Webinar Date: March 23,2013
In this webinar, Dr. Tim Hudson shares insights about leveraging technology to improve student learning. At a time when schools are exploring “flipped” and “blended” learning models, it’s important to deeply understand how to design effective learning experiences, curriculum, and differentiation approaches. The quality of students’ digital learning experiences is just as important as the quality of their educational experiences inside the classroom. Having worked for over 10 years in public education as a teacher and administrator, Dr. Hudson has worked with students, parents, and teachers to improve learning outcomes for all students. As Curriculum Director at DreamBox Learning, he provides an overview of Intelligent Adaptive Learning, a next generation technology available to schools that uses sound pedagogy to tailor learning to each student’s unique needs. This webinar focuses on how administrators and teachers can make true differentiation a reality by focusing on learning goals and strategic use of technology.
TH: Hi everyone. I’m Dr. Tim Hudson with Dreambox Learning and I really appreciate you joining the webinar today about “Intelligent Adaptive Learning: A Powerful Element for 21st Century Learning and Differentiation.” I’m really excited that you’re here because we’re going to talk quite a bit about learning and also share with you a little near the end about Dreambox. So, thank you very much for being here and we’ll get started.
A couple of webinar tips that you can see there, the bulleted list. Um, some tweeting there at the last bullet, about the CE certificates, how to maximize your screen. So take a quick look at those to make sure that this webinar works really well for you today.
So, I am the Senior Director of Curriculum Design at Dreambox Learning. We do Elementary Mathematics Software on an Intelligent Adaptive Engine that I’ll you more about later but I did spend 10 years, over 10 years in public education as a high school mathematics teacher and as a K–12 Math Curriculum Coordinator, did some strategic planning as well. I’ve consulted for Authentic Education, that’s Grant Wiggins’ company. He’s the author, one of the authors of “Understanding by Design” and “Schooling by Design.” I’ve got my Doctorate in Educational Leadership and co-authored this book here from NCTM about Math Intervention Models. I’d encourage you check it out if you’re at all doing any work with math intervention. I don’t get any royalties but there should some ideas in there that, that you would find quite valuable. So, that’s a little about me.
And really our question, main question today, our essential question: “How can we leverage technology to improve student learning?” And as you probably read in the description, “Flipping Classrooms” is popular idea today, “Blended Learning” as well. A lot of schools that use DreamBox are using Blended Learning models. There’s a lot apps, iPads, tablets, just crazy amounts of different software applications out there to look through and to try out. Online videos, of course, teachers making their own, you can you know find those YouTube channels everywhere.
So, what I like to start with is when we talk about leveraging technology for learning. We need to talk about some learning outcomes and as a first question you know, which schedule would you say is better as a high school teacher, some high schools use block scheduling where you can see there are eight courses a semester, four courses a day, every class meets every other day for 90-minute periods. But then there’s the more traditional schedule where you have those same eight courses for the semester but you have eight courses for the day. Every class meets every day, there’s 45-minute periods. It was always interesting when schools would take a look at block scheduling. The school I taught in actually had a hybrid but when they would talk about these things, you can’t make a judgment call as to which of those schedules is better just based on this information. Because the scheduling is secondary, really what is happening during class, and that is the key thing. We need the work in the classroom to drive the schedule. We can’t know how to schedule learning until we’ve decided what we’re trying to accomplish and know what sort of time structures we need to get there.
So, really our question is, okay: “Scheduling is a means to what ends?” Similarly then, if you’ve done any work with Blended Learning, you perhaps read Michael Horn and Heather Stakers’ publication from the Interside Institute about different Blended Learning Models and here’s two snapshots, there’s the Flipped Classroom from the Stillwater Area Public Schools, and you see there’s practice and projects at school, then computer work at home, online instruction, and content. And they have another one called the “Enriched Virtual Model” used by Albuquerque E-cademy, where you have computer work at home, online instruction, and content and you see how the school is set up. There are some computers. There’s some face-to-face supplementation. And so kind of the same question applies as to the block scheduling you know which of these models is better. And just judging based on this information, you can’t actually reach an answer because the key questions, what’s happening in those face-to-face interactions? What’s happening in class? And what’s happening on the computers also a key question. You know these both say online instruction and content but that means, could mean quite a lot of different things.
So, again, same as with the scheduling, blending is a means to what ends? So, as was in the description, you know the quality of digital learning experiences is just as important as the quality of classroom learning experiences. And that’s a key thing moving forward in the 21st century, there’s a lot of technologies and software out there to help reach and teach all learners, that’s a key benefit of technology—being able to connect with more kids great learning, connect more kids to great learning. But digital content quality’s just as important as classroom content quality and classroom learning experiences.
So, I want to start today with a couple of quotes from Schooling by Design. If you’ve done any work in your schools and districts on, you know setting your school’s mission and things, Schooling by Design is a fantastic book to reference. And one of things they point out, I think very insightfully is that contemporary school reform efforts typically focus too much on various means, structures, schedules programs, PD, curriculum, and instructional practices like cooperative learning. And based on a couple of slides I’ve just shown, I would also add Blended Learning, Flip Learning, iPads, hardware. A lot of times, I remember being at my high school, I was the representative for the district technology committee and my assignment was to go back to our 140 high school math teachers and find out whether a one-to-one laptop initiative would be a good idea would be successful. And in working with the department leaders, and talking to the teachers, we realized the capacity really wasn’t there to know exactly what to do with them.
So we decided let’s not buy the hardware and then figure out um what we’re going to do with it. Let’s make sure that we’re buying hardware to fit an actual classroom need. So we need to make sure we’re focusing on the means, clarifying what the means are and clarifying the ends, because as they say in Schooling by Design, these reforms like Flip Learning, like Blended Learning. These are the fuel for the school improvement engine. These are things we should be trying, risks we should be taking—calculated risks to reach all learners and help all students succeed. But, we need to be sure that these aren’t the destination. These aren’t confused with the destination. The destination is improved learning.
So we’re playing curriculum backwards if you’ve done with any work with Understanding by Design. It’s uh, they would tell you it’s not new ideas, it’s simply three things: what do you want, what are your desired results, how will you know kids have learned it, and what are the learning experiences and instruction that are going to help you make sure all students can show evidence of their learning toward the goals that you want. Too often, the reason they call it backwards is because too often, like I did my first year of teaching, I thought a bunch of things and then I looked at what I taught and wrote a test and who knows if I actually hit the goals that I wanted. It’s more of a, they call it “teach, test, hope for the best.” But we don’t want to do that for our kids. They deserve better.
Another way to ask these questions is to, say, “You know what you want students to accomplish? What are your learning outcomes? How will you know that they have achieved it?” And then here’s where the technology comes in. Here’s where the flipping as a strategy or the blending as a strategy, or apps, iPads, DreamBox, what technologies can help students meet these goals? That’s planning backwards.
The first stage in the design process calls for clarity about priorities expressed as achievements. We need to talk a whole lot about what we expect students to achieve. And we need to take a look at the assessments, how will we know this is actually works. Too often schools that switch to a block schedule or stick with the traditional schedule, they haven’t necessarily put into place the assessments that will let you know whether one schedule is better than the other. So let’s start with a poll. As we talk about desired results and we talk about the 21st century learning, this first poll, if you are familiar with WolframAlpha, have you ever used it? And we’ll see the live update happening right now.
All right, give me about two more seconds and I’ll close the voting. Numbers are still coming in. All right, I’ll go ahead and close it. So it looks like we have, nope. The 71 percent of you have never heard of it and five of you have used, use it regularly, which is pretty cool. A couple of you used it a few times but the majority of you, over 80 percent of you has not yet, well, yeah, haven’t ever used it.
So let’s go ahead to the next slide and I’ll show you some things that WolframAlpha will do. So with WolframAlpha, you can type in something. I taught high school math. I would assume most of you took high school math and you can type this question: “What is the equation of the line perpendicular to y = 2x + 1 through the point 2, 4?” I’m sure you remember fondly all of your high school math days doing problems like this, graphing them by hand. Well, now you can type that in into WolframAlpha and the output it gives you is this. It’s interesting how you can take a look at the input, the “normal line” is another word for perpendicular and you see how it interprets your input. Then it gives you the equation that probably took you quite a while to figure out by hand. And it also shows you a graph. Interesting thing about the graph is those lines aren’t actually perpendicular but it tells you at the bottom that it’s not necessarily to scale. But in any case, something that high schoolers that have been spending a lot of time doing on their homework assignments, one through thirty evens kind of thing. WolframAlpha will do it fairly instantly.
But it’s not just high school math anymore. As of about a year ago I think, they added this functionality in. So if you’re a first grader you could type in: I have two cookies, you give me three cookies, how many cookies do I have? And WolframAlpha interprets what you said, “You give three cookies to me.” That’s some interesting language parsing there and it tells you, “I have five cookies.” And it shows you: 2 + 3 = 5. It even shows you a representation of it using what they say are manipulatives; using those blocks. So it’s critical that we know about things like this that are out there in order for us to help define what our goals for student learning should be.
So, another poll, now that you’ve just learned several of you, 70 percent of you just saw WolframAlpha for the first time. Here’s a question: “If computers like calculators WolframAlpha can solve math problems so efficiently, why do we drill our students in answering them?” There’s four multiple choice actions there. Feel free to click “Other” type here in the chat as you can see there.
All right, it’ll take about two more seconds. And I’ll close the voting now. So, 64 percent of you saying its evidence that students understand important ideas. Several of you, you know you won’t always have access to computers, you know these are ideas that I threw in sort of is as thought-provoking possibilities, good practice, just some things you have to know. And this is really where we have to be clear, if you’re teaching any math course, we need to make sure kids know at the end of that course exactly what the learning goals were. It’s amazing how many adults can’t define what mathematics is, can’t define what algebra is. And yet they spent 13 years of their life doing mathematics.
So we actually have an answer to this question from David Bressoud who writes a monthly column for the Mathematical Association of America. And back in 2009, he wrote an article about WolframAlpha where he actually posed this exact same question that I posed you just now: “If computers can solve math problems so efficiently, why do we drill our students in answering them?” His answer shared by many of you, “Because there were important mathematical ideas behind these methods and showing one knows how to solve these problems is one way of exhibiting working knowledge of these ideas.” “Working knowledge of the ideas” I think is a key phrase. He went on to say this, “The existence of WolframAlpha …”—I would include in calculators as we have been dealing with those for even longer—“… does push instructors to be more honest about their use of standard procedures executed by memorizing algorithmic procedures.” And here’s really the kicker, “If a student feels that he or she has learned nothing that cannot be pulled directly from WolframAlpha, and then the course really has been a waste of time.” And I think for us as educators that should be a compelling question: “Is this course a waste of time if students can get something so quickly from a calculator, from WolframAlpha?” We really need to think about that and I actually, I facilitated our new teacher induction when I was the Math Director in my school district. You know we’d always have about ten, ten to twelve new middle and high school math teachers, and I would pose this question to them, walk them through this exercise, What do we offer students in our classroom that they can’t get online for free?”
So, let’s talk a little about some actual math problems that a calculator could do but what would be evidence of important mathematical ideas that we could glean based on how students solve it. So, here’s a problem: 3,998 + 4,247. And I’ve been giving parents; I’ve been giving teachers, students, different audiences this problem for several years just as a way to start the conversation about what number sense is, about what it means to be fluent, to think mathematically. And it’s a, it’s a pretty fun problem because a lot of times, maybe as you’re looking at it right now, maybe you’re thinking, “Those numbers aren’t even lined up properly. They need to be on top of each other. So, I don’t know if I can solve it in my head. I need a pencil. I need to stack them so I could see the columns.”
Usually, there’s a few folks, probably some of you out there, are looking at the first number and realizing, “Wow! That’s really close to 4,000.” And so a lot of times people say, “Oh, I’ll just round that up to 4,000 and then I’ll also take two away from this. Now, I’m just adding 4,000 plus 4,245. That’s a pretty simple problem, pretty simple to do in my head. I don’t need a calculator, I don’t need a pencil because I can use the numbers and break them apart.” That’s actually a strategy called “Compensation,” and usually there’s people in the audience who haven’t ever sort of learned that idea or realized that idea that you can break those numbers apart. But that’s really evidence of fluency. And so for younger students, if we want them to look at problems and not immediately grab a pencil or a calculator—we want them to have good mental math strategies, we have to start working on those ideas earlier. So even something like: 288 + 77.
And then even when the kids are amazingly young, even when they’re five years old, six years old, starting to learn addition, you know we engage kids usually in thinking about: 8 + 7, as doubles plus one, doubles minus one strategy. But we need to make sure that we’re also teaching them how to break seven into two and five or even break eight into three and five, but the goal being to make ten, ten’s a friendly number to add. Just like in that first problem, 4,000 is a friendly number to add. And we want kids to be able to break numbers apart even when they‘re really small numbers—especially when they’re small numbers, because it’s about transferability. It’s about students transferring an idea. If you only have one strategy for solving this first problem up here, you’re thinking, “Oh, eight plus seven, there’s a five in the ones place of the answer,” and you’ve got a whole lot of carrying ahead of you. And I would say that if you have a fourth grader who doesn’t look at these numbers: 3,998 at 4,247 and think oh, may I can break those apart then perhaps we’ve only trained to solve problems in the one way which will not serve us for the 21st century.
So you know these problems bring up great mathematical questions, you know what is a good strategy. There are a lot of different ways to prove the Pythagorean Theorem, a lot of different ways to prove different mathematical theorems. And what are good strategies? What are elegant proofs? What is fluency really mean? How is fluency learned? Maybe you know, just as important of a question and can you get this from WolframAlpha? I would imagine students who walk out of class having learned “Compensation” and understanding it. They wouldn’t see WolframAlpha as necessarily being where they would go to get that because it is not just about information that I’m getting, it’s about a way I’m thinking about numbers.
In Dreambox, we teach this idea of “Compensation” using what we call the “Compensation Buckets.” So we have first graders who see a problem like 29 plus 72 and for a lot of adults, their eyes immediately gravitate to the nine and the two to start probably doing, probably doing the addition algorithm. So you’ve got a one in the ones’ place. But, using these “Compensation Buckets” where students intentionally can’t count the counters that are in the buckets, they can use the plus and minus arrows to adjust values to find friendly numbers. So, oh, that’s really the same thing as 30 plus 71, which is a much easier problem to solve mentally. The key is students are developing number sense. They are able to look at one problem and adjust the values to make it a friendlier problem. That’s what good mathematicians do. And every student can be doing this on their own, thinking through it and arriving at the answer, developing their own fluency.
This brings us to a learning principle that’s highlighted in Schooling by Design that is probably my favorite learning principle from their work. There’s actually two of them. An “understanding” is a learner realization about the power of an idea. You know the idea of those compensation buckets, that’s helping develop understanding about how numbers work. It’s not just one skill or one strategy. It’s about having multiple strategies which is evidence of understanding. And then this next piece really describes what DreamBox, what we do at DreamBox. Understandings can’t be given. They have to be engineered so that learners see for themselves the power of an idea from making sense of things. As teachers, we love those light bulb moments. We love it when an idea finally clicks. And I think sometimes in mathematics, especially, we tend to think that those light bulb moments happen when kids realize, “Oh, I have to do the same thing to both sides of the equation,”or “Oh, so I have to carry a one now.” And we often think of those light bulbs as being sort of catching onto a procedure as opposed to really having a deep understanding of how numbers work. And you know ideas, things like that don’t always happen in eureka moments. It’s really a long process of thinking about numbers, of playing with numbers, and figuring out for themselves is a key thing. So at DreamBox, we don’t, you know explicitly tell students how to do a particular problem, we build manipulative so they can make sense of things for themselves. And I’ll give you several examples of that later.
So what do you remember about math from when you were in middle and high school? I give audiences this question a lot because we sometimes forget how things were for us when we were students, now that we’re adults. And there’s a great book, if you haven’t ever picked it up or read it, I would highly recommend it. It was published in 2001 by Silver, Strong & Perini called “Teaching What Matters Most.” It’s published by ASCD. And in the course of the work that they’re doing, they talk about authenticity, they talk about rigor, they talk about thought, and they’ve got great rubrics and things, fantastic book. I can’t recommend it enough. But, there’s a great quote in there that they have from fifth grade teacher in New York, and as you think about your own middle and high school Math experience, what this teacher said was this, “I had a lot of good people teaching me Math when I was student, earnest and funny and caring. But the math that they taught me wasn’t good math. Every class was the same for eight years. Get out your homework, go over the homework, here’s the new set of exercises, here’s how to do them. Now get started, I’ll be around.” And as I’ve shared this quote with many adult audiences over the past several years, there’s a lot of nodding going on, that this really does represent a lot of people’s experience with middle and high school mathematics.
And it kind of reflects this sort of typical teaching cycle that I think a lot of us are familiar with, that you have some whole class or some small group instruction. And there’s a key distinction between instruction and learning experiences, you might have noticed that on the slide about the “Understanding by Design,” 3 stages, that there’s learning experiences and then there’s instruction, and both are instructive in terms of causing student learning, but they distinct and different, and we’ll talk about that more.
After this whole class or small group instruction what that fifth grade teacher I just quoted, went and did some guided practice. That’s pretty much how it went. After that there would be a whole class assessment, maybe it’s a quiz, maybe it’s a teacher looking over a homework assignment, maybe it’s a unit test. And then after that assessment, you know, one of two things happens, maybe the teacher goes back and says, you know what, folks didn’t get this. So, I’m going to have to go back. I’m going to use that information, I’m going to go back and do some more whole class or small group instruction. Or depending on the constraints, time, or whatever, maybe the data are used summatively and it’s you know, the test is done and we have to move on.
And I would call this in a lot of ways, teaching as “Content Delivery.” That’s kind of what that teacher was alluding to in the quote that I mentioned, that, the teacher, that’s the whole stage on the stage idea where I’m trying to transmit information to students and that’s exactly how I taught early in my career in the classroom. My frame of thinking was this, I thought, you know, I understand a lot of math and I understand how to teach it. My job to help kids understand things that I understand and explain it in ways that aren’t boring; in ways that are fun.
And then I came across an elementary teacher’s reflection where she said, the biggest challenge I have every day is hearing so many great ideas coming up from my students, coming out of our discussions in classroom, and as a teacher, the hardest part for me is choosing which of those ideas I should follow through with, I should highlight, I should emphasize, because those are the ideas that the whole class needs to hear and really think about. And that really shifted how I thought about my own classroom practice, where I wasn’t thinking, I’m just delivering information to students.
So, now it’s time for another poll. How old were you when you decided whether or not you were a math person? And I bring this poll up now because as you think about your own experience in middle or high school mathematics, you may have decided early on whether or not you’re a math person.
So, I’ve got half of the audience. Yeah, over half of the audience right now is saying by sixth grade you kind of knew. A few folks after college, I’ll go ahead and close the voting now. And can we bring up the next slide.
So, pretty early on, you have your mental models of what math is and whether you are good at it. And philosopher Lichtenberg said, “We accumulate our opinions at an age when our understanding as it its weakest.” So, if you’re a third grader and you’re deciding I’m not a math person. Arguably you’re understanding of what Mathematics was, was pretty weak understanding. All you knew about math is how it was taught and how it was learned. Probably you might have envisioned that you chose to be an educator, you might have envisioned this as how teaching mathematics goes or how teaching anything goes. That’s a real transmission view. I taught high school, so I kind of taught I’d have a bigger audience, maybe a few more stuffed animals in the back rows passed out. Is that a manatee? Yeah, I think that’s a manatee.
But, thinking mathematically, this is what DreamBox is all about. It’s not so much about every single procedure; this is going back that teacher’s quote. It’s about learning how to think mathematically and so what DreamBox does, like you saw with the compensation buckets, and as you’ll see here in a couple, in a few slides, is it’s about kids thinking mathematically. It’s not about telling them what to do and having some steps to follow. It’s about engaging them in the process of mathematizing their world.
So, we have a lot of people who decided, you know, okay I did well in math, but I never understood what I was doing. I remember a hundred of procedures but not a single mathematical idea. So, at DreamBox we build around mathematical ideas and then our Intelligent Adaptive Engine actually responds to the ideas that students are having in real time. So, instead of saying let me show how to do this, now you go do this, can you do it on your own, maybe you need to be shown this again or okay, you know X. We don’t want schooling to merely be content delivery and I see a lot of articles these days where people say teachers do so much more than just tell kids things, and that’s true. And, Flip Learning Models, Blended Learning Models, they’re ways to help elevate and highlight the value that teachers add—that there are things that you can’t get for free on the Internet. You also can’t give understandings this way, all you can show is how to do a particular process.
“Blended Learning,” if you look at the definitions from Innosite Institute report, it talks about a formal education program where at least part of the work is online delivery and part of work is a supervised brick-and-mortar location away from home. And a key phrase here is online delivery of content and instructions, that’s the key, going back to what I said earlier, the quality of the content—the digital learning experiences is just as important as the quality of the experiences in schools. And one thing that they point out as well, Blended Learning and sort of self-directed learning is about time, place, path, and pace, and you can read these things a little closer when you get the recording.
But one of the things they point out learning is no longer restricted to the pedagogy used by the teacher and they’re talking about the pedagogy of a teacher in a particular classroom, because you can go online and sort of experience different pedagogies. But the key thing is that learning is restricted and impacted by the pedagogy used by the online teacher, in the online instruction, or in the designs of the learning software. You don’t get away from pedagogy. Pedagogy is critical. The difference between instruction and learning experiences is critical. So, I’ve changed this graphic just a little bit to show that typical cycle where, okay, it starts with at school, you have explicit instruction and problem solving, then at home you have the practice problems, then the assessment, then, okay maybe you need to be shown X again, and when I come back to school, I’ll be shown that again. And sometimes I worry that Flipped Learning, all it does is sort of just switch this, this first step is still explicit instructional videos with some online practice at home, and then you go to school and have the guided practice and problem solving. And, instead it’s okay maybe you need to watch the video again. I worry that as you look at these two slides. I’ll go back to the other one. If we start both of these models, I’m worried, start with explicit instruction first regardless of where it takes place. So, you think about the compensation idea. What we did with DreamBox, where we have those two buckets. You don’t have to start by showing students here’s how you compensate with two numbers, instead you engage them in a problem solving activity that’s kind of meaningful for a first grader. So, sort of in summary, the benefit of Flipping and Blending is that we’re becoming far more thoughtful and strategic about the use of precious class time. That’s something Grant Wiggins was saying long before YouTube, that you know, if you need kids to be together doing something, don’t necessarily have it a lecture, put that in the library on a video cassette, and let kids access it when they need to.
When kids are together, we should be doing meaningful things that require them to be together. So, that’s a great benefit of Blending and Flipping. But the danger is, I think we sometimes become less thoughtful and strategic about how students learn and make sense of things. We think that we can sort of just explain things better or more optimally, going back to this “Transmission View of Learning.” Just sort of watching on a screen that Y = MX + B doesn’t cause understanding, you can’t give understandings.
We often think if I cover it clearly, they will get it. But we have to realize as they point out in “Understanding by Design” that, “Presentation of an explanation, no matter how brilliantly worded will not connect ideas unless students have had ample opportunities to wrestle with examples.” This is also echoed in the work of “How People Learn,” which was published in 2000 or 2001. And it was, one of the lead authors was John Bransford, who was an early adviser on the DreamBox board. He’s out here at the University of Washington and one of the key quotes from that work. I got a little bit ahead of myself.
As an example of, you know, no matter how brilliantly we word something, I’d encourage you to go to YouTube and just search for “Kid Snippets: Math Class,” if you haven’t seen it, it will be the funniest thing that you see all day. What they do is they have two children that were; one was explaining how to do math to another one, another student. And then they go ahead and have adult actors act out a scene with the children’s voices dubbed over it. It is hilarious, and when you think about, you know, we say things to a first grader like one ten is the same thing as ten ones. That just sounds kind of crazy to a young learner, so to the point that we can’t just clearly word things and hope that kids will understand. And then here’s the quote from “How People Learn.” “Providing students with opportunities to first grapple with specific information relevant to a topic has been found to create a time for telling that enables students to learn much more from an organizing lecture,” so that explicit instruction doesn’t have to come first.
Alright, so, next poll as we get a little closer to talking about how DreamBox Intelligent Adaptive works and differentiates for students. Are you currently working on differentiated instructions in your school or district? Oh, quite a few of you, 90 percent, excellent. So, the majority of you are, that’s great. So, you’re probably familiar with some of the works of Tomlinson and Imbeau, “Leading and Managing a Differentiated Classroom.” They have 10 different bullet points about what differentiated instruction is, and here are four of them. We know that scaffolding is critical. Teachers need to make continually evolving plans. As I look at these things, especially this last point, what does this student need at this moment in order to be able to progress with this key content and what do I need to do to make that happen? That is I believe, wholly unmanageable for a single teacher to do with a classroom of 20, 25, 30, 35 students. There’s not enough time, energy, or data in order for a single teacher to do that, and that’s where DreamBox comes in to help partner with classroom and teachers to differentiate for students in real time.
We have teachers on staff who work with our designers and our programmers to build tools that enable that differentiation on our Intelligent Adaptive Engine, and one thing about differentiation, that’s key to know is that I think our mental models of learning, especially if we think about that transmission view of learning, we tend to differentiate in two wrong ways.
The first is around knowledge, skills, and procedures rather than around ideas, understanding, and complex performance, you know, we might be inclined to say, okay these students are having trouble multiplying fractions, these students are having trouble adding fractions, and often we’re focused too much on the skill as opposed to maybe some of the underlying ideas about fractions, that, if you try to add fraction by adding the numerators together and the denominators together, that’s not really a point of differentiating around the skill. You’re doing something that doesn’t make any sense for, if you understanding anything about fractions. So, we need to think more about the underlying ideas.
And then secondly, we tend to differentiate in response to student knowledge after being shown a skill instead of differentiating around student thinking when they’re solving an unfamiliar problem, or at the point at which they’re forming a conception. So, if I explain to you how to multiply fractions and then differentiate while you’re practicing, that’s a different thing than if I give you a problem in context where you need to multiply fractions and then I watch what you do. The difference is I have no idea really how a student is going to attack the problem in the second instance.
So, in DreamBox, what we do, we have a formative assessment that’s embedded that happens in real time with our adaptive engine. So, for a problem like 29 + 62, there are a lot of possible wrong answers that a student might get. They might answer 81, there could be a couple of reasons why they’re answering 81. They might answer 811. They’re just adding the columns independently, 2 and 6 is 8 and 9 and 2 is 11. Maybe they just make a small arithmetic error; maybe they think it’s a subtraction problem. But, you know, these are just five examples of mistakes kids could be making, all of them requiring different differentiated approaches and all of them are things that are Intelligent Adaptive Learning engine is aware of because our teachers write lessons to look for these problems.
So, conversations we have all the time at DreamBox, as I’m sure you and your staff would as well. How would you score each of these errors? Are some of them more egregious, more troublesome than others? Definitely, yeah, how would you respond to each error, do each of these students who’ve answered, you know, with these different mistakes, do they need different things or the same things? What lessons need to come before or after? What would you do next if you were working one on one with a student who was doing one of these errors? And what I think is one of the most fascinating questions is which of these errors is naturally occurring, that if you had a student who understood numbers pretty well and hadn’t ever been shown how to add numbers, using any algorithm or really any approach, and you said 29 + 62? I don’t really think a student would answer, you know, 811. Kind of like with adding the fraction example that I just mentioned. But, we could have a great conversation about naturally occurring errors and errors, and errors that occur because students are misremembering a process they’ve been told rather than a process they understand.
So, I’m going to real quick go through this slide, there’s a, I would normally have you do it. In the interest of time, I’ll just sort of show it to you. This is a problem where you’re computing with time, you’ve got a bike race and two people have different times and it’s really kind of a subtraction problem, you could add up if you could up if you wanted to solve it, but the key thing is, is that most adults when they take a look at this problem, knowing that they haven’t been taught any process any procedure for adding and subtracting time, that’s usually something that’s not taught, that means, that adults attack this problem with their own intuitive understanding. And, a lot of times adults say, you know what, they’re both right around 3 hours. They use 3 hours as a landmark, so they see that Donald was 4 minutes and 11 seconds above 3 hours and now, I just need to figure out far below 3 hours Keena was and solve the problem there.
But, no adults actually attack the problem by borrowing. Oh, I got to borrow a minute and now I’ve got 71 seconds, I’ve got to do some borrowing here. I’ll just click through these, you know, this strategy for this problem is completely laughable. It’s not appropriate, it’s not good thinking, and it’s something that would be easily confused and confusing to students. But, the analogous problem for our purposes today is 304 – 298. If you have number sense, if you understand how numbers work, you shouldn’t need a pencil for 304 – 298. But I’ve met plenty of third, fourth, and fifth graders who feel they have grab a pencil to that solve problem even though they could use 300 as a landmark.
If you don’t have number sense, any deviation from the known path may rapidly lead to being totally lost. There’s a great research study from 1992, Ann Dowker, about how mathematicians solve problems mentally, and this work informs what we do at DreamBox. So, in DreamBox we have a constant difference lessons where kids use the number line and they make jumps on the number line to turn a problem like, let me go back, to turn a problem like 175 – 137. If you shift it on the number line, it can become 178 – 140, that’s, you see the constant difference there on the number line and then those answers are the same. That’s an element of number sense. That’s a strategy not a lot of adults experience. So, real quick, I’d like to know did you learn the constant difference strategy for subtracting when you were in school?
So, we have a lot of, we have some not rememberings. Totally fair answer, that’s why I threw it in there, but, yeah, definitely over half of you, pretty confident that you did not learn it. I know that I personally didn’t. And so at DreamBox, I mean, we build tools in there to do that because it’s not just about subtracting; how you get the answer is as important as, that you do get the right answer, because when we see how you solve the problem, we have insights into what you’re thinking, into how you understanding numbers, and you will get the right answer, that’s a key thing to do. But, until I see how you’re thinking, until I see how you’re actually playing with the numbers and working with numbers, I’m not a 100 percent sure what you need to do next in order to deepen your understanding.
So, now back to our original question. How can we leverage technology to improve student learning, and so my four points would be this, we need to improve our learning goals, we need to clear about what students need to do that they can’t just simply get from technology. We need a guaranteed curriculum, you know, when kids are absent or you know, you’ve heard of the squeeze cheese curriculum before that kids move from grade level to grade level that guaranteed curriculum as a former math director, I know the importance of that. We need to require student thinking, that’s a key piece, whether it’s in the classroom or on software, kids need to be the ones doing the thinking. Kids need to be the ones who actually doing the doing, they need to be actively making sense of ideas, and that’s how we build DreamBox.
So, this is the DreamBox Pedagogical Design, let’s sort of go back to that flow chart that I showed earlier. But even though it says DreamBox Pedagogical Design, this is great classroom Pedagogical Design as well. Student engages within a context. So for a first grader or a second grader those compensation buckets, that’s a context. There are counters in there and you need to find friendly numbers. If you’ve done any work with Cathy Fosnot where she has contexts for learning, is what her units are called, and they’re great classroom problems that are spread out over multiple days for kids to really sink their teeth into and to develop a mathematical community around. This is the sort of thing that Dan Myer is doing with his 101 questions, his three-act math problems where you give kids a context, give them a compelling problem to solve, something that they can throw out a guess for, and so what happens is students transfer and predict. They make a prediction, they start clicking the buttons to see how the compensation buckets work.
And that transfer piece, too often we think of transfer as the end result of learning, when in reality students are transferring, using their prior knowledge at every point of the learning process. It’s not just a static thing that happens at the end. Then the student receives feedback in DreamBox as to whether their answer was correct or incorrect, as to whether their strategy was optimal or not optimal and then our intelligent adaptive engine differentiates for students based on what they’ve done, and then once students are outside of DreamBox, they use their number sense to do work in the classroom as well. And this is how we engineer things for realization, so if you recall that quote that I shared earlier. Understandings can’t be given, they have to engineered so that students realize for themselves the power of an idea for making sense of things. That’s what DreamBox is about.
So, here’s an example of “Division of Remainders.” Problem is 5,916 divided by 4. So, we give kids these gumballs and what we called the bag-o-matic machine that’s going to bag the gumballs for them. And you can see there’s a lot of gum balls there, you can’t count them individually, that’s of course purposeful, and students are able to pack gumballs into bags of four however they would like. So, my next poll question to you, for 5,916, how many gumballs would you pack first? Again, the problem is 5,916 divided by 4. I apologize for not putting that in the question, that was, I meant to do that.
So, as you can see we have, you know, we have dealt with educators on this webinar, all of whom have some different strategies. I’ve thrown a couple in there, the, you know, most widely used choice is 4,000 gumballs into a thousand bags. There’re, you know, other options as well. I’m going to go ahead and close the voting there.
And as you see here, here’s how DreamBox works, if a student decided to do 400 divided by 4 first, that bag-o-matic animates automatically and a student can see sort of what sort of progress they’ve made toward that answer. You see there’s four red dots there on the machine; those represent how many moves that they have left. We don’t just let, want them to say, 40, 40, 40, 40, 40. They have to be sort of limits so that we know that they’re thinking optimally and we can engage them in better thinking by limiting the number of moves they can take.
But, for this problem, if you were doing long division, you know, the first thing you would do is 4 into 5 is 1 kind of thing, but before you get there, it’s helpful to understand division even better and to say, you know what, 16, that’s a good first step, so that I get a round 5900 that I have to then pack. So, all of these are valid first steps that the student can make sense of, and our Intelligent Adaptive Engine responds appropriately. I’ve shared an addition problem and a division problem, and a subtraction problem and I wanted to throw in some of multiplication as well so that you understand how DreamBox works in terms of multiplication.
There’s the Ma and Pa Kettle math video, again, go check it out on YouTube, if you haven’t already where he’s able to prove in three different ways that 25 divided by 5 is 14, pretty funny video from awhile ago. But, so in DreamBox and students when they’re learning multiplication in third grade, we don’t just focus on 6 times 6 equals 36. We want students to also know that 6 times 6, is equal to 2 times 6 plus 4 times 6, if you want students to be able to use the distributive property, if you want them to understand partial products and the multiplication algorithms. These are understandings that need to be developed early, just like with 8 plus 7, breaking it into 10 and 5. So, in third grade, students work that out on the number line, but they also work with it in the array, because multiplication is scaling and representing multiplication on the number line has its limits with fractions. So, students use an array and they have to find 9 times 11 and compose that from a 9 by 9 array and they use these rulers here to build their own second partial product to complete that array.
Then in later grades, 12 times 24; students unfold a virtual map—again like with the gumballs creating their own strategies to make sense of the idea and we have some concrete representation there as they drag the map outward and keeping track of all of their work. So, that they come to realize that, you know, the fewer helper problems, the fewer partial products I make the easier it’s going to be in the end when I have to add them all up.
We also have what I think might be the only two-dimensional open array digitally that anyone’s built where we ask students to build 38 times 78 and they’re able to see, okay, well, that’s 10 times 10 plus 10 times 10. These are things they’ve constructed, that they have created, and we have very specific learning progressions that kids go through as we limit the number of times that they can drag out these arrays to get to the answer. But we allow them to start the learning by making sense of it for themselves, that’s the key thing, and even in this lesson, we do the multiplication for them, because we want them to focus on the bigger idea, which is the partial products.
That moves through a couple of other lessons eventually to get to the multiplication algorithm where we have students first make an estimate of what 2,552 times 18 would be, then they solve each of the partial products, Part B there, and then they add those partial products to get the answer, and our adaptive engine knows when students are making mistakes on the estimate, making mistakes on the partial products and making mistakes on the sum there at the bottom. If you get problem A, and problem B correct, but you don’t get problem C correct, it’s not necessarily a multiplication problem, it’s an addition problem, and so, we would recommend for students that they go back and work on the addition algorithm lessons.
With multiplying fractions, you can play this one online at our website where students, we again use the array because a lot of times students don’t have a mental model, a conception of how fraction multiplication works visually.
So, DreamBox focuses on kids’ original independent strategic thinking. We start with an engaging learning experience with a context. We give kids accessible problems, they make a prediction, they take a guess, they get feedback on their guesses, and that data, and those data inform what students get next. It’s self-directed, it’s coherent, there’s connected paths, and it’s really helping differentiate in ways that really support classroom teachers. The lessons, the practice, and the assessments look identical to students. We don’t just have simply banks of practice items. The numbers that we choose in lessons are very specifically tailored to helping kids realize things about mathematics. And kids don’t really need prior instruction to engage in the lessons. If you’re a fourth grade math teacher, and you have some students working at the second grade level and some at the sixth grade level, some below grade, and some above grade level. It’s incredibly difficult to help connect those kids with those resources given limited time, data, and energy.
So, what DreamBox is able to do is work at every student’s individual level and making sure teachers can be confident kids are engaging with the ideas conceptually, and it’s not just, if I have a fourth grader who knows fractions pretty well that the software would just say for dividing fractions, oh, just invert and multiply, now go practice it. In DreamBox, we don’t do that, we engage students in sense-making, so that they come to realize things rather than just be executing some skills. So, DreamBox works kind of like this, you have a unit pre-test, again it looks just like a game or a lesson to students, and if they understand it then they don’t have to complete the lessons associated with it. But, if students don’t pass that unit pretest with proficiency, then a series of lessons are unlocked for them. And it’s not linear, there are multiple units, unit pretests open to students at any different time, because you can be working on second grade place value and early skip counting, and maybe even early fractions at the same time, if you’re a third grader or a second grader. So, it’s not linear and there’s always multiple options presented to students.
This is what our primary engagement environment looks like, you’ll see children have an avatar and there’s three lessons that they can choose from. Our lessons in early grades look like this, we have students build number like six using these counters, and the fact that we restrict the counters that they can choose from helps us capture their strategies. We eventually want them to build six optimally which is one group of five and one group of one, because grouping is an important idea. We use quick images so kids look for structure, so this card displays for a short time, students have to figure out what number is and a lot of times kids get frustrated because they say the card isn’t up there long enough for me to count them, and that’s actually the point. The goal is not for you to count them, the goal is for you to see that you used this manipulative quite a bit and there’s 10 reds and 10 whites, and if you see 3 missing, that’s 17, that’s [“subatizing”] type operation. We don’t get to be looking for structure, if you’re in a state that is implementing the Common Core.
Here’s our intermediate engagement environment. It definitely looks different; kids have eight choices available to them. Some of them are practice, some of them are some concept videos that help kids think about ideas in a fun way. We have fluency games, we have, there’s that multiplication algorithm lesson you saw, as of, coming up very shortly in the next couple of weeks. We’re going to be releasing some lessons actually at the seventh grade level, even though we’re a pre-K to 5 product, where, you know, want to make sure that our fifth graders who are working at above grade level, have plenty of headroom and where kids are going to be adding and subtracting negative decimals. So, actually understanding why a negative minus a negative is a positive, that sort of thing.
In this lesson this is a sequence challenge, you can see here where actually a student can subtract a negative in order to make a positive move on the number line, so in this particular lesson, kids are moving a pirate ship to help find some treasure and they click the addition or the subtraction buttons in order to create their own equations to make some moves on the number line. We also have a really fun miniature golf game, it’s for fluency, but as you look as that score card there, students start with their, you see their patter there, they start with the ball on zero and they have to hit negative 8.3 and you see on that score card not only is there a mix of positives, and they, actually, oh, those are all negatives. Not only is there a mix of decimals and fractions, but, also, you have this windmill here, so you have to use your estimation and reasoning skills to not take numbers where your golf ball would get lost, and it’s going to take you two putts, so, fluency and critical thinking in a fun, engaging way for students.
We provide a significant amount of students’ feedback. This is a, it’s a kindergarten data report. You’ll see there the grade al says one. It was actually like on the first day of first grade from my prior district, and the names have been changed, but these were the students. They hadn’t played DreamBox in first grade yet, but it was the first week of school, and this is what they have accomplished in kindergarten, and you can see there are students already working on quite a bit of first grade content, we have flags and notifications for teachers, links to more detailed reporting and really, you know, when you think about “Flip Models” and “Blended Models” and trying to separate time from proficiency, you can see how kids with different amount time make different amounts of progress in DreamBox.
We also have student reporting by proficiency, so at one click teachers can see which students have completed something with proficiency which are working on it in DreamBox and who hasn’t started it in DreamBox. So, with our rigorous elementary, our motivating learning environment and our intelligent adaptive engine, we are really helping schools differentiate, meet the needs of learners, whether it’s a “Blended Model” or “Flip Model,’ we’re a technology that’s partnering with schools to ’cause great student outcomes.
And now, I’ll open it up to some questions. Okay, so we have a question from, let’s see from Mary and, oh, all right, great. How does DreamBox make Math real, how does it bring Math into everyday life? Great question, so, the point of context is a key one. Obviously DreamBox doesn’t know every community, every school, and every context and there’s great Mathematics that teachers are connecting kids with in their own communities and in their own lived worlds. So, what we do in DreamBox, I mean, you saw the miniature golf example, you saw the pirate ship example, there are, you know, it’s a context that kids can access. We have a lesson where kids are recycling bottles. We have lessons where kids are sharing things equally and evenly in fair sharing context. We have number lines, and race courses for kids to understand the location of fractions on the number line.
But as far as, so bringing Math in every life, its number sense that kids are going to be able to apply in their everyday life and the context that they’re experience in DreamBox is a way for them to make sense of it. So, I hope that kind of answers the question.
Learning is contextual and not every context is necessarily real life, and we rolled over the classrooms, and the schools, and the teachers that we worked with and that use DreamBox, you know, they connect kids with things in their sort of everyday life, in their community as well, and we help with the number sense.
If a lesson is passed in the pre-test, this is from Dennis in Costa Mesa, but work afterward is weak, would the lesson be delivered? So, if I understand this question correctly, you know, we have pretty rigorous standards in DreamBox for when the student should move on when we know that they’re proficient. But we require kids to do quite a few problems before we’re able to say, okay, they’re fluent. So, you saw that miniature golf lesson for example. There’s a lot of problems kids have to do before we’d say, okay, you’re fluent with adding and subtracting fractions and decimals. If a student along that path, you know, exhibits some weakness as you say, yeah, our adaptive engine does send them back to earlier lesson when necessary. I think that probably answers the question.
If a lesson is too hard, kids can click too hard. Sometimes, if the kids click hint too many times or if they exit the lesson, we’re able to figure out what they should have next, and yeah, sort of move backwards as it were.
Mike’s question from Brooking, South Dakota, how long does an average lesson in DreamBox last? So, average lesson in DreamBox, they sort of very widely depending on the grade level and the content involved. Average lesson is usually somewhere between, somewhere around 4 to 6 minutes. Some of the once like you saw that multiplication algorithm one, sometimes that students are entering all the partial products and making the estimates, that one sometimes takes a little bit longer. Sometimes if there’s a lot of manipulation going on and the first and second grade place value lessons, it does take some time. But we try to design the lessons so that, you know, to keep the kid’s attention within about the 4 – 6 minute range. Others they might finish in 3 minutes, and it all depends on how the students, or thinking as well.
All right, let me see if there are any other questions that had been highlighted by my great team here.
Okay, let’s see. All right, so Mary’s follow up question. The biggest complaint I hear is that students do not understand why they need to learn these different Math operations; they do not see a connection to real life. It’s not that they can’t do the problems, they don’t understand why they need to, and can DreamBox help with that?
So, yeah, I think DreamBox, seems I can’t help with that. The main thing about the way we present Mathematics, I think sometimes, I’ll use an example from high school when I was teaching and working with pre-calculus teachers, and Algebra 2 teachers, when we were looking at Parabolas, we were often trying to figure out, okay, what’s the context that kids will really enjoy, and can really connect with and I think one of the best things we came up with was, like the t-shirt cannon at a baseball game or at a football game and you’re in the 3rd level of bleachers, can a t-shirt actually reach you? And that does seem kind of contrived, I think kids can definitely sniff that out and say okay, this isn’t really something that’s important; you’re just trying to wrap it in something that you think I will like.
Kids are very intuitive that when we’re, maybe a little disingenuous. So, what we do in DreamBox is we try… we don’t try to create something that we think will personally connect with every single student, because that’s not necessarily possible, that’s what the teachers are great at doing. What we do is we create the mathematical models and the manipulative that really kids need in order to understand things. So, if you’re talking about multiplying fraction, I showed you that example. If you’re not working with an array and thinking of fractions of fractions in that sort of way that 1/3 or 1/6 is 1/18, and able to see that and visualize that, then I would argue you don’t actually understand how to multiply fractions.
Now, whether we have to take that into a recipe context or something to connect it to the real world, that’s a conversation probably for about another time about when in mathematics we can say this is real world applicable as opposed to, this is the model that you need and the context that you need in order to make sense of the mathematical ideas. What we do is we create context where kids can make sense of it and we don’t find kids using DreamBox who say when do I need this or when do I use this, because of the way that we crafted the context in the game and the environment kind of answers that question. It’s fun enough to do.
When you enroll kids in DreamBox, this is Lynn White’s question. When you enroll kids in DreamBox, do they have access to multiple grade levels of content? Many are lacking are foundational skills from several years prior in the common core curriculum?
Yes, they do. Most students in DreamBox at any given time are working across multiple grade levels of content based on the nature of mathematics and mathematical understanding, like a lot of students have trouble with place value, but they can do skip counting before that. But, once they do skip counting and learning some multiplication, well, now they understand things about multiplication that help them better understand place value which has underlying multiplicative structures.
So, what DreamBox does is, yeah, it does help fill in those gaps for students who might be working below grade level and might have some missing conceptual ideas underlying that. That’s really one of the values that we provide to schools, especially given the time constrains on teachers for doing that.
So, wow, it is exactly 1:00. I zipped through those questions pretty fast. Thank you all very much for your questions, for you time, for your attention, for your interest in this topic. I hope that you have gotten some good, you know, ideas and insights into, not just “Blended Learning” and “Flip Learning,” but also about learning itself, and that’s what we are, that’s what we’re all about here at DreamBox Learning. We want, you know, the future of learning is going to be better learning, not necessarily digitizing things that had been done in the past. So, with that, here’s the last bit of housekeeping from edWeb.net. You’ll receive your CE certificate. And join the “Blended Learning Community” as well to access this resource and others and then I’ll be doing another webinar in April along with one of our UX designer, one of our game designers here at DreamBox, about games for learning, those design principles, not just for digital engagement, but also for classroom engagement.
Thank you very much, to edWeb.net and thank you all very much for your time.