Khan Academy Ed Talks with Ted Coe, PhD - Tuesday, March 15

Hello and welcome to Ed Talks with Khan Academy. I'm Kristen Decervo, the Chief Learning Officer at Khan Academy, and today I am looking forward to talking to Dr. Ted Co., who is with NWEA, one of our key partners. Here he is the Director of Content Advocacy and Design. We're looking forward to talking about math and math learning today.

Before we get started, I want to remind you that Khan Academy is a non-profit organization and relies on donations from folks like you to help us continue to be able to do our work. So, if you go to khanacademy.org/donate, you will find a place where you can help keep us supported and work in continuing our work, so thank you for that.

I also want to acknowledge our corporate supporters who have come and helped us work through the COVID-19 crisis, including AT&T, General Motors, and Fastly. If you would like to listen to this conversation or any of our previous conversations, you can, wherever you get your podcasts, find the Homeroom LaSalle podcast, and you'll be able to listen wherever you prefer—at the gym, on your walks, doing the dishes. We hope you enjoy it.

So, before we get started, let me introduce Dr. Ted Co. He has over 25 years of experience as a teacher, professor, department chair, administrator, and a non-profit director, basically weaving together perspectives about mathematics education. He has experience as a full-time high school mathematics teacher, community college faculty member, a mathematics chair at two community colleges, and an assistant dean at the university level. And I will say, as someone who is a graduate of Arizona State University, he is a fellow Sun Devil as well.

Welcome!

Dr. Ted Co: "Thank you!"

Kristen Decervo: "We are glad to have you. So first of all, tell us what your role is specifically. What do you do at NWEA?"

Dr. Ted Co: "This is an interesting question, right? The answer is the title is Director of Content Advocacy and Design for Mathematics. The interesting side to it is, well, I get to advocate for content, and so, for many ways for me, it's like this dream world to be in where I get to kind of envision the kinds of things that we want to work on—how do we want to grow? How do we want to move beyond a focus on assessment to teaching and learning and getting into the classroom? Getting in there and rolling up our sleeves. That's a lot of what I get to do, and to me, it's just great fun."

Kristen Decervo: "That's fantastic! How did you come to be in this role? What was your path that you took to end up here?"

Dr. Ted Co: "It was so bizarre! It's a crazy career path, you know. I started out as a high school teacher; things were going well, I loved high school teaching. Highlighted the career! Then I ended up getting pulled over to one of the community colleges, and I actually walked right in. My first job in the community college system was as a chair at Rio Salado College in the Maricopa Community College District. It was like this big leap. I did end up switching over to another community college in the district, Scottsdale. While I was there, I ended up being involved deeply in the mathematics around the whole community college system. I was involved in the articulation conversations around the state and everything.

Somewhere along that time, the PARCC consortium came along, and you might remember this; it was in the early years of the Common Core. There were the two major consortia—PARCC and Smarter Balanced—and I was brought on to the PARCC side of the conversation. Arizona was one of the governing states at the time. I was brought in as one of the higher ed content experts.

So, I was dropped into this space suddenly, where I had been a college professor and high school teacher, and now I was in this space where they wanted my input on this multi-state assessment based on these new standards. I thought, 'Well, let's read the standards, let's see what this is about.' I got really involved in it. So much so that eventually, PARCC split off from the group that was running it, which was called Achieve. Achieve was responsible for the American Diploma Project, helping to lead the Common Core and all that. They hired me to be their math director. For the years prior to coming into this space at NWEA, where I've been for two years, I was the Director of Mathematics at PARCC and got to work all around the country with great folks everywhere and really get to see how the policies worked out and the difficulties people were encountering. It was great fun and has been one surprise after the other, and here I am today!"

Kristen Decervo: "Fantastic! That sounds great. So let's talk a little bit about teaching and learning math. Historically, there's been all this discussion about should I teach procedural skills? Should we teach conceptual understanding? There's almost these wars about which one is more important. Is that the right conversation to be having?"

Dr. Ted Co: "You know, it's a tale as old as time—at least as old as my career has been—that there's always this tension. I've had to sit back and think to myself, you know, how would I describe teaching and learning mathematics after my 20 years of teaching? I was like, well, how do I sum this up? It was just this idea that part of what we do in mathematics is we teach these ways of doing, but we also want to make sure that you're picking up ways of thinking.

I'll give you an example that goes alongside habits of thinking. You have these ways of thinking at a core that are bolstered by these habits of thinking, which are then in turn really brought to the table in certain ways through the ways of doing that allow you to go on and do something bigger with the mathematics. An example I use frequently is with this idea of ways of thinking. I can ask a room full of adults this question: how many years was it from December 7, 1941, to December 7, 2014? That's how long I've been using this question.

And, you know, by and large, you ask this to a room full of even adults, and they start chewing on an answer. They start going, 'Oh, what can I do?' They go through some mechanizations in their head. They might think, 'Well, that's actually kind of a subtraction problem, but I don't really need to use subtraction to get it because in this case, I'm just trying to find this additive comparison and this gap between the two numbers.'

There are different ways to do it. Maybe I want to go from 1941 to 2000 and cut it off, or maybe I want to take the whole thing starting in 1941 and slide it back a year to start in 1940. Either way, all of these things are fair. But the majority of the folks I ask that question don't answer it with the way of doing. They don't do the 15 steps of setting up the algorithm. They bring ways of thinking to bear. I think part of why we get into this mess is in the standards we have—learning progressions that look like the way of doing is the goal. The way of doing is just one more piece.

So, I really like to emphasize this idea of ways of thinking with regards to habits of thinking, all of those things working together. Because if you have just the ways of doing, all you've got is a box of puzzle pieces."

Kristen Decervo: "Right."

Dr. Ted Co: "So I'm going to keep going with a year example here to push on this. My sister got married in 1997, and I have to say I somehow never made this connection. But she said to me one day, 'If you want to know how long we've been married, just take what current year it is and add three.' Because from 1997 to 2003, and then I was like, 'Oh, yeah, why did I not think of that?' So that's a way of doing it. That raises the question: how does that relate to ways of thinking?"

Kristen Decervo: "And are there many different ways to do it and the ways of thinking around it?"

Dr. Ted Co: "Yes, the idea is that when I ask that question, there are so many different ways you could approach it. You could approach it with a way of doing, and there's no harm no foul. You could approach it with ways of thinking. But if you only have the way of doing and you don't have those ways of thinking, then when you get to somewhere in real life, it doesn't illuminate. A place where this happens is you can ask an adult where in real life have they ever actually used division of fractions. I'll be surprised if you actually get any valid answer back from anybody. Why is that?

If we reflect back on it, when we were taught the division of fractions, those of us of a certain age, we were taught a way of doing, and it was totally disconnected from ways of thinking grounded in what it means to divide. Yes, you were just taught the procedure."

Kristen Decervo: "Yeah, that totally makes sense. So if I'm a teacher, how do I start thinking about encouraging these ways of thinking?"

Dr. Ted Co: "That's the tricky part. One really cool thing that happened in the last couple of years, Peter Liljedahl came out with a book called Building Thinking Classrooms, and I'm like, 'Yes, yes, we're seeing this emphasis shifting to thinking!' At the same time, my colleagues and I at NWEA have been trying to work on this idea of how can we get ways of thinking moving and talked about in classrooms in ways that are non-threatening, that feel safe for both teachers and students. There's awkwardness when we ask ourselves thinking questions. There’s vulnerability involved.

So we built out some formative conversation starters. The role of these formative conversation starters is to help shape how you might enter into some of these kinds of conversations designed to focus on ways of thinking. You also know what you're trying to listen for at the same time. We've got these bins of big ideas to nurture sense-making, and we've got clusters of questions that unfold as you ask them. They might lead to moments of cognitive dissonance or might lead to places where you step back and think, ‘I really do need to think more deeply about that.’

There are different options to do it. You can put in classroom routines, you can build tools like we're trying to build out, but the idea is what can we do? There are all kinds of things Peter's got in his book, as well as what I'm offering here today."

Kristen Decervo: "I like this idea of conversation starters. I think we're putting a link to them in the chat for folks that are following along. Can you give us some examples? What are some examples of what these are, and how do they work?"

Dr. Ted Co: "Yeah! Sometimes the idea, like with fractions, we might start with a question like, 'Where's a fraction?' And then the next question might be, 'Is a fraction one number or two numbers?' Just sitting back and talking about it. Then ask, 'What do you think of when you think of 3/8? What do you think of when you see 8/3?' Because if you're thinking of 3/8 as three out of eight, then 8/3 becomes something that's a little tougher to think about.

Another example would be a question that starts off with, 'What does division do for us?' You want to sit back and think about it. Then the next question wouldn't be a division problem, but to ask, 'How can you think about the outcome without computing? How can you think about a problem like 10 divided by one-half?' Then you'd go on to the next one: 'How would you think about negative 10 divided by one-half?'

If your ways of thinking are consistent and robust, then your answers will gel together. But if you have fractures and if they're disjoint, you kind of have to go back and figure out how to make your thinking work with your answers."

Kristen Decervo: "From the teacher's perspective, what should you be listening for in these conversations?"

Dr. Ted Co: "We know we want fractions; we want students to think of a fraction as a single number, as a point on the number line. But if I'm talking to my students about fractions and they're only thinking of it as two numbers, then I'm talking about something different from what they're hearing. It's this idea of getting these meanings out in the open so that we know we're all talking about the same thing.

When I say 'fraction,' you're thinking the same thing. Or when I talk about 'division,' you've got a notion of thinking of it as a multiplicative comparison. When you have those ways of thinking, a question like 'How can you think of 10 divided by one-half?' doesn't become something out of bounds; it becomes something you can hang under one of those previous existing meanings of division."

Kristen Decervo: "Right, yeah, that makes sense. Do these starters work across ages and grade levels? What kinds of things can younger kids do, and what kinds of things can older kids start to get?"

Dr. Ted Co: "The interesting thing about them is that the entry point is through an item—through a standard. We've spent so many years unpacking standards. We want to find out what all of the other mathematics is that has to come in alongside it to help get through this.

We tend to look at an item and go, 'This is the standard.' So if they get it wrong, we want to go back to the nearest connecting standard. What we're saying here is maybe you want to ask them, 'What's a fraction?' So let's talk about what a fraction is. It might not be the nearest neighbor standard that's the issue; it could be some ways of thinking that endure.

Yes, they grow a little bit, but they endure. They grew out of these things, actually, because we thought about what we could provide teachers. Students are coming with such uneven opportunities to learn. Philosophically, I say I can't just focus on ways of doing; I have to focus on ways of thinking and the habits that come alongside it.

So if you're a third-grade teacher, you could probably pull stuff from fourth and fifth grade. If you're a high school teacher, you can pull from any grades. They will still be fair game because they're things to talk about—ideas that endure that go on and on."

Kristen Decervo: "If you're in high school, you might dig into functions, you might dig into comparison, you might dig into operations. I hope you dig into proportional relationships!"

Dr. Ted Co: "Yes!"

Kristen Decervo: "This whole conversation is making me think about something we talked a lot about at Khan Academy, which is what Sal used to call Swiss cheese gaps but now everyone calls unfinished learning. It's the idea that lots of folks didn’t get some of those foundational concepts when they were first presented, and so trying to build the next level without those concepts creates a shaky foundation. It just keeps going.

So it sounds like these conversation starters are starting to think about, almost diagnosing, in some ways, like do you have these foundational ways of thinking?"

Dr. Ted Co: "Yes, the foundational ways of thinking are, when you look at your state standards, you're going to see them kind of in there. You might not see them as explicitly as we're trying to call them out here, like these are some core ones that you want to keep coming back to, revisiting, and keep listening to.

The kinds of questions we're putting out here can be used however you want—full class, one-on-one, in small groups, for turn-and-talks. The idea is to sit back and listen. What we would point out is this isn't intended to be a teachable moment; we want to set up the culture so that this is free, so we can open up and say, 'Okay, tell me what you're thinking.'

And what's interesting about that is if you keep asking more questions, you might realize that your first thoughts about what people like to call misconceptions could be completely wrong—that you've misconceived their misconceptions. I don't like using the word 'misconceptions' because the students have the conceptions. This is all about what conceptions they might have, and let's build from that."

Kristen Decervo: "That connects back very closely to Sal's house-building analogy."

Dr. Ted Co: "Yes! These are foundational threads that wind their way through—not just on the bottom floor. You just keep strengthening them like the trunk of a tree; it just keeps getting stronger and stronger."

Kristen Decervo: "Yeah, absolutely! That totally makes sense. How do you think about the idea of mastery in this world? We talk a lot about mastery learning here, and you know, it’s easier when you're talking about skills and having mastery of skills. But mastery of some of these things is somewhat different. Is that even a valid concept?"

Dr. Ted Co: "It’s so hard to say what mastery is. We always want to draw every right; we always want to draw a line for proficiency somewhere—for what it means to have learned something. But in this mindset, you know, on the ways of thinking, these are things that continue to grow, change, morph, and evolve. As I think about how we ever establish whether students have mastered this, I'm not sure that's even the right type of question!

Rather than thinking about some arbitrary endpoint, it's just how is that schema building up in their minds as they continue to grow? How can I leverage that to go on to the next place?"

Kristen Decervo: "Right, you can think about how you would expect some of these answers to change as you learn new things, changing your ideas about what these are as you move up through new skills and things developing."

Dr. Ted Co: "Interesting."

Kristen Decervo: "So we obviously do a lot with technology, but we also want to think about how teachers and technology work together. I’m interested in your thoughts about the interplay between the two. As I think about what you're talking about, I see that technology is never going to be good at doing what you're talking about—it's the thing that teachers need to be in classrooms. So I’m interested in your thoughts about how teachers and technology can work together."

Dr. Ted Co: "I'd be happy to hear you see it that way because that's how I see it too. It's the idea that computers are really good at helping with ways of doing things, but to put these complex ways of thinking together, that takes some time and working and developing. I see a very human side to that, and that would be very hard to capture with a computer system.

Each can play a very important role. So the computers provide an unprecedented access to content in ways that we've never seen before. Building that up, the next task is how do we take that access we've now shared and build in these kinds of things?

What can we have the computers help to support us with when it comes to engaging in these kinds of conversations that will help sort of listen to students and figure out exactly what might be the next move based on ways of thinking—not just the ways of doing as you might see pointed out in standards and standards mappings?"

Kristen Decervo: "Absolutely! So another piece that we try to encourage is helping students become owners of their own learning and developing those metacognitive skills, being able to understand what they know and evaluate what they understand and need to learn, and how those fit in. I'm interested in how you think about that within this framework too."

Dr. Ted Co: "That might land in that habits space, right? Being able to reflect on—am I being honest with myself? Am I making sense? In my own mind, can I justify what I'm saying? Or am I just starting off spouting things? With this approach, you get the opportunity for the student to reflect.

It's by not correcting right away; by continuing to ask questions, you're providing the student opportunities to reflect. And maybe once it's done, you sit down with the student and help them have some metacognitive reflection on it: what would you like to work on? What questions did you answer that you felt like you wanted to take back or think through again?"

Kristen Decervo: "Right. To help them spot where they need to strengthen their ways of thinking."

Dr. Ted Co: "Yes! I think that idea of helping—thinking about your confidence in your answers is something we talk about too. Like after students give answers, having them think about how sure they are of that answer and what they did or didn’t know. I could see them doing it with this too. So asking students, 'What do you think a fraction is?' and then 'How comfortable are you in that answer?' and 'Where does that fit in?' would be interesting to think about."

Kristen Decervo: "Definitely! So, as you think about advising teachers on next steps, they've had this conversation. What is the next thing they should do based on the information they get?"

Dr. Ted Co: "As you're listening, you're trying to interpret what you're noticing, and what work do you want to notice with what the students are saying? Reflect back on it. In these things that we're putting together, we've got some videos, and it's very fun to sit and watch because you can get even more out of it.

Teachers will need to do some pretty good interpretation of what they're hearing and then think about what the understandings look like that they need to either fill in or move forward. It’s also self-reflective. The questions we put down made us stop and think: How would we answer this?

We weren't putting down questions to get right or wrong answers. We were coming up with questions that uncover thinking. You don’t have to ask right or wrong questions to uncover thinking."

Kristen Decervo: "Right! That's another piece in math education that we’re coming to better understand—that math education isn't just about writing wrong answers. Math does have right and wrong answers to some things, but we aren't minimizing that need for right or wrong answers.

There’s a time for that and a time for listening to thinking."

Dr. Ted Co: "That really makes sense. Excellent!"

Kristen Decervo: "So anything that you're excited about? I'll close on this question: anything you're excited about in the coming couple of months that your team is working on?"

Dr. Ted Co: "Oh, yeah! One of the things I’m working on—I just did a talk on this at South by Southwest last week—and it's called 'Mathematical Rigor Mortis: The Quest for a Cure.' It’s about how many ways math kills things. Math kills dreams, it kills joy, it kills progress.

I think it also kills moving policy in the right directions and making changes at systemic levels. These things are so entrenched in the way we do things. But I'm encouraged; I hope the folks out there are encouraged. I think we're moving in the right direction to cure mathematical rigor mortis.

If you ever want to reach out and talk to me about it, I'd love to. That’s something I’m encouraged about—the sort of spaces I’m seeing across the country to help address that."

Kristen Decervo: "We had a question from Angel on YouTube, who's a student and wants to know how they can stay motivated when doing work. We hear this question a lot from folks needing to stay motivated to do their work. What do you think?"

Dr. Ted Co: "Angel, I hope this is helpful! If math is being presented to you as a collection of puzzle pieces, that you're just shaking the box and can kind of hear them rattling, the challenge would be to know that there is a big, beautiful picture that all of the pieces fit together. No piece stands alone.

If you find that encouraging, then you can figure out how these pieces fit together and understand the story that's being told here. It’s a great one! There are lots of places where you could go to help get support for that as well, but don’t let the current situations you might be in or the happenings around us destroy the joy mathematics has for you."

Kristen Decervo: "I like it! And putting together that puzzle could be a challenge—a fun challenge to undertake and think through."

Dr. Ted Co: "Excellent!"

Kristen Decervo: "Well, thank you so much for joining us today. It was a pleasure to talk to you, and I hope some of our folks out there dive into those conversation starters."

Dr. Ted Co: "Well, thank you! This is great."