often feels good. Um I wonder if that was something that was being played into there. um you were noticing about like the actual like values of the points because we have the scales here. So we can kind of interpret that and we're left with these questions not just about what does this line represent or what does this line represent but if we're thinking about healthcare spending all of these countries have different currencies not all of them but like many have different currencies does it account for that or I notice that this is not the number of points that is equal to the number of countries in the world so what countries are included what countries aren't included And I'm seeing that there's countries from different continents, but I'm seeing a lot of European countries. Seeing like a couple Asian, not a ton of Asian countries. So, how did they determine which countries they were going to use? How did they determine what numbers to use? There's all these questions that we have. There's all this interpretation you had to do. And you're all math teachers, future math teachers. You have experience dealing with this data. And students when handed this can get overwhelmed by the visuals. They can get overwhelmed by all of these points and what do the points mean? What do the numbers mean? And it can be difficult to come up with that takeaway. How many of you came up with that takeaway that it was like the US is spending a lot for not great outcomes? That's wonderful. Beautiful analysis. How do we get there? So part of this is the statistical problem solving process. This is from gay which is the guidelines for assessment and instruction statistical education also endorsed by NCTM and ASA which is the American statistical association. So in order to go through this whole data process first you start by formulating a statistical question then you collect consider the data try to think about your own biases you analyze the data and you interpret the results. Maybe as you analyze the data, you realize you need to go back and change something about your collection process. Maybe after you interpret the results, you're like, I should have asked a different question. So, it can have this very like circular cyclical thing going on. Um, but today we're not going to talk about the whole process where you have to actually form the question, collect the data. We're actually going to start towards the end, which might feel a little funny, but we're going to talk about how someone has already done the work of asking the question, collecting the data, and how do we support students as they're analyzing what was done and interpreting those results and also thinking really critically about the bias in those results. So, with what is data literacy, I meant to have those pop up after. Does it not do that? No. Uh so there's a million infographics super super meta about what is data literacy and people don't really seem to agree but there were some themes that emerge around the kinds of quantitative skills you need to have the kinds of visual literacy skills you need to have that it's a lot of it is about communication because someone has gone through this whole process they've gone through this whole thing and you are now only seeing the very end so how do you deal with that and understanding it. So the components of data literacy that we're going to talk about today are graphing skills, data analysis, and critical literacy. The graphing skills are the reading and understanding of that graphical features like that's when sorry I didn't catch your name actually. Logan. Logan after we had Logan beautifully describe all of the thinking. Um that like remember thinking about like oh is that the mean line? Is that the median line? understanding the graphical features, understanding what the numbers mean, understanding what each individual points mean. Those are graphing skills that are mathematical. Then there's data analysis. That's what all of you were starting to do as you were coming to conclusions about how the US spends too much for the poor outcomes it gets. That's the analyzing, the interpreting, the evaluating of information. And often this is where classroom work ends. And I'm going to say that we should push ourselves a little bit more into a domain that in literacy education is called critical lit critical literacy. And that is considering the context for the data. Thinking about what's missing, thinking about multiple viewpoints, thinking about the bias that went into the question and the collection of that data and that our students are very capable of this work. It does hinge on all of this. So, as we're thinking about what does it mean to be data literate, we are now going to think about what these things look like as we're developing data literacy for our students in the classroom. We are going to talk the most about an instruction routine called flow reveal graphs. You just do like a meu if you're familiar with it so I can see how many of you have actually done it in a classroom or is this new. That's really exciting. I'm excited to be with you. So slow reveal graphs it's an instructional routine that is about graphs and data literacy. Instructional routines have very predictable structures to them but the context is new and novel. So that as you're repeating this instructional activity they've done before, you're swapping out the graph or you're swapping out the story, swapping out whatever, they can then focus on the new context. They can focus more on the mathematics because they know exactly how they're going to interact with it. they know the kinds of questions that are going to be asked. Um, and they know the structure also for their own participation. And I think you'll understand more of what the structure looks like if we actually do one. So, we're going to start with just this in a slow reveal graph. The first slide is always totally stripped down. This graph has more information to it, but I hid it and we'll reveal it later slowly. So, first I just want you to think about what do you notice and what do you wonder? Just going to give you 10 seconds for that and then we'll do a turn and talk. So, what do you notice? What do you wonder? All right. Turn and talk with someone near you about what you notice and what you wonder. Hey guys, [Music] where are you seeing the outliers? What I assume are outliers. Which ones? bag though. I see the the two in particular. One, two, three, four, the furthest out from the cluster. But I it feels like you could almost lasso them, right? I almost wonder if the one closest to the Y ais there that's by itself. Like you get the cluster and then there's a space and there's a one and then there's one way above. Then that first one above that has the space between. I almost wonder is that an outlier? See, I think it's, you know, myself. All right, I'm gonna have you finish up what you were talking about in three, finish up in two. finish up in one. That's always the hardest part of interrupting all of your beautiful conversations. Um, and also getting us all to be able to hear each other at the end. So, what were some things that either you or your partner said about what you noticed and what you wondered and we're going to need the microphone again for Zoom. Have you guys share what you were talking about? we we were trying to determine outliers and so and then even what the outliers could be. So the four that if you were to do basically negativex starting down look like outliers, but then if it's um a logarithmic or a parabolic, those two on the bottom right might be just fine. You know, just trying to figure out and so then it's like the only one we feel confident is an outlier is one practically in the center. That one. Yeah, that one feels different. Yeah. Or how about yeah like if this one if it were logarithmic this way I guess but if it were logarithmic this way then this would be like a huge outlier. So we definitely need some more information about scale before we can determine anything about outliers. Were other people also talking I heard a little bit about if you thought this was like a linear relationship or you thought it what were you guys talking about? We talked about how like a different shape I'm supposed to wait. Sorry. Yeah. We just talked about how the shape could be conducive to a couple of different shape or different types of like regression and how based on the type of regression that we're going to use that will determine which points are outliers. Yeah, it makes a huge difference. This information that we're missing can change everything about this graph. Go ahead. Yeah. So we we also talked about patterns like negative relationship right with those points that are kind of going downwards kind of like a negative slope but then also looking at the cluster of points that are going straight vertical next to the axis that are overlapping. Um, and then there's um commenting about viscerally, initially, for me at least, being bothered by not knowing what the X and the Y axis was, right? Like it being um like decontextualized, but then as I reflect, not having a context forced me to look just at the dots, right? as opposed to like wander if it was about food, you know, like oh well now I'm hungry and you know or whatever context you'd have there, right? So I was bothered by it, but now I kind of like it from a teacher perspective. I guess the first time you do this with students, they're always kind of like what what what's the right answer? Um it's this very open and ambiguous space. And I actually think that once they're used to that structure, there's a huge amount of power in it. There's nothing wrong here. If you argue that you think all these four are outliers, you can make that argument. If you want to argue like, well, what if this is like logarithmic on the x scale? That one's a crazy outlier. You can make lots and lots of different arguments. They actually bring in a lot of different mathematical topics. So, there's this big space like this big canvas we have for the kind of mathematical thinking that we're going to do. I think what I'm going to do is give you some more information. So, here's what we have next. All right. So, what information did we just learn? Something popped up. It might be changing your thinking. But what did we just learn? John, what did we just learn? What's a linear scale on X, for example? So, John just said it's a linear scale on X. Um, and what is making you say that that's a linear scale? We're gonna have the mic come over. It's a linear scale on X because it goes 5,000 10,000 15. It's an equidistant kind of markings. So, it's not not logarithmically scaled, for example. Yeah. So, now we know the scale along the x-axis and we can like rule out certain things. Um, we can rule out maybe some outliers. Maybe we're like changing our mind about the outliers. This is the total number of the US stores. Does this change your thinking at all about what this could be about? What are you now thinking? Turn and talk with someone else quickly to see how this changes. I'm thinking something I was going to put as a wonder before you pulled this up because I've done these before is so the bottom was big gaps, right? And there was five of them and the side doesn't have anything. So I was like, "Okay, these are obviously going to be five different things, but I don't know what this one's going to be at all." And so that I was like, I just want to know like how the gaps are because normally they don't give you those gaps. You're ready. Come here for a second. I saw that, but I didn't know that was going to be it. Okay. Okay. Yeah. Yeah. All right. Voice is off in three. Voice is off in two. Voice is off in one. All right. Everyone back here. I love how much you guys have to say. I think that's beautiful. So, some of the things that I heard were now we're now thinking about stores. So, people are thinking about different kinds of stores. Um, one interesting thing, it was Marissa, right? You had noticed that like you saw the little tick marks like, "All right, there's five there, but like can't really see the back." So, if you're really close, you can see these thin gray lines in the back. You might not be able to see it or like it's hard. Um, but you can start to see like the increments that we're dealing with. reveal to you some more information. So, just because the microphone's a little tricky, the question I usually ask then is like, what new information did we just learn? Some of you are starting to jump to inferences. You're starting to jump to like making sense of it all. I actually like to just point out like this is the information that just popped up. That way we're all on the same page and no one misses it. So the thing that just popped up is that this point is Dunkin Donuts. This point that everyone thought was an outlier is McDonald's. This is Starbucks. And this is question mark. Any predictions about what you think question mark might be? Dollar. Subway. All right. Dollar General came up. Subway came up. Waffle House. Starbucks. I try my [Laughter] best. Oh, I think I wasn't listening closely to the conversation, but you were talking about like your wage at Starbucks. So, actually, in my misering, ready, voices off in three, two, one, zero. So, as I was listening, you know how like sometimes as a teacher, we're like listening for certain answers. Um, I wasn't listening for specific answers. I was looking for just like store names. Um, so I heard Dollar General, Subway, Waffle House. Interesting that two of them are food, one of them is Do they sell food at all? It's not like a food place. Dollar General. You can I guess so. I don't know if I've like I guess we have some around me. Um some of these are like of course very regional. Um so for example, I am from Boston. Do you know how many Dunkin Donuts we have? So many. Then Affleck filmed that commercial right down the street from my house like two blocks away. Um we got a lot of Dunkin Donuts. Um, but like I have not necessarily been in a Dollar General. So, Subway, Waffle House, Dollar General. And then when I heard you say Starbucks, you started talking about how much you made. And we don't yet have another axis here. And if we knew more about that y-axis, we might have better predictions for this one. Any last predictions that I maybe missed for this one? All right, let me reveal it to you. It is Subway. So, now that you know that that's Subway, does this change any of your thinking about what the Y ais might be measuring? What are some things you think the Y-axis is measuring? You think profits? So, I'm going to write it over here because this looks more y axy to me. Profits. Wages. Wages. Coffee. Coffee like number of coffees sold. International stores. International stores. Amount of employees. Number of employees. Yeah, Jacob. I wonder if it's like the number of people like in the store per hour or per like time frame. And that really top one is Chick-fil-A. Oh, the really top one. I'm going to write down like Chick Are you saying in part? No, no, no. You're saying that there's a lot of people. There's not a lot of them, but they're always crowded. Yeah. But I've never been into a subway where I've been in a line more than two or three people. Yeah, I see what you're saying. I was thinking like the time it takes to get them through the line. No, no. Like, so you're saying not that. So the density of customers, the number of people in a particular time frame, we have density of customers. [Music] All right. Three, two, one. Paul, I wanted to tell you something really funny, which is that point is Chick-fil-A. Um, usually when people make a wild guess like that, it's like totally off, but that one is Chick-fil-A. So, we have all of these different predictions for the Y-axis. Maybe it's measuring profits. Maybe it's measuring wages. Maybe it's measuring number of coffee cups sold, which like McDonald's actually does sell a ton of coffee, I think. Um, number of international stores. I've ever been in like international versions of these stores before. It's kind of funny, right, that like it's like slightly off, slightly different. Um, they like targeted towards different different tastes in different countries. Um, the number of employees, the number of people in that particular time frame, like the density of people within a store. Um, it is true that like subways like they do kind of seem lonely sometimes, a little empty. Then I reveal to you some more information. What new information did we just learn? For the sake of the microphone, I'm just going to say it, which is revenue per restaurant unit. So that's slightly different than the profits. slightly different because this is revenue per restaurant unit. Could we use that to figure out the profit? Maybe this is revenue per unit. So with that in mind what does let's see what are you now thinking like how does that change your thinking? So I know with students sometimes they'll ask about specific points but did this change your thinking to learn that y- axis? Are you surprised by the future reveal of Paul's Chick-fil-A being Right. Do you have any They're making a lot of money. Correct. Right. And I I think revenue is kind of related to the density. Right. Here, bring him a microphone. Paul, hold on just a second. Sorry. Oh, I was saying that I think revenue is related to density because if you think about the ones that are revealed on average, you're going to spend about the same amount of lunch, right? So then all of a sudden then if you're spending about the same amount of money at for lunch at each restaurant, then revenue is then determined by the number of people kind of going through and buying that so to speak, right? That like a lunch at Subway is probably comparable to a lunch at McDonald's. Yeah. But that Yeah. And at Chick-fil-A, I don't know. I haven't been to like a lot of these places. I feel a little out of my depths. Um but that like here they're making so much more and have fewer stores. here. They're making tennis stores, not as much. Reveal to you a little bit more information, which is all of these. Oh, did you have something you wanted to say? Go ahead. Um, if I wanted to own a, you know, say, okay, I want to get into business as a franchisee, I still might be very, very interested in a Subway or a Starbucks because my startup cost for a Subway might be 200,000 where my startup cost for a McDonald's might be 3 million. Yeah. And I might not have 200 I might have 200,000 but not three million. I actually have that data later for like That's perfect. You guys are so good. Um I actually I just have it like slightly in an article you're going to see. Um but yeah, no there might still be attractive things that are make because someone wants to start Subways somewhere clearly, right? Like you're not going to have more than 20,000 Subways if someone didn't want to start a Subway franchise. How many Subways do you think there are by the way? Any ideas? 22,000. 22,000. 21,000. With students, we do a lot more work estimating. Um, you'll see a picture later of kids running up with rulers. I often get kids running up with rulers in this because they want to be correct. They'll be like, "Oh, yeah, it's like 21,000, like 22,000." There's always a couple kids that are like, "I'm going to get a better answer than you." And they will come up and they'll be like, "All right, that's 8 in and that is one and a half inches in to my 8 in." And they will be doing all of these things and they're pushing themselves to do more math than you ask them to do because they want to beat their friends. So, it's always kind of interesting to see when that happens. Um, so we have tons and tons of Subways. We did reveal then quickly at the end. That is Chick-fil-A. Isn't that crazy? You got that? Um, raising canes. Kids are always like, "Oh, the chicken places make a ton of money. Maybe chicken is expensive to sell." I don't know. I think McDonald's sells chicken for like a dollar or something. Um, crispy creams in Boston. That was funny for my students because Krispy K Cream had a big big opening. They made a big deal about Krispy K Cream coming to Boston and then it lasted like two years and was like pushed out by Dunkin Donuts which isn't even very good. But like we're just very loyal to Duncan in Boston for whatever reason. Five Guys, Chipotle, Wendy's, Taco Bell. So we have a bunch labeled. You also notice we have a bunch that aren't labeled which is kind of curious. I kind of wonder what those are. Especially for ones that we maybe have or we maybe don't have. Do you guys have access to all of these ones or are some of these less Michigany? You don't have raising chains. It's pretty good. They're super popular. So, the ones in Boston are always like located right next to universities. Maybe not why there is many locations. Um, but they'll like put them right on a university campus and it'll just be like line out the door. You walk in and it's just like mobbed. Um, very strategic, different strategies. So this is the headline. So as we're thinking about data literacy often we're coming up with like our big takeaway. So in like the big takeaway of like that USA like spending graph that was like the big takeaway is that the US spends a crazy amount on healthcare and we don't survive longer despite all of our medical marvels. That's the big takeaway. Here we have a headline that I have obscured part of and I will reveal the rest to you in a minute. First, let's see if you can come up with it. Blank has the largest fast food footprint, but low blank. Turn and talk with someone. How would you fill in that sentence? What do you think? All my friends are turning and talking away from me. I'm texting a friend about this. That's so funny. That's so cute. Which is uh seventh and eighth grade. Like a tier 2 kind of class like But she uses these a lot. So, I was like, you would love this. That's amazing. I'm so excited for you. You know, I know the K35 one was earlier. I can send you that one. We did like looking at some second grade work. It was fun. Fun. Yeah. So, I'm thinking Subway has the largest, right? But low. Yeah, exactly. Nailed it. All right. Eyes up here in three, Voices off in two. Voices off in one. So, I think this is often what people came up with. You might have had a different word for the last one. So Subway has the largest fast food footprint but low sales per store because it's not quite just like low profit. It is specific to a singular store. And you can see the average Chick-fil-A sells more than $6 million in food and drink every year where Subway has over 21,000 restaurants. Revenue per unit sits at just $440,000. Um, also the source popped up and I want to pay attention to that. I'm big into looking at where the data is coming from. That is a different thing than where the visualization came from. This visualization came from Charter, which is like a really cool data company. They publish these newsletters that are just like fun with lots of graphs. So, I love Charter. Um, but the source where the data came from is a QSR 2022 top 50. That is the quarterly sales revenue magazine 2022. And those are the top 50 fast food restaurants. So, this is not all fast food restaurants. And I wonder if even some hyper local chains would be really difficult to see on this scale. They'd be like touching the y-axis, but maybe make more or maybe not. Maybe those hyper local chains, they'd be all over the place. We're just we're missing some data always. So, as we're thinking about what just happened, we started with this entirely scripted down thing where I have concealed all of the quantitative information and you were just looking at also we didn't ever really come back sorry before we get back to that to like the regression and people were talking about like oh like the negative correlation or like it looks like you could draw some sort of line right that you can actually make predictions about this but that there are some that are just above and beyond and I wonder like how marketing plays into that because it kind of looks like looks like Chick-fil-A actually is really an outlier. Chick-fil-A has done something right. It looks like McDonald's is a huge outlier. They have great brand name recognition. My husband is from North Africa in a country that does not have chain stores at all from other countries, but they all try to copy McDonald's because there's also no copyright law. So everything is like mix something. They all want to capitalize on McDonald's. Sometimes their stores are called McDonald's because McDonald's is not going to sue anyone in Algeria for that. So huge brand name recognition and it's working for them. I don't think their food is necessarily better than everyone else. But you can kind of see here like is Subway an outlier? Like depending on the curve we're drawing, it's maybe not an outlier, but if you're thinking more linearly, it might be a huge outlier. That's how we make sense of this. So, how did this work? We started with the stripped down graph, everything removed except for the points. Then we revealed the x-axis. That did change your thinking. Also revealed that it was now about stores. Um, students got very hung up on like stores. They were like, "Okay, store stores." Like, and they like thought it had to be like a CVS at first. Then they were like, "No, it's going to be stores, like those like clothing stores." And turns out fast food didn't feel like stores to them. Oh, yeah. tricky. And then we identified a few points before revealing the y-axis to start to generate like what would we even measure about these things. This was actually we did this in eighth grade as the introduction to bariate data. They had never seen a scatter plot necessarily and we were using this to introduce it. So first we were spending a long time just talking about one variable so that then we could build up the idea that like actually graphs can measure two variables. And that's the crazy thing about eighth grade. Um, and so it it actually worked beautifully that then when we did the lesson in the curriculum next, they were all like, "This is super easy." And we're like, "Right." Because like we'd already talked about all of these concepts and we built it up as a story. Um, it was fantastic. I swear. So then we identified a few points so that we could dwell in that story of what is what are we measuring? And then we revealed the y- axis. And then we discuss that headline. This is the data analysis part where we're like what is your big takeaway? So this focus on graphing skills. This is more like data analysis parts. And then you can go off into the critical literacy which we will discuss in a moment. So the questions that propelled this routine are what do you notice? What do you wonder? And then you could ask that every time but it starts to feel a little redundant. So building off of that is what new information did we just learn? Being as low inference as possible just explicitly say what happened and then how does that change our thinking? So thinking back to these components of data literacy I think we hit on most of them. We had a lot of graphing skills where you had to think about what these different points were meaning. There was some interpretation data analysis and we'll talk about some of the critical literacy that the eighth graders got into with this. So here's quick case study of an eighth grade class last year. So in eighth grade we had started with this graph. The graphing skills came in here. This was the first reveal. You can see they actually were kind of thinking about like oh is it like a curvy thing? But they don't really understand quadratics very well. So they were like I don't know what's happening but it looks like a curve. um just revealed the y-axis. You can see there's no headline. There's like not all of the points are labeled in. We got to the McDonald's. They were really fixated on that because it felt like an outlier to them. But it's like what does that point mean? Because they kept talking about as like the $13,000 or 13,000 stores McDonald's. They were so focused on this one axis, this one variable. And we needed them to understand that that point is representing two different variables. So when asked what does this point represent they were able to say again that 13,000 we had to that's why you can see like I drew that on like the smartboard that like all right and now we have the um it also is representing that 32ish 3.3ish million dollars per store and then a kid did ask can we figure out the total revenue because they really really wanted the y-axis to be the total revenue overall they didn't like that it was by store and it's like ratios again and they're like no. Um, but could we use this point to figure out the total revenue at least in the US because that doesn't take into account all the international McDonald's in thinking about how they also created conjectures. This is that data analysis part. You were all doing that pretty quickly. You were quick to jump into generalizations. Um, for students, sometimes the buildup of that slow reveal helps them get to those generalizations. It might take a little bit longer, but without even asking, sometimes they're starting to notice these patterns and make sense of it because they've had that processing time that we don't often give them. When we're giving them the full graph at the beginning, if they're not experienced with lots of graphs like that, it can be visually overwhelming and overstimulating. They have to process so much. But here they've been able to process all along. They're coming up with these things. So Oscar had started, this was in my eighth grade class last year, as the number of stores goes up. And then he kind of trailed off. And he didn't want to be wrong. He was always so concerned about being wrong. So Marissa popped in and said the total revenue per store goes down. What do we think of that as a conjecture? As the number of stores goes up, the total number of store goes down. Is that a reasonable conjecture? Are there points that follow that conjecture? Are there points that do not fit within that? And I helped them revise it with slightly more mathematical language. So it was as the number of stores increases, the revenue per store decreases, which is about the same. But again, are there stores where that's not true? And like what's causing them to not fit our conjecture? Because we like our conjecture. Lastly, for the critical literacy part, these were some of the questions that came up. What contributes to revenue? Are employee wages already taken out? They were very concerned about this revenue part where they're like, is it profits? Is it how much you make, but you have to take out your how much you pay the employees? And then people are like, what about you have to pay for like electricity? What's included in this revenue? They were like very confused by that. They don't usually read QSR 2022 magazine. um what does it take for a chain to be included on this list? So that is in the source when that question comes up of like what bias is already baked in here. Well, we know that it's only the top 50 chains in the US, but what would this look like? What if we tried to make this with a different scale for just local chains? Boston's like very into being super local. We like it's our thing. Um, so like lots of things only exist in New England and then Florida where everyone goes to retire. But um, like what would it look like for New England things? What would it look like for like Midwestern for Michigan? Um, and what chain is the most desirable to own? So the teacher and I posed out the question, which one would you want to own? If you were going to go into a franchise, which one would you want to do? And everyone was like, Chick-fil-A. And then one kid was like, no, because they can't work on Sundays. And I'm like, that doesn't seem to be a problem for them. But whenever the kids like have it came there was another graph that had a Chick-fil-A in it and the kids were like Sundays and I'm like got it we we know. Um so as we were discussing what chain is the most desirable to own. I had looked up some articles about Subway and we discovered that there's actually a huge social justice component hidden in this graph. So Subway got too big and franchises paid a price. There's too many Subway stores. You know how you go in and there's not anyone there. There's too many of them. They're not really like you know how Starbucks will like do a lot of marketing research before they move into a place like Whole Foods does the same thing. They kind of follow each other around trying to find the next pocket of wealth. There's just too many subways for them to be super profitable. Sabotage meatballs. The wrong soap says like it's all crazy. However, I also found this a lot of YouTube videos about reasons not to buy a Subway franchise. It's like, why are they having to convince me on YouTube? That's very curious. From Franchise City, if you guys want to look up that YouTube account. Um, and this was the most striking. Subway exploited immigrants and victimized franchises. Bombshell lawsuit alleges. It is much cheaper to buy into a subway. And they actually target specifically immigrants from the Middle East and India to try and purchase these subway not so profitable franchises where they want to get you to like start your own. So here's from that article which was from the New York Post in 2021. Subway is allowing its BDAS to profit off the backs of minorities, Indian-Americans, and Indian immigrants who oftentimes have invested their entire life savings. And here's where some of that Subway on average charges $15,000 fees to open up a new store. And McDonald's and Burger King will charge around 45,000 for franchise startup fees. But that's just the fee you pay to the company. That has nothing to do with getting the lease, nothing to do with making everything look like a McDonald's. You know, they all look the same. Um, this is just like you pay to get the name and then you have to pass all their inspections. Um, so way cheaper to start a subway and probably easier to get that aesthetic too. Um, and so given that they are targeting specifically immigrants, um, this then started to feel much more insidious to the kids. Um, so we were able to actually have then a discussion about some of the quotes from this article and like how they felt about it. Quickly, I want to give another example from a graph. So, critical literacy tends to be the like the toughest part to teach of graphing because when I've done projects where kids like have to come up with a question and collect all the data, they're often not attuned to the kinds of bias that can be embedded in there. Um, and the only way to really get them really good at it, I find, is to actually have them doing things like this more regularly where they're thinking about bias and they're thinking about how this is in. So this is the graph that we had done. At this point they had noticed the green lines were always higher, the yellow was always the lowest, black was always in the middle. And they had done some estimating about the different values of the percentages. And then we revealed this. Black is all participants, yellow is men, green is women. And they were like interesting. The women are always the highest. The men are always the lowest. And then a kid brought up, well, but what about non-binary people? They aren't included. So, in fact, if you're thinking about who's not included here, you can see non-binary and other genders are not included. How would that change the data and the shape of the data? Kids are not included. Do adults have one opinion and kids have another? We'll never know because they're not included in this data. And as we were starting to predict like, all right, so women were always the highest bar. What if non-binary is actually going to be even higher? Oh, but what if they're higher, but then the black bar is the average of all the participants? Will that change the mean? Would adding other genders change the mean? Or would it not even make much of a difference because there's like a thousand men and like 990 women and then like 20 non-binary people and like this is just not enough to change a mean. So, that was all the conversation in the middle of it. We don't even know what this graph is about yet. And then we reveal this. Well, it's actually these countries, Canada, US, Germany, and Russia. And in fact, I'll show with you that the headline of this graph from Statista is Canada is the most comfortable with female leaders. Well, that's only four countries. How do we know for certain that Canada is the most comfortable with female leaders given that there's only four? Um, and I think the kids that said European, they had focused on that. These are not European countries. Um, they're white though, so I think they were like fixated on like the western. You see, I wrote Western. Why these countries? But like there's a lot of countries missing, right? And I wonder what the data would look like for that. And is this just anti-Russian propaganda? We could not tell. So I did find this graph. My husband is Arab. So I went to Arab barometer and found that this was the graph for a woman can be head of state. You agree or disagree. Look at Lebanon. Pink is the men, purple is the women. That is noticeably higher than Canada. And in fact, all of these are way higher than Germany and Russia. And so I will say there's a little bit of nuance to this. The kids were first of all they were like what? We did not expect that from the Middle East. These are all Middle Eastern and North African countries. And I was like whis well I don't know what to tell you. That's maybe your own bias. However, this does say strongly agreeing or agreeing. And this one says very comfortable. So we can't fully compare these two data sets because different questions were asked and the representation slightly different. However, it can give us a general sense that like maybe we're not the only progressive people in the world, especially given that like half of America, like this is the women 59%. Um, and that I feel like these are the opportunities in math class where you can really get deep into the social justice work in a mathematical context where you're making your arguments based around mathematical information and we're thinking about percentages. This says how many people across 11 countries what is the sampling technique like often I get this question where can these fit into the curriculum I don't have time to teach this um sometimes we do it to introduce topics we'll do it to introduce percentages or somewhere around the beginning of the percentage unit exponents logarithmic scale there's a couple I've tagged with logarithmic scale um that one that we just did we use to introduce bariate data And it was really interesting. I've done it in collaboration with other content areas. I'll go into a social studies class and do a slow reveal graph to like launch a social studies thing in the context of data. You can do it in data statistics unit also like you can do it like oh there's a holiday coming up and I need to change my schedule a little bit or like there's some current events that like this is a way to ground that current event issue in mathematics. You just have to be very mindful about like classroom community and culture and like talking about difficult topics or like when I was teaching sixth grade, it's so annoying when like the sixth grade sections are like out of sync with each other. Let one of them do a slow reveal graph while the other ones are like finishing up the other lesson. Um, but before we conclude, I did want to say that I know sometimes people can be uncomfortable with that beginning stretch of the still real graph where it's so ambiguous, but I think it really does invite discourse. It gets kids thinking. It's a safe space and everyone's going to have to revise their thinking. No one looks at the initial graph and is like, I know exactly what that is. It has no axis label. It has no title and I know totally what that is. unless they've done that one before. But it's it's a really fertile space for mathematical discourse and argumentation. So this first of all, I just like this picture. This was I was doing a slow real graph in a third grade class and the kid was like um because like often even in secondary I can almost always get every kid to say something out loud. You might have to be a little judicious with how you're calling on kids, but you can usually get people to say things, especially in those early stages. Um, so it's like the enthusiasm, the mathematical disagreements, like making the argumentation and justifying your thinking. And these are all really important elements of this routine. Again, this routine is not the only thing you can do to promote data literacy and work on it with students. And in fact, I do other things as well, but these are ones that like even if you just do it a couple times a year, they catch on to the routine really fast and it's fun. And you can see here, these were some fourth graders who were running up to the board. Um, they had done this graph that was about, this was actually in their social studies class, and we were introducing like the start of slavery. Very dark topic. This was a super dark graph. And this was about the Tyino population in Puerto Rico, the indigenous group, um, and how there had been about three million of them until European settlers arrived and then not so much. And because we had spent so much time, they had noticed that this does not actually hit the bottom. It kind of looks like it's plummeting towards the bottom, but it kind of stays stable here. And they wanted to know so badly what that number is before it was revealed that they were running up with like rulers to be like, "All right, so this is a million." And we actually got super lucky. It was 10 inches to go from like 1 million, like zero to 1 million. They were like, "Okay, and it's about an inch." They were like trying their best to like figure it out. Um, so as we're thinking about what future do you envision for today's students, I think you can facilitate a graph in ways that will like lead you towards this or towards this. You have to be really careful with your questions and like dealing with the ideas that come out. I have had ideas come out where you're like that's not good. But I think that all of these things are contributing to helping us create students who are going to go off and be better adults. So we're thinking about helping students to make decisions using data, thinking about their graph skills, their data literacy, and critical literacy. and that that makes the world a better place, which I know sounds so cheesy, but like it I've seen kids really grow and develop and get really really emotional and mad about graphs in ways that I haven't necessarily seen in the past. Um, and I remember launching one with sixth graders and one of the kids, we' done a couple of them at this point, she was like, "Is this going to be another really sad graph where I want to cry?" I'm like, "No, maybe. I don't I don't It depends. And like they're just not used to necessarily getting so emotional in math class, but like even that one that looks innocent about like the subway. When you start to realize that like one of the reasons there's so many of them is because they're like targeting and manipulating immigrants who are spending entire life savings because they don't necessarily immigrants are not less smart. They don't know the American system. That's not fair. So, there's a lot of things to get not get upset about in math class. So, in our last like minute that we have here, I did create a website. I did want to also credit friends that have helped me with this along the way. So, while I made the website and I kind of like made this into a routine, um my friend Brian Bushart was really instrumental in this. Dan Meyer and Casio Wedkin and Heidi Fessendon all helped out. Um, and so what it looks like now is that on the website silvergraphs.com, I post lots of different graphs. Sometimes I make them, sometimes they're sent to me from like literally around the world. People from Australia tend to like do more. I've got Sweden, all sorts. Um, if you click on a graph, this is just one of the more recent ones. If you click on the first part, it will show you the graph. And you can see I actually write in questions to make it particularly friendly, but you can ask whatever questions you want. These are just some ideas. Um, because a graph does tell a story, so at least this like helps you. This is the story I made. You could do a totally different order of reveals on your own because you want to tell a different story. Um, I list the visualization source. That's where the graph visual is from. The data source again is different. That's where the data comes from. and that there's bias in decision- making all along the way. And then often I'll list like how this connects to sustainability. Here are some texts that like bring out the critical literacy component. And you can also sort by if you go up here to classroom resources like if you want all of the scatter plots, here's all the scatter plots. Or if you want like all the social justicey ones, here are ones that are more likely to be social justicey like nepotism. That one's an interesting one. And that one is actually about logarithmic scales. So it's really fun. Um so this is the site you can go on here at the bottom you will see hold on let me just restart that slowregraphs.comvsu. I've made a landing page for you where it lists just like a couple introductory resources, either articles or blog posts, some short videos um to help you out. I also put the slides up there and I also put a quick survey. Um the survey is just basically like what's one thing you want to remember from today? and it emails it back to you so that you don't have to like try really really hard to remember something that like you can come back to this in a week and be like, "Ah, that's what I liked or that's what I want to remember or that's a question I had or I hated that don't do that." Um, whatever it is you want to remember, you email it back. And with that, I want to say thank you so much for coming. We cut it again very close, but we did it. I had one minute left. So again, my name is Jenna. This is my email and my Twitter, which I know is dying, but I still really love it. Um, this is my blog. I don't just write about graphs. I actually usually write about lots of other things that are happening in K through8 world in my classroom. Not my classroom, but like all my colleagues classrooms because I work with the whole school. Um, and then slowrevealgraphs.com is the website for that. All right. So, thank you so much