Nicki Newton: This is what I tell people. The standard tells you this is what kids have to do, but the trajectory tells you what are different pathways that they can take to get there. The standard doesn’t tell you that. The standard says, “By the end of kindergarten, kids have to compose and decompose numbers within 10.” But the learning trajectory tells you there are five levels of composing and decomposing numbers, and what I think is fascinating about this is level three is [crosstalk 00:00:29]-

Jon Orr: Well, do we have a treat for you today. We’re speaking with the great and obviously passionate Dr. Nicki Newton. By the end of this episode, you’ll wish Nicki could just keep on going. She shares with us today her passion about learning trajectories and why we all need to learn them for various mathematical ideas.

Kyle Pearce: She fills us in on why we should be using running records and she gives you many knowledge bombs as well as resources during this one hour episode. Let’s get to it. Hit it.

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 engagement-

Jon Orr: Fuel learning-

Kyle Pearce: And ignite teacher action. Jon, we are ready for episode number 62 with Nicki Newton. Are you ready to get this thing going?

Jon Orr: Of course, Kyle, of course. We are super pumped to bring you this episode for sure.

Kyle Pearce: Awesome. Well before we get in there and start talking with Nicki, we want to thank you for listening to us where ever you are, 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 episodes and got some value out of it, we would love to hear about it. We read all of the reviews from this podcast from all over the world, and right now we want to share one of those reviews with you. This one is from [Tiger A 00:00:02:32] which I don’t know. Is that maybe a Canadian? Tiger A on Apple Podcasts?

Jon Orr: Tiger A? Yeah. Tiger EH?

Kyle Pearce: Yeah. Tiger A. What does Tiger A say there Jon?

Jon Orr: Tiger A says, “Love having a dedicated math podcast. I am so happy that I stumbled across this podcast. It is the best math PD I have experienced. I appreciate the vulnerability of the guests and wide variety of strategies discussed.”

Kyle Pearce: Awesome stuff. So again, a Canadian salute to Tiger A on Apple podcasts. Where are you listening from? Are you listening from the United States? Are you one of our wonderful math moment makers from Australia or New Zealand? We’ve got a lot of listeners there, or are you listening from a country that isn’t listening or isn’t well represented on the Making Math Moments That Matter Podcast yet? If so, we encourage you to go leave a rating and a review and we will be more than happy to share it on an upcoming episode.

Jon Orr: Our podcast here’s big why is providing teachers across the globe high quality professional development on your schedule. We often talk about our three part framework on how to create resilient problem solvers, and a must for us when developing lessons using the framework, is having our students discuss, reason, defend mathematical ideas. A few routines that we use regularly in our classroom help us start these things called math fights.

Jon Orr: No, they’re not a real fight. There are math fight. It’s a little better. Get your students actively estimating, questioning, discussing, and defending their insights, not with the teacher, but with each other. We’ve put together a one-page resource for you that shows you how to encourage discourse and discussion in your classroom while giving you the resources to make it happen. Be prepared to have your students argue about math. Head over to makemathmoments.com/math-fight.

Kyle Pearce: If you haven’t checked out the math fight resources, you definitely have to. So go ahead, go grab that cheat sheet, makemathmoments.com/math-fight.

Jon Orr: And now let’s get into our chat with Dr. Nicki.

Kyle Pearce: Hey there, Nicki. Good morning and welcome to the Making Math Moments That Matter Podcast. We are super excited to have you on the show today. Now that all of our technical glitches are behind us, we spent a good chunk of time together. I can tell your resilient. You could tell Jon and I are resilient. We did not let the technology take us down. How are you doing today?

Nicki Newton: I’m doing wonderful.

Jon Orr: It was a pleasure to see you at the CAMT conference, the Conference for Advancement Of Mathematics Teaching in San Antonio, Texas this summer. Nicki, could you help our listeners understand a little bit about yourself, where are you coming from and how did you get into education?

Nicki Newton: Well, I’ve been teaching for 30 years, 31 years, something like that. I started out as a bilingual teacher in California, bilingual Spanish, and I actually started out because I wanted to travel and sub and then I was a Spanish speaker and they were like, “Oh, we’ve got something for you to do. Do you want to hang around a little longer than a month?” I got into an emergency teaching program and I became a bilingual teacher, then I taught for 10 years and then I decided I wanted to get my doctorate. So I packed up my bags, moved to New York City and went to teacher’s college at Columbia and I was a literacy social studies person.

Nicki Newton: Then one day, an Australian company came to town and they were hiring literacy consultants, and so my friends were like, “Oh, go get a job.” So I went down there and then that day they were like, “All literacy is closed. We’re only doing math.” And I was like, “I hate math. I don’t want to do math. I can’t believe this.” And then they were like, “That’s all we have.” So I was like, “Okay, I guess I’ll do it for a year, and then I’m out.” And then the rest is history. I fell in love with math. All my friends say I’ve come to the dark side.

Kyle Pearce: Oh that’s so fantastic. You’re actually … Your story, we had Andrew Stadel on a little while back and there was some similarities there. He was supplied teaching and he was playing in a band. You wanted to do some traveling, you get in there and then all of a sudden, you’re falling in love with math, which I think is exactly what keeps us there, right? Is trying to help make sure that students are falling in love with math as well.

Nicki Newton: Yeah.

Kyle Pearce: I’m wondering Nicki, at first it sounded it wasn’t your dream job in your mind. I’m wondering maybe you could share a little bit of your experience from math class. We always ask people about memorable math moments from their own mathematical experience. Sometimes they’re positive, sometimes they’re not, but I think it’s really important that we hear about them so that we can all learn and grow from them. So do you mind sharing one with us?

Nicki Newton: For me, my dad was an engineer. So my dad loved math, he could do math and he taught me how to do math so I could always do math, but I didn’t understand what I was doing. I just knew how to get the answer, so math was always very frustrating for me growing up. I did not like it, I could do it, my dad could help me every night with my homework, so I didn’t like math. I loved literacy and history. And then when I was going to be a math person, I was like, “Well I guess I better learn some math, not just how to do it, but how to explain it and how to teach people how to do it.”

Nicki Newton: And so that’s how I fell in love with it, actually learning how to do it and what it means and how you get the answer and how you help kids to learn it, then how you help teachers to learn it. Because really, in the elementary school, a lot of teachers are uncomfortable with math. So I fell in love with learning how to do it when I learned how to do it in ways that made sense.

Jon Orr: Nicki your experience is very interesting to me because I was a high school math teacher and I had the experience of always being quick to the procedures and understanding the math very well. Actually I shouldn’t say very well because I was the big memorizer and the rule follower and I think that had an effect on me as a teacher because I went into teaching math, not like you. You went in thinking about like, “I have to learn this well and understand it,” whereas I thought I understood because I would grasp concepts by those procedures in high school very easily, but I think that hurt me as a math teacher early in my career because I was like, “I know the way to do this,” but I didn’t.

Jon Orr: I think you had this nice perspective of coming at it from the other end, which is what most teachers, especially in the elementary field, are coming in. They’re either scared of math or weren’t sure exactly how will all this math fit together and I think you’ve got to unique perspective and advantage in your career for sure.

Nicki Newton: Yes. I totally agree with that. I came in like, “All right. How do you actually get people to understand this? How does it make sense? What are the models?” One of my really good friends, Christine King, she was a middle school teacher and I would say, “Okay, I know how to do this, but I don’t understand how you model it in a bunch of different ways,” so she would teach me. So I think mentors are invaluable.

Kyle Pearce: Yes. I was just going to ask you that. So it sounded like right away, as Jon was talking and just reflecting on your experience, I was just thinking to myself, “Okay, so I bet there’s a bunch of people who are listening to this who are either new to teaching or maybe new to teaching math,” or maybe they’re like Jon and I who it took us a long time to realize what we didn’t really understand. Where might they start? Is going to a mentor and trying to go to someone with a little more expertise in a building, is that somewhere where you think might be a good place to start?

Nicki Newton: I think as elementary teachers, we have to do a lot more of, and I do this a lot as a consultant, we need to do a lot more content PD just in general and say, “Let’s learn a lot about this thing,” and to really give people the opportunity to stretch their own pedagogy and to say, “I’m not so comfortable with this.” I’m in a PDs a lot where the kindergarten teachers are like, “Oh no, you’re in third grade math. We’re out.”

Kyle Pearce: In reality, it’s like you’ve got to understand where kids are coming from and where they’re going. Jon and I talk about this all the time in our online workshop about, “How am I going to be an effective kindergarten teacher if I don’t understand how they learn early on or where they’re going?” These are really important details there and we’ve talked about this as well. I tip my caps to all elementary teachers because you need to know a lot about a lot of different things that aren’t connected like history and math or literacy and math even. You need to learn a lot of things and that’s a lot of hard work. So I really appreciate you saying that.

Nicki Newton: It is, and well more and more schools are going towards departmentalization so that you have people at elementary schools who want to teach math teaching it, which is really a good thing because there’s some people that just don’t want to teach it, but we should all be numerate. Whether we’re teaching math or not, whether we’re in a departmentalized school or not, we all have to get much more comfortable with math. I think that has become part of my mission in life. My tagline is always happy mathing because I want people to like math. I want people to think, “Even if I’m in a departmentalized school, I need to know some math.” Math is fun. It’s a fun thing to do.

Nicki Newton: We wrote a book recently, I wrote a book with a school district called Mathematizing Your School. What I loved about the book is I went into their school and they were doing all this fun stuff and I was like, “I want to write a book with you guys. I don’t want to write a book about what you’re doing. I want to write a book with you.” So it was nine teachers and they all wrote chapters and they have mathematized their district. It’s Pasadena school district in the Houston area. When you walk in their school, they have mathematicians row that has Boggle and Tic-Tac-Toe and What Doesn’t Belong, all down the hall in big forms. They are living and breathing and loving math. Their goal in their district is that people love math, and so they are trying to shift the discourse.

Nicki Newton: I see that as something that I want to do. I want people to love the math. I want people to think math is fun. I want people, and there’s a book written by Charlesworth and she says in the book, the little boy goes, “How was it?” And the other little boy goes, “It was hard fun.”

Jon Orr: Hard fun.

Kyle Pearce: Yeah.

Nicki Newton: And I want that for the kids. I want kids to be having hard fun, enjoying it. And so I love the math. So here it is. I remember how long I’ve been teaching now, 31 years, because I did literacy for 15 and then now I’m on my 16th year of teaching math. So now I’ve taught math longer than I’ve taught literacy, but I brought a lot of my structures when I switched. So the first book I wrote was Guided Math because I knew a lot about guided reading. I was at Columbia where Lucy Caulkins is. So I wrote a book on guided math then I wrote a book on math workshop, then I wrote a book on math workstations because I knew a lot, not that they’re the same, but I knew a lot of the structures. And so I’m really interested in that … My degree actually is in curriculum design and teacher ed, so I believe there’s a lot of different ways to teach math.

Nicki Newton: There’s no one right way. I don’t think it’s a dichotomy. It’s either or. I think it’s always both and. I think there’s so many strategies and I think we as teachers, we need a knowledge base that is expansive just like in literacy. In literacy, everybody knows the research. Schools do whole book studies all the time on the research. In math, not so much. I want us to know a lot about math research because there’s so much good research out there but the way researchers write it, nobody wants to read it.

Kyle Pearce: Exactly. You need that sort of the middle people. I would consider you being one of those great people who takes that research and you bundle it into a nice present for someone to dive into. I just want to go back and talk a little bit about content specializing or departmentalizing. I’m really happy that you had mentioned first off that that’s a good thing, I think, if teachers are able to gain expertise in an area, but we can’t shut the door on those other subject areas. We are together trying to help students learn all of these different areas, all of these different subject areas, so we don’t want only a handful of teachers becoming experts in an area. However, I also think it’s really, really tough for a teacher to do a really good job teaching all kinds of different subject areas right?

Nicki Newton: Absolutely.

Kyle Pearce: I think there’s that balance there, just like you’re saying. It’s not a one or the other, but finding that happy medium I think is really important.

Nicki Newton: Yeah, and I think I’m really interested in it because I want to … I’ve started a research project where I’m going around interviewing people about it because I see a lot of people moving towards it. I am on the road probably around 200 days a year. I go to probably at least 40 States a year, and then I do work in Canada actually and in-

Kyle Pearce: Oh, Canada?

Nicki Newton: I do. I do a lot of work in Alberta. I’ve done other parts of Canada as well, but I do a lot … I go to Canada at least probably twice a year for a week at a time, but this is what I see in the States especially. I see everybody’s doing it. Some people are doing it from kindergarten, some people are doing it only in the testing grades. And so I am right now getting ready to start doing a bunch of research on what works about departmentalization, what doesn’t work about departmentalization because I want to do it like a voices from the field. What do you think? What advice would you give somebody that’s trying to do it? I don’t want to just do it from research papers, so I’m actually out there interviewing school districts that are doing it.

Nicki Newton: You get all kinds of really interesting answers. Some schools it totally works because the scores were really low and people didn’t understand the math and they didn’t want to teach it. Now you’ve got people that absolutely love it doing it. The kids love it and then everybody’s teaching at their genius, but then you have other schools where it’s not working. So we’ve got to figure out what’s working and what’s not and how do we fix it. I think, like I said earlier, regardless of whether you departmentalize or not, everybody at your school should have basic numeracy. As teachers, we should all want to have basic numeracy. I don’t understand why anybody would not want to have a basic level of numeracy.

Jon Orr: It goes to show, like you just mentioned, that in a departmentalized school, you’d want those teachers to show their passion if they’re passionate about that subject and it shines through to the kids. It’s sometimes hard to think about a teacher who is say not passionate about math and then now has to teach math.

Jon Orr: It makes me think about that school that you were talking about who it’s their mission was to, everyone was going to enjoy math. I think it’s so important whether you’re departmentalized or not, that everybody, like you just said, has to have some foundation and also get that out of their mindset that math is the punishment or math is this subject like I don’t want to do it. I’ll let you do it.

Nicki Newton: Right.

Jon Orr: That’s why you’re departmentalizing? Then something else has to change about the school. We can’t just say, “You do it because you like it.” We all have to have that say mentality. Maybe that’s why those schools aren’t working. It’s like that person’s doing it and I’m the naysayer over in the other grade. And so the other thing to think about is sometimes teachers are moving grades or schools. There’s a lot of fluctuation based on school numbers. I’m interested to hear your thoughts about that. You were studying like what is working, what’s not? Do you have any ideas on that?

Nicki Newton: There is the school of thought that, just in general, when you have adults working in their genius that you have a lot more productivity. I think if we celebrated math in general in schools, just as a school, if you’re celebrating math and then you have people that are working in their genius, if you just love math and science and you want to teach the math and science section and do things that nobody else would ever want to do with, [inaudible 00:18:47] go for it. I think there’s a really powerful stuff to be said for that model, and then other people are teaching literacy and social studies.

Nicki Newton: But I’ve also seen it fail miserably, and there’s also the developmental argument. It’s like some people say, “Well, kindergarten, first grade, second grade, you really shouldn’t do that because of these reasons.” But then other people are like, “Oh no. We do it from kindergarten up and we’re having all kinds of success.” So that’s why I’m so interested in it because it just depends who you ask, and there’s not a lot of research on it. There’s some, but not a lot, so I am in the middle of gathering what research we have on it and then out there interviewing schools in different states about how is it working, looking at models of success, because I think that’s really what we have to do.

Nicki Newton: At CAMT, I was talking about equity and one of the things that they talked when they talked about equity is that they say, Gutierrez says that, she says, “We spend so much time gazing at the gap that we don’t do anything.” We can’t just gaze at the-

Kyle Pearce: Right. It’s paralyzing.

Nicki Newton: Yeah. We can’t just gaze at the gap. Do something. Look at what can be done. Look at places that are closing the gap, because we do have people that are closing the gap. We have people that are succeeding. Let’s look at those models and figure out what they’re doing well and do that, right?

Jon Orr: Right.

Kyle Pearce: Right, right.

Nicki Newton: So I am really for that, like looking at who’s teaching kids math in ways that is succeeding and then how do we model that? How do we take some of that good stuff? Instead of just talking about how everything sucks and it’s not good, what is good? There is good somewhere. Somebody is doing good right now in this moment. I want to find that person and how do I replicate this or how does this inform what I’m doing here in my location? With kids that I work with?

Kyle Pearce: I think some of that gap gazing that you referenced, I think that’s so true. It’s a great expression, but I guess for the wrong reason, because we don’t want to be gap gazing. We want to actually do something about it. And then for districts, it’s like sometimes they just get paralyzed from how hard it is to change structural items or structural things, the way we’ve always done things. If I do this, then how’s that going to change staffing? How’s it going to do this? But at the end of the day, it’s like we do it all for the kids anyway, so it is so worth it.

Kyle Pearce: Then you had mentioned about the genius zone and I think that right there is genius because we want teachers in that genius zone, but I almost worry that some might interpret that maybe differently than you or I are thinking about genius zone because again, you could have that teacher that has learned procedurally their entire life and they were told they were good at it and then they’re going to perpetuate that speed and memorization and the lack of conceptual understanding.

Nicki Newton: My friend Annalise, she always says that speed and accuracy have hijacked fluency. I love that because it’s true.

Kyle Pearce: You are so right, and Annalise is actually a part of our Make Math Moments Academy. We have some great discussions in there and she’s a phenomenal, phenomenal math speaker as well as just has that expertise under her belt, so that’s fantastic. I love that expression. I want to go a little deeper here because I think all three of us have agreed that understanding the content is so important. What I’ve been promoting in my district and districts I visit is this idea of I don’t need a teacher to come out of their pre-service being a subject specialist, but what I do want is to give them the opportunity to gain expertise because they have more than one opportunity a day to teach mathematics. So I think at the core of that is content.

Kyle Pearce: Something I heard at your CAMT session that I really liked and you really focused in on the idea of counting and quantity. And at one point you had said how many different stages, I think you referenced them as stages, how many different stages of counting are there? I made a very intentional move to look around the room to see not only listening at my table about what people were saying but to listen around the room, because when I present about counting and quantity, we get all kinds of different ideas.

Kyle Pearce: Some are like, “What are you talking about? It’s just one, two, three, four. I’m there.” Other people are saying, “Well,” they’re including all kinds of things that aren’t even necessarily connected to the early counting. So tell us a little bit about that. Why are you so passionate about counting and why teachers have to understand the progression through how students learn counting?

Nicki Newton: I love this question. I love the learning trajectories. I don’t know if it’s possible to be in love with a learning trajectory, but I am.

Kyle Pearce: Oh it is. It is.

Jon Orr: Yeah. Yeah.

Nicki Newton: Why I love learning trajectories is this. Remember I have a literacy background. So what I know about literacy is that when teachers sit down to talk about where kids are, they have the language to describe what kids are doing. What I know about math is we say, “Oh well they can count and they can’t,” but that’s the end of the conversation. So what a trajectory does is it gives us the descriptions of what kids are doing in different spaces. The Clements and Sarama, everything. You can critique anything you want and say, “Well it’s this or it’s that,” but what it does is it gives us a trajectory. If you look at any of the NAEYC stuff, they’re mentioned a million times. They’re the grandfathers of it. Well, the grandfather and the grandmother of it, but I agree with that. It gives us a space to talk about what kids are doing.

Nicki Newton: It’s not to say that, “Oh my gosh, it’s lock step. You’re going to do the …” No. What it is is it’s a trajectory. It tells us these are all the things that kids can do when they’re counting. Now, given the live child that you have in front of you, what are they doing? Where are their strengths and where are their weaknesses? I think that everybody should know their trajectories and you know why? Because the more we know, the better we can do for our kids and so I love it.

Nicki Newton: There’s 20 levels. I taught for … I don’t know. I just learned about the trajectories three or four years ago. So I’m like, “How did I even teach without knowing these things?” Because now I can look at a child and I can say, “You know what? They’re doing really well here and this is where they’re struggling. I’m going to have to do some things to help them right here where they’re struggling.” Or when I’m trying to differentiate, they’re good descriptions that help you when you’re trying to differentiate for kids. It’s not that you’re saying, “Oh, he’s a level one and he’s going to be a level two next.” It’s not that. It’s a description that tells you where kids possibly could be because some kids are like they’re going, going, going and then they’re at level eight but no, they can do stuff on level 12 too.

Nicki Newton: What it does is it gives you a description that is a beautiful description of what the landscape looks like, and I think if teachers knew that more, just strategies. It is amazing to me. In the States, the common core has been out for … Not everybody did it, but most people did or did some version of it in their state. It’s been out for 10 years. Do you know to this day there are still teachers, and it’s no fault of their own, but there are still teachers that are like, “I don’t know the names of the strategies.” They don’t know the names of the strategies like doubles, doubles plus one, doubles plus two, make 10. If we’re trying to teach kids, you need to know those, you know what I mean?

Nicki Newton: And so I just wrote a book called Leveling Math Workstations, and a lot of people misinterpreted the title. I should have just called it differentiating math workstations because I’m not talking about leveling like, “Oh these are the bluebirds and these are the lizards.” I’m not talking about tracking kids. I’m talking about what are the descriptions, where are kids along a trajectory and how do we know how to move them, how to help them? How do we help kids? You know what I mean? And so that’s why I love the trajectories because they allow us to differentiate. They allow us to know what the trajectories are.

Nicki Newton: I also I’m really a big fan of concrete pictorial abstract and really looking at that cycle of engagement so that I know like, “Oh, I’m going to teach these guys with the rec and rec and we’re going to build it on the rec and rec, and then we’re going to draw it and then we’re going to take it to the symbolic.” But I also know that I’m not going to rush kids to symbolic because I need them to actually understand it conceptually. Most programs rush through concrete pictorial abstract. They rush through concrete and they’re rushing to symbolic.

Nicki Newton: Why? So you rush to symbolic and you get there and then kids have no understanding of what they’re doing. So all they know is, “Oh flip it and multiply,” but they can’t tell you a story about it because they don’t understand what it is. They can’t even really tell you if their answers right because they don’t know for sure. They just know the procedure. So we’ve got to slow down in order to actually speed up. You need a knowledge base.

Jon Orr: Yeah. We’ve said that many times we call it rushing to the algorithm, being more high schooly, the middle schooly for us, but we tend to, like you said, we skip all the concrete and go straight to the abstract and it’s a wonder why kids are like “What?” Where do these letters come from?” Why all of a sudden are we doing these procedures when they didn’t understand that, the concrete from before?

Nicki Newton: One of things also I wanted to say about learning trajectories and really understanding what those descriptions are is they allow us to pull guided math groups. I’m a big believer in guided math, not as a way of tracking kids because I definitely don’t think you should do that, but as a way of pulling a focus group and working with kids in their zone of proximal development, I think that’s really important. I think there’s lots of strategies for teaching kids and so I don’t believe in either or, I believe in both and, but I believe there’s room to pull kids in homogeneous groups and in heterogeneous groups.

Nicki Newton: Because I really believe that sometimes when you got a bunch of folks who can’t count to five and then you got a bunch of other folks who can count to a thousand, that sometimes you pull those folks that can’t count to five and you work on counting to five. And other times, you work in a heterogeneous group. I believe that there is room for homogeneous grouping, not as like, “You’re going to be in this group all year long,” but as a focus group of differentiation for a time. I believe after 31 years of teaching, a lot of kids and a lot of places, that if you pull kids in small guided math groups, teach them in their zone of proximal development, you can help them to get to where they need to go.

Jon Orr: I think especially about that, teachers get bogged down and be like, “Oh we should not do homogeneous groups so we’ll never do them.” Like you said before, the more you know, the better teacher you can be and the more strategies you know, the better teacher you can be. I think we get that tunnel vision. It’s like, “Oh I can’t do that because someone said something bad about it.” All we want to do is help kids, and if that strategy is going to help that group of kids or in that timeframe, then great. But yeah, who wants to be and all of a sudden like, “I’m in the eagle group for the whole year?”

Jon Orr: It goes back, I remember what I want to say about the learning trajectories too. I think you mentioned this too, but the biggest thing for me was seeing what a kid can produce or where they are on that trajectory allows you to be where you can help push them to go next. And if you don’t know that, if you don’t know that trajectory and you’re just maybe relying on a textbook, you’re just going to be like, “Okay, we’ll wait till tomorrow when I do the next lesson all together,” which is the homogeneous group.

Nicki Newton: Yes, you’ve totally named it. And that’s what I love about the trajectory because you then know exactly where kids are and you can say in your teacher mind, “This is what I need to do,” because this is what I tell people. The standard tells you this is what kids have to do, but the trajectory tells you what are different pathways that they can take to get there. The standard doesn’t tell you that. The standard says, “By the end of kindergarten, kids have to compose and decompose numbers within 10.” But the learning trajectory tells you there are five levels of composing and decomposing numbers. And what I think is fascinating about this is like level three is okay, kids are going to compose and decompose within four and five. Then level four is that kids compose within seven. So seven is like one of the benchmarks that you compose up to seven and then finally you compose up to 10.

Nicki Newton: So although the standard is to compose and decompose to 10, we now know what the trajectory looks like, and so we can talk about where kids are, where they need to go next, where are they really strong, where are they struggling? The trajectory allows you to plan better, and that’s what my book on leveled math workstations actually does. It says, “Hey, here’s some trajectories and then here’s some activities that you can do along the way for these trajectories.” It doesn’t say, “You have to be here and do this and then go here and do …” No. It says, “Here’s a whole variety of things that you can do with kids that are working at this stage of learning this thing.” Some kind of way in math ed. It’d be really interesting to do and people have done this, a study, teachers and ask them, “What did you learn in your math ed class?” Because most of them did not learn how to teach math.

Kyle Pearce: I know I didn’t. It was about classroom management and even that didn’t work for me. I came out and I felt like I didn’t know anything. That’s, a lot of it’s my lack of expertise, but I think something that just came out through the discussion, both in your comments and Jon’s comments about whether homogeneous, heterogeneous groupings and really like you were saying before, it’s not a dichotomy. It’s not a one versus the other. It’s a when is it appropriate. You’re referencing Nicki, specific math skills that we want students to be able to achieve over time and you’re targeting them through this trajectory.

Kyle Pearce: I think where some of the heterogeneous groupings that we’ve used in the past, the reason why those weren’t effective and why they were not so helpful for students’ confidence and so on and so forth is because like Jon said, you’re stuck in that group all year. And the reason why was likely because I as the teacher didn’t know how to help you in that group get to the next place along your journey. It was just like you were just there and I’m just going to give you stuff that you can do already. That’s not helpful at all. That’s that horizontal mathematizing. We’re not helping you push up and I can’t help push you up because the only thing I know is the next step is the standard, which you’re nowhere near the standard.

Kyle Pearce: So I think that’s super helpful. Jon and I talk about heterogeneous groupings and mixing students up. We’re typically talking about when we’re problem solving. We’re not looking at a very specific skill. We have a learning goal in mind, absolutely, but then from that problem solving, that’s where we can go, “Okay, wait a second. I’ve got three students over here that are at this place, and that’s where those small groups can be so helpful.” Going with heterogeneous grouping in a small group is not going to be very helpful because you’re going to have kids that are at very different places. So I’m hoping people at home are listening to this and going, “Okay, so there’s a time and a place depending on what we’re trying to do here in our classroom.”

Nicki Newton: Absolutely, and I think that’s just so important. I also think like, I love that you brought up problem solving. There’s so many cool things now going on with problem solving. You’re right. A lot of it has been in the literature forever, but people are actually saying this is what it looks like in the 21st century. So I love it because I think doing numberless word problems, doing three-act tasks, doing three read protocols, all of that is so wonderful and we need to have rich experiences, heterogeneous experiences with our kids. There’s also the traditional routine problems, the 15 levels of single-step, the five level of two step. I think that stuff’s really important because schema-based problem solving has been around forever, but we do know that it goes from simple to complex.

Nicki Newton: So what I find is a lot of times teachers, for independent practice, having kids work on really complex problems and they can’t even solve the simple ones. So I think it’s important that teachers know, “Hey, this goes on a root of a simple to complex problems. Where are my kids on this? Which ones can they solve really well and where are they struggling? They can really solve the change unknown, but they can’t solve the compare it all.” It’s important that teachers know those things. It goes back to teacher knowledge. What do you know as a teacher? And then how can you use that to get everybody where they need to be by the end of the year? See, I think teacher knowledge is very tied into equity because you can’t give your kids a fantastic experience if you don’t have the knowledge base to do that.

Kyle Pearce: Great. Exactly.

Jon Orr: Amen to that. That is so true, and we are loving that and trajectories. We’ve been talking about that for a good amount of time and we should be, but say I’m at home right now Nicki, listening in, I’m thinking about trajectories and where should people go to learn more about this?

Nicki Newton: Well everybody has critiques of whoever and they can, but I think some of the foundational trajectories that are referenced the most in NAEYC and in NCTM are the Clements and Sarama. Remember, they got a Bill and Melinda Gates grant, and so now they have a website called LTLT or LT squared. You just put it in LT squared Clements and Sarama, and they have all of the trajectories up there with videos of the kids and all kinds of free resources. If you have children and they’re at this level, here are some activities that you could do with kids at this level. It’s a phenomenal site. It builds teacher knowledge, which is what we’re always trying to do. So I would start there, and then there’s NCTM just put out a really good book on learning trajectories.

Kyle Pearce: Oh, we’ll look that up for the show notes for sure. So I’m going to make a note of that here as well because I did not know that. I just pasted in a link in our notes as well for the learningtrajectories.org. That’ll take you over to that LT squared site. In our district, in particular, one of our early years consultants, Angeline [Humber 00:36:29], she is a huge fan of that and I’ve been through there and it is super, super in-depth. I love. Like you said, you can read something and go, “Okay so here’s the trajectory. I’m not quite sure what this one means.” Well, hey just click this video here and you’ll get to see what that looks like in action, which I think is so fantastic. And as you mentioned, it’s supported by the Gates Foundation and you are good to go to get yourself an account and log in and it’s all free and open to the public.

Kyle Pearce: So that is such a fantastic tool. I’m learning so much about it every single day. And I think for people listening, and I think Nicki, you’ll probably agree, this is not something where you go, “Okay. Starting today, I’m going to learn this and by tomorrow I’m going to have this down.” This is a process and you have to continue refining that process and refining your understanding and you’re going to change some of your understanding along the way. You’re going to think you have something and you’re like, “Oh no, it means this. And then over time, that’s going to continue to shape into something a little bit different, a little bit more in-depth.

Kyle Pearce: Think about teachers. We already talked about how some teachers are teaching multiple subject areas. Some are teaching up to six, maybe even eight different subject areas in a week, and overwhelm is something that is so easy to run into. Jon and I run into it all the time and it’s hard to take our own advice. So obviously, we have a website to start at. What might be some tips for someone to just start scratching the surface so that they don’t get too overwhelmed and they either go on stress leave or they just shut the browser down.

Nicki Newton: Well, I wrote a book recently just called Leveling Math Workstations, which is about learning trajectories and differentiation. The book is a really good place to start. The website is a really good place to start, but more importantly this. I think people should start, if you’re in the primary grade, start with the counting trajectory because it’s amazing. It gives you the language to talk about where kids are. And then I would do composing and decomposing and supetizing, because the supetizing trajectory is so important because what I see happening is I see teachers flashing those dots all year long. Same dots, all year long. But if you understood the learning of supetizing, you would understand this is about … Supetizing lays the foundation of arithmetic.

Nicki Newton: There is so much place value in it and there are 10 levels of supetizing. Kids should be supetizing all the way through fifth grade where they’re doing the quick images of the decimals, and where the kids are saying, “Oh, I saw five tenths in 200. I saw 50 200s,” or in first grade where they’re doing the hundred grid. Remember Kathy Richardson made those Tell Me Fast cards where you flash it really fast and then kids have to tell you the place value. They say like, “Oh, it’s a hundred grid but it’s blank.” And then say there’s two unshaded and they say, “Oh,” the kids say, “There’s 98 shaded because two are not shaded and two more would make 100.” There’s so much place value, but if you didn’t know the trajectory, you wouldn’t know that you leave the dots. You leave the land of the dots at some point, the dots, the ten frame, the dice and the domino, and you go onto the hundred grid. Then you start supetizing by groups.

Nicki Newton: Graham Fletcher has done some really good work around supetizing by groups. That’s a part of the learning trajectory, that the kids started seeing 20 but they see it in groups. “Oh, I saw four groups of five.” Graham has a good video up of a little girl doing that, and then you go on you know what I mean? There’s a trajectory for supetizing. Teachers need to know that trajectory. So I would say counting and then composing, decomposing and supetizing as a place to start. But I also want to stay, because I think it’s really important, understanding what are the phases or the levels of sophistication or whatever you want to call them, the continuum, these are all the different things it’s called in the research, for teaching basic math facts.

Nicki Newton: That teachers really need to understand there’s doubles and you’re probably going to learn, make ten before you learn doubles, although kids love doubles, but you’re going to learn how to bridge 10 you know what I mean? That you learn usually plus zero plus one before you learn how to count within five and subtract within five.

Nicki Newton: So that idea of like, how do we scaffold access with math facts within 20, there’s a really good book, I don’t know it’s called. It came out by Bay-Williams and Kling recently that has a bunch of games on the levels. I just wrote a book with Annalise actually and Alison Mello, Dr. Alison Mello, on Fluency Doesn’t Just Happen and it’s coming out in about three or four months, and it’s all just a bunch of games for different strategy levels. So I think that’s really important. I do a lot of work in Canada on guided math but also on running records because I wrote a book on running records and the book on running records is, again, it’s about you give the kids a clinical interview because you know the thing is, this is the thing. We don’t do this nearly enough in math. You find out the darnedest things when you ask kids questions.

Kyle Pearce: When you actually ask and listen?

Jon Orr: Right.

Kyle Pearce: Who knew?

Nicki Newton: Yes. If you ask a question, and it’s so hilarious because I’ve done thousands of running records at this point. Running records aren’t new. They’ve been around in the research for years, but the way they did them, they did them on a PowerPoint slide and they did 96 slides and then the kids have to answer in three seconds, whatever. So anyway, I took the basis of a lot of the running records and then I would ask kids questions. I took a lot of that and made into something we could do in schools and people do it.

Nicki Newton: Literally, I kid you not, I know of at least seven countries that are doing running records, including Mexico, Bahrain, Japan. People call me all the time and I Skype with them about doing running records, but I do a lot of work in Alberta running records as well. This idea that if you actually ask kids questions, you find out all kinds of amazing stuff. Some of the best times I’ve had with kids doing running records is I’ll go to a school and they’ll be like, “Oh, Dr. Nicki, this is a high kid.” They’ll say that. “This is a high kid. [inaudible 00:42:13] don’t know everything.” And you sit down and you start asking the kid about numbers and they don’t know everything.

Kyle Pearce: Right. They know lots of answers, but they don’t know what the heck is happening.

Nicki Newton: Yes, and the teachers are like, “Oh my gosh,” and then somebody else will come in and they’ll be like, “Oh, he doesn’t really know much,” and he knows all kinds of stuff, and they’re like, “What?” It’s amazing. He’s a C student and he’s blowing everybody away. He’s just doing, saying all kinds of stuff. Just doing all this amazing stuff. I always tell people, “You got to talk to them. You got to ask them questions.” That’s why I love the running record because it allows you to see the thinking behind the thinking.

Nicki Newton: The running record has three parts. So the first part is just quick, kind of an overview, but the second part you ask kids, “What are you doing to get the answer? What do you do with those numbers? What’s your strategy?” Then traditionally in math research, there’s codes for what kids tell you to do, so that part is incorporated in the running records. So then when you’re done, you have a portrait of where kids are in terms of the different aspects of fluency. Yes. You understand if they’re automatic or not, but that’s not the important … That’s just some information we need. But you really get to see are they flexible? Are they efficient? You get to see are they strategic? Do they have accuracy? How do they think about numbers?

Nicki Newton: Then another part of their running record, I ask kids, “Do you like math and what do you do when you get stuck?” And people say, “Oh they’re just going to tell you anything because you’re the teacher. They’re going to say nice things.” No. They tell you the truth. Kids say, “I hate math,” or, “I love math. I can’t get enough of it,” or, “It’s okay.” You know what I mean? Because disposition is really important. It tells us what kids, how they’re going to engage by their disposition. So I just think we have to do more of that in math. We’ve got to ask kids questions.

Nicki Newton: Speaking of mentors, at the beginning we had talked about mentors. There’s a woman named Coco Aguirre and she taught me how to teach basically. And on the threshold of her door when you walked in it said, and I’ve never forgotten this. From the moment I met Coco at a summer workshop, a Madeline Hunter mastery teaching summer workshop when I first started teaching. On the threshold of her door it said, “If kids don’t learn the way you teach, then teach the way they learn.” I have never forgotten that. If kids don’t learn the way you teach, then teach the way they learn.

Kyle Pearce: It’s so interesting too because in some places in the world, the word teaching and learning is the same word. So if I’m teaching but they’re not learning, then I’m not teaching. You’re just talking. So that is so powerful. I’m so happy. We were looking at the time worried that we wouldn’t get to running records. So I’m so happy that we got a brief overview there of what running records are. We’ve been doing a lot in our district with different tools. One’s called prime and a another is leaps and bounds. Leaps and bounds, a little bit more targeted, and really involves sitting down with kids and asking them questions. It’s crazy, like you had said, the students that are like, oh this student tends to struggle and some of the things they say, they’re like, “Well, when you asked me this question, my teacher wants me to do it this way but I think it should go this way.”

Kyle Pearce: And then they show you something that’s completely fantastic the way they’ve approached this problem. You learn so much about where they are and where you need to go next. And basically from that last little bit, we were talking about where do people start and it sounds like to me one of the big takeaways I have is we just have to get started. So starting with counting, like you had said, seems like a great place to start. I’ll put a link to one of our … We have a principles of counting cheat sheet, but like you were saying, that’s just the start. You look at that and it continues to grow. I continue to modify it as I continue to learn and grasp and understand how kids learn and develop. These are things that you’re never going to get there, but it’s just get on that journey is what I’m hearing and I think your book would be another great place for people to start as well, so we’ll be sure to link that up.

Nicki Newton: I think that it’s about the journey. It’s about constantly being a learner. I love this profession that we’re in because I’m constantly learning. I research all the time. I read research, I love research, and then I love to see that I know what the article said, but that kid’s not doing that. I love to be in classrooms. I try to be in classrooms at least one or two weeks out of every month so that I can actually see what kids are doing. I know what the research says, I know what kids are doing. I want to put those two things together and see how we help and contribute to the field. How do we all get better? How do we learn more? I had a mentor, Heidi Hayes Jacobs was one of my mentors. Heidi used to always say this.

Nicki Newton: She would pull out an empty chair and she would tell whoever she’s doing PD with, and she still does it to this day. She says, “We’re going to have a conversation and it’s about Maria.” Maria would be the empty chair. She said, “Because everything that we’re going to talk about in here is about Maria.” Because you have teachers sometimes who don’t want to get on the train and learn new stuff. But Heidi would say, “We’re not talking about what individual teachers want to do and don’t want to do, we’re talking about Maria. Any decision we make in here, is it best for Maria? Does it help Maria learn what she needs to learn in her journey at this school?”

Nicki Newton: I love that because when I’m working with teachers, you guys know because you do a lot of PD. When I’m working with teachers, I always try to center the child and say, “Yes, we’re going to learn about learning trajectories. Yes, we’re going to learn about differentiating this or differentiating that. And you know why we’re learning about it? Because the question always is, does this help us, help Maria learn what she wants to learn?” And you know the new Hattie stuff.

Nicki Newton: Hattie’s shifted to the, I think it’s eight dispositions or something, and remember one of the things that he said, “I’m not so concerned anymore about what teachers are teaching. I want to know what kids are learning.” That mirrors something that one of you just said. I think Jon just said it. Not what are we teaching, but what are kids learning? That’s the question. In the guided math group, whether it’s heterogeneous or homogeneous, in the whole group, in the middle, what did they learn? Not what did you teach but what did they learn? Because sometimes those are different things even though they shouldn’t be.

Jon Orr: And I think that goes to also like putting a face on your day. You’re talking in generalities, but you have to put a face on it so that everyone could imagine their kids in their room. It’s almost like … I get this image of everyone critiques doctors for not being … When you go into a hospital, you’re a number. But then you get that one doctor who treats you like a human being and you’re like, “Ah, thank goodness,” and it’s like we’ve got to think about that too as teachers sometimes during PD sessions. We’re not doing that.

Jon Orr: The saying we have in our district is similar to that mantra. It’s why this learning for this student at this time? And that’s our guide in our district right now. Why are we doing this for that kid at this time? So that’s been our mantra. This has been a fantastic conversation. I’m sure that we could talk all day about stuff but we know you’re busy. So right before we say our goodbyes, we want to know what current projects are you working on just before we wrap up here?

Nicki Newton: Well, I am working on a couple of things. I am working on a big book of pre-K, K and first grade math, hands on math activities. Again, trying to build teacher knowledge around the learning trajectories. I also am working on a book called Equity, Getting the Conversation Started, really looking at what can we do in schools every day, what do we have the power to change and then why are … I just want to say this one thing. This is one example of it. This is what we know. We know that we have a lot of kids in a lot of our schools that are impoverished. We know that kids that come from low socioeconomic backgrounds, not all of them, but some of them, most of them actually are a couple of years behind in math. We know that kindergarten is a foundational grade. We know that if you do intervention in kindergarten that you can change the trajectory of a child’s mathematical and reading career, but nobody does intervention in math in kindergarten. Very few States.

Nicki Newton: Why? If we know all that stuff, why? Why are we not doing intervention in kindergarten? It’s a question of equity. That’s just one example, but I’m really passionate about this idea of not gazing at the gap, but what are things that we can change, because we can change that. We can say we’re going to do intervention in kindergarten because we understand it’s one of the most important grades. We are not going to do pull out programs for our bilingual kids in math class because we understand that there is no matrix out there. We are the matrix. We control the schedule. We can say we’re not going to pull the kids out during math class. You know what I mean?

Nicki Newton: So I want to write a book about Equity, get it … I’m in the middle of writing that book. Actually, on my podcast, that’s the new series, it’s equity, getting the conversation started. I think we have to do that. And then the last thing is I write a couple of books at a time. I’m writing a book called the Eight Habits of a Productive Disposition, because mathematical disposition from the research, there are eight elements. Nobody ever talks about them. The only one we talk about is mindset. Did you know that growth mindset is one of eight elements? Do you know what some of the other elements are? I’m just going to tell you. Curiosity is one of the elements. Creativity and curiosity. We don’t ever talk about that in math class. Imagine if we did.

Kyle Pearce: Yeah, and those are central discussions for Jon and I and I’m so happy that you’re bringing that up to the forefront and I’m super excited to hear what you come up with when you finish that book off.

Nicki Newton: Yes. I think we have to talk about some, and you know who really inspired me? Was when I was in Alberta, and I say Alberta, I know it’s a state, but I go, usually when I go, I go to five stops in the state. So I take basically almost a train, a plane and an automobile on that trip.

Kyle Pearce: And maybe horseback.

Nicki Newton: Yes. What I see in the classrooms is part of the Alberta standards is curiosity in math. In some of the classrooms, they have these tables and I’m like, “What’s happening here?” And they’re like, “Oh, we want kids to be curious.” And so it’s like what do they see and what do they … I am like … I love the fact that curiosity is part of the Alberta math standards. You know what I mean? That’s important.

Kyle Pearce: Yeah. If kids don’t have that productive disposition, if we don’t have their attention, and we’re not just talking about engagement is that buzzword, it just gets thrown around. But if we don’t have kids leaning in, excited to learn, what are we doing? What are we even trying to accomplish here? We’ve got to get them leaning in, we got to get them thinking positively about mathematics, feeling like there’s a reason we’re doing it. Not just because of my job I’m going to have later, I want to make money, or we as adults think those things are important, but most kids don’t. They want to learn and they’re naturally curious. I think it’s because we usually stomp out curiosity. We stomp out creativity and we get them all single file and all of these things. I’m so happy to hear that you’re bringing that to the forefront. So that’s fantastic.

Kyle Pearce: Listen Nicki, we do not want to hold you up but Jon here, we’re talking in a Google doc, and he’s saying, “I think we’re going to have to do another episode here,” so hopefully you’ll be open to that maybe in another six to eight months. Maybe we could bring you back in, talk about your progress on these new projects and continue and dive deeper into some of these discussions. What do you say to that?

Nicki Newton: No that’s, I would be honored to come back and talk about it.

Jon Orr: Awesome stuff. Awesome stuff. We want to thank you Nicki and maybe one more thing is where can all of our listeners go to learn more about you and also your podcast?

Nicki Newton: If they go to drNickinewton.com, we’re in the middle of building it out even more, but they’ll see a lot of links on there. Because I have a podcast, I have a math academy as well that has about 20 courses, some which they can take for credit. There’s a guided math course and the running records course and stuff that they can take university credits for. And then what’s important also is that the, in my books, the Blackline Masters, you can pull them off the website. Then I can also send you guys some links because I have the link for the equity talk that I did at CAMT. I have that link and it’s a padlet that has tons of resources. I also have a link for renting records. You know what I mean? Maybe I can send you guys a couple of links.

Jon Orr: Perfect. Yeah, if you send them to us we will put them in the show notes and everyone listening right now will be able to just click them.

Nicki Newton: You guys, I want to thank you so much for inviting me to do this. This has been really fun and I just appreciate you having me on and I’m very honored to have been invited.

Kyle Pearce: Oh, thank you so much.

Jon Orr: All the pleasure is ours.

Kyle Pearce: Yes Nicki. It was an honor and hopefully it won’t be too long before we bump into you at another conference sometime soon.

Nicki Newton: Okay. I’ll be in Alberta in October.

Kyle Pearce: In October. Not October for Jon and I, but you know what? If you’re in Canada, you’ll feel our warm welcome. Hopefully we’ll see you soon there, Nicki. Stay in touch and we will talk soon, I’m sure. Thanks so much for your time.

Nicki Newton: Okay, thank you.

Kyle Pearce: We want to thank Dr. Nicki Newton again for spending some time with us and the math moment maker community so she can share her insights with all math educators around the globe.

Jon Orr: As always, how will you reflect on the learning you’ve heard here in this episode? Have you written ideas down? Have you drawn a Sketchnote? Have you sent out any tweaks? Called a colleague? Be sure to engage in some form of reflection to ensure that the learning you’ve here sticks.

Kyle Pearce: And in order to ensure you don’t miss out on new episodes as they come out each week, be sure to take that phone out of your pocket and smash the subscribe button on Apple podcasts, Google podcasts, Spotify, or whatever your favorite podcasting platform might be. And if you have some time, definitely leave us a rating and review because that ensures that people around the globe have a better chance of bumping into this particular podcast.

Jon Orr: Show notes and links to resources from this episode can be found at makemathmoments.com/episode62. Again, that’s makemathmoments.com/episode62.

Kyle Pearce: Well, until next time, I’m Kyle Pearce.

Jon Orr: And I’m Jon Orr.

Kyle Pearce: High fives for us.

Jon Orr: And high fives for you.