#103: Learning and Teaching Early Math: An Interview With Dr. Doug Clements

So to build visual imagery of these things and the like, understanding more about that is really important so people see the richness of it. Otherwise, all of us, all of us, me included, tends to default back to the math we were taught, which means, oh, I'd better get a worksheet or something. And we're trying to show them that a lot of the mathematics you missed or think you're no good at is the fault of your experiences. And in this episode, we speak with author and early years math curriculum designer and researcher Doug Clements. Doug is a professor at the University of Denver. He's published over 130 research studies, authored 23 books. And actually, we might even need to double check that number because it seems like there's always something new coming out on early years mathematics. He served on the US president's national mathematics advisory panel and he's a co-founder of learningtugectories.org with his partner Julie Sarama. Stick around while we dive into why we need to think deeply about early elementary mathematics, how we can help students think conceptually about mathematical ideas at a young age, learn about resources you can use at home to strengthen earlier as mathematics and how a parental mindset is a key in a student's progression of mathematics. Hey, cue up that music! Welcome to the Making Math Moments at Matter Podcast. I'm Kyle Pierce from tapintoteenmines.com. And I'm John O'R from Mr. O'R-Izzagheek.com. We are two math teachers who together with you, the community of math moment makers worldwide who want to build and deliver math lessons that spark curiosity, fuel sense making, and ignite your teacher moves. John, are you ready to dive into this awesome chat with the ever so friendly, ever so awesome Doug Clements? Yeah, we've been waiting for this one for a while now and I can't believe we get to chat with Doug. It's going to be an awesome discussion. Before we dive in and get to our talk with Doug, we want to thank you for listening to us wherever you are, wherever you're from, whether it's in the car, at the gym, in the kitchen washing dishes, or maybe on your prep time. If you've listened to us before and enjoyed the episode and got some value out of it, you don't know how much it means to us for you to let us know how things are going. We read all of the reviews of this podcast and right now, we want to share one of those reviews with you from K-squared_math. And this one comes from Apple Podcasts. K-squared_math says, "Keep the math conversation going. This podcast is allowing me to learn and reflect on all things math on my commute. Cannot recommend this podcast enough. John and Kyle bring together so many amazing math heroes that share so many great ideas. Thank you for helping make me a better math educator." Huge shout out from Nova Scotia. Awesome stuff. We've got friends out on the east coast of Canada and Nova Scotia, what a beautiful place. So great that they're listening out there. Thank you so much to K-squared_math. As I mentioned, nothing energizes us more, but even more importantly, by adding to those ratings and reviews, it helps more educators find the podcast and be able to dive into the same learning that you're getting right here. If you haven't taken 10 seconds, hit the pause, scroll down in your podcast app and tap on the five stars. Okay, okay, don't tap five stars if you don't think it's accurate, but please do hit a star rating to give us that quick feedback. If you want to be a math moment maker hero, then take the extra minute or two to just write a short one to three sentence review. This goes a huge way in helping the podcast reach more math moment makers from all over the world. All right, my friends, enough from us. Let's get on to this fantastic conversation with Doug. Hey there, Doug. Welcome to the Making Math Moments that Matter podcast. We are, as always, excited to talk to you and have you on the show today. How are you doing today? Wherever you are, let us know where you're coming from. Hey, I'm in Denver, Colorado. I'm at the University of Denver here, but of course, I only am in the university for an hour or two weeks and working out at my home where I'm talking to you from now. Oh fantastic, fantastic. We are, as John mentioned, very excited to have you on the show. I've learned so much from yourself from Julie. And let's make sure that math moment makers. I'm sure there's many out there who are saying and looking forward just seeing your name on this episode, especially our K-through or I guess birth through grade three to maybe grade five age teachers would probably know you more than maybe some of our secondary educators. But tell us a little more about yourself. What's your role in math education and how did you land in this crazy place we call math education? I'll take the second question first, I put. I was in early childhood in elementary education, loved math and science, right as a kid and in high school and everything else. But saw myself as a generalist teacher. However, when I was starting my PhD, I happened to get a graduate research assistantship with a Dr. Radaisal in a project called Project Vanguard where we actually drove a van around and showed people exciting new developments in mathematics education, the materials, the approaches, videos and everything like that. Because of that, I started just diving into math more and more. And then when I taught kindergarten, I taught with a wonderful program by a now deceased Mary Goretta Lorton called, well, Workjobs 2 was one of first but math their way was the premiere kind of thing. It really showed me the potential of young children's thinking. So I taught kindergarten for five years of preschool for one year and just became fascinated with younger and younger kids all the time. And how not only the surprising potential of those young children, but the really surprising lack of resources that we sometimes have to realize their potential. And so long story short, I did my dissertation with preschoolers and everything and then kept my interest in elementary for sure. But now I have extended it to birth through grade six in a desirable for equity reasons because some kids don't have the resources geographically that other kids have in this country. And that's a shame. But also for all kids, there's a certain gap in our availability of really high quality mathematics, professional development for teachers, curriculum for teachers and kids and so forth. Interesting. You know, that's a great intro into what your role looks like and how you even got into that role. And I eager just when you said Van around the country, like I just had that image Doug that you just pulled up on a beach somewhere and you just opened this van up and you were like, hey guys, come check out this math. This awesome math going on right here. I had that image, but the other thing that really intrigues me is most researchers are professional development experts are focused on school age children and I find it very interesting that you've got resources for birth through first year and second year all the way up. Do you even getting in school? How did you come across that as a focus? Yeah, that's interesting. That's a great question. Again, as I said, I taught preschool and kindergarten. But then got involved with the National Science Foundation projects that developed curriculum for elementary school. Very much child-centered children's thinking, constructionism kind of focus and thought that was very interesting. And then the National Science Foundation and the South put out a request for proposals for really childhood years. Then Julian I who had met actually writing the elementary math curriculum, my wife and college, Julie Sarama, met doing that activity. We decided to put in a grant because we hadn't done as much with the kindergarten and nothing below kindergarten previously. And since I have experience and the like, you know, that seemed interesting, that started things off. More than two decades ago, we are funded by MSF to do a preschool curriculum. Originally it was supposed to be preschool through grade two. But they kind of didn't have enough money when it came to fund. So all the projects that they wanted to fund were just reduced in scope. So Julian I decided technology will do that preschool through grade two like original plan. But the whole curriculum, people have kindergarten through grade two. So we're going to concentrate on preschool. That's the genesis of the building Blacks project, the preschool curriculum for mathematics. But then dominated Julie's and my life right up through decades afterwards. God is excited about it. We were funded by the US Department of Education Institute of Educational Studies to do what's called efficacy trials that is see if under good conditions you get a good statistically impractical particularly significant, effective your approach we did, and then scale up. Can you bring it to an entire district, and we did, and each of these two years and years and years, but it got us into the domains of professional development and all that. And then because we followed these kids up in the large scale study into the primary grades, we extended our approach, which is based on learning trajectories beyond just preschool, to go up and up. And then I better stop and have you ask a question, because I can go on like this over here. But then we developed tools for professional development and have people telling us, what about three-year-olds? What about toddlers? And so we took the time to look at the research, because we don't make these things up. We create them based on the shoulders of giants, of hundreds and make sometimes thousands of studies of people from early childhood, math, that cognitive psychology, developmental site, and other fields, where we put that research together and then test these so-called learning trajectories. So now we've developed them really birth through grade two or three, and have done research with them, still developing tools for them and the light. I think that gives people a really good sense of who you are and your journey along. So that's fantastic. And I know, for me, John and I, and I know you know this from when we had chatted at, I think it was NCTM, that we've come out of the secondary classroom. And for us, it was sort of a similar scenario. You were just referencing how people were saying, well, then what about this? What about that and backing all the way up and kind of going right out of those elementary grades right into those preschool grades or ages? And it was the same for us as well. You know, we were looking at our grade nine courses. We were teaching and seeing students struggle. And then the next thing you know, we're going, well, if they don't understand or aren't feeling successful with this idea, I've got to go back and figure out where the struggle is. And the next thing you know, you're down in the middle grades and you're down in the elementary grades. And I've been very fascinated as I've been digging, kind of starting from the high school grades and working my way back. And I'm fascinated, as obviously you've mentioned earlier, at how interesting and how different students' responses can be based on the types of questions we're asking them and where they are along those learning trajectories. And it's really fascinating. I also want to mention before we get into our next chunk here, we've had Dr. Nikki Newton on a previous episode. And she was raving about your learning trajectories.org site. And I'm sure we'll get into that later as well. So I'm sure many are really excited to dive in. And John and I are really excited to ask you this question, which is one that we ask all of our guests. And it's to describe a memorable math moment from your past. This episode or this podcast is making math moments that matter. And we're wondering if you think back to your experience in school as a student, it's usually from when you're a student, but you're welcome to go outside of that if you'd like, what would be a math moment, a memorable math moment that pops into your mind when you think back to those memories? Boy, that is so cool. I will probably think of many, many, especially you hang up with me. Where it goes. I forgot this one. One pops into my mind right away. This is early on when I taught kindergarten. I just got my first job. I had taught three school, but that was with my sister. And there was plenty of-- a guy, this was my first unmatched child loan job. And then I had already been through this project fan guide. I already had thought a lot about math. I started teaching with math there, like I said. But this just hit me. I had this really bright boy. He was called Eric. And he loved school. And he walked to school. I taught in a fairly rural area. And he was just down the street. And he'd come in early and he'd stay late, because he'd fool around with stuff. And so one day he came in and he said to me, can I play with your calculator? Because he knows, Tom, it does. Of course, that's-- of course, I say. And then he comes back to me and he says, what is this? And he points to the square key. And I said, boy, Eric, that's really complicated to talk about. Well, it gives me really big numbers. Yeah. Can you explain it? OK, Eric, look here. Let me try this. Take these tiles, you make a square, and then make a bigger square, and then make a bigger square, and then come back to me. Because I'm trying to get ready for the day. He comes in before-- I get ready for all the other 23 kids. And he comes back and says, OK, I made it. I said, now put in this one. How big is that square? Well, that's four. Yeah, but how many on a side? Two on a side, two on a side, he says. So press two and then the square key. Oh, yeah. That's how many in there. Well, try it for the other ones. And I just left again, because I'm frantic trying to get my room ready for the other kids. He comes back and says, so every time I press a square key, however big a side is, that's how many tiles are in the middle. And I said, yep, and that makes the squares. And he went off and come back to me with this long chart. Two and four and three and nine and way up to 10 and 100 and beyond. And close home tells his mom, it made me think, holy macro. I was willing to say a bright kid. This was beyond them, not necessarily. And to me, that was fundamentally changed my view of what kids could be capable of and why we should always keep our minds open to the possibility that it's done in a way that makes sense to them. I'm thinking back to Old Seymour Papper, who said when he talked about deers and how deers taught him ratio at a very early age, if you can assimilate some mathematical idea to models that are meaningful for you, you can understand way more than people think usually. And so it's surprising what people can learn if they can assimilate to meaningful models for themselves. Hey there, mouth mommakers. We know that the most impactful way we can be better educators, better prepared to deliver the engaging thinking type lessons, the moves that allow students to have the appropriate discourse, or show their reasoning, attempt to solve problems at a deeper level, develop their own fluency skills. We know that in order for us to have the right moves to engage our students in these ways, we have to have a strong understanding of our own mathematical proficiencies, our own mathematical understanding, at a conceptual level, we have to know how to connect that to the procedural and the algorithms that we eventually are going to be introducing to our students. We know that strengthening our roots of our classroom trees, our own mathematical proficiencies is the key to being more flexible in the classroom. And it's hard because we have to find the time to do that. One of the things that we want to do here for our listeners of the podcast, for the people who are on our email list, and regularly read those emails is every single month we are hosting a free webinar on often misunderstood or hard to teach topics for your grades. We're going to introduce lessons, ideas, activities, walk through the mathematics, and in understanding that conceptual understanding of that mathematics, so that you are prepared to engage in those flexible teaching strategies. We're doing it every single month. Go ahead and check it out. Make mathmoments.com/training, see what lesson, see what topic, see what grade level we are focusing on this month. Go check out makemathmoments.com/training and get yourself registered. You won't regret it. We're going to strengthen the roots of your classroom tree. Let's do this. I really love that story as that memorable moment for you. And it makes a lot of connections. And one thing that I often think about is because this is going to be two pieces of this comment that I want to make to this, because you're making awesome connections with this student too. And we've said this on the podcast before that those connections about square numbers in particular, I don't remember having discussions like that in elementary school. I don't even remember having discussions like that in high school. It wasn't until I was a teacher that I remember. It wasn't even that long ago where I was in a position where I'm like, wait a minute, a square, when I hit the squared button, or just when I squared numbers. And this became more clear when I was working with completing the square in algebra and expressions that we were actually dealing with squares and the areas of squares. Like the Pythagorean theorem, we've said that over here before on this podcast, A squared plus B squared equals C squared is something I memorized in school. However, it didn't occur to me that we're really taking the area of squares off the triangle. It blew my mind. And I really love the fact that you had this conversation with a kindergartener and it goes to show this point that you just hit home with is like a lot of the time we don't give kids credit for having deep level thinking on concepts just because you know, oh that's a high school concept or that's a great eight kind of concept. It's mind-boggling that we don't have more of those and you know this is the other part that I wanted to touch on is that like I have some really great elementary school teachers and it's possible that I did have that conversation a long time ago but however maybe throughout the ranks you know of going from grade to grade to grade. It became lost by the time I got to high school when I started to work with square numbers where it was just like yeah we just hit this button and now we move on and and it was like this wonder is kind of gone and I feel like that happened to me is that maybe there were some great wondering moments in elementary school but by the time you get to high school it feels like and this is why Kyle and I create the podcast initially is because it feels like high school we're just kind of crunching through content where it's not exploring these concepts that connect so far in the future to higher level but also so in the past to younger grades I really love that story you just shared. Hey and you mentioned Pythagorean theorem and another math moment for me but this was for me not for me with kids it was a math moment for me but it was also math moment for Eric one of my own was when I opened my eyes to the Pythagorean theorem you could put any shape side triangle as long as they're similar it works out it doesn't have to be a square you know how many years have I thought that it had to be a square the only way you could do that kind of thing and not understood the basic mathematics you might keep going on this yeah for sure so will you keep breathing me correct? I referred to group and I had an email conversation where one of them wrote and said why is it that we found in our research and I'll simplify his research finding but the amount of preparation of kids coming to kindergarten these are very small kits this is a research group that works with very small kits did not depend on the educational level of appearance and he thought it would it doesn't almost everything else literacy naming letters language ability and stuff like that education level of parents gives the kids a resource that not all kids share equally it's a shame but there it is and he wrote back and said does anybody have any ideas and I wrote back saying yeah the US education system misrepresents and misteaches math to so many with little emphasis on conceptual understanding that the educational level parents mean not have the same kind of influence and he wrote back and said I guess so but come on this is just three and four year olds were talking about they understand the math for three and four year olds and I wrote hmm no I don't think so look at fractions as an example many a high school teacher who knows fraction procedures expert was consult problems applies right from knowledge health brake manipulations are pre and these teachers are pretty dismissive I gotta say a early childhood educators and elementary educators they know the mathematics but ask them why the procedure for the division of fractions work can you show me with a model can you give me three real world examples of division of fractions you often see that these teachers and adults similarly who know their mathematics can't justify them or model them with concrete materials that is for teachers they may know but they not have the basic foundational understanding that can exist in well taught intermediate grade students yeah what you're sharing is something that is always always top of mind for us and that is one of our major pieces when we share and we talk about a three part framework like we're all about curiosity but the second piece is fueling sense making to our three part framework and it's just that and John and I both came to realize much later in our career than I wish we ever realized I wish we knew it going in but we knew how to crank through problems procedurally but we did not understand and we use that division of fractions as a great example another great question that's really interesting is you know to ask middle school teachers like what is a proportional relationship to define it to really get an understanding or to ask someone about division and a lot of people will define division as one of the two types of scenarios like partitive or quotitive usually if you ask a teacher of a younger student they tend to kind of default to that fair sharing right that part of the scenario and then we head into this long division world and all of a sudden we do the guzzenthing you know that whole quotitive scenario and as you mentioned like to me that makes perfect sense why the education level of parents is probably not that important is because so many are either procedural or maybe didn't like math even though they have a high level of education right or didn't feel successful as a math student growing up in that procedural world so that definitely connects to us so much so and I think it leads us to dive into this conversation a little deeper I'm wondering if that's the case because I think back to parents and most parents I think they treat early math education similar to like the alphabet but all of it's the alphabet it's like you've got to learn your alphabet you've got to learn how to count and we're going to have you memorize some basic facts but not that conceptual piece and that to me is so important and I wonder if someone's listening and they're saying okay I have young children at home or maybe I teach young children and I'm not sure about the conceptual understanding especially with young children in the early years what would be important for parents and educators of young children to add mathematics to their I guess their regular diet at home like are there any tips any suggestions where they might start to kind of dive in here yeah that's a great one if I can still keep going with what I was saying before to get myself there absolutely I'll give an early childhood example Julian and I in our learning trajectories for counting I always have to talk about counter small numbers where kids know how to recite the numbers counting numbers they also know how to finally keep one to one correspondence between reciting those numbers and objects they are counting but only at this counter small numbers level do they understand that the last counting word when you count a set one two three four five the five tells how many now you might say wait college educated people can count they know that the last word tells how many they understand no we ask them we talk to them well I guess so but they don't understand that as a conceptual principle they haven't reflected on it they don't notice when kids don't know it because they just add is given they don't understand that you move from an ordinal kind of counting one two three four to a cardinal or how many this kind of understanding as a clear conceptual shift that's a different kind of understanding number than counting as performance just like you just said the fractional kind of division or procedures are a different kind of understanding from why that works mathematically so we really work to try to provide parents and teachers early childhood teachers parents families with resources that'll help them with both here's some cool places to start here some cool activities and here is why those are important so find to get into your question here where do they start if I can suggest a resource and then maybe we can give some examples of it julie and I over the decades that we develop building blocks and professional development for developed a website that taught teachers learning trajectories and we are funded by the bill and the linda gates foundation and the heisen assignments foundation to make that free to everyone not tie to the building blocks copyrighted materials which the original website was original website was 20 year old it finally had to go it's a security risk at this point 20 year old code it's a little outdated so the learning trajectories dot org website learning trajectories one compound word dot org provides a view of kids development and then what you can do to promote that development it teases it out by age so if you've got a toddler you can click on that age and it will highlight the kind of levels for each of the topics that's there we don't quite have what we want I realize which is what we're trying to get to is that you put in information about your kid or kids and it will guide you through a full sequence we're working on that but we're not there yet but it does try to provide videos of kids thinking and a little video of us talking about the mathematics because to us a learning trajectories three parts right there's a mathematical goal you have to understand the math there's a developmental progression you have to understand the levels of thinking through which kids pass and learning that math and then there are the instructional activities approaches strategies and the like that help you move from each level to the next so that's what's provided on the website and what can people do you're asking me I think mostly at home at this point so becoming more aware of all the mathematics that are around people and becoming aware of how that mathematics develops but really the depths of that mathematics like your fractions example like Elge, for example, like last member counting a kind of example, you really get a richer picture of this. So we provide on the website and through publications and everything an attempt to give people all those parts but we realize especially for parents sometimes it comes because we get a more rich activity. Some of the activities are right online so they don't have to do much but help the kid get online and do it and then they'll notice the interesting things kids can do that they didn't think. Some activities again they have to do with the kids but we try to design them so that it makes salient the kind of mathematical thinking of which kids are capable. And some of the activities you know we try to do little videos that explain why this is important and the like. So what kind of activities? I mean rather than a geometry rather than just holding up a square or a triangle all his base horizontal and saying this is a triangle what is this a triangle? We try to go way beyond that to look at geometry and spatial thinking dynamically. So we do a lot with shape composition where kids are putting together different shapes to make pictures first and then ultimately to understand the shapes can compose to make units of units to make other shapes. We do a lot with spatial visualization where kids make mental images in a game called snapshots where the metaphor is take a picture with your eyes but I'll only show you this composed set of three shapes for two seconds and then you have the same shapes but you have to put them together to match that so to build visual imagery of these things in the like understanding more about that is really important so people see the richness of it otherwise all of us all of us me included tends to default back to the math we were taught which means oh I better get a worksheet or something. Yeah put facts on a page and do that and we're trying to show them that a lot of the parents a lot of the mathematics you missed a lot of the reason you don't even like math that much or think you're no good at it is the fault of your experiences and Julie has this wonderful phrase she always tells teachers and parents we have to stop the cycle of abuse we have to stop saying rectangles have two long sides and two short sides when math is one of the worst definitions at all of mathematics you've opened my eyes for a lot of different things but I really appreciate you sharing the resources and where people can go to learn more about this but getting at the deeper level of conceptual understanding I think is so important and I think when you said parents you know like I'm just going to go back to a worksheet because that's what I was used to like I had that exact thought with my own daughters I wanted them to have a more of a rich mathematical experience at home and we were told this is how you become a good parent and a good educator as a parent is that you had a read to your sons or daughters at night before bed and we did that we got to follow that every night we're going to get a book we're going to read together before we go to sleep and then I remember having a book by Christopher Danielson called which one doesn't belong a shapes book and we were oh I loved them yeah it's such a great book isn't it and with that book like we would lay in bed after we read a story we're like you know what let's pull his book out we'll do one which one doesn't belong page and we just share our thinking why we think that doesn't belong and it would open that discussion up to talk about so many things that we'd normally not talk about that book is is you know Doug is pretty sure in the sense that you wish you had more yes yes yes so I was like oh I'd love to get some more so I asked my daughter at the time and I said well what are you working on in school like what does your book look like and so all I get is when I look at like let's continue these rich conversations at night every single night and the sheets were just kind of worksheets on adding or multiplying or whatever it was at the time it wasn't a discussion starter it wasn't a rich kind of conversation that we can have about it where it like the shapes book and actually I started to create my own it and it ended up being math before bed the website that I've used with my own daughters is that it was just a collection of rich pictures of things around the home or which one doesn't belong or just a prompt to get us to have a discussion before we went to sleep to talk about math numeracy and connect it to literacy really but I really appreciated you saying that because I think it's so important to name to understand as parents the conceptual understanding behind these things so that we can have those deep conversations with our kids absolutely and your guys resources in website obviously I assume people who listen to these podcast know about them all but those are so valuable to change people's view of what mathematical thinking is about everybody knows that the goal of literacy and language is to speak and listen and think and read and write and things like that and mathematics it's like fill out of worksheet compute at the right answer and I love a physicist I think and the name escapes me I said years and years ago if we ever taught literacy and language like we teach mathematics kids would spend all their time up to grade 12 and never read a story and never write a poem you know we've got to change that view there's so much creativity and interest and thinking and mathematics you know mathematics predicts school success early mathematics kindergarten entries mathematics predicts later school success better than any other factor they found executive function social skills reading ability nope mathematics comes out on top it's not like things aren't related and all those are important but we deny the importance or we minimize what's interesting about mathematics to our peril because it's also a way of thinking that's fundamental to all domains you raise so many interesting points there in particular this idea and this is something that we've mentioned as well the idea of how highly correlated success in early math and then success in school in general including in literacy and reading I was shocked by that but then when you actually think about it it seems so obvious but the way we teach math typically or at least in kind of the old North American style that we would do things procedurally we're robbing so many students of the opportunity to have that success simply because it is so procedural and it really only leaves the door open for some students who are able to memorize and maybe recognize a pattern to kind of get them through but early on the part that I wish I made a comment on earlier you had referenced that experience with that young kindergartener who was working with the square numbers and I think one of the biggest shifts that we can make is just by asking kids not only to construct their own learning like you were doing there but to give them the opportunity to reason through problems and we tend to do less and less of it with every passing day with every passing grade it becomes more procedural so you bring up this idea of mindset and we've heard about mindset from so many great researchers and educators Joel Boehler obviously comes to mind for the math moment maker community I'm sure and those negative mindsets are something that we obviously need to overcome and we appreciate the positive comment there about our goal here at MakeMathMulments is to try to do some of those things what sort of research might you be able to share with us that kind of links between this mindset and I guess do you have any research about that parent mindset and how that may shift what they do with their children at a young age around mathematics and I guess sort of the response would be how does that impact their future success in math and maybe in some other subject there is as well yeah it's so important the kind of discouraging research is that especially parents of children who come from low resource communities and don't have the mathematical charging stations in their homes community in schools that other kids are pretty too part of the lack of resources that they have is they have a lack of vision because of their own experiences of this kind of you and mathematics that you just so eloquently described and thus what they think is that I'm not very good at math so I shouldn't talk and kids will learn that in school and so they're even less of a emphasis on the mathematics in the home less of an intentional emphasis they might read to their kids but they're not going to mess up the kids on math because you know math is one right way of doing it yet if you don't do it that way you're wrong so they don't want to mess up their kids so that's unfortunate the good news is that with resources and it can be training programs it can be websites such as yours it can be a lot of things you know that everybody's making it really can change their view of the mathematics it can give them games and activities that like I say sometimes you need to dive into it and come in at the ground level you two probably know the theory of inhalase where they say learning geometry you start a visual level and then you start thinking about the properties of of shapes and then you can become logical with them. In a similar way, it's a big conceptual or metaphorically, but in a similar way, I think teachers and parents and all sometimes have to start at the visual level. Do this with your kids. What do you notice? Now let's think about it and talk about the mathematics. Coming at them saying, "Your math is incomplete. I'm going to teach you all this mathematics." And then finally you can go teach your kids. Probably not going to be a motivating or necessarily even the most understandable approach. If I'm a listener right now, Doug, or an educator or a parent, where might someone go to learn more about what you're offering? You and Julie, and I know that you've already mentioned learning trajectories.org. Is that where everyone here should go right next to learn more about what we're talking about here? Is there a book that we can be sharing out here so that they can say, take that on and read and deeper into these ideas of, "What would you recommend as people's next step after listening to this episode?" Yeah, I think the learning trajectories are a lot to offer because what we've tried to do again is not just give words, words, words, but videos of kids at each level it's thinking. So you can say, "Yeah, that's kind of the way my child or my children, my students or whatever think." So I can start there and move to that next level. And then you can see videos and write-ups and resources for kids at the next level. We do have a book that's out too. Can you believe that I had a book across the room to figure out what it would be like? That's awesome. You probably be referring to learning and teaching early math. I have the learning trajectories from which everyone around me calls the blue book. So that doesn't help us remember the precise title. I just finished it last week, the third edition of it, but it'll be a good year before that one's out. And that's the one I featured your website in, guys, which is always at much communication first, you know, because I loved your website. It's so consistent with our approach and everything we've been talking about today, which is a new vision of mathematics go forward. And what a great book is. Well, the picture in my mind is it is the blue book I did before the episode. I had to relook it up because I didn't want to get, you know, sometimes I put teaching first learning for I don't I don't remember what the order was, but it's a great resource. And also I do want to speak to the learning trajectories dot org website. We have used it in our district quite a bit. I've used it obviously for my own personal learning and trying to better understand the progression, the trajectories that students work through in the early years, but also that website is just fantastic to be able to move through each stage. If you're sitting there thinking you go to learning trajectories dot org and you start, it's great. You can start wherever you'd like. But let's say you start at the beginning and sometimes the name of the stage might be a little daunting because you may have never heard it described in that way. But there's a video there. It will show asking questions to students. So you can see what it looks like and what maybe it doesn't look like. And then also some activities that you can actually try for your own classroom or with your own students, your own children at home. I guess parents right now during the COVID situation are saying, yeah, they are my students at home now. So that's a great resource for people to lean on as well. So at this point, Doug, we want to be so respectful of your time. We really appreciate you taking the time. We also appreciate you supporting the work that John and I are trying to do it in the math community as well. So keep on doing the great things you're doing. I hope Julie next time we get you on, we'll be able to join you. I know she's on the road right now. And we just want to wish you an awesome day. We'll include all those links and resources and we'll also share back when this episode goes live. So that's you and Julie can listen and share as you see fit. That would be great. And Julie would be happy to. And once you have Julie on air, you'll not come back to me. I can promise you. But I love make math moments. I think the work you guys are doing is really, really important. I appreciate being a little part of it. Well, in YouTube. So thanks so much for saying that. All we can't thank Doug again for spending some time with us to share his ideas and insights with us and you, the math moment maker community. Also, if you're liking what you're hearing, please share the podcast with the colleague and help us reach a wider audience by leaving us a review on Apple podcasts and tweet us at Makebeth Moments on Twitter, Instagram and Facebook. Show notes and links to resources from this episode can be found at makemathmoments.com/episode103. That is makemathmoments.com/episode103. You can also find full transcripts there and download them right to your computer. So we've got transcripts and notes all at makemathmoments.com/episode103. Well, my friends, until next time, I'm Kyle Pierce and I'm John Orr. High fives for us and high fives for you. How can you take your district math program to the next level? Take the makemath moments 12-minute assessment and it will reveal what is going right with your district math program and what needs work. At Makemath Moments, we believe that an effective mathematics program should be developed like a strong, healthy and balanced tree. The trunk represents leadership. Without strong and capable leaders, nobody knows the vision for the district, why it's important or how to make the vision a reality. The roots of the tree represent mathematics, content knowledge and what it means to be mathematically proficient. If your district math program doesn't consist of educators with deep mathematical content knowledge and promote instruction that develops all five strands of mathematical proficiency, the tree will not get necessary water and nutrients to thrive. Like a tree requires soil, water and sunlight, your district mathematics program requires a productive educator mindset and the belief that all students can achieve at high levels. 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If a mathematics program doesn't have the necessary resources to do the work, the tree will starve, wilt and eventually die. Are the six parts of your district mathematics program developing perfectly? Does your tree look like this? Or this? Our free assessment will help you grow a strong, healthy and balanced mathematics program that works for all students in your district. Take the assessment and start growing your mathematics program today. And to take that assessment, head on over to makemathmoments.com/grow. That's makemathmoments.com/grow. And when you know how to strengthen those six parts of an effective math program, you're going to create a program that grows strong and wide.