Christina: I think our end goal is that we want kids to be proficient in mathematics. The difference starts to happen in how we get there. How do we help children develop that proficiency? And to me, the foundation of that proficiency is number sense. That, if you don’t do the things to make the foundation solid, the rest of it will start to be affected by-

Jon Orr: You’re listening to the Recovering Traditionalist, Christina Tondevold. Christina previously taught mathematics traditionally, much like we did. And now, she gets to work with teachers in developing students’ mathematical understandings in ways that are not limited by the traditional ways of doing and teaching mathematics, through her site, buildmathminds.com.

Kyle Pearce: Stick with us as we learn what is number sense, and why is it so important? What are the four early numeracy concepts that all teachers should know and understand how to teach? What are the four number relationships? And finally, what steps you can take to ensure you learn fundamental mathematical progressions that underlie students’ understandings of number sense.

Jon Orr: All right, Kyle. Let’s dive into this conversation.

Kyle Pearce: 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 educators worldwide, who want to build and deliver math lessons that spark engagement-

Jon Orr: Fuel learning-

Kyle Pearce: And ignite teacher action. You are listening to episode number 44, Building Math Minds, an interview with Christina Tondevold. Are you ready to get started, Jon?

Jon Orr: Of course. Of course, Kyle. As always, we are super pumped to bring you this resource-rich episode. But before we get into this episode with Christina, we want to let you know that Christina herself is running a free webinar training called Math is Not a Worksheet. Registration is open now until October 12th, 2019. You can register by visiting makemathmoments.com/buildmathminds. Again, that’s makemathmoments.com/buildmathminds. If you’re listening to this after October 12th, 2019, then that link will either bring you to the webinar replay or a wait list for the next webinar.

Kyle Pearce: We also want to take a minute here to give a quick shout out to a math moment maker who left us amazing five-star review on iTunes. Passagenpiaffe says, “Truly paradigm-shifting. I always struggled with math, and I am now teaching third through sixth grade. I found this podcast to be truly life-changing and so helpful for me to fall in love with math. It has done wonders to shift my own personal mindset regarding math, as well. Thank you for all you do to bring revolutionary ideas and resources to the table.”

Jon Orr: Thanks to Passagenpiaffe for the amazing five-star rating and review. You don’t know how excited we get when we see that amazing math moment makers like yourself take a quick moment out of their day to show some love on Apple Podcasts, Google Podcasts, Spotify, or whatever platform you’re listening on.

Kyle Pearce: Well, that’s enough of that, Jon. Let’s get into our chat with Christina to build our math minds.

Kyle Pearce: Hey there, Christina. Welcome to the Making Math Moments That Matter podcast. We are extremely excited to have you on the show today. How are things going in your neck of the woods?

Christina: Well, what I like to say is better than I deserve. Right?

Jon Orr: Wow.

Christina: Any Dave Ramsey fans out there? Right? Things are so much better.

Jon Orr: Oh, awesome. Tell us more about that. Where abouts are you, and what is going well on that side?

Christina: Well, I live in Idaho, in a very rural town. I have four kids and life is just always busy. And I have to remind myself sometimes that it is better than I deserve, because being a working momma with four kids, you tend to feel like, “Ah! I’m just so stressed out,” and, “How am I ever going to do all the things that I need to do?” And the reality is, it never happens, but I just try to do the best that we can and say, “You know what? If the laundry doesn’t get done, that’s okay.” Or whatever it might be. That’s the hard part is, I do a lot of stuff. People see my face, I do a lot of videos, I do podcast stuff, too. And they see the front side, so everything always looks so great. But then what you don’t see is the piles of laundry and piles of dishes and all of that fun stuff of life that we don’t always see about people that we interact with online.

Kyle Pearce: Yeah, for sure.

Christina: So I am better than I deserve, but life is always full of trials and good things. And all four of my kids are in school, sports, a very active family, always out doing stuff and just love to be together. And luckily here in Idaho, we have beautiful country to be able to do all that stuff in. So I always feel blessed, that’s for sure.

Kyle Pearce: Amazing, amazing. I think we can mimic that. We’ve all got some struggles on a day-to-day basis, but I think for me, also, Christina, I can’t complain about many things. So let’s dive in a little bit more about who Christina is. We’ve met a couple of times, but Christina can help our listeners a little more about yourself, and maybe more specifically now that you’ve let us in on kind of a little bit of your family life and where you are, some background on how you got into math education and then also what is your role in math education right now, for our listeners who don’t know?

Christina: So I got into education by default, I say. I actually went to school to become an interior designer. And my first year of college, I had to take art classes that I failed at miserably. And I did not have a growth mindset at all. And I said, “Failed at miserably,” I got a C in. Which is not failing, but it felt like a failure to me. I was always that kid who did well and never really struggled in school. And when I did struggle, I didn’t know what to do. I look back at that point in my life, and I thought, “Oh my gosh. You really ran from that because you got a C?” But I am so thankful I did because I turned to education. And I say it was by default because my mother was a junior high teacher. And I didn’t know what else to go into, because I thought I was going to be an interior designer.

Christina: So I went into education. And during my student teaching, it was a year-long student teaching, and I wanted to junior high, because that’s what my mom was. And I thought I was going to be an English teacher like my mom, and they had us do a rotation where we did a quarter teaching every subject; the four main subjects. So I did English then science, I think was next; then the math, and then social studies. And when I was done, I thought to myself, “I want to teach math.” Because math was the easiest thing to teach. There’s one answer. It’s easy to correct. When I did my student teaching, it was still … The kids would bring home the homework, I would say the answers out loud … They would switch papers, you would grade each other’s papers, switch the papers. So I would say the answers, then when you’re done, they would tell you their percentage out loud, and you would put it into the computer. Right? You remember those days?

Kyle Pearce: Right. Yeah, oh yeah.

Christina: [crosstalk 00:07:53] did that.

Kyle Pearce: Quick and efficient, right?

Christina: [crosstalk 00:07:56]

Jon Orr: Christina, I did that.

Christina: Yeah. I think back and think, “Oh my gosh, what did I do to my … ” That’s just one of the things that you think back of and think, “Oh my goodness, I was such a horrible teacher.” But in my mind, that was the easiest thing to teach. And I thought English was the worst because there was no back and white answer. I felt like teaching English and writing was subjective, and I didn’t really get it. Math had rules that I could follow, and I was very much a rule follower. So I just felt comfortable in that mathematics space. And so I applied for teaching positions that were just mathematics in junior high setting. And I taught sixth, seventh, eighth grade in Nevada, Oregon, Idaho, just kind of all around this area. And it wasn’t until I went back and got my master’s degree that things opened up for me. And that was when I was introduced to cognitively-guided instruction and was introduced to this whole idea of a different way to view mathematics and to teach mathematics.

Christina: That led me into doing professional development. While I was doing my master’s degree, my supervising professor had a grant to do professional development with teachers. And after I had that whole experience of doing professional development while getting my master’s degree, I went back into the classroom, had my own class again, and got pulled back into professional development world. So it was kind of a long journey, but I’ve been doing lots of professional development with people all over the United States. And traveling around while also having four children, four young children.

Christina: And now I have the fortune to be able to do a lot of the trainings online. My focus is all on elementary education and laying a foundation for kids and what that looks like; what are the big pieces that our kids are missing? Because I saw it in junior high, where they had no sense of numbers. I was trying to teach them these rules and procedures, but it just wasn’t making sense because they didn’t understand how numbers worked. I could teach and have them memorize these rules and procedures, but it just didn’t sit right. That’s how I learned, and when I got my eyes opened to the different way of seeing mathematics, I didn’t want my other students to be going through what I went through. I made it through, but I look back now and think, “Man, it could have been so much different.” And that’s really what I want for all of our elementary kids out there, is to have a different experience than what we had with mathematics.

Jon Orr: That’s some big goals, some honorable goals, too. And you know, we always love asking our guests to think back to their experiences and what helped them, what molded them to get into what we’re doing now. And there’s a lot of similarities. And I know Kyle’s got similarities to what you’ve done. What you’ve just outlined is almost kind of my path, also. My dad was a math teacher, not an English teacher. But I remember thinking I wanted to get into teaching. I wasn’t going to be a teacher when I went to university, but when I was in university, I was like, “You know what? My dad’s life was pretty good.” You know? My dad was the ultimate high school traditional math teacher from the ’80s and into the ’90s. And I was like, “Yeah, he’s got life pretty good. All summer he’s sitting around reading books and swimming.” And I’m like, “Maybe I can do something like that.” And I kind of wanted to be a gym teacher, because I was like, “That’s even better.” And I was good at math, and I kind of fell in there, too. And I was that traditional math teacher, much like you said. And it’s not until you realize that you were that rule follower and when you meet kids for the first time who are not that, you know that something has to be different.

Jon Orr: I didn’t see those kids. You probably didn’t see those kids, too, in your classes. We were in different classes. You know? We didn’t hang out, maybe, with some of those kids that were just … I know I didn’t. All my friends were like me, you know? The math was pretty easy for us because we were all good rule followers and memorizers. And it wasn’t until that I saw students that were not like me that it’s like, “Something’s got to change. I can’t do it like this anymore.” So I’m glad you pointed it out.

Jon Orr: But it’s always amazing to me that many of our stories sound very similar and lead us to where we are now, helping this change happen. Because we all came from kind of that same place, but realized it has to change.

Christina: Yeah. It’s definitely … I don’t tell the story too much of really why I chose math, but it was totally a selfish reason. Right? I thought it was going to be the easiest thing. You wanted to go into education because you’re like, “Eh, it seems like a pretty [crosstalk 00:12:36] job.”

Jon Orr: Right.

Christina: And then you get in there and you realize, “Oh my goodness. This is not what I thought it was going to be.”

Jon Orr: Right.

Christina: And you know, I could have continued to have taught mathematics in that traditional way. But there was definitely points along the way that made me realize it opened your eyes. Right? Like you said, you have those moments that your eyes get opened and there’s no turning back from that point on. Once you know that there’s a different way out there, it’s like, “I can’t go back.”

Jon Orr: Part of it for me, too, was is that I’m sure … It’s only when I reflect back on it that I’m sure no matter what I decided to do, I wanted to do a really good job. And I felt like seeing what I saw in my classroom, I’m not doing a really good job if I just keep teaching the way my dad taught.

Christina: Yeah.

Jon Orr: So my dad looks at what I’m doing in class now and he’s awestruck because it’s different than what he would have done, and we have many conversations about what mathematics education should be. And sometimes they get a little bit heated, but …

Kyle Pearce: They no longer speak anymore.

Jon Orr: Yeah. No, it’s …

Christina: So at the dinner table, it’s not just … You can’t talk about politics and religion, you can’t talk about math education anymore at your house?

Jon Orr: Right. Partly. Partly, for sure.

Kyle Pearce: Well, you know what? Something that was interesting to me, just kind of backing up. You had named it really early on in the conversation, about having a fixed mindset. And I think it’s really, really hard for us as adults when we’re learning about this idea of growth mindset and so many people have seen Jo Boaler’s work or Carol Dweck’s work, and we read about it, but oftentimes it’s easy for us to not actually reflect on our own experiences. And I know I was one of those students, as well. Luckily, it didn’t affect me because I got through. But when things did get tough, it was like I played the blame game. I would blame the teacher. I would blame anything and everything but my own work ethic or my own perseverance, or any of those pieces. And it’s like when you have this stumbling block, and for you, it was going through when you were intending to go through as an interior decorator, or was it … Sorry.

Christina: Designer.

Kyle Pearce: Designer.

Christina: Same.

Kyle Pearce: Interior designer.

Christina: Similar, yeah.

Kyle Pearce: Apologies. As you were going through and you got that C, again, you’re so used to kind of “getting it.” It’s almost like sometimes it’s that perfectionist in us that sometimes that traditional math approach can sort of instill some of those perfectionist behaviors when we’re only looking for right answers and we don’t want to ever fail, we really struggle to dust ourselves off and pick ourselves back up. So I’m happy that we are making such a shift in math education right now.

Kyle Pearce: So what I’d love to do is, I want to back up and get you thinking a little bit more about your memorable math moment. When you think of your experience in math class, you had already said that you kind of were a rule follower. So I’m sort of guessing that you might have been like Jon and I where maybe math class wasn’t extremely difficult for you, or at least at the way it was being taught and what you had to do; the hoops you had to jump through. But go ahead. Is there a memorable moment that you can share with us for folks to think about?

Christina: Yeah. I definitely feel like math, when I was going through school, wasn’t that memorable. I wasn’t one of those kids who struggled, was traumatized by the timed tests. I definitely remember all of those things, but it wasn’t a bad experience for me, so it didn’t pop out. I also didn’t have any wonderful experiences. You know? Our moments that stick in our minds are usually at the ends of the spectrum. They’re either really negative experiences or really positive experiences. And through my education as a student, I had none of those. It wasn’t like math was so awesome and I had these great activities that we were doing and things that we explored, but it also wasn’t negative. So I don’t have those big moments through being a student.

Christina: However, when I went back and got my master’s degree, that’s when things started opening up for me. And the moment that I remember distinctly was through my master’s degree, my supervising professor was very much about cognitively guided instruction. So I got introduced to CGI, and we were watching these videos of kids solving problems with no instruction, and they had these amazing strategies. And to me, that was the math world opening up to me. And at the time …A little bit of a backstory here. My boyfriend at the time, who is now my husband, we dated for like 10 years, but we had known each other our whole lives. We had gone through the same school. We live in a small town, we had some of the same teachers, same education, same upbringing. And I started telling him about these strategies that I saw kids using. And it was things as simple as 9 + 8. And I would say, “And this kid took one from the eight and gave it to the nine and made it a ten and had 10 + 7.” And I was like, “Did you know you could do that stuff?” And he looks at me like, “Duh.”

Kyle Pearce: I just did it.

Christina: Doesn’t everybody know how to do that?

Kyle Pearce: Yeah.

Christina: Yeah. He’d say, “That’s natural to me.” And I was like, “What? Are you kidding me?” I never saw things like that. Even on things like something that we see as basic, 9 + 8. To me, that was just memorized. It was something that I just knew.

Kyle Pearce: Right.

Christina: It wasn’t something that I could manipulate and make sense of. And my husband, boyfriend at the time, was like, “That’s how I’ve always seen it.” And it was just like the world opening up to me of, there are people who see things differently and who see mathematics as making sense, not as rules to be followed. I look back now and can totally see that, because he is such the rule breaker. He gets told, “This is the way you do it,” and he’s like, “Nah, I see a better way.”

Christina: So there’s all of these kids out there that are doing that, even if we aren’t open to it; they’re doing it. But then on their paper, they’re writing down the way that we want it. And I could look back now at a lot of my students and think, “Oh my goodness. I did such a disservice to them.” Because they were trying to tell me that they saw it differently, but I wasn’t open to hearing it. It wasn’t open to allowing them to share their strategies. I would just be like, “Eh, I don’t get it. Do it this way.” And then when they couldn’t do it that way, I was just repeating it; saying it louder, slower for them.

Kyle Pearce: Say it louder, yeah. Yeah. Exactly. This is how you do it again.

Christina: I was totally that teacher. Oh my goodness. But that time when I was talking to my husband and saying like, “Did you know you could do this?” And he was like, “Yeah.” I was like, “Why wasn’t I allowed to see that?” My mind was not opened to this until I was 30 years old, 20 something. I don’t want to date myself, how long ago that was and how old I was.

Kyle Pearce: Yeah. It’s such an interesting idea, to think about. Not necessarily just in math, too, about rule followers and rule breakers. I, myself, growing up was always enforced to become a rule follower. I had so many rules. It was like, “We have to do this, and then you do this, and then you do this.” You know? It was also like, “You’re going to graduate high school and then you’re going to go to university, and then you’re going to get a job.” And think about our parenting techniques. It’s like, are we creating rule followers? Is that what we want to create? Or do we want to create kids who are thinkers and will break rules? And it’s a double-edged sword sometimes, because you’re seeing some of these kids who sometimes don’t have the boundaries that you think that they should have. And then you’re like, “Am I creating a rule follower, or am I creating the kid who is going to break the rules and think out of the box?” Because you want both, kind of. And it’s like I don’t know where the line is.

Kyle Pearce: As a parent, it’s hard to think about that. And as a math teacher-

Christina: I struggle with it.

Kyle Pearce: Yeah. And as a math teacher, it’s so clear to us now, in math class. But then to open the doors to just life in general, is scary.

Christina: Totally. I have one of my kids who is totally the questioner of everything. “Why do I need to do that? Why do I … ” And in school, that’s kind of, in the past has been … We stamp that out. Right? There’s been research done of how many times kids will ask the question why at certain grade levels. And the higher up they go in grade levels, the less they ask that question. Because they learned that that’s not something you ask in school. You just listen and learn and do what you’re told to do. And I struggle with my one child, because he is such a, “Why do I need to do that? That doesn’t make sense,” kind of a kid, and we want to encourage that, because later in his life, that is going to be such a powerful trait; to be a questioner and thinker and go outside that box and don’t follow the rules. But man, sometimes in school, they need to follow those rules.

Jon Orr: Yeah, exactly. It’s like everything in life, and I feel like on this show if you were to go and do Wordle of everything we’ve ever said, I feel like some words come up quite often, and balance is one of them. We always say … Depends, we say a lot. Like, “It depends on the scenario.”

Christina: Yeah.

Jon Orr: And this is one of those cases, where it’s really hard. You had already referenced … You said you have children at home. I think you said four children at home. Am I right with that?

Kyle Pearce: Mm-hmm (affirmative), yeah.

Christina: Mm-hmm (affirmative). Yep.

Jon Orr: Four children at home.

Christina: Yeah.

Jon Orr: So you’re trying to engage in your work life and you’re trying to engage with your family. And sometimes you get tired. And usually when we show our true selves, how we’re truly feeling if we’re run down or if we’re tired, it’s usually at home. And our poor children sometimes, when they are inquisitive and they’re asking us these questions, sometimes it’s not even what we say, it’s just how we react.

Christina: Yeah.

Jon Orr: Even if it’s a look or the tone of our voice.

Christina: The tone, yes.

Jon Orr: Yeah! It’s like it’s sending that message to them that, “You know what? Stop doing this. You’re annoying people.” And we don’t want to send that message, because we want them thinking outside the box. So I think this is a great way to segue into some of the questions that we had for you, because we’re really curious. We know you are very passionate about working with some of our youngest learning in K through five especially.

Christina: Mm-hmm (affirmative).

Jon Orr: And as you mentioned, number sense, it kind of blew your mind when you started focusing on cognitively guided instruction and seeing these strategies and why we need them. What we’d really like to do is kind of dive into it. So if I’m a K through five teacher, why do we need to invest time in learning, deeply, how cognitively guided instruction can help us with number sense, or even compared to patterning or measurement, or any other topic? But in particular, why is it important that we focus in on cognitively guided instruction and really do our own homework since many of us didn’t have the chance to learn this way? Why should we do that work for our students?

Christina: Okay so I like to start off a lot with … You know, our end goal is really all the same. We want kids to become fluent and flexible thinkers of mathematics. Now, if you talk to anybody, even parents who are talking about how we shouldn’t be doing this new math or anything like that, when they’re kind of combative about how we’re teaching mathematics, I think our end goal is that we want kids to be proficient in mathematics. The difference starts to happen in how we get there. How do we help children develop that proficiency? And to me, the foundation of that proficiency is number sense. I equate it to … My husband is a contractor. He builds homes, does remodels, all that fun stuff. So he has seen lots of times where things are not done right the first time. Right? You can come back and fix it later. And oftentimes, we try to jump ahead; we will skip steps. Sometimes he sees that a lot in building; that if you don’t do the things to make the foundation solid, the rest of it will start to be affected by that later on. It might not be right away. Right?

Christina: And so in education, sometimes we try to skip the foundational piece of building kids’ sense of numbers and we jump straight into the operation of numbers. We want kids to add and subtract, multiply and divide as they move up in grade level, but without a foundation of a sense of what numbers are, how they work, how they relate to each other, then those operations that we’re trying to teach them become just rules and procedures. If we can lay a foundation of numbers sense, we see kids who are flexible thinkers and can manipulate numbers. I mean, just this morning, we were headed to school this morning. I drop off my children every day, because it’s just a good time to chat, also. You know? I like spending time with my kids. And we’re talking, and my youngest son, he’s really young for his age. He just turned six in August, but he’s in first grade. So he’s a young first grader.

Christina: And we’re headed down and we’re talking about math, and some of my kids don’t like math. They’ve said before, “I don’t like math.” And my first grader, he goes, “I like math. I’m a master of math.” And I go, “Oh, you’re a master of it?” He goes, “Yeah. I’ve mastered addition and subtraction.” And actually, he said, “I’ve mastered pluses and minuses.” That’s what he calls it. And I go, “Oh, really?” And my daughter goes, “Okay. If you think you’ve mastered it, what’s 20 + 20?” And he instantly says, “40.” And she was about to say something else. And I go, “Hold on. I got one for him. Okay. What’s 20 + 19?” And he thinks for about three seconds and he goes, “39.” I’m like, “What? How did you do that?”

Christina: “Because it’s just one less.” Right? He sees those relationships. He’s not lining it up, doing the traditional algorithm like I used to do. If I would’ve been asked that question 20 some years ago, it would have been lining it up. Seriously. I could not do mental math. I had to do everything as … We line it up and we add our ones, add our tens kind of a thing. And he just sees relationships. He doesn’t have to be taught strategies. He sees how it works.

Christina: And I know that all the kids who have this foundation of number sense … And yes, my children come from a very unique home life because they have me as their mother and we have done stuff that builds number sense, but I’ve never taught a strategy to them. We talk about numbers, we play with numbers. We play lots of games. You know? We don’t know number talks, like teachers call number talks, but we definitely talk about numbers. We play lots of games where kids will see things, and they earn money for stuff.

Christina: Even just earning money for chores nowadays, kids don’t have those experiences to think about how much they have, how much more they need to get something that they want. And thinking about, “I want this thing on Fortnite, the pack.” My kids are all into Fortnite and they want to buy the V-Bucks or whatever it is, and how much money they need to earn in order to do that and how far away they are and how many chores they need that will get them there; all of those experiences. It’s all built upon experiences. And a lot of our students don’t have those experiences before they come to school. And in school, our standards jump us straight into thinking that our kids have to be able to add and subtract in kindergarten and first grade, so we focus on that end product of getting our kids to add and subtract, and we don’t think about all the foundational pieces.

Christina: It’s like jumping straight to putting in the cabinetry and stuff in your house, picking out the light fixtures and the doorknobs. Right? That’s the end product that everybody sees, but what’s the foundation of how we get that house to look nice? And that’s what we got to think about in mathematics is, yeah, our standards may say we need to be able to get kids to do this certain thing, but what’s the foundation of their understanding of numbers that get them there? That is such a huge piece.

Jon Orr: Yeah, and that’s a huge analogy. That’s really eye-opening. You know? I think it’s a great analogy to start with, when you were talking about why we should be doing this. Because everyone would agree with you. We would never start with the finishing touches on a house, we’d start with the foundation. And you’ve made it very clear that number sense can be a great foundation.

Jon Orr: And one of the other things that related to both Kyle and I right now is that when you had talked about … So many of us are rushing to that algorithm, and that’s one of the phrases that we use quite regularly, when we go right to formulas. We go right to, “Let’s just stack these numbers and let me show you how to add them,” like the finishing touches. Sometimes we just say we rush way too much just to that algorithm and the abstract representation of numbers, or abstract representations of math concepts in general. We, too often, especially in middle school and high school, too, being from that area myself, jump right to just saying, “This is the formula. Let’s apply it, let’s move on.” And we don’t do enough foundational building there.

Jon Orr: And you know what? I think there’s some teachers probably leaning in right now and thinking like, “This is awesome, and you’re right, but where do I start?” What are some early numeracy concepts that all teachers should know, and then how to teach those?

Christina: Yeah. So I want to say a little bit to what you just talked about, about rushing to algorithms too quickly. Because I would say that I believe teachers even rush to strategies too quickly. Because the knew swing has been, “Okay, we’re not going to teach the kind of traditional “algorithm,” quote, unquote. You can’t see me, there … of stacking [inaudible 00:30:26] and doing that traditional algorithm. So instead, we’re going to work on these strategies, like the one that my son came up with, that if it’s 20 + 19, they think in their head, “20 + 20, and then I just subtract one.”

Christina: So our textbooks, even, will have those strategies as something we need to directly teach kids. And I feel like we jump to those strategies even way too soon; that we try to directly teach these strategies and the kids … If a kid doesn’t understand how 19 relates to 20, that strategy makes no sense to them whatsoever. Right? So we need them to be able to have that foundation of number sense.

Christina: And none of this is my own. I learn from others, I compile it together. I’m curator of knowledge, basically. So I’ve taken what I’ve read. I love to research things. I’m just a researcher at heart, so none of this is my own research. I’m pulling from other people. But the first one that I ever discovered was from John Van de Walle’s book. And we all call it John Van de Walle, but there’s so many other authors who worked to compile that information together. So the book’s Teaching Student-Centered Mathematics. And those books are … I call them my math Bible. So there’s one for pre-K to second grade, and there’s one for third through fifth, and one for six through eight.

Christina: And the thing that I discovered in the pre-K to two, even though I was teaching middle school, I found it in the pre-K to two book. And it was the four number relationships that they talked about. And it’s a little tiny part of that book. The book is really big. It talks about all kinds of content, but it was one little piece that I have just grabbed onto because I see the power of it. And I see the power of it not just for our pre-K to two kids, but all the way through. And these four number relationships were … In the first version of Van de Walle’s book, they talked about it as needed these for the numbers up to 12, but I feel like you need it for all numbers.

Christina: And one of the first one is spatial relationships. We need to have a visual of a quantity so that we can see the other relationships that we’re about to talk about. All right? If I ask you to close your eyes and picture eight, most of us … If you just asked a random stranger, right? And I know this is a different crowd, so you may have a different visual of eight. But if you picked a random stranger walking down a street, and said, “Close your eyes and picture eight,” most everybody sees the digit eight. They see the abstract version of eight. And that abstract version of eight does nothing to tell me how eight relates to any other numbers. How does eight relate to five? How does eight relate to seven? How does eight relate to ten? You don’t see that when the only thing you see is the numeral eight.

Christina: So the first thing is, we need to have some kind of visual of eight. And the hard part is, though, that it’s hard to visualize anything past five, even for adults. You have to have some kind of grouping piece of it. So like your fingers, right? You may see finger pattern of eight. You may be able to see it in a 10-frame. Right? Lots of teachers are familiar with 10-frames. You may see tally marks. You are grouping it in some way that helps you be able to visualize eight things. But most of the time, you’re seeing some kind of a group of five and then three more. Or other people may see a group of four and four. Right? But having some kind of visual of that eight then helps us with all the other relationships.

Christina: So the next relationship is knowing what is one and two more and less. When you see eight, you should also instantly know that it is one less than nine; it’s two less than ten; it’s one more than seven, two more than six. Right? All of those should just be natural things that we understand about the number eight.

Christina: Another one is how it relates to the benchmarks. So one of the number relationships in that book, they talked about how does a number relate to a benchmark number? Well, for our small numbers, the benchmarks are a five and a ten. We need to instantly know how eight relates to five and how it relates to ten. Because those help with all those strategies, right? We talked about how, if I was adding 8 + 9, how my husband goes, “Well, you just pluck one off and give it to the nine to make it 10.” If a kid doesn’t instantly know how those numbers relate to each other, how much to make the next ten, what happens when I pluck one off of the eight, all of that stuff goes into those strategies that we’re trying to teach kids.

Christina: And the final one is that knowing … Well, it’s called part-part-whole, and that’s the idea of knowing that if I take a whole amount and break it into parts, I still have that whole amount. For our young learners, that is a weird thing. If I count out eight things, but then I move three of them off to the side just a little bit, and I say, “How many do I have now?” They have to back and recount it. They don’t understand that you can break apart the whole into different chunks, and it still stays the same. That’s a weird thing for kids. And that’s essential for kids to be able to do those strategies where they’re breaking apart numbers and making new numbers. Because to them, it’s making something new. But yet, we need them to understand it stayed the same.

Christina: So those four number relationships became a huge piece of my work with kids in helping them understand those strategies that we were trying to get them to do. And the work of cognitively guided instruction is you lay those foundation of number sense and kids will intuitively use these strategies when they’re solving problems; that we don’t have to directly teach strategies if they have this foundation of number sense, and we give them problems that are in a context that they care about. Right? If you just give them 8 + 9, there’s nothing there for them to really latch onto, but if you turn it into a story that they care about; that you create a story problem out of it, that kids can solve those kinds of problems without any direct instruction, they have this intuitive knowledge about the situation that you’ve given them, and their information about numbers, and then they go from there.

Christina: And where kids will normally start is they’ll count everything. So then that took me onto a journey of, “Okay, we get these kids who are just counting everything.” Right? And everybody’s like, “How do I get my kids to get past counting?” But counting is such a foundational piece. There are a time and a place where kids need to count, but there’s definitely a time where we want kids to start using their grouping pieces. And I like to relate this to the idea of … When we teach kids to read, we teach them to sound out certain letters. You sound it out letter by letter. It’s B-A-T. We want them to sound it out … But when they start getting into bigger words, we don’t want them sounding it out. Right? So if it’s something like baseball. My kid’s still playing baseball right now. That’s top of mind to me. I don’t want kids at that stage to be reading it out sound by sound. We want them to see groups.

Christina: And so when kids are adding and they’re adding 2 + 3, sure they’re going to count. That’s fine. But if they’re doing 8 + 9 and they’re still counting one by one on their fingers, that becomes an issue. Instead, we want them to use their number sense, the relationships of numbers and see groupings that they can use to make that problem friendlier, just like we want them to see groupings when they’re reading; to see those chunks. Right? Chunking is so huge.

Christina: So there’s a lot of work by Clements and Sarama of whole learning trajectories there, of getting kids to do more with subitizing. Counting is super important, but also being able to see groupings where you don’t have to count; that is another foundational piece that helps kids be able to work in that number sense atmosphere instead of having to just count one by one by one all the time.

Kyle Pearce: Well, that was a huge value bomb for everyone who’s listening. And over here, my math heart is full because when I hear you talking about Clements and you’re talking about Van de Walle, for me, a book that’s very similar to what you had been focusing on, but more or less kind of stretching a little further into the middle school mathematics is Elementary and Middle School Mathematics: Teaching Developmentally by John Van de Walle and friends, as you mentioned. Not just John Van de Walle, there was group in there; Karen Karp and Jennifer Bay-Williams and Lynn McGarvey.

Kyle Pearce: So there is so much goodness out there. And the part I really love about this, because Jon and I, while we work with K through 12 educators, we tend to kind of play a lot in that middle school land. And the work that you’re doing to build that foundation in the K through five years is so important. When we go back and kind of recap here, talking about spatial relationships, we talk about spatial reasoning so often and the importance of it. And I’m hearing this idea, the visual [inaudible 00:39:32], that conceptual subitizing is popping out there. And when you’re thinking of these ideas of one or two more or less, I’m hearing additive thinking, which is so important as we take that journey on our way to proportional reasoning and relationships, and your benchmark numbers as well as that part-part-whole relationship, which is so important as we get into fractions and we start thinking proportionally.

Kyle Pearce: So folks who are listening to this episode who are in our academy and taking our proportional reasoning mini course, they’re going to touch on some of these ideas. But if they are teaching in some of the younger grades, they might want to take an even deeper dive into some of these ideas. So I’m really excited to continue this conversation.

Kyle Pearce: So these are four big ideas. You’ve referenced an amazing book which again, we will link up in the show notes. I’m wondering, do you have any sort of go-to ways to help teachers as they start learning about these number relationships, for example? How do they go about avoiding that, I love it, the rush to the strategy, we’ll call it. We call it the rush to the algorithm, but you’re even kind of going before that and saying, “We don’t want kids to just memorize these strategies,” like straight up memorize and repeat. “We want them to build automaticity with these strategies.”

Kyle Pearce: So what are some ways teachers might be able to sort of embed this learning in their classroom?

Christina: Yeah, I think one of the biggest ones is having lots of visuals. Mathematics for so long has been very abstract. It’s just a bunch of symbols on a piece of paper. And I think in the early grades, we tend to do that more. We bring out the manipulatives, we have lots of visuals. And as we go up in grade level, we start to kind of not use that as much. But it is so important. Any time that we are learning something new, even as adults, we have to go back to that concrete phase. So having great manipulatives, wonderful visuals; we need to be including those in everything that we do. Because no matter what grade level you teach, you have kids who are in need of seeing those visuals, having hands-on manipulatives to work with as well as attaching that to the symbols. You have some kids who are rushing to those symbols because that’s their comfort zone. But a lot of those kids sometimes still need the work with having a visual and using the manipulatives to really make sense of what they’re doing. There are some kids who are still great memorizers, great rule followers.

Christina: I mean, when we hear a parent say things like, “Yeah, my kid can do multiplication,” and they’re in first grade. Sure, they’ve memorized this, they can do the symbolic version of mathematics, but we need to have lots and lots of visuals and lots of manipulatives. So I’m a big proponent of the subitizing cards. I mean, if you just Google subitizing cards, tons are going to come up. I’m a big fan of the Rekenrek. Other people will call it a math rack, arithmetic rack. I even love that even for doing multiplication because you can get ones that will go up to 12 rows of 10 and even 10 rows of 10, to help kids create arrays for multiplication and see those groupings. I think that those are wonderful tools.

Christina: I mean, even 10-frames as you get up into the working with multi-digit stuff; it still helps to see those groupings. Kids need to see groupings of things. So I say it’s not just about using manipulatives, it’s about using structured manipulatives and being very intentional about giving kids groupings and helping them see those groupings. Not all the kids will see the groupings. Even if you hand the kid a Rekenrek. If you’ve used a Rekenrek, it is colored with those red and white beads, and we assume that kids will see these groupings that we adults see, but it’s not. You have to bring that out for them sometimes and ask the kids, “Well, what do you see? When seven beads are you pushed over, what do you see?” Some of our kids still see it one by one by one. But letting the other kids talk about, “Well, I see three that are left over. I see five red ones and two white ones.” Right? We want them to have those conversations.

Christina: So lots of visuals, lots of manipulatives and getting out of the way. It has been forever that we have been seen as the bearer of all the knowledge. I’m a firm believer of putting a problem out there and just seeing what your kids do with it. I’ve been on a kick lately about talking about how to teach math without a textbook. Because I feel like our textbooks make us feel like we have to provide all of this information; we have to give them the step-by-step stuff, where are kids actually come to us with a lot of intuitive math knowledge. And if we could just take a step back and feel like we don’t have to be the bearer of knowledge, our roles is more of a guide and we need to listen to what our students have right now, and then build upon that. And the only way we do that is by posing a problem. Give them a problem. I love cognitively guided instruction. And it’s as simple as creating a story problem and using numbers that are appropriate for your students. Right?

Christina: So if you work in the early grades, it might be something where you are … I’m going to use Fortnite, because that’s what my kids are all about, but you replace this whatever you want to. But my kids do it naturally. I hear them talk about it. And they will say things like … I know it’s all about guns and there’s some violence and all of that kind of stuff. We talk about all that stuff with our children. So but they collect guns, and they’re talking about, “Well, I have three of this kind of gun, and … ” I know you can’t use this in your classroom, but I’m totally just doing it because we talk about it in our personal home. But I know you could not have this as a story problem you would use in your classroom.

Christina: But they collect this many of the ARs, whatever it might be. Whatever fits for your kids. That’s the other piece, is you could leave them open. I’m getting totally sidetracked, so reign me back in, here. But the whole idea of story problems is you use a context that matters to your kids, but you don’t have to dictate the numbers.

Christina: So I also love Brian Bushart’s stuff about the numberless word problems, where you can leave it empty. You don’t put the numbers in there, and then you can give the kids the number afterwards. And you may give some kids a different set of numbers. Some kids may still be working on something like 19 + 20, where some of your other students in your classroom are ready for something like 28 + 39. Right? And you can decide what numbers to put in there, and you let them solve it any way they can. It’s the exact same story problem, it’s just different sets of numbers, and then you watch how are they approaching it. You have them discuss and talk about the strategy that they used, and then you can look to see how many of your students are recognizing that they could make a 10, make one of those numbers friendly. How many of your kids are grouping their tens first and their ones first, and then putting it together? How many of your kids are still drawing the pictures? Right? They have to draw out 19 things and 20 things and then count them all, because we still see that, even when kids are working with multi-digit numbers.

Christina: But we only see that if we first give them a problem, and then to say, “Go for it.” And if we first say, “Okay, here is how you solve a multi-digit addition problem. First, you add your ones together. Next, you add your tens together. Then we add those together, and that’s our answer.” Every kid in that classroom then, on every problem you give them, thinks that’s the way they need to solve it, even if that doesn’t make sense to them at that time. If the problem was 29 + 38, a ton of your kids want to take one from the 38 and give it to the 29 and say that’s 30 and 37. But if we first dictate, “Our book says that we need to solve these problems in this way, and you’re going to solve the next 20 problems in that way,” they solve it that way thinking that’s the way they need to do it, but it doesn’t make sense to them to solve it that way.

Christina: So lots of visuals, give them manipulatives, but start with just giving them a problem and listen and see what they’re doing.

Jon Orr: You know, you wanted us to reign you in a little bit. We just wanted you to keep talking, because you keep telling us great strategies to do with our students. And you know what I really love hearing you talk, and I know that we’re talking right now about early numeracy number sense skills. But it’s amazing what you’re saying about give our students problems, see what they can do with that problem, and then that shapes your lesson. That’s exactly what we’re doing in trying to talk about in middle school and high school lessons, too. It’s all about listening to our students. So it’s great to hear that you’re trying to make that happen in lower grades. We’re trying to make all of that happen in higher grades. All the same stuff. It’s all about what you just said. It’s listening to your students, giving them a chance to showcase what they have, and then you can build on that using the strategies you’ve talked about for number sense, and then many other strategies for any other topic or grade level.

Jon Orr: I think it’s awesome that that connection is right there. So if we think about this, and think about all these strategies, Christina, how can teachers become more comfortable with learning the strategies you’re talking about and specifically, teaching with number sense?

Christina: Yeah. Okay I have two suggestions, here. Number one is the way I started out, which was … I had no clue what I was doing, so I just tried it. And it failed miserably. Right? I put a problem out there, and I said, “Okay, just try it. Solve it. Let’s see what you can do.” And they all looked back at me like deer in headlights; big-eyed, just frozen. I don’t know what to do. I don’t know where to start. And I started to reflect because all of the other instruction I was doing up to that point had been, “Here’s what we’re going to do. Here’s what I need you to do. Here’s the steps I want you to take. Now, you go do it.” It was the I do, you do kind of scenario where I do [inaudible 00:49:38] you do scenario. And so when I first opened it up and said, “Hey, go for it,” they’re like, “What? What do you want us to do? This is so foreign.”

Christina: So I just had to kind of feel my way and try to figure it out. And it depends upon your kids. I feel like … When I think back, I’m going to date myself, but I mean that was 16 years ago, after I had graduated with my master’s degree and I’m trying to put this stuff in place and I think, “Things have come a long way, now.” I think it’s a lot easier now because more and more people are starting to do this, to where kids are getting more used to it. When I first started doing this, it was still very much we taught the traditional way. And kids weren’t used to any other way than that. And so I feel like it’s a little bit more easy to get started.

Christina: And so my advice to get started, too, is just when you open it up, just play dumb. It is very hard to do because as a teacher, we went into this to help our students. And I still will get sweaty sometimes when I open it up and the kids will just sit there like, “Tell me what to do.” And I’m like, “Well, what do you think? What do you think this is telling us to think about?” All the questioning, and I start getting nervous about it because I feel like, “I just want to tell you. I just want to help you. I just want to tell you what to do, because I want to help.” But sometimes giving that help is not actually helping. You have to think about that.

Christina: I love an analogy that I heard from Robert Kaplinsky once, was that you’re like the spotter of a weightlifter. If you try to lift all of that weight for the person who’s lifting the weights, they will never get stronger. You have to just be there to help support, you can’t do it for them. So you’ve got to open it up and just play like you don’t know. And it’s so hard because we do know, and we are sitting there thinking, “Man, just do this one little thing, and it’ll make it so much easier.” But instead of telling, we have to question. And we have to dig deep and try to figure out what do they know about these numbers and how can I lead them based upon what they know?

Christina: So that brings me to the other piece that I didn’t start out with, was really understanding the progression of how kids build their knowledge. And so for me, my focus has been on the progression of number sense. How do kids build this number sense? And it’s very helpful when you understand that progression, because you can see like, “Oh, they’re not getting this strategy because they don’t understand that 49 is like making a 10.” There are so many of our kids who can make a 10 when it’s 9 + 8, but then magically, somehow when it’s 49 + 38, they don’t see how to make a 10. What’s the issue? Do they understand that they can break apart numbers? Are they not able to visualize what benchmark number is close to 49? You have to understand and know what those number sense components are to figure out what the gap is.

Christina: And so oftentimes, we just see they aren’t able to do the strategy, but really, what’s at the root of that. Why can’t they do that strategy? What is their misunderstanding or lack of understanding about the numbers that’s causing them to not be able to understand that strategy that we’re trying to get them to see?

Christina: So it takes a little bit of playing dumb but also being very knowledgeable about the progression of how kids develop their understanding so that you know how to question when you are playing dumb.

Kyle Pearce: Yeah. And that’s fantastic. I think those two pieces are so important. And the second piece, I really want to talk about, and this idea of the progression; the trajectory. Jon and I are constantly talking about … We have to understand where are students are coming from. Not only where they’re coming from in their own experiences in math class, their home life and all of those pieces; those are very important, as well. But also our mathematical understanding of where they’re coming from and what they’re bringing with them. So the mathematics that we’re hoping that they’re coming to us with … And we all know that a lot of students don’t have that, even though our curriculum standards or expectations ask of certain things, often times not all students are at the same place.

Kyle Pearce: And if we don’t understand the actual progression or that trajectory, then it’s really difficult for us to be able to actually help our students. So I have to know where students are coming from. I have to know my own understanding of the current content that I have to teach. And then I also have to know where it is that they’re heading to so that I can get them prepared.

Kyle Pearce: And you referenced some books earlier. I’ve referenced a book, as well. There’s also some websites, as well. The Clements website, learningtrajectories.org is great. But I’m telling you, there is a lot of content for us to learn. So I’m wondering, someone sitting at home, and maybe they might even be getting a little bit antsy or anxious thinking about all of the work that we as math educators really have, because it’s an ongoing process. I’m wondering if someone’s looking for some way to get sort of a little bit of extra help, a little bit of guidance, what else could they do in order to kind of get a kickstart on doing some of that early math learning?

Christina: Yeah, so I definitely feel like we can spend our time digging into research and that’s what I got to do. That was my job for a long time, was digging into that research and learning it as a math coach so that I could help the teachers that I work with. Because let’s be honest, especially as elementary teachers, we got a lot. You have to do not just math. You’ve got all subjects. So I think it’s a hard job right now as elementary teachers, because we are expecting teachers to understand all of this for mathematics plus all of this for reading and writing and all the other subjects. And it’s a lot. And I really am a proponent of having coaches at schools. I think that’s one of the biggest things that can make a difference longterm, is to advocate for coaches in the schools. Because they can focus on those particular subject areas and be able to help provide professional development.

Christina: The biggest piece of … The only way we change is by learning. Right? We grow when we learn something different. So you can go and do it on your own, which is difficult because you have a lot of responsibilities as a teacher. We can hopefully start bringing in more math coaches into our schools so that we can have them consume that information and then find the pieces that are important to bring out to our teachers that they work with, their teachers they work with. And I feel like that was my role for a really long time. And luckily, I’ve been able to put what I’ve done into an online course.

Christina: So I’m also going to just kind of plug the course here a little bit. Because I know it’s hard work to find the time to read all of that stuff that we’ve talked about. So I put it all together. That was my job, was to go out and do professional development in schools. But the hard part was that not everybody has access to that. It is very difficult to get professional development people to your schools. It costs a lot of money. It’s hard to have time off for everybody to be there to come listen to the professional development speaker. Have a person there for one day and then have to go back and implement it.

Christina: So I put all of my stuff that I would do in an in-person training into an online course, which I know you guys have done the same thing with Making Math Moments That Matter. It’s just a great way to be able to learn from somebody who has gone through this, who has sifted through all of the content that’s out there. And I’ll be honest; all of this stuff is in books. You can Google all this stuff and you can spend time doing it, but the power of having someone who has done that before you and can narrow it down to the biggest pieces that you need to know and understand, it’s a powerful thing to do. And to be able to learn from that person who’s taken the information and compiled it for you.

Christina: So I’m also going to give a little plug for, come take my number sense courses. Because I have one that is for what those number sense concepts look like in pre-K to two, but also three through five. Those same relationships hold true and help kids even as they get into fractions. Think about if you’re adding three and a half plus five and three-fourths. We can have kids just memorize the procedure for doing that, but if we have built number sense, these strategies that kids have for things like 9 + 8 still apply when they’re doing things like three and a half plus five and three-fourths. Kids can chunk numbers off, make numbers friendly. They can manipulate numbers and become really flexible if we lay the foundation of number sense. And it makes the operations so much easier on all kinds of numbers; decimals, fractions, multiplication, division. It is all so much easier if kids have this number sense.

Christina: But if we try to jump straight to the algorithms, to the strategies without the foundation of number sense, it really does … It’s like beating your head against the wall. You’re like, “Why aren’t these kids getting it?” And it’s because we don’t have the foundation set for them, first.

Jon Orr: You know, in building a course for any teacher anywhere to access from home, from school, on their own time, is an amazing thing. And you’ve built, and that’s exactly why we’ve built our workshop, too. It’s we can provide the PD to teachers wherever they want, and you can help change the students in those classrooms, because you’re helping those teachers do those same things. And the moment that this is going live, if you’re listening to this right now on the day it went live, that’s when we’ve started a third cohort of our workshop.

Jon Orr: So I’m wondering, is your workshop similar to ours? Can you tell us a little bit about how it looks, and structured? How it’s structured online? We’re pretty curious about that, about yours, and what would … Say a teacher enrolled in your workshop or in your course. What would that look like?

Christina: Yeah, so I want to back up really quickly. Because I hate only providing things that have to be paid for. So when this goes live, there’s also … I think we’re going to have a button on the show notes page …

Kyle Pearce: Yeah, yeah. We got lots of your information to put it in the show notes.

Christina: For the free webinar. Yeah. So I’m going to do a webinar all about that math is not a worksheet, and it’s going to go way more in detail about how do we actually do this, and what are these number sense concepts. And so how do we take those number sense concepts and do it without relying upon worksheets, our textbook, that kind of thing. How do we provide kids these experiences? So there’s definitely the free workshop that you can come and attend. You don’t have to take the full course. But if you want to go deeper, there is the full course.

Christina: So in the course, we not only talk about those four number relationships from Van de Walle and friends, but we talk about the four early numeracy concepts that Clements and Sarama talk about.

Christina: And so each week when the course starts, I do what’s called dripping. So you don’t get the full course at once, because it’s like drinking from a fire hose if you try to have it all at once. So the first week, you learn about the first early numeracy concept. And the first one we talk about is subitizing. Whether you take the pre-K to second grade course or you do the third through fifth grade course, because even kids in third through fifth, if they aren’t able to subitize, we need to build that. So we talk about what it is, why it’s important, and I provide you with a way to assess students. And my assessments are not like, “Sit down and let’s do this worksheet.” It’s through observations. How can you observe this through the natural progression of the day? Right? What should you be watching for to see if kids can subitize, and how do we do that?

Christina: So we talk about how to look for that early numeracy concept. And then I give you a lot of my favorite experiences. I don’t say activities. I call them experiences. Kids need to experience numbers and their relationships. So we talk about experiences, and then I’m big proponent of games. So every week, you get a download of my favorite games. And also, I’m a big proponent of including your parents and getting parents informed about what we’re doing and why we’re doing it. Because I think back to … And I know you guys have talked about this before. I call myself the recovering traditionalist. And I think about if I didn’t know this stuff, I would’ve been that parent who’s like, “Why aren’t you just drilling and killing them,” basically. Like, “Let’s give them flashcards. They just need to memorize this stuff.” So at the end of each module for the week, you get letters that you can send home to the parents about what you’re doing, why it’s important, and some quick little things that they can do at home to help build that relationships or that early numeracy concept.

Christina: So each week, they get that for all of the eight number sense concepts. And so what it is, why it’s important, how to assess it, activities or experiences and games and a resource to give your parents. So they go through it week-by-week, and then you have three months that you can access it, which is so amazing. Because that’s what always bugged me about going out to do professional development, is I’m there for a day and then I’m gone. So I love the power of this online learning, because you get it continuously for three months. As you’re trying it out, you can come in there and you can add comments underneath each video and talk about what you’re trying, get advice from me and the others in the group. It’s just a fun way to be able to find people who are working on the same thing. Because sometimes you might be the only person out there at your school who’s wanting to try this, and so it’s fun to be able to be online and learn with others.

Christina: And you don’t have to be online at the same time. I used to call it PD in your PJs, but I was told somebody else uses that phrase, so I can’t actually use that phrase. But you can do PD wherever you want in whatever …

Jon Orr: In your most comfy clothes you have.

Kyle Pearce: In your [crosstalk 01:03:56]

Christina: … clothes you want. Clothing of your choice and the drink of your choice. Right?

Jon Orr: Right.

Kyle Pearce: Oh.

Jon Orr: Right.

Christina: how often do you get professional development like that?

Kyle Pearce: That’s up my alley.

Jon Orr: Yeah, for sure.

Kyle Pearce: Fantastic. Well, Christina, this has been such a great conversation. I know that some, especially our K through five educators who are listening, but again, the big idea is the same. It doesn’t matter what grade level you’re at. It’s all about foundations. Whether it’s a foundational skill for a grad nine course or a 12 course, it’s all about the building blocks and making sure that we do it and we do it right. And I really liked the metaphor of your husband and going and building homes and looking at jobs that aren’t done right the first time and the problem that that causes. You can still fix it, but it’s going to take more time and effort and probably a lot more planning to try to figure out, “Okay, where is the problem? How do we help address this and help move on from here?”

Kyle Pearce: So this conversation has been huge. I know, because Jon and I, having met you and having seen your content online for so long, you provide all kinds of great value bombs with the vlogs you do. You also have a Build Math Minds podcast that’s out, so we will definitely link that up. Are there any other website links that you’d like to share with folks before we say goodbye? We will be sure to post them in the show notes page.

Christina: I would definitely be looking up for those progressions. I know you guys are going to link my stuff, but for those who want to go learn on their own, looking up Clements and Sarama’s learning trajectories, Cathy Fosnot, she calls them landscapes for learning; understanding those progression of how kids develop their understanding is super huge. So even just going and Googling math progression, or a specific content; fraction progression, multiplication progression, you’re going to find lots of info out there to help you build your understanding of where kids were, where they hopefully were before they came to you, where they are now, and where we need to get them.

Christina: So that’s just a big piece. And I don’t care if you learn that from me or somewhere else. It’s an important thing that all of our students need.

Kyle Pearce: All right. And I want to make sure that folks check out that webinar that Christina was talking about. It is wide open for everyone to participate in, so be sure to share with your colleagues. Makemathmoments.com/buildmathminds will bring you straight to that page. Again, makemathmoments.com/buildmathminds will bring you to the registration page, but we’ll also link it up in the show notes.

Kyle Pearce: Christina, we are so happy to have had the time to have a chat with you tonight here on the Making Math Moments That Matter podcast. We’re hoping that you have a fantastic evening, and we will be certain to check you out in that webinar coming up real soon.

Christina: Thank you Kyle and Jon. It was a pleasure to be on here, and thank you for letting me spread this message even further.

Jon Orr: Awesome. Thanks so much, Christina.

Kyle Pearce: We want to thank Christina again for spending some time with us to share her insights with us and you, the Math Moment Maker community.

Jon Orr: As always, how will you reflect on what you’ve heard from this episode? Have you written ideas down, drawn a sketch note, sent out a Tweet, called a colleague? Be sure to engage in some form of reflection to ensure that the learning sticks.

Kyle Pearce: Like you heard in the episode, Christina is offering a free webinar training called Math Is Not A Worksheet. Registration is open now, as this episode is released until October 12th, 2019. You can register by visiting makemathmoments.com/buildmathminds. Again, that’s makemathmoments.com/buildmathminds. If you’re listening to this after October 12th, 2019, don’t worry. That link will either bring you to the webinar replay or a wait list for her next free webinar.

Jon Orr: In order to ensure you don’t miss out on our new episodes as they come out each week, be sure to subscribe on iTunes or your favorite podcast platform.

Kyle Pearce: Also if you’re liking what you’re hearing, please share the podcast with a college and help us reach an even wider audience by leaving us a review on iTunes, Google Play or Spotify, and Tweet us your biggest takeaway by tagging @makemathmometns on Twitter and Instagram.

Jon Orr: Show notes and links to resources from this episode can be found at makemathmometns.com/44. Again, that’s makemathmoments.com/44.

Kyle Pearce: Well, you know what that means, Jon. Until next time, I’m Kyle Pearce.

Jon Orr: And I’m Jon Orr.

Kyle Pearce: High fives for us. And high fives for you.