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 better get a worksheet or something." We're trying to show them, a lot of parents, a lot of the mathematics you missed or think you're no good at it is the fault of your experiences.

Kyle Pearce: 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 U.S. President's National Mathematics Advisory Panel. And he's a co-founder of learningtrajectories.org with his partner, Julie, Sarama.

Jon Orr: 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 early years mathematics and how parental mindset is a key in a student's progression of mathematics. Hey, cue up that music. (Music playing).

Kyle Pearce: Welcome To The Making Math Moments That Matter Podcast. I'm Kyle Pearce from tapintoteenminds.com.

Jon Orr: And I'm Jon Orr from mrorr-isageek.com. We are two math teachers who together-

Kyle Pearce: With you, the community of math moment makers worldwide, who want to build and deliver math lessons that spark curiosity-

Jon Orr: Fuel sense-making.

Kyle Pearce: And ignite your teacher moves. Jon, are you ready to dive into this awesome chat with the ever so friendly ever so awesome, Doug Clements.

Jon Orr: 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.

Kyle Pearce: 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.

Jon Orr: 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. I cannot recommend this podcast enough. Jon 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 for Nova Scotia.

Kyle Pearce: 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.

Jon Orr: 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.

Kyle Pearce: 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 onto this fantastic conversation with Doug.

Jon Orr: 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.

Doug Clements: 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 in a week. And working out at my home where I'm talking to you from now.

Kyle Pearce: Oh, fantastic. We are, as Jon 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 thorougher, 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?

Doug Clements: I'll take the second question. First as I put, I was in early childhood and 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. inaudible 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 the now deceased inaudible well, workshops too was one at 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, preschool for one year and just became fascinated with younger and younger kids all the time and 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 pre-schoolers and everything, and then kept my interest in elementary for sure, but now have extended it to birth through grade six in a desire both 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.

Kyle Pearce: Interesting. Yeah, that's a great intro into what your role looks like and how you even got into that role. And I'm eager just, when you said van around the country, 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 or 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?

Doug Clements: 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, NSF, put out a request for proposals for early childhood years. Then Julie and I, who had met actually writing the elementary math curriculum, my wife and colleague 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 had experience and the like, that seemed interesting. That started things off. More than two decades ago, we were funded by NSF to do a preschool curriculum.
Originally it was supposed to be preschool through grade two, but they 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 Julie and I decided technology, we'll do that preschool through grade two 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 blocks project of preschool curriculum for mathematics that then dominated Julie's and my life right up for decades afterwards. It got us excited about it. We were funded by the US Department of Education's Institute of Educational Studies to do what's called efficacy trials.
That is inaudible under good conditions, you get a good statistically and practically significant effect of your approach we did and then scale up, can you bring it to an entire district? And we did. And each of these 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'd better stop and have you ask a question because I can go on like this forever. But then we developed tools for professional development and had 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 sometimes thousands of studies of people from early childhood math, cognitive psychology, developmental psych, and other fields, where we put that research together and then test these so-called learning trajectories. So now we've developed a really birth through grade two or three and have done research with them, still developing tools for them, and the like.

Kyle Pearce: 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, Jon 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 going right out of those elementary grades, right into those preschool grades or ages. And it was the same for us as well. 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, starting from this, 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 learningtrajectories.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 Jon 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.

Doug Clements: Boy, that is so cool. I will probably think of many, many, especially after you hang up with me.

Jon Orr: That's the way it goes.

Doug Clements: I'll probably be like, "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 preschool, but that was with my sister and there was plenty of guys. This was my first I'm in charge alone job. And then I had already been through this project, Vanguard. I already had thought a lot about math. I started teaching with Math Their Way, like I said, but this just hit me. I had this really bright boy, he was called Eric and he loved school. 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 noticed inaudible of course I said. And then he comes back to me and he says, "What is this?" And he points to the square key. And I said, "Oh boy, Eric, that's really complicated to talk about. Well, it gives me really big numbers." Yeah, can you explain it. "Okay, Eric. Here, let me try this, take these tiles, make 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 got to get ready for all the other 23 kids. And he comes back and says, "Okay, 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 said. So press two and then the square key. Oh! Yeah, that's how many are in there. Well, try that 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 this 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 comes back to me with this long chart, two and four and three and nine and way up to 10 and 100 and beyond. And goes home, tells his mom, it made me think, holy macro, I was willing to say a bright kid, this was beyond them, not necessarily, right?
And to me, that has fundamentally changed my view of what kids could be capable of and why we should always keep our minds open to the possibility that if done in a way that makes sense to them, I'm thinking back to old Seymour Packard, who said, when he talked about gears and how gears 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.

Jon Orr: 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, because this is going to be two pieces to this comment that I want to make to this because you're making awesome connections with this student who, 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, when I hit the squared button or just when I square numbers, and this became more clear when I was working with completing the square in algebra and in expressions that we were actually dealing with squares in the areas of squares, like the Pythagorean Theorem, we said that over here before on this podcast, A2+B2=C2 is something I memorized in school.
However, it didn't occur to me that we're really talking of the area of squares off the triangle. It blew my mind and I really love the fact that you had this conversation with a kindergartner and it goes to show this point that you just hit home is a lot of the times we don't give kids credit for having deep level thinking on concepts, just because, "Oh, that's a high school concept or that's a grade eight concept." It's mind boggling that we don't have more of those. This is the other part that I wanted to touch on is that I had some really great elementary school teachers and it's possible that I did have that conversation a long time ago, but whoever, it may be throughout the ranks 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 it was like this wonder is gone. And I feel like that happened to me is that maybe there was 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 created the podcast initially is because it feels like high school we're just crunching through content, we're not exploring these concepts that connect so far in the future to higher level, but also in the past to younger grades. I really love that story you just shared.

Doug Clements: Hey, and you mentioned Pythagorean Theorem and another massive moment for me, but this was for me, not for me with kids, it was a math moment for me, but it was also a math moment for Eric. One of my own was when I opened my eyes to the Pythagorean Theorem, you can put any shape on the side of a inaudible triangle, as long as they're similar, it works out. It doesn't have to be a square. 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 mind me keep going on this?

Jon Orr: Yeah, for sure.

Kyle Pearce: Oh, absolutely, you keep riffing my friend.

Doug Clements: inaudible 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 kids, there's a research group that works with very small kids, did not depend on the educational level of the parents." And he thought it would, it does in almost everything else, literacy, naming letters, language ability and stuff like that. Education level of the 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 U.S. education system misrepresents and misteaches math to so many with little emphasis on conceptual understanding that the education level of the parents may not have the same cognitive influence." And he wrote back and said, "I guess so, but come on, this is just three and four year olds we're talking about, they understand the math for three and four year olds." And I wrote, "No, I don't think so. Look at fractions as an example, many, a high school teacher who knows fraction procedures, expertless consults problems, applies fractional knowledge algebraic manipulations, these teachers are pretty dismissive. I got to say of early childhood educators and elementary educators, they know the inaudible."
But ask them, why does 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'll often see that these teachers and adults similarly, who know their mathematics can't justify them or model them with concrete materials, that is procedures, they may know, but they may not have the basic foundational understanding that can exist in well-taught intermediate grade students.

Kyle Pearce: 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. We're all about curiosity, but the second piece is fueling sense-making to our three-part framework. And it's just that and Jon 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, to ask middle school teachers, 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 part of, or quoted of usually if you ask a teacher of a younger student, they tend to 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 goes into thing, that whole quoted of scenario. And as you mentioned, 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 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? Are there any tips, any suggestions where they might start to dive in here?

Doug Clements: Yeah, that's a great one. If I can still keep going with what I was saying before to get myself there?

Kyle Pearce: Absolutely.

Doug Clements: I'll give an early childhood example, Julie and I, and our learning trajectories for counting always have to talk about counter small numbers where kids know how to recite the numbers, counting numbers, they also know how to finally chip 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 it as a conceptual principle, they haven't reflected on it.
They don't know this when kids don't know it, because that 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 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 will help them with both. Here's some cool places to start, here's some cool activities, and here is why those are important. So finally getting to 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 developed building blocks and professional development for developed a website that taught teachers learning trajectories and we were funded by The Bill & Melinda Gates Foundation and The Heising Simons Foundation to make that free to everyone, not tie to the building blocks copyrighted materials, which the original website was. The original website was 20 year old, it finally had to go, it's a security risk at this point. A 20 year old code is a little outdated. So the learning trajectories.org website, learningtrajectoriesonecompoundword.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 trajectory is 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 depth of that mathematics, like your fractions example, like algebra example, like by last number counting 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 realized especially for parents, sometimes this comes because we give them a 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, we try to do little videos that explain why this is important and the like. So what kind of activities, I mean, rather than in geometry, rather than just holding up a square or a triangle always 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 spacial 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 inaudible can compose to make units of units, to make other shapes. We do a lot with spatial virtualization where kids will make mental images and 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'll have the same shapes, but you have to put them together to match that. 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 better get a worksheet or something. Put facts on a page and do that." And we're trying to show them, 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 that is one of the worst definitions in all of mathematics."

Jon Orr: 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, "I'm just going to go back to a worksheet because that's what I was used to." I had that exact thought with my own daughters. I wanted them to have 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 have to 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-

Doug Clements: Oh, I love that book.

Jon Orr: Yeah, it's such a great book, isn't it? And with that book, we would lay in bed after we'd read a story. We were like, "You know what, let's pull his book out and we'll do one, Which One Doesn't Belong page." And then we just share our thinking of why we think that doesn't belong. And it would open that discussion up to talk about so many things that we would normally not talk about. That book as you know, Doug is pretty short in the sense that you wish you had more.

Doug Clements: Yes.

Jon Orr: So I was like, "Well, 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? What does your book look like?" And so all I get is like, "Let's continue these rich conversations at night, every single night." And the sheets were just worksheets on adding or multiplying or whatever it was at the time, it wasn't a discussion starter. It wasn't a rich conversation that we can have about it like the shapes book. And actually, I started to create my own and that's what 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.

Doug Clements: Absolutely. And you guys' resources and website, obviously I assume people who listen to these podcasts 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. In mathematics, it's like, fill out a worksheet, compute, get the right answer. And I love a physicist, I think and the name escapes me, 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."

Jon Orr: Wow.

Kyle Pearce: Oh, boy.

Doug Clements: We've got to change that view. There's so much creativity and interest in thinking and mathematics. Mathematics predicts schools 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 of 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.

Kyle Pearce: You raise so many interesting points there. And 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 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 kindergartner 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, Jo Boaler, 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 Make Math Moments is to try to do some of those things. What sort of research might you be able to share with us that 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 areas as well?

Doug Clements: Yeah, it's so important. The kind of discouraging or searches that especially parents of children who come from low resource communities and don't have the mathematical charging stations in their homes, community, and schools that other kids are privy to, 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 view of 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 there's even less of an 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 has 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 that everybody's making, it really can change their view of the mathematics and 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 van Hiele's where they say learning geometry, you start at visual level and then you start thinking about the properties of shapes, and then you can become logical with them.
In a similar way, it's big conceptual or metaphorical leap, but in a similar way, I think teachers and parents 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 is probably not going to be a motivating or necessarily even the most understandable approach.

Jon Orr: 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? 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 in deeper into these ideas? What would you recommend as people's next step after listening to this episode?

Doug Clements: Yeah, I think the learningtrajectories.org has 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 of thinking. So you can say, "Yeah, that's 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 it? I got to go across the room to figure out what the title is.

Kyle Pearce: That's awesome. You'll probably be referring to learning and teaching early math.

Doug Clements: That's it. The learning inaudible approach, which of course, everyone around me calls the blue book, so it 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 inaudible communication first, 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 going forward.

Kyle Pearce: And what a great book as well. The picture in my mind is it is the blue book. Before the episode, I had to re look it up because I didn't want to get, sometimes I put teaching first, learning for, 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.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.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 Jon and I are trying to do 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, will 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 you and Julie can listen and share as you see fit.

Doug Clements: And that'd 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. crosstalk. Thanks, 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.

Jon Orr: And you too. So thanks so much for saying that.

Kyle Pearce: All we can thank, Doug again for spending some time with us to share his ideas and insights with us and you, the math moment maker community.

Jon Orr: Also, if you're liking what you're hearing, please share the podcast with a colleague and help us reach a wider audience by leaving us a review on Apple Podcasts and tweet us @MakeMathMoments on Twitter, Instagram, and Facebook.

Kyle Pearce: Show notes and links to resources from this episode can be found at makemathmoments.com/episode 103. That is, makemathmoments.com/episode 103.

Jon Orr: 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/episode 103.

Kyle Pearce: Well, my friends, until next time. I'm Kyle Pearce.

Jon Orr: And I'm Jon Orr.

Kyle Pearce: High fives for us and high five for you. (Music playing).