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Join us for an insightful conversation with Bill McCallum, co-founder of Illustrative Mathematics and a distinguished professor in mathematics education. In this episode, we delve into important topics that will transform your perspective on mathematics instruction. 

Discover what makes a problem engaging for students and how to identify struggling learners through classroom observations. Explore the power of connecting mathematics to meaningful stories and the potential impact of A.I. and digital tools on future instruction. 

Most importantly, learn why empowering students to do the math is crucial for their success. Don’t miss this thought-provoking episode with one of the influential voices in mathematics education, Bill McCallum.

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Illustrative Mathematics

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FULL TRANSCRIPT

Bill:

I think it’s different for different kids, too, and that’s something you just have to face up to. Not every problem is engaging for all kids, which means you need some diversity in the types of problems, in the types of contexts, that you use, so that at any given time, some of the students are going to be finding the problem engaging. We did a lot of… The Illustrative Mathematics project was to…

Jon:

Join us for an insightful conversation with that person you just heard right there. That’s Bill McCallum, co-founder of Illustrative Mathematics and a distinguished mathematics professor and also co-writer of the Common Core State Standards.

Kyle:

In this episode, we’re going to delve into some important topics that we’ve been discussing for quite some time, and it’s just so great to hear Bill reiterating some of what we believe here at Make Math Moments to be transformative practices in the mathematics classroom. We’re going to be talking about what makes a problem engaging for students and even how we can identify some of our struggling learners through our classroom observations.

Jon:

We’re also going to explore how mathematics is itself like a story. We’re going to talk about the impact of AI and the digital tools on the future of math instruction. Most importantly though, folks, we’re going to learn why empowering students to do the math is crucial for their success. Don’t miss this thought-provoking episode with one of the most influential voices in math education, Bill McCallum.

Kyle:

Let’s do it. Welcome to the Making Math Moments That Matter podcast. I’m Kyle Pearce.

Jon:

And I’m Jon Orr. We are from makemathmoments.com.

Kyle:

This is the only podcast that coaches you through a six-step plan to grow your mathematics program, whether at the classroom level or at the district level.

Jon:

And we do that by helping you cultivate and foster your mathematics program like a strong, healthy, and balanced tree.

Kyle:

The trunk of your tree represents leadership in your organization or the classroom pillars in your math classroom. The roots of the tree represent mathematics content knowledge and what it means to be mathematically proficient.

Jon:

Like a tree, it requires soil, water, and sunlight. Your math program requires productive educator mindset and the belief that all students can achieve at high levels. Your professional learning structure or your professional learning plan is represented by the limbs of the tree.

Kyle:

The branches of your tree represent the development of educator pedagogical content knowledge, including effective teaching and equity-based teaching practices. And the final section is the leaves, and we’re going to explore those leaves in depth here today, representing the resources, the tools, and your classroom environment.

Jon:

And if you master the six parts of an effective mathematics program, the impact of your math program will grow and reach far and wide.

Kyle:

Every week, you’ll get the insight that you need to stop feeling overwhelmed, gain back your confidence, and get back to enjoying the planning and facilitation of your mathematics program for the students or the educators that you serve.

Jon:

All right, let’s jump right into the conversation with Bill.

Kyle:

Here we go. Hey, hey, there, Bill, how are you doing in your neck of the woods? It’s awesome to finally catch up here on the podcast. Been a long while since I ran into you in Arizona. Hey, tell us a little more, for those who don’t know you, I feel like a lot of people in the math community are aware of many of your wonderful math education accomplishments, but give it a whirl anyway. Who are we talking to? Where are you coming to us from?

Bill:

Sure. Yeah, and thanks for having me on the podcast. Yeah. Well, my career started out as a normal math researcher, math professor. I got involved in education early on through thinking about calculus reform. Then of course there was the moment in 2009-2010 when I was involved in writing the Common Core State Standards. Illustrative Mathematics was a project at the University of Arizona that was going last a year or so. By 2013, I guess we decided there was something more than that, and so I left the university and started the nonprofit Illustrative Mathematics. I’ve been involved in math education at all levels, undergraduate, K through 12. Yeah, that’s me.

Jon:

Awesome. Yeah, we speak with a lot of educators across the country, across the world, actually, and many of them are raving about illustrative mathematics. So we’re big fans over here. I’m curious, though. We ask a lot of our guests about their start in education. There’s some curiosity that I have, Kyle has, I think a lot of teachers and then listeners have, about comparing or thinking about how they started versus the guests that we have. So, you said you kind of were a normal math researcher, but I’m wondering what drove you initially. So you’re going in the Wayback, right? The Wayback Machine right now of going like, what sparked this love of mathematics? What got you going down that route versus, say, going into something else?

Bill:

I think the thing that sparked me… Let me think about this. I’ll give you an episode from actually when I was quite young, when I was a kid. My mother used to take me down to the local public library, get puzzle books, and I don’t know if you know these coin weighing puzzles where you have a counterfeit coin and you have… The mother of all coin weighing problems is you have 12 coins. One of them is either lighter or heavier, and you have to figure out which it is and whether it’s lighter or heavier in three ways.

So, I was working on this problem. My mother used to work the problems, too. I couldn’t figure it out. My mother solved the problem. She came into my room, and I was crying with frustration. She said, “Do you want me to tell you how to do it?” And I said, “No.” And so, she just turned around and walked out, and I think that’s a teaching move on my mother’s part, which I thought was awesome. Because her child is crying. She said, “Okay, you want to work it out? You can work it out.” Then eventually, I did work it out. But yeah, that for me is probably a moment I think of when I saw both the love of mathematics but also saw an awesome teaching move.

Kyle:

And it’s interesting, because I want to ask a follow-up to that in just how that… I’m guessing it did help to shape who you are not only as a math student, as a learner, but then obviously into the world of research and curriculum writing and all of those things. But talk about the ultimate productive struggle.

Bill:

Yes.

Kyle:

Now, some might say crying is not exactly what we want for our students.

Bill:

Yeah, [inaudible 00:06:52] there.

Jon:

How much productive it was.

Kyle:

However, what really jumps out at me is just your almost willingness… So, not only did your mother have a fantastic move to actually give you the option, and you then took the option to figure it out yourself. So if you talking about perseverance, if you’re talking about grit, determination, and rigor, that is exactly what it’s all about. Again, minus the crying, we want more of our students to be math…

Bill:

Yeah. We do not want our students crying, just to be clear.

Kyle:

Yeah, we don’t want them crying, but…

Bill:

That is not a Illustrative Mathematics belief.

Kyle:

Yeah, I think that is so fantastic. So, I’m wondering, now let’s extrapolate, because you hit the math moment question. I think you knocked it out of the park there, and you’ve painted us a really clear picture. Now my wonder for you is I want you to go maybe a little later in your journey when you were a researcher. So before, I’m wondering, or guessing, how did that impact or influence who you were as a researcher? But then, I’m guessing you probably also did some teaching as well as part of your role.

Bill:

Yeah. I’ve got another story about that. This is when I was a graduate student in mathematics at Harvard University, and I did teach calculus. And at Harvard, it wasn’t just that… you actually had a section of calculus that you taught yourself. And I remember very clearly the moment when… I used to give these fantastic lectures. They were beautiful, they were clear. I was explaining everything so well. And then, I remember the moment when a student came in with a question and I probed a little bit, and I was like, “Oh. Nothing I’ve been saying, my beautiful lectures, they’re just landing somewhere out in the ether. It’s not working.”

I think every mathematician has that moment, but I think a lot of them just sort of close the book, look the other way. You lift up, you look inside the student’s head, “Oh, my God, I’m going to pretend I didn’t see that.” And they just keep on giving these beautiful lectures. But I guess for me it was an interesting problem. Okay. So, that was probably when I first started thinking about, what is a way of teaching that engages students that get some thinking? And it’s not just about perfect explanations, although perfect explanations do have a role. They usually have a role after you’ve got the students engaged in thinking rather than the first thing you say, which is sort basically the problem based instructional approach that we take with Illustrative Mathematics.

Jon:

I’m curious, you made this shift towards Illustrative Mathematics. What would you say? Maybe elaborate a little bit more on the turning point between, “I’m a researcher, and then I realized that maybe the resources that are at our teachers’ disposal, maybe the access needs to be changed, or maybe what happens in the classroom is needing to be changed.” What made you think that, “Hey, I’ve got to go here and make this dent here, because it’s so much needed.” I’m curious about that transition.

Kyle:

And can I ask a question that maybe is just a scooch before that? And that’s when you recognized you were teaching as you articulated these fantastic lectures, which again, you nailed it. I felt that way. I’m sure Jon’s felt that way. Many people have felt that way, until you actually look for feedback, right? You actually assess the situation, go, “Oh, my gosh, you have no idea what I’m talking about.” But I’m wondering, as you’re making that transition, I’m like, what did you do in order to get better? Because I’m sure you didn’t just leapfrog to, “My lectures are not landing as I thought.” You obviously had to do some learning there. There had to be some reflection. And I’m going to guess that your faculty wasn’t… Or maybe I’m wrong, but I’m going to guess maybe the faculty may not have had the same epiphany that you had. So I’m wondering, what did that look like and sound like, especially in post-secondary or post high school education?

Bill:

Yeah. Yeah. I had a great mentor in teaching at Harvard, Deborah Hughes Hallett, who was in charge of all the calculus TAs. This is after I finished my PhD, and it was at Berkeley. She put together a consortium of universities to think about how to teach calculus in a different way from the way it was currently being taught. More conceptual, more application based, and that was the Harvard Calculus Project. And I learned a lot from that about teaching. I learned a lot from watching Deb teach. I learned a lot about getting students engaged. I don’t think it was some sudden transition where I suddenly switched to a different pedagogical philosophy. I think it was a slow learning process, but I guess I learned, which I know sort of sounds obvious, but you do have to have engaging problems, and you have to give students a chance to work on them and to think about them before you just tell them how to do them.

Kyle:

Yeah.

Bill:

You do, of course, have to in the end make sure that they have achieved the learning you wanted to achieve. That does involve some sort of explicit instruction sometimes, but it happens after you’ve gotten students working and thinking. It happens when they feel the need for the notation or the definition or whatever it is that you were going to teach them. It comes up, like, “Oh, we really need a word for this,” or, “Wow, wouldn’t it be nice if we had a formula for that?” You wanted that to be happening out of the problem solving process, not something you feed into the problem solving process at the beginning.

Jon:

Yeah, that’s perfect, because I think that goes hand in hand. We’ve been calling that the real flip classroom of transitioning, like saying, hey, normally for 10 years I taught by doing the opposite of that by saying, “This is what we’re going to do. This is how you get formulas. These are all the definitions. Now let’s go and apply them,” versus bringing that shared experience to light with the students through an engaging problem and then formalizing after that. I think for me, that was a career defining change and we’ve been calling that the real flip classroom, because it’s a…

Bill:

I like it. Yeah.

Jon:

Now, you did say it all starts with an engaging problem, and I wanted to get your take on this. What would you say is necessary for that engaging problem to exist? Because I think a lot of our listeners, a lot of teachers ask us, “We’ve got resources here, we’ve got textbook resources here. There are lessons we can find online,” but a lot of teachers want to figure out how to make some of these engaging problems or transform those engaging problems into engaging problems. What would you say, in your opinion, makes a true engaging type problem to start a lesson with?

Bill:

We have a lot of ideas around that. In the writing of our curriculum, we have these warmups. I also think that engaging problems can be mathematical problems. They don’t have to be contextual problems. I think a lot of what makes a problem engaging is the launch of the problem and how you frame it rather than just the problem itself. I think problems can be dressed up in different ways to be more or less engaging. They can be launched in different ways. I think it’s different for different kids, too, and that’s something you just have to face up to. So not every problem is engaging for all kids, which means you need some diversity in the types of problems, in the types of contexts, that you use, so that at any given time, some of the students are going to be finding the problem engaging.

A lot of the Illustrative Mathematics project was to illustrate the standards. We wrote three or four problems, and “we” was a huge group of people contributing to illustrate each standard. It’s an empirical question, partly, whether a problem’s engaging. And from the way people were downloading these problems and using them in the classroom, we figured, “Okay, we’ve got something going here. We seem to have the authoring capacity to produce engaging problems.” And then, that got us thinking about writing curriculum.

Kyle:

I love it. And one of the big takeaways I took from that chunk, which we 100% agree, is it really comes down to how you launch the problem. And it can look very different. And a lot of people look at the work that John and I do, is we tend to try starting with context, but sometimes, there are topics that are very difficult to find, especially as you go further down the road, right? As you continue down the abstraction train, we’ll call it, or track, things sometimes become a little bit more and then maybe contrived in some ways. So, it’s like, how do we then launch this problem in a way that’s just about the curiosity of it all? How is this possible? And these are great strategies to have in our bag of tricks. And then, you also connected that, which I think is really important, to this fact that we have so many different students in our classrooms with different backgrounds, different experiences, different memories of mathematics.

And that in and of itself is going to require that we don’t want to just be sort of reliant on one strategy moving forward. So, that’s just fantastic. So I want to extrapolate a little bit from here. I’m getting a really cool sense, like a timeline in my mind, of your journey. Tell us more about… And this is something that I’ve always known, having had an influence in writing and helping to write the Common Core, what was going on there? And my understanding would be, or at least how I thought of it in my mind, was that Common Core happened first and then Illustrative after, or was Illustrative something that you were already working on, and it sort of led into your connection to the Common Core? Tell us more about that.

Bill:

It’s the former. Illustrative Mathematics… We finished the standards in June of 2010, and everybody was like, “You need to explain. You need some document explaining the meaning of these standards. Illustrations. You need illustrations of the standards.” And so, that was really the genesis of the Illustrative Mathematics Project. That’s why the company is called Illustrative Mathematics, because we were illustrating the standards. It was a great project, because what we were trying to do there was coordinate the expertise of lots of different people, mathematicians, teachers, math educators. I’ve always believed that… And I think this is partly what is good about Illustrative Mathematics, is our ability to coordinate expertise. It’s not like we have only mathematicians or only teachers or only math educators. We are sort of taking what we can from the expertise of all these different groups of people and using it to forge something coherent.

And that’s what we did with the Illustrative Mathematics Project. For that matter, I would say that’s what we did with the Common Core, and that’s what we did with our curriculum. People ask me, what’s the secret sauce in our curriculum? There isn’t any secret sauce, it’s just a well-coordinated… It’s like if you think about the Boeing 747 or something like that, which was at the time an engineering marvel, it’s not like they invented wings. They didn’t invent some new way of flying. They had an engineering team that coordinated a lot of expertise and made sure that that expertise was sort of balanced, coherent, collaborative, so you came out with something that worked, and I feel like that’s what we’ve done with Illustrative Mathematics.

Jon:

Yeah. By saying that you have brought a lot of great minds together to kind of build on and work as a team, and you had to do that with the Common Core, I’m curious actually about two things. One is, what would you say a big challenge has been, or maybe still is a big challenge, when working with teams? You’re putting a bunch of writers together to kind of come up with standards, and you’ve said that you’re working with a team. You had worked with a team to build Illustrative Mathematics. Many educators consistently have to work in teams, especially our district leaders who are working with coaches and consultants and people who are helping implement the curriculum. So, I’m thinking what would be one of the biggest challenges you faced when working with these teams, in either writing the Common Core or working at Illustrative? And then how did you overcome that? And maybe you have some advice around that when working with large teams of large individuals.

Bill:

A lot of it is just listening to people, especially people who are maybe unhappy about something. Listening to them. And listening doesn’t mean automatically doing what they want, because you’ve got lots of different people wanting to do lots of different things, so you can’t satisfy everybody. But there’s a balance between listening, showing that you’ve listened. If somebody makes a suggestion that actually, “Oh, we could change something to satisfy that request without doing any harm,” then you make those changes. You make it visible that you’ve responded, but you also don’t try and be everything to everybody.

So there’s some sort of balance there you need, and that’s what I mean about coordination of expertise. The coordination part is you’re not just listening to everybody and doing whatever they want. You’re taking it in, and you’re balancing it. And also, I’m going to say there’s a huge element of luck here. We just had awesome people wanted to come and work for us. And so, I think they were excited by the prospect, and we had some great people say, “Hey, count me in.” Those are people who had been downloading our tasks, who liked Illustrative Mathematics, and they had sort of come to think, “Now that this is a company that’s going to start writing curriculum, I want to go and work for them.” So, we just have some awesome people.

Kyle:

I love it. I love it. That’s fantastic. I love that difference between coordinating and listening to what everyone says and doing that, right? You’re compiling all this information, then we still need to make decisions, right? And sometimes some people might not like the decisions, but they have to be made. And you’ve got me thinking also. We’ve talked about this idea of allowing students to engage with the mathematics, to explore mathematics, to investigate mathematics, but ultimately, at the end, we need to make sure that students got what we were trying to help them understand in that process.

You talked about explicit instruction and how that still has a role. Sometimes hear people say, “Oh, there’s these people that believe math is this and you always do that, but you never do this,” and vice versa. And in reality, this theme of balance is really key. I think ultimately, through the process of allowing students to investigate, and truly investigate, requires an amount of content knowledge from the educator to be able to take that experience, put it into play, taking it and seeing that and truly facilitating so that it actually lands in the way that was intended or that is helpful for students to truly engage with the mathematics. And then, they have to also be monitoring that work in order to determine, “Where do I help them to tie the loose ends? Sometimes it’s a lot of me, sometimes it’s a little bit of me.”

My thought is when we look at a lot of classrooms, where we still see a bit of this struggle is we see a lot of people who learned the way I learned, and I think a lot of people listening have expressed this, that they learned procedurally. And that is obviously a massive, massive challenge in education. It’s what you knew as a child learning mathematics, and it’s like you have to relearn all kinds of concepts that you thought you knew, but you only partially knew. You only knew the procedure.

So, I guess my wonder for you is, and this might be within your work in Illustrative or the Common Core or maybe just in general and how you’re feeling now, how do we… And we’re speaking to district leaders. A lot of district leaders listen to this podcast. How do they support the educators that they’re working with in order to allow them to be able to teach math more conceptually? Given the fact that especially in our younger or earlier grade bands, a lot of these teachers don’t actually have the mathematical background. They don’t have the background to prepare them. What might be some thoughts, some encouragement, or maybe pointing them in a direction to help them so that they can help more of the educators that they’re serving.

Bill:

Any district leader who wants that help has already made a huge step in the right direction.

Kyle:

100%.

Jon:

Right.

Bill:

That is to say, understanding that it’s more than the curriculum, it’s more than the professional learning you give the teachers. It’s the district, it’s the community support. That entire thing is what we call the IM classroom. An IM classroom is not just a classroom using our curriculum. It’s a classroom where the teachers have access to that professional learning, where the district leaders support them in making the transformations they need to make, where the district leaders also understand this is not happening overnight, that it’s a learning experience, a multi-year learning experience for teachers to change their practice. And there’s that sort of community aspect to it as well, with family support. All of that is what you need.

We tend to think, if someone asks, comes to you and says their kid’s struggling in math, you say, “What curriculum are they using?” And I always think that’s the wrong question. The question is, what does the classroom look like? Go and look at a classroom and see what’s happening in the classroom. And that’s actually our next journey at Illustrative Mathematics, is having built this curriculum, we’re trying to build out those supports. We have the professional learning, but implementation supports, supports for district leaders. We’re trying to sort of build that whole coherent set of supports around the curriculum for districts wanting to make that change. But like I said, wanting to make that change is a big step.

Jon:

Yeah, and that’s super interesting. You’re moving down that path, and amazing to support those district leaders. And oftentimes, in the discussions we’re having with those leaders, and I’m sure that you’ve had those with leaders as well, is this barrier about change. I think when we talk about what a good math classroom should look like to achieve balance, to achieve instruction that changes students’ ability to understand mathematics at that deeper level, oftentimes our district leaders are feeling frustrated because they’re trying to invoke change in teachers, but there’s resistance to change with certain teachers. I would say, what have you learned so far to help address that barrier of a resistant teacher to that change? We’ve had lots of chats, but I’m just curious on your take on that.

Bill:

I think for one thing, you need to meet teachers where they are. I don’t know. If you remember back in the days of reform curricula in the ’90s and the ’00s, you’d go to all these lectures, and people would say, “We have to change everything they’re doing overnight.” Oh, it’s not going to happen. That doesn’t happen in human cultures. And we’ve seen schools where there’s a range of integrity of implementation and also a progression of learning. In fact, we have a tool called the Implementation Reflection Tool to help school districts sort of look and see where they are on a journey. Not, “Have you figured it out or not?” But, “Are you just starting out? Are you just aware of the instructional routines? Are you beginning to try some of them out? Are you beginning to master the instructional routines?” Different teachers have different phases in their journey, and I think you sort of have to be welcoming to all of them.

I think it helps when you have that variability, because anecdotally, we’ve heard that the teachers who are… and I’ve seen it happening in classrooms, the teachers who are really implementing the problem-based instructional model are having more success, and their kids are enjoying themselves. And you can see that they’re enjoying mathematics, learning mathematics. And so, I think that becomes a sort of a gradient of rumors spreading. “Hey, you know what? Actually, if you do this the way it’s supposed to be done, it actually works.” And then, the teachers who are maybe not there yet begin to think, “Maybe I’ll do some warmups.”

It’s a process of learning. We designed the curriculum to have actually built-in educative features for teachers, both on the mathematics and on the instructional practice. We have these instructional routines, which you can learn how to do. We didn’t invent any of them, but… Most of them, we didn’t invent. Think peer share, number talks, whatever. But the point is they’re routines, and we describe the routines, and the teachers and the students come to know the routine. So, it’s not like you’re saying you have to change everything about the way you teach. It’s more like, “Hey, try this routine out. See how it goes for you.”

Kyle:

Right. I love it. I love it. Something that we’ve been thinking a lot about, and Jon and I have been chatting with a lot of district leaders over the last while, and something that sort of dawned on us was that, and I’m curious to get your perspective on this, but when we get teachers and help teachers build their content knowledge in mathematics, what we’ve noticed in our experience is that we focus on things like strategies and the models and looking at developmentally, where might these come out in the learning continuum or trajectory? And something that we’ve noticed is that once teachers are learning the math themselves, and they’re able to see it and be more flexible with the mathematics, we notice that their actual practice does change. We’ve never met a teacher that’s like, “Now that I know these strategies and these models and I see how they all connect, I still want to only teach the procedure.”

What we see is it’s like the opposite, right? They go, “Oh, my gosh, I wish I knew this back when I was blank,” and then things start to shift. I guess I’m wondering is, what are you seeing in that world in terms of, where do you see the epiphany for most teachers? Because something I have noticed is that sometimes I try to teach… Routines are helpful to shake things up, but if the routine is done, and I’m still focusing too procedurally on things, I don’t see the changes in my students, and then I might actually think the routine isn’t working so hot.

Bill:

Isn’t working, yeah.

Kyle:

When in reality it’s like, “Oh, actually it was the delivery of the content that’s the issue here, not necessarily the routine.” What are your thoughts on that? Does that connect to some of your experiences, or have you seen something different?

Bill:

I mean, that’s another thing that I want to talk about, which I think I’ve heard teachers mention in our curriculum. We designed it to be coherent. I know this sounds sort of obvious, but I think the progression of ideas across grade levels in mathematics is a story that has meaning, and you’re doing one thing because you’re going to move the next thing, and those things are connected. The idea that mathematics is a subject that has meaning, that tells a story, that has progression, most adults did not experience that as students. Most adults experience math as just one thing after another. “This, then I do this, then I do this.” The idea that there’s a connected, coherent sequence of ideas, that the procedures you’re executing… You should learn the procedures, but they’re meaningful procedures. They’re doing them for a reason. They work for a reason.

And so, the idea that there’s meaning and that there’s a story is hugely important to me. That’s the way I’ve always thought about mathematics, of course. And I was really interested in, can you actually tell that story in a curriculum? I’ve heard teachers say that, especially teachers who maybe at first there was a lesson which had some progression of activities, and they’re like, “Why are we doing this? Why don’t we just tell them how to do it?” And then, they realize that three lessons later, you reach that culmination, and they realize, “Oh, okay, I need to trust the story here,” and they really have that epiphany. I’ve heard teachers say that they understood mathematics in a way that they never understood before because of that idea that there’s a meaningful story being told.

Jon:

Yeah, in thinking about the story of mathematics, I’m just imagining kind of continuing the story of mathematics, and I just wanted to get your thoughts on the future of math instruction. Thinking about… I’m leaning towards, obviously, digital technology is becoming more prevalent in our classrooms. It’s been like that for a while now, but also there’s AI is making kind of a big jump lately into the classrooms, and there’s probably some hesitation on teachers on what’s really important versus what can be automated. So, I’m curious about where you see the story of mathematics kind of evolving, thinking about, and bringing in, say, technology in the digital age.

Bill:

The original design of our curriculum was as a paper and pencil curriculum simply because we wanted to write a curriculum that was usable by everybody, that didn’t have any demands, that if you were a school district that didn’t have a lot of technology, you could still use it. There’s really interesting work being done on how to translate this ideal paper and pencil classroom where kids are talking to each other, teachers walking around the room observing, looking at student work. The five practices, where orchestrating productive mathematical discussions are being executed. There’s a lot of sort of selecting, observing, presenting, going on in that classroom. How do you make that happen on a digital platform?

I think early digital platforms were not really there. I mean, a lot of early digital platforms were just like kid in front of computer doing stuff, but I think we’re beginning to see now emerging people taking seriously the idea of implementing the instructional model digitally. The pandemic had something to do with that, because it’s a made it obvious that the traditional instructional model was not working. If you think having a teacher stand up in front of the classroom and tell everybody how to do the problem and then do worksheets, if you think that’s not a good idea, it’s even more obviously not a good idea when it’s on a Zoom call. And so, I think this has caused some thinking around, how do we implement the problem-based instructional model or something like it digitally? I don’t think we’re there yet, by the way.

Kyle:

No, right. No, neither do we, for sure.

Bill:

You see a few efforts where people are thinking about that, and I think that’s the next big challenge.

Kyle:

I agree. Where our heads have been going, and of course this evolves as well, like your own thinking and perspective, especially with technology, you start to envision… And I wonder about just maybe certain aspects of the class where, “Hey, a digital tool might be helpful to aid in this portion in order to maybe free up or open up opportunities to do some of the things that are maybe more challenging for a teacher to do or a facilitator to do in the classroom,” and so on and so forth. So, great answer there. We really appreciate it.

We are looking at the time, and we don’t want to hold you up on the rest of your day. We have a really important question, and we want to know, for those people who are listening right now, we’ve covered a lot. We’ve covered all the way down to amazing instructional practices we can use at home that lead to your children crying, like your mother did, or all the way up into post-secondary and beyond, writing curriculum and looking at how we can shift so that mathematics is taught in a way that tells a meaningful story, which I love how you articulated that. The question we have is, if someone is listening to this podcast, and they can only take one big idea with them today, and they want to know, “Hey, Bill wants me to know this piece of information, or to think on this piece of information,” what would that big idea be for them to take with them here today off the podcast?

Bill:

I think you need to let kids do the math. I think you need… People don’t learn mathematics by watching somebody else do mathematics. You need to give kids something to chew on and let them chew on it. Don’t make them cry. Help them when they need help, but they need a chance. It’s just like learning to ride a bicycle. You don’t learn to ride a bicycle by watching somebody else ride a bicycle. You learn by getting on the bicycle and giving it a try. And any approach to instruction, which is letting kids try and solve the problem first, letting them do the math. You learn math by doing math, not by watching somebody else do math.

Jon:

Love it. Thanks for that, Bill, and thanks for joining us here to chat all things math. I’m just going to give you kind of a shout-out here about Illustrative Mathematics, but also, where would folks go to learn more about the work you’re doing, or where could we steer them to go, “Hey, let’s give folks this resource or give them this next step”?

Bill:

Yeah, go to illustrativemathematics.org. The curriculum is freely available. You can download it there. It’s free for anybody to use, and you can learn about our professional learning there. You can learn about lots of stuff, and you can access the curriculum.

Kyle:

Hey, thanks for taking the time for the Math Moment Maker community, and I’m hoping we’ll get a chance to bump into you sometime soon at a upcoming conference.

Bill:

Yeah, [inaudible 00:34:45] be fun.

Jon:

Awesome.

Bill:

Thanks a lot. I appreciate it.

Kyle:

Take care. Have a great day.

Bill:

Bye.

Kyle:

All right, my friends. In today’s episode, we were digging into many areas of our math program tree, but in particular, we’re going to start with the easy one, and that is the leaves, because Bill and his team at Illustrative has put together a fantastic curriculum. It’s a curriculum that we are big fans of. Some of the district partners that we have leverage that tool, and they are offering it online for you to check out, so definitely something for you to be exploring. Jon, what else? What other parts of the tree did you hear sort of coming out in this conversation with Bill?

Jon:

Couple other parts, but one in particular was the branches of the tree, which the branches are our pedagogical content knowledge. These are the teacher moves. Bill talked about different routines to use in the classroom. He had mentioned number talks. He had mentioned the five practices as a routine, a structure that we use to engage our students in mathematical thinking and discussion. Thinking about how to best approach your branches of your tree, you want to strengthen your teacher moves, and Bill talked about that here today and gave you some insight on how to strengthen that up.

Kyle:

I love it. And dare I mention, I’m going to sneak the roots in here, Jon, just because…

Jon:

Ooh, I think so. Roots were there.

Kyle:

We brought up this idea of content knowledge, and that is honestly one of the pieces that we’ve had coming out in a lot of the conversations we’re having these days, is that content knowledge is so key. Pedagogical practices are great. The routines are great, doing those in our classroom, but if we don’t have the content knowledge, the confidence to help students better understand the mathematics from a conceptual standpoint, it really makes it difficult for us to move away from simply presenting algorithms and procedures. It makes it hard for us to put students in a situation where they’re just left to problem solve as he was left to do when he was a young child in his math moment that he had shared. So, we hit three pieces or three parts of the tree, but let’s not forget, the other three are equally important, and they do work together. They do work together, just like the mathematical proficiencies. And all six parts of your math program tree are important to ensure that we keep things healthy, strong, and flourishing in your mathematics programs.

Jon:

Hey folks, we want to thank you for listening to the Make Math Moments That Matter podcast. We couldn’t keep going and do this. This is episode 237, and without listeners just like you, we wouldn’t keep producing these. We couldn’t keep having discussions like we’ve had here with Bill or with other classroom educators, which actually helps us become better educators. So, we want to thank you for helping us do that. If you need to head on over to the show notes page, that’s where the links from today’s episode, any resources that we talked about here, you can find that over at makemathmoments.com/episode237. Again, that’s makemathmoments.com/episode237.

Kyle:

Hey, friends, if you want to find out about your own math program, your own tree, where things are flourishing, and where you can start taking next steps to grow that tree, head on over to makemathmoments.com/grow, and you can go ahead and take our short assessment, which will produce a customized report with next steps that you can take to grow that area of your math program tree. Well, until next time, I’m Kyle Pearce.

Jon:

And I’m Jon Orr.

Kyle:

High fives for us.

Jon:

And a high… Ooh, did you hear that? Five…

Kyle:

They shouldn’t.

Jon:

For you.

Kyle:

Vibrato.

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