Future of Math Learning: Fosnot, Fennell, Mills

Webinar Date: January 14,2013

Webinar Description

Join National Council of Supervisors of Mathematics (NCSM) President Valerie Mills, renowned educator and author Cathy Fosnot, and past NCTM and AMTE President Francis (Skip) Fennell for a conversation about the future of mathematics education. Everyone interested in the success of all students in learning mathematics—educators and community members—will gain valuable insights from these leaders.

Topics will include:

  • Formative assessment
  • Meeting the diverse needs of all students
  • Common Core State Standards
  • Digital learning technologies

Webinar Presenters

  • Cathy Fosnot - Professor Emeritus of Childhood Education at City College of New York, New York
  • Francis (Skip) Fennell - Professor at McDaniel College, Maryland
  • Valerie Mills -

View Transcription

TH: Hello, everyone and welcome to today’s webinar, “The Future of Math Education: A Panel Discussion of Promising Practices.” My name is Tim Hudson, and I’ll be moderating the panel and webinar today. I’m the Senior Director of Curriculum Design at DreamBox Learning, and I’ve spent over 10 years in public schools as a high school math teacher and K–12 math coordinator. Our team at DreamBox Learning is sponsoring today’s webinar.

DreamBox offers a rigorous PreK through grade five mathematics programs that empowers students to think mathematically. Built on our unique intelligent adaptive learning platform, DreamBox lessons fully differentiate for each student in real time no matter where they are in their learning to compliment the work classroom teachers are doing and to support schools using innovative technologies and blended learning models. I’ll share more about it at the end.

It’s always an interesting time in the world of mathematics education it seems, but for many reasons, it kind of seems as though we’re in uniquely unprecedented times. And at times like these, it’s valuable to gain insights from experts such as our distinguished panel today. So I’m pleased to introduce our three panelists who are going to share their thoughts about formative assessment, ensuring the success of all learners, the Common Core State Standards, and digital learning technologies.

Today, I am joined by Francis “Skip” Fennell, Professor of Education at McDaniel College and past president of both NCTM and AMTE. Also joining us is Cathy Fosnot, Professor Emeritus of Childhood Education, City College of New York and the Founding Director of Math in the City. And Valerie Mills, the current president of the National Council of Supervisors of Mathematics and a supervisor and mathematics education consultant with the Oakland Schools in Michigan. And both Skip and Cathy serve on our Curriculum Advisory Board at DreamBox Learning.

So a couple of quick technical things before we begin. It’s a good time to review some technical aspects first. Please check the audio setting on your computer and speaker volume if you’re having any audio trouble. If you’re still having some issues, check out the detailed audio troubleshooting file available in the handouts folder at the bottom of the screen. There’s also some other icons there at the bottom that open some additional feature panels in the webinar console. You can read about today’s speakers in the bio panel or click the handouts panel to download a copy of today’s slides. And for joining us today, you’ll later receive two related white papers, one authored by Cathy Fosnot and one authored by Skip Fennell. An on-demand archive of today’s presentation will be available in the next 24 hours, and both the archive and a free download version of the PowerPoint slides will be accessible through edweek.org. And finally, if you’d like to discuss this webinar on Twitter, use the hashtag #edweekmath.

So without further ado, let’s get started. As I mentioned previously, the agenda’s going to be a little bit of conversation about formative assessment, then success for all students with Common Core and learning resources, then selecting and implementing digital learning resources and then a Q&A at the end. So feel free to use that ask a question box as well as the chat window.

So we’ll go ahead and get started. A couple of questions that we had discussed from planning this panel about formative assessment were: how do we ensure this is not just another thing to be done that needs to be done? And how do we really ensure that formative assessment is an integral component of learning rather than just seen as another assessment or testing type opportunity? Those are a couple of the guiding questions, and I’m going to go ahead and we’re going to hear first from Valerie Mills. So Valerie, over to you.

VM: Hi. Thank you, Tim. I appreciate your introduction, and I’m looking forward to today’s webinar. I think it’s an interesting interactive sort of arrangement for one of these things. There are a lot of things called formative assessment these days, so I thought it might be helpful to begin with some characteristics that I think are, in fact, central to formative assessment.

So let’s take off. What you see on the screen right now comes from a paper entitled “Learning Progressions: Supporting Instruction and Formative Assessment” from Margaret Heritage of the CRESST Center. She lists these three key strategies as being critical to formative assessment strategies. And if you look at them, in some ways, they’re pretty typical of what we see: elicit evidence about learning, adjust instruction based on the evidence that you’ve gathered, and then of course, involving students in this process. But if you look—I want to draw everyone’s one attention to the very first one and to look a little bit more closely at it, because I think this is a place where we can all push our own thinking about formative assessment.

It says, “elicit evidence about learning to close the gap between current and desired performance.” When she refers to this gap, and she does as she goes through her other three components, what she’s really suggesting is that not only do we have to pay attention to our mathematical goals per lesson, but the tasks that the we use and the goals that we set need to identify not just where we want students to end up but to also identify where students are in their current thinking, so that by embedded formative assessment, we need to include both ends of this learning gap as we gather evidence and gather information so that when we find out where students currently are, we can work from that point, adjusting instruction, moving it towards our desired outcomes.

Another piece that I would like to add to this sort of look at formative assessment is to think a little bit more carefully about the nature of the goals that we frame, and then of course, what follows from that is the evidence that we collect. Goals, as most folks begin to frame them, focus on whether or not students are able to perform particular kinds of skills. And what I would like to suggest is that, in addition, that for formative assessment strategies to really be effective, we need to frame our goals not so they focus just on whether or not students got particular kinds of problems correct or incorrect but rather that we focus our look at students understanding on their knowledge of concepts and practices in ways that are understood to fit within a trajectory or a constellation of other mathematical goals.

So let me offer a couple of quick examples for us to take a look at to explain a little bit more about my thinking on this one. So here’s a really standard task from an algebra textbook with very straightforward directions. So we’re asking students here to simplify some mathematical expressions each involving exponents. So if you think about this and you think about what you’ll collect from this as a formative assessment task, the data that you can collect is going to be pretty limited to an understanding of whether or not kids were able to, in fact, just simplify. But if, on the other hand, you shift this so that your learning goals focus more on whether or not students are able to influence—or excuse me, to understand the influences of the sign of simplified mathematical expressions so that the goal—or excuse me, the description of what you want students to do looks more like—if I can get it there—this. As you can see you have an opportunity to collect data that is quite different and can lead to an analysis of the gap of learning in your classroom.

So the description here is look closely at this set of problems to identify the solutions that will be positive and those that will be negative without fully simplifying each task. Describe the important features of an expression that help you make this decision. So what you’re looking for is for kids to notice that, in this case, the ones that are now boxed in green are going to end up being—I’ll let you take a look yourself for just a minute. They’re going to end up being negative, and the ones gray will end up positive of course because of the even or oddness of the simplified power.

So one example. Let’s take a look at another one. This example comes from the PISA study in 2009, and it’s a slightly modified version of the task as it appeared, but I’ll give you a minute to read it. This is a task we used in a project in Oakland County a couple of years ago, and you can see that this task sets up two patterns actually: the number of apple trees in the apple orchard and then the number of pine trees around that apple orchard. The two patterns—one of the two patterns—one is linear and one is quadratic. The tasks that students are given associated with these patterns of change are here on this slide. So you can see the first step, of course, is to fill in a table of values, so just think about how the number of apple trees grows and the number of pine trees. And I’ve got those—the answers there in blue. And then the students are asked to examine those patterns of change and to describe the patterns either in words or in symbols so that you can find the number of apple trees for any stage of the pattern. That’s the first one, and then the second one being the pine trees.

I’d like to focus just on this sort of second one to give us, again, another look at what a formative assessment look can be with a task like this and how it might advantage instruction for it. If we think about the data that you can gather just by checking to see whether or not students have this—have any of these portions correct or incorrect, I think you can see that it’s a—it’ll be a pretty limited set of information that may suggest that you should do some more problem types, but it doesn’t really help you with that sort of gap analysis that Margaret Heritage was talking about. What I’d like to suggest is that, again, instead of focusing on right or wrong, that we take a look instead at what concepts students have developed and where they are in terms of, in this case, thinking about describing in words or symbols the patterns that they see.

So for example, the pine trees here, you can look at this pattern of change and notice that it goes up by eight. So you can get, excuse me, a recursive kind of growth pattern going on in your head. You can express that with words, “The number of pine trees increases by eight.” Or you can think about it as now/next expression, so a symbolic representation of a recursive description. Or you could think of it explicitly. It is—the number of pine trees will equal eight times the term number or the orchard number. So if I’m focusing instead of on right or wrong, if I’m focusing on the kinds of strategies that students use and I organize my data collection around strategy, I get a very different look and I get a very different collection of information that I think can inform instruction in ways that are extremely productive.

So here you can see what we’ve done, it’s to take about a thousand pieces of student work from—collected from classrooms as young as fifth or sixth grade, same task all the way up to advanced algebra, and we’ve organized them by the kind of strategy they use to describe, a representation they use to describe the pattern of the pine trees growing and the apple task. And what you see are opportunities here, I think, for teachers to not only understand what students know and don’t know but also what they—what you might do next. So for example, you can see an awful lot of students at very early grades as well as beyond algebra have a recursive sort of description described. But the question then might be: what can a teacher do to help move students from this recursive understanding to either being able to describe that with equations or from the recursive descriptions to a more explicit sort of representation, comparing and contrasting those kinds of pattern growth.

So two things. Number one, I think, as we think about formative assessment, the opportunity exists for us to think really carefully about the kinds of data that we collect in the classroom and whether we focus on concepts and skills or simply on right and wrong and the way that sort of leverage greater growth from formative assessment work. Cathy?

CF: Thanks, Valerie. It’s nice to hear all the points that you made there—we’re so in line and aligned in our thinking. I want to start with a slide of a quote by Jennifer James. She says, “Tapestry is that body of assumptions, beliefs, customs, and practices that we accept as foundational. They define who we are. In this time of great change, the tapestry is being torn rapidly and everywhere, and we begin to fall apart, becoming anxious and losing belief in who we are. We look backward. We become pessimistic about the present and the future, because we can’t envision a new tapestry.” I’d wanted to start with that quote because in many of the schools where I’m working, there’s such worry about this whole focus on assessment and data-based instruction, and yet, I see this a real nice possibility.

And if we go to the next slide, Heraclitus said, “You cannot step twice into the same stream. For as you are stepping in, other waters are ever flowing.” And if we think about assessment as not static, where it is a one-shot test so we can monitor how kids are doing, but instead, we think about it as formative, then it needs to guide teaching. It’s not about the kind of evaluation where we assess whether a kid gets it or knows it. It’s more about thinking about pathways. It needs to be continuous. It should provide information about the zone of proximal development that Vygotsky tells us about. To do so, it needs to foresee where and how one can anticipate the strategies, the big ideas that are just coming into view in the distance by the child. It needs to capture genuine mathematizing, children’s strategies, their ways of modeling realistic problems and their understanding of key mathematical ideas. Bottom line, it needs to capture where the child is and my term, as I call it, the landscape of learning, where the child has been, what her strategies are, what her struggles are, and where she is going. It has to be dynamic.

So if we go to the next slide, this is just a piece. Those of you familiar with my work, you know I’ve written about several landscapes of learning. This is a portion of the landscape of learning for multiplication. So if you look towards the bottom left, that children begin developmentally as they’re constructing ideas about multiplication with strategies such as counting and skip counting and repeated addition, but they then develop ideas about grouping and that groups can even be regrouped. And these ideas about groups can be regrouped lead children to then be able to develop doubling, partial products, generalize the big idea of distributive property. We start seeing doubling and halving develop, generalized eventually to the associative property. And this is what I’m calling a developmental pathway.

So when we do formative assessment, we’re not looking at how is the child going to do on my summative evaluation by the end of the year. We’re trying to pick up information that’s going to help us monitor and inform our instruction, and that means that it’s got to be developmental. We’ve got to get in there and try to understand where the child is and where the child is going. And how we do that is we have to be picking up continuous data all the time, in the moment when we’re conferring with children, when we’re analyzing their work, when we’re kid-watching as they are at work with a partner. We can be picking up continuous information by a digital technology. For example, DreamBox has worked out ways to mine the data not just in looking at children’s answers but looking also at children’s strategies by developing digital manipulatives that allow us to pick up what the children are doing with the manipulatives and then to have all of that data at the disposal of the teacher. So this is yet a second leg of the assessments tool. And then, of course, the third leg is we can design formal items so that we pick up information on an individual child’s thinking. But there are ways to do it where we can really capture more than answers, as Valerie pointed out, and I want to give a couple of more examples of ways that we’ve been looking at assessment, formative assessment.

This is a two-pen assessment. They’re a bunch of related problems. It’s just a page of multiplication problems, but note that there are relationships here. So if you look at these problems where the arrows are, you’ll see relationships immediately. There are several other relationships that we’ve built into this task. There you see partial products. Here you see the associative property and the use of the ten times. Here you see doubling and halving in both cases. So we give the child two minutes with a red pen to do all the ones that he or she thinks are easy, and then we ask children to switch colors and to chose a second color and then finish. That allows us to pick up information on the ones that the child thought were easy.

So let’s go back. Let’s look at the relationship we have right now on the screen. If the child thinks 12 times 9 is easy or 12 times 12 is easy, but he doesn’t think 6 times 18 is easy or she doesn’t think that 6 times 24 is easy, then we can infer that the child does not know the relationship between the problems and cannot substitute one expression for another. So this is allowing us to pick up information that is directly aligned with that landscape of learning that I showed you. We know then that a child may not yet got to the idea of doubling or halving or the use of partial products if we click back up, if we go back—no, and we get the black lines up there. There we go, and one more click will see a different—there. And so if a child knows 10 times 13 is easy and 2 times 13 is easy but doesn’t think 12 times 13 is easy and does it with the second color pen, then we can infer that that child does not construct partial products.

Okay. Let’s go to the next slide. This is an open-ended assessment item. Notice that at the bottom, we do not say after the child has an answer, put in words, pictures, and symbols what you did, nor do we say explain your thinking, which is the common language used in performance-based items. That confounds literacy with mathematics. We want to capture all the math—we give pens they can’t erase. We give them the work space. That way, we’re capturing all false starts, what they do first, what they do second, what they scratched out. We’re picking up all kinds of information about where they were on the landscape and where they’re heading.

This item is actually, and several other ones, that we couple with it where we can also pick up partial products. This one gives us the chance to look at how a child might solve 4 times 16 by either skip counting, repeated addition, or regrouping groups. But then we could look at 5 times 16 and then ask about 9 seats with 16—9 rows with 16 seats to find out whether or not a child uses the two prior problems.

All right. Over to you, Skip.

FF: Thank you, Cathy. I want to reflect a bit on some work that my project partners and I have been doing for now, I guess, it’s close to a year and a half, Jon Wray of the Howard County Public Schools and a current member of the board of directors of NCTM, and Beth Kobett, a faculty member of Stevenson University and I do a lot of work with and for elementary math specialists. And our project site, mathspecialist.org, frankly has all of our stuff if you want to go ahead and grab it.

But I guess the important thing I want to talk about is our journey relative to assessment, and it comes from a couple of spoke points, if you will. One is the sort of look at assessment literacy and both a peek at preparation and how that plays out in professional development. And one of the things we found actually, no surprise to many of you, is that as people prepare to be teachers, there isn’t a whole lot necessarily in that pre-service background experience prior to getting in the classroom. I’m struck by, as we interview teachers, how many say that, “You know, the first time I really got hold of assessment is when I had to sort of deal with it all by myself.” So that’s, if you will, one spoke of the challenge that we’ve mentioned that we kind of got into.

And the other thing that came out in both what Valerie said and Cathy said is the sort of understanding that we all need to have is that we can’t divorce planning from teaching from assessing. In other words, every time someone sits down to create that lesson, that person is involved not only in, if you will, the mathematics but also how that’s going to roll out and frankly, along the way, how we assess what kids are doing. That’s the formative assessment piece. And so our work, we’ve created these—as you can see on the slide pathway, and I think Cathy used the word pathways in one of her introductory slides. And what we’ve done is frankly distill, if you will, formative assessment. There’s a lot of stuff out there on formative assessment. And let’s face it, this isn’t new stuff.

Allow me a couple of direct quotes. “Observation of public pupils’ oral and written work is a very important assessment procedure and should be encouraged. Closely associated with the use of observation is the interview with the pupil regarding his or her daily work or his solution or attempted solutions of items of a test,” from Herb Spitzer, 1951. I’ve got quotes actually earlier than that. And let me give you one that’s a little more sort of in your face, if you will. “Aside from teacher-made classroom tests, the integration of assessment and learning as an interacting system has been too little explored,” this is from Glaser and Silver in 1994. So we’ve known about formative assessment, and many of you on the line can grab seemingly tons of books about this. And so what we try to do, if you will, with our pathways is distill formative assessment techniques and strategies into, frankly, five we like and also, to use a carrot. And in this case, the carrot is formative—excuse me, summative assessments and particularly, those summative assessments that are very soon to get our attention, Smarter Balance and the PARCC assessments.

And so as we look at the next slide, we’ll say a little bit about where we’re going here. As I think about a lesson rolling out, one of the formative assessment strategies that teachers use all the time, sometimes not recognizing it, is observation. They observe what kids are doing. They’ll be able to see how the lesson is moving forward, what they might have to shift seemingly at that very moment and so forth. So this is a common sort of a formative assessment tactic but maybe one that ought to be paid attention to as we think about the lesson as well. Closely connected, are two formative assessment techniques that we discuss and use a lot with both our math specialists and those folks with classroom teachers, is the interview technique.

The interview technique popularized way long time ago by Jay Fred Weaver and earlier and others in mathematics education where—and many of you have done this where you’ve actually found the need to essentially grab a child one on one or a small group of kids and talk to them about the mathematics that they’re doing, get some insight into where that’s going and what misconceptions may have occurred and/or frankly, because I think sometimes we think about the interview as a deficit model, and it should not be. This is, maybe that interview is to push somebody forward because he/she just—they picked that up in 30 seconds even though we thought it would take 30 minutes, so the notion of this one on one and small group.

And then the Show Me is a tactic that we use. It’s a little bit different than that and asks simply, “Show me how you doing that. Can you use an array to show me how you align those problems? Can you use the number line to show me how you’re representing that,” again, keeping in mind that we are using a lot of the work of Wiggins and McTighe in terms of back mapping from those targets I mentioned earlier, whether it’s Smarter Balance or PARRC or your own state of assessments or similar formative assessments that you may be tied to in a school or school district.

Dylan William, frankly, one of my heroes in the assessment world, is popularized, at least by me, of using what he refers as a hinge point question or hinge questions which sort of are the questions that, as you’re implementing that lesson, you want to raise to your class to see essentially where they are with that major concept or set of skills or whatever the focus of that lesson is. I often say—and one of my colleagues gives me hard time about this, but I often call this the deal breaker. If they do pretty well with that question, all right, we can move forward. If they don’t, then you know there’s some back mapping you might have might have to do that day, that next day or what have you. It’s interesting, we were testing—we test these pathways with our project teachers. And one of them in actually doing some of this work referred to the hinge question as, if you will, sort of, for those of you who like to cook, is like testing the meat as you’re cooking it. So it’s, okay, this is doing pretty well and so forth. It gives some sense about where that lesson is going. So that’s the work that’s sort of indicated. That’s pretty hard to do, to actually get those hinge questions in line and really raise good questions to sort of test that lesson to help you, frankly, consider the next step.

And then finally and sort of related to that, among these pathways is an exit task. And an exit task to us—and I’m being very deliberate here to not call this an exit ticket, because a ticket takes you somewhere. And exit task to us sort of defines, okay, this is the culmination of a lesson, and this task sort of gives you the indication about where this is going. And so the difference between that, of course, and the hinge question is that the hinge question is for the class and that’s going to help again test that lesson for you, whereas the exit task gives you a tangible response from all the students, and that’s kind of connected to the last slide that Cathy used.

So that work kind of summarizes what for our project has been a two-plus-year initiative, but we’re really trying to grapple hard with formative assessment. Now, this webinar is sort of for the future. So I’m going to make some comments relative to the future before we skip to the second phase of this, and that is that point one is that teacher education program really needs to do a better job in assessment literacy for all candidates. For those of you who work in an administrative capacity or those of you who are specialists or coaches, however that’s defined, we would want beginning teachers to feel better about their background in assessment and so forth. So that’s one piece of the future from my perspective.

And then here’s another one, and that is I think our profession really must determine how much accountability and when. We’re overboard, in my opinion, with accountability. Certainly, the profession and schools and school districts and teachers and students need to be held accountable. I’ve got, frankly, no problem with that, but what’s then the really long range plan to do that and not go over what I’ve think we’ve done, and that is go over the tipping point with regard to all the assessments that we’re holding children and, frankly, teachers accountable for. So to me, a partial response to that is in the power and consistent use of formative assessment. I’ll stop there so we can move on to our second slice of this webinar.

TH: All right. Thanks Skip. Yeah, and we’re a little bit over halfway through our time with a couple more questions to go. So as the moderator, I’ll just ask all three of you to keep speed through these next pieces appropriately so that we still have time for Q&A at the end.

Our next part that we’ll have folks talking about is success for all students. Reaching all learners is an absolute must. So in this Common Core era, here are a couple of questions: How can we wisely choose and create learning resources? And what can we learn from past initiatives, past standards, implementations, and resource adoption? So for this one, we’ll go right back to Skip to start it off.

FF: Okay. Thanks, Tim. One of the things that I think is really important as we think about that question is, frankly, what’s before us. And what’s before us, at least for 45 states and the District of Columbia, are the Common Core State Standards in Mathematics. And I’d like to say a little bit about that, and that is there’s tremendous opportunity here with regard to the Common Core and curriculum and instruction, and I’m going to just raise a couple of them. One is that here, here we have a set of standards that, for the first time in a very consistent way, are calling for kids to frankly understand what they’re doing as they learn the subject. Now, I cherry-picked some of the standards from the fourth grade level across three different content domains. And for those of you—and I’m sure some of you on the line almost have the Common Core sort of like memorized, so you know the level of import and the level of understanding that’s suggested in the Common Core.

And this next slide also teases out what I refer to as representation. So not only do we have an expectation that kids understand the math that they’re learning, but within the standards are actual references to particular representations, represent fractions on a number line diagram and using equations, rectangular arrays and/or area models using cubic centimeters, cubic inches, cubic feet, and improvised units, tables of equivalent ratios, tape diagrams, double line diagrams or equations. So representation coupled with understanding kind of takes me to a point that’s pretty important to me. One is the computational understanding is not an option. It’s an expectation, and frankly, it’s about time. We have a set of standards that presents targets that calls for kids to understand the mathematics they’re learning, and it also calls very directly and using particular tools, representational tools to sort of help that process.

So as we think about resources and discuss their importance as well raise what you’re using—here many are on the line who are sort of perhaps in the position of thinking but, okay, what do I use, what do I think about as resources as I deliver a curriculum in the classroom every day that calls for kids to understand and use a variety of representational tools and so forth. So in my opinion, one of the things we have to do is look differently. And the other thing that we have to do, and it’s not among the three slides that are here, is the standards of mathematical practice provide for us this wonderful opportunity to, along the way of developing mathematical content, kind of get at the important habits of mind that also bring about kids engaged in problem solving and reasoning and using tools and being precise and using structure every day in every classroom in every lesson. My take for the future on this, and then I’ll move forward to the next speaker, is that the access to online curricular opportunities is going to continue to expand.

Let me also say that paper is good; we’re not going to throw that away—not be totally thrown away. But hopefully, the emphasis on the Common Core, which is really a less-is-more story, will be sustained and a long-term goal of mathematical opportunities that truly provide the college and career readiness will be continually updated and sustained for every child in every classroom.

So now, let me shift to Valerie who’s going to pick up her points on the same issue.

VM: Thanks, Skip. I think you’re absolutely right, focus on representation and on conceptual understanding and on looking for ways in which our instructional materials can help make clear the structure in mathematics. That structure to build new ideas is a really critical piece of the puzzle. I think a lot of folks have gotten very used to aligning textbooks with content standards, but I think there are some pieces in the Common Core that are, while they’re very similar when you’re looking to see whether or not textbooks are going to get you where they need to go, there are also some ways in which I think we’re going to need to do some things differently, and I want to point to two of those very quickly.

A couple of years ago, the Council of Chief State School Officers supported the development of a mathematics curriculum materials analysis tool kit, and you can find it online at the NCSM website. Bill Bush led the team, which I was a member of, and the materials themselves are quite comprehensive. There’s a lot of materials in them. I want to look really just at two of the tools to point out a couple of things that I think that are different for folks to start paying attention to as they look to find the right instructional resources.

The first one is going to be that first tool that looks at content. The content in the Common Core State Standards are laid out along teaching and learning progressions. I think a lot of people are aware of that, but I think what they may not be thinking very carefully about is the importance of these progressions when they think about looking at textbooks and how textbooks take up additional material over multiple grades.

So what I would like for us to do is to begin to think about ways in which when we look to see how standards are included or not included in textbooks, I’d like us to—instead of looking at one grade, I’d like us to start thinking about looking at multiple grades. So when I pulled a tool from that very first set of tools, this one happens to be in kindergarten, first, and second grade, and you can see that the way the tool is organized. We’ve laid out constant standards that are related across multiple grades so that the analysis isn’t just, “Does my textbook cover place value in kindergarten, first grade, and second grade,” but instead, “How does the textbook develop those ideas over multiple grades.” And so the organization of this particular tool is designed to emphasize that for folks.

The second piece that I think might be really helpful for folks to start paying attention to picks up on what you were talking about, Skip, and that’s the role of the practices in textbooks. First, it’s one thing to take a look at the practices and to say, “Oh yeah, I see opportunities for students to attempt precision or look for and use tools appropriately,” but it’s another thing to think about how these practices are used to develop the important mathematics ideas.

So the second tool in the tool kit that I referred to is the designed to sort of push that point. So you don’t just look to see if the textbooks—I’ll pick up on your example, Skip, which is look for and make use of mathematical structure. The question is, how do they use mathematical structure to develop the ideas of place value? And how do we help students build this bigger and bigger sense of what mathematics is in a much more coherent way? In which structure becomes an underlying feature important for us to attend to no matter what mathematical ideas we’re looking at. So two things there, one, to begin to start to look at materials not for just what they do in a single grade but how related mathematical ideas are built over multiple grades. And then the second thing is not just to look to see if your textbook has interesting problems or any of the other practices, but in fact, how these practices are used to develop specific mathematical ideas.

Cathy, I think you’re up next.

CF: Yup. Thanks, Valerie. Both of you, Skip and you, Valerie, have made a lot of the points I wanted to make, so I’m going to be very brief in my comments. I just have one slide. I want to make sure we have time for questions and answers at the end too so we can hear from participants.

So basically, I just want to throw out three criteria that I think are absolutely critical in anything that one chooses to purchase, and that is that whatever is purchased, that curriculum needs to take the standards of practice seriously. So many of the things I see on the market are not doing that. They’re just a series of activities. They’re not really giving kids a chance to develop viable arguments to convince others. They’re not necessarily asking children to choose an appropriate model. They’re introducing a lot of models but not giving kids opportunities to really do the choosing themselves and to model their thinking.

So the first criteria is that we have to make sure that the resources take the standards of practice seriously. The second one is that the material should provide professional development within. It’s not that one should buy the materials and then provide PD to help teachers learn how to use those materials. There should be a lot of professional development built within the materials. I don’t just mean telling teachers what to do. I mean providing teachers with sample dialogue boxes with analysis of sample children’s work with understanding of the development, deepening teachers’ understanding of that landscape of learning that I had up as an earlier slide. That kind of material should be built in as a very important piece of the resource.

And then lastly, we need to have crafted sequences that are going to support progressive development over time, not just good nice problems. And I see a lot of materials that come out with just good nice problems, but they’re not really supporting children’s growth and development over time along learning trajectories like the landscapes I mentioned.

So over to you, Tim.

TH: All right, great. So we’ll make our last section even more quick as we look digital learning. Here are some of the questions that we talked about here on the panel and things we can address, and each one of our panelists has about one slide to share their thoughts about how we can wisely use and select technology to meet the needs of all learners and really, what sort of PD is necessary in order to help educators be wise consumers. And before we go, and we’ll go back to Cathy here in a second, I wanted to bring a couple of other voices into this conversation. As we look at blended learning opportunities and technologies, like Skip has kind of mentioned, we are seeing more and more tweets like this, “Why am I sitting in a history lecture at 7:20 in the morning? Is there any other way that I could access this lecture?” And that’s the student voice, and we need to be, as math educators, very cognizant of how we’re spending our class time and what that means for our students.

We also had Michael Fullan back in July wrote his report with Katelyn Donnelly about—called “Alive in the Swamp,” where they basically point out: “A lot of technology-enabled innovations don’t focus enough on pedagogy and outcomes.” And we see there in orange, “Most often, technologies are introducing concepts with video instruction, following up with a series progression exercises and tests or just doing the same age-old practices in a digital format.” Sean Junkins kind of captured that idea in a pretty short and sweet image here, you can follow him on Twitter there, that the iPad was not made to be a worksheet. And in math education, sometimes we really love our worksheets and what if they could be on an iPad, and we need to think better for students. And another—a physics educator on Twitter as well, Frank Noschese, points out something really great. The biggest shift in teaching is not replacing textbooks with iPads but replacing textbooks with experiences and questions.

So with a couple other voices coming in, I want to pass it off now Cathy Fosnot for her thoughts on digital learning.

CF: Thanks, Tim. Once again, I just have one slide here. And I’ve already discussed what’s on the left of it, so I’m going to focus on what’s on the right. With digital environments, to me, we’re leaving in a time period of revolution. I see it as an incredibly exciting time to be an educator. That’s why I mentioned when I started the webinar that I see this as not a place where the tapestry is being torn, but it’s what the openings become. It’s the openings that excite me, because digital technology affords the power to really begin to change what we’re doing in the classroom. A lot of programs talk about adaptive learning. I first got involved with DreamBox because—we like calling it intelligent adaptive learning. It’s more than just looking at children’s answers. It’s more than just looking at linear pathways. It’s taking seriously those landscapes of learning which are landed behind the DreamBox platform and really having seamless formative assessment going on continuously and a seamless home/school connection. It’s not just that learning takes place in the classroom but that learning can take place anytime, anywhere.

It’s involving parents in the learning. It’s giving parents information about what children are learning. It’s giving children choice in personalizing the learning. It’s also, I think—although I don’t have this on my slide, I think over time, what we’re really going to see is a total integration of blended learning where the computer is going to be an integral part of math workshop. It’s going to be a tool for teachers. Kids are going to be back and forth from the computer to the real-time in the classroom, discussing, using the tools and the digital environment to defend their thinking, to develop viable arguments, to represent their thinking, to do problem solving. And here you see the standards of practice really being taken seriously.

TH: All right. Thanks, Cathy. Skip?

FF: Thanks, Tim. The slide that I’m presenting is essentially, again, for those who have memorized everything related to the Common Core know this: practice to the appropriate tools strategically. And I think in many ways, it kind of gets at question that Tim posts at this section of our talk. So questions that I might have are: What types of tools are we and will we be using then, if you will? Will they be fully electronic? Will we use paper/pencil drawings to help accommodate that? What are the roles of tools as we know them? And part of that slice, actually number two here, might be embedded on number one: what’s the role of manipulative materials? Again, go back to what all of us were saying I think in related ways to the prior question is if we want kids to understand mathematics, how we represent particular concepts is pretty critical foundationally for such understanding. So you could argue that the tools help drive that and that includes other, if you will, more general tools like drawings and number lines and so forth and so on that work with that.

And of course, the question, what technological tools might we have? And I have this wonderful opportunity to, on a regular basis—I have nine grandchildren. I see seven of them all the time. I am amazed at how facile they have become with pretty much everything technological. Don’t you dare have me play them at any sort of a game activity. I can’t compete. So the notion of kids being facile with such tools but then on my side is what’s the appropriate use of such tools and so forth. And I look at sort of where we are with two things, the bottom two points on this slide and one is the flipped classroom. And there are probably people online who do this on a regular basis, who use the power of technology, however you define this, to have such activities done at the home. And then you’re flipping that, providing more opportunities to do one-on-one student kind of formative assessment stuff I was talking about earlier, with regard to our pathways, when you have then, if you will, in their seats. So I think that is an important consideration. I also raised the sort of statement in the last two words of that as a question, and that is: is that for everybody? And where does that become more evident? That is the flipped classroom then others in terms of levels, whether that’s the secondary level versus elementary, or levels of kids and so forth and so on.

And then I’m continually learning about transmedia. And I say that because of a role I play with public broadcasting where I’m advising the television show that if you’re an early childhood educator, hopefully youknow about, that’s Peg Plus Cat which just set a record last month for the greatest viewership of anything public broadcasting has done for children in the last several years. And so with Peg Plus Cat and with public broadcasting, what they’re trying to do is say, “Hey, we’re a lot more than TV. We have apps to go with this. We have a website to go with this. We have all kinds of opportunities.” And if you take the notion of transmedia and take that notion from the classroom to the home, to parents, to community centers, to churches, YMCAs, and other places where kids actually have access to technology—and access, by the way, still continues to be a key issue and very important for us. And so it looks, in my mind, all those are dynamics in terms of what we’re going to be with such tools.

So my take on the future is, frankly, Tim, who knows. How I would love to say that this will be all seamless from preschools through higher education, and I use home technology with all kinds of tools and the like. However, to achieve the seamlessness, we must figure out how we can access and engage everyone, and by everyone I mean all children, who are our responsibility for the next generation.

Let me stop there and turn this over to Valerie.

VM: Thanks, Skip. I think I’m going to do something a little bit different than what we had thought—I had originally thought about. What you see on the screen right now I can simply tell you is some design features that you might want to consider as you begin to look at digital learning resources. There are a lot of very new and exciting features that I think we all need to learn about. But at the same time, I think it’s important for us to take what we’ve already learned about effective instruction and think about how those features of effective instruction live in the electronic digital learning environment. So things that have always been important are things that we’ve learned are important around multiple representations and the use of high demand tasks and mathematical discourse are features that we ought to see and look for opportunities for in the electronic learning materials that we purchase as well.

I think what I like to sort of conclude with are some questions that I think a lot of the field is just starting to pay attention to, and I’d love to hear kind of other folks’ thinking on this. In addition to learning to do a better job of evaluating digital learning resources, I think another thing that, as educators, we need to begin to think harder about is what, in fact, are those features of digital learning that we can leverage? And where might digital learning [inaudible] learning experiences for kids? Another kind of question is what kind of professional development are teachers going to need and administrators as [inaudible]? What does the support need to look like over time? And how might we need to help kids learn to adapt to this learning environment as well?

So Tim, I leave you where you started out with: a whole new set of questions.

TH: All right. Well, thank you, and thank you to all three of you. We have tons of questions coming in on the panel. Let me real quick go to the slide with contact information for the presenters that will—let’s see here—let me, before we do it, we only have time for a few questions and answers, but here’s where you find contact information for our three panelists today with some Twitter handles there and a couple of websites so that you can follow up. And you see Cathy just typed into the chat some ideas about two-pen assessments. There are several ideas coming in about the two-question assessment. So get that contact information down there. There were a couple other slides. Just download the handouts, and you’ll be able to see those.

I think if I can ask each of you one overarching question; at the start of this item, it’s—it does seem like an interesting time in the Common Core area. And one question that came in, wanted to hear from all of you, if you could just give us a 30-second thought about it, how is this era perhaps different than the 1980s and the 90s and past decades in terms of what NCTM had done in the past and NCFM as well? How would you describe the differences today? And we’ll leave it at that one question, and then I’ll wrap it up. And we’ll start with Skip.

FF: Thanks, Tim. Where NCTM is going today, I think that what’s perhaps different now, as those on the call probably know, we used to be in charge of standards. We wrote the Curriculum Evaluation Standards in 1989, the Principles and Standards for School Mathematics in 2000, Focal Points in 2006 and so forth. And the Common Core is orchestrated by, respectively, the National Governors’ Association, the Council of Chief State School Officers. That doesn’t mean that NCTM is not involved. Many NCTM members were part of the writing group. I happen to be one of them who were involved with working on the Common Core State Standards. And I think NCTM has made a concerted effort to be behind and supportive of teachers and the challenges related to implementation and providing support relative to materials that are out now and forthcoming and so forth.

Similarly, you know, support is beginning to be provided for consortial assessments and frankly, in a sort of indirect way, for the ten states that have decided not to be part of the consortial assessment. So I see the Council—and again, I’m a past president, and current board members and current president, Linda Gojak, and incoming president, Diane Briars would be more attuned to this on a daily basis than I am and frankly, it’s their responsibility. But I see the Council as an organization, the largest organization for math educators in the world, I might add, who has stepped up, who is ready to be supportive of its membership, and that’s classroom teachers and mathematics educators at every level around this country and internationally. Valerie?

TH: Great. Yeah, Valerie?

VM: What do I see as different from the 1980s? One of the things that I think is a really high leverage feature of the Common Core State Standards is their focus on content trajectories and this idea of building coherence within the standards themselves as a way to help students develop an understanding and a knowledge of mathematics that is, in fact, a more clearly connected set of mathematics ideas. The standards, as we’ve looked at them for the last 30 years, certainly had these features of these trajectories, but I think this newer set of standards and the attention that it’s garnered around assessment as well in connection to assessment is a really critically important thing for folks to start paying attention to.

I think another thing that’s quite different for a lot of folks is the focus on high-demand tasks that move us beyond skills. It’s not that standards in the 80s didn’t tell us to pay attention to conceptual understanding. It didn’t talk to us about a balance of problem solving and skills. But I think, again, this new set of standards pushes that bar up just enough higher. And now, the other big feature, and that’s this common assessment that will be coming down the road with the new consortium in a year and a half, I think pushes all of these things, a focus on high-demand tasks and the use of trajectories in teaching and learning and developing a more coherent set of mathematical ideas to a new level.

TH: Thanks, Valerie. And then last, Cathy?

CF: I’ll be brief. I think that the standards of practice are actually not process standards, and back in the 1980s, we were seeing no such process standards. And from what I’ve heard—and Valerie and Skip, you probably know better than I—but from what I’ve heard, about 60 percent of the items PARCC assessment are attempts to actually pick up the standards of practice as an integral part of what it means to do mathematics. I’ll stop there.

TH: All right. Thanks, Cathy, and thank you to everyone for attending. Real quickly, I’d like to take the next couple of minutes to tell you a little bit about DreamBox Learning if you don’t already know DreamBox. Here at DreamBox, we combine three essential elements to accelerate student learning: a rigorous elementary mathematics program, a motivating learning environment, and our intelligent adaptive learning technology. It differentiates uniquely for each student. DreamBox is available for gradse PreK up to grade five, and it’s available both online and on the iPad. We also have some middle schools using DreamBox for intervention.

We believe at DreamBox that the quality of the math software is just as important as the quality of classroom learning experiences, and schools use DreamBox as a partner with those great classroom experiences, supporting so much of what you’ve heard today. There are other math programs and technologies that are collecting data and monitoring and even adapting in some regards to students but none of them really at the level of DreamBox, where we build our tools from the ground up to be adaptive, keeping in mind learning trajectories, accounting for student strategies, watching how they’re thinking about problems and modeling, and adapting in real time. Kids have to think critically. It’s not just practice. It’s conceptual. And we don’t start our lessons by saying, “Here’s how to do this. Now go do it.”

We also provide tons of reporting for teachers and administrators. We intelligently adapt and individualize to students’ intuitive strategies, the ideas that they have that naturally occur to them about mathematics. And we respond based on the kinds of mistakes, the efficiency of the strategy, the scaffolding that’s needed. We have experienced classroom teachers on staff who work with game designers and programmers, skilled programmers, to create really engaging learning experiences to make sense of mathematics.

We also empower students to move at their own pace and learn to progress at their own pace. This is an actual classroom report from a group of first graders in the first week of school having used DreamBox in kindergarten. And you can see that students are in very different places like we all know, but now, we have teachers are empowered with far more rich information than they might have had access to in the past. And we empower teachers with data reports that let them know which of their students might not need to hear the lesson tomorrow, because it’s something that they already know. It’s really about personalizing education more and empowering students as mathematicians.

If you’re interested in seeing some of our virtual manipulatives, you can access a lot of them for free at dreambox.com/teachertools. Any teacher anywhere who has an interactive whiteboard or any computer really and would like to engage students with some of our manipulatives, you can go to that website. And lastly, if you’d like a free trial, we offer free school-wide trials. Just go to dreambox.com, and up at the green, up at the top, you’ll see an orange bar that will give you some information on how to sign up to use DreamBox with your students.

So in conclusion, we’d like to remind you that if you want to watch this presentation again, an on-demand archive will be made available through edweek.org within the next 24 hours. You can also visit edweek.org to find articles that explore today’s topic, and you will be receiving a couple of white papers from DreamBox. So once again, thank you very much to our distinguished panel, lot of great …