Deconstructing Calculus with Amy Langville and Kathryn Pedings-Behling

Picture a calculus textbook. You’re probably picturing a hardback book an inch and a half thick, full of mathematical notation. The traditional calculus textbook can be intimidating for students, like five and a half pounds of pure confusion.

On today’s episode, I’m excited to share a conversation with two mathematics faculty at the College of Charleston: Amy Langville, professor of mathematics, and Kathryn Pedings-Behling, adjunct instructor of mathematics. Amy and Kathryn have designed a very different calculus textbook which they call Deconstruct Calculus. It’s one part textbook, one part journal, and part activity book, and I’ve never seen anything like it in higher ed. 

Amy and Kathryn share the inspiration for Deconstruct Calculus, the activities and visual design the book uses to engage students and help them learn, and teaching principles from Deconstruct Calculus that can apply to any discipline. 

Episode Resources:

·       Deconstruct Calculus, https://www.deconstructcalc.com/ 

·       Wreck This Journal by Keri Smith, https://kerismith.squarespace.com/books 

·       Small Teaching by James Lang, https://www.jamesmlang.com/books 

·       Leading Lines interview with Remi Kalir about annotation, https://leadinglinespod.com/uncategorized/episode-114remi-kalir/ 

Podcast Links:

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Transcript

Derek Bruff  0:07  
Welcome to Intentional Teaching, a podcast aimed at educators to help them develop foundational teaching skills and explore new ideas in teaching. I'm your host, Derek Bruff. I hope this podcast helps you be more intentional in how you teach, and how you develop as a teacher over time. 

Derek Bruff  0:24  
Picture a calculus textbook, you're probably picturing a hardback book an inch and a half thick, full of mathematical notation. The traditional calculus textbook can be intimidating for students. It's like five and a half pounds of pure confusion. Now give that textbook to a student taking calculus online, asynchronously, and well, that's not how I would want to learn calculus. On today's episode, I'm excited to share a conversation with two mathematics faculty at the College of Charleston, Amy Langville, professor of mathematics, and Kathryn Pedings-Behling, adjunct instructor of mathematics. Amy and Kathryn have designed a very different calculus textbook which they call Deconstruct Calculus. It's one part textbook, one part journal, and one part activity book. And I've never seen anything like it in higher ed. For instance, one of the activities in the book instructs students to rip out the next page and make a paper airplane.

Derek Bruff  1:21  
In our conversation, Amy and Kathryn share the inspiration for Deconstruct Calculus, the activities and visual design that the book uses to engage students and help them learn and teaching principles from Deconstruct Calculus that can apply to any discipline. 

Derek Bruff  1:38  
Amy and Kathryn, thanks so much for being on the intentional teaching podcast. Very excited to talk to you today about the Deconstruct Calculus project.

Kathryn Pedings-Behling  1:47  
Yeah, thanks for having us, Derek. 

Amy Langville  1:49  
Thanks, Derek.

Derek Bruff  1:50  
Yeah. Glad to have you on. Before we jump into the project, I'm going to ask a question I asked most of my guests, which is this, can you tell us about a time when you realized you wanted to be an educator?

Kathryn Pedings-Behling  2:05  
All right, Amy, I'll take this first. So I I've always known I wanted to be a teacher. But I thought very strongly that I wanted to be a, an elementary school teacher. So my whole high school career, that's what I was going to do. And I had a very influential math teacher in high school, who really encouraged me to take some higher level classes that I hadn't necessarily intended. And so when I came to the College of Charleston for my orientation, I had picked all of these math courses out in the catalog that I wanted to sign up for. And I went and met with my advisor. And I said, Okay, well, I definitely want I want these classes, because I had, I had gotten all my my calculus one, and calculus two behind me with AP credits. And I said, I really want to take these. And she said, Well, your math credits are done. You know, if you're going to be an elementary education major like you, there are other things you need to take. And I looked at her and said, What do I need to tell you today to be able to sign up for these math courses?

Kathryn Pedings-Behling  3:12  
And she said, why don't you just become a math teacher instead? I said, you know, that sounds great. And while on the spot, I changed my major to math education, and signed up for all the math classes I wanted. That was sort of the end of that. Yeah.

Derek Bruff  3:29  
Well, I love that story. Sometimes it just takes, you know, one day, first year of college and there you go, here we go. How about you, Amy?

Amy Langville  3:41  
So my mind is a different story in that. I never saw myself being an educator. I had taught desktop many things, but they were usually physical. I was a basketball camp counselor, through junior high high school, college. And when I was in graduate school, there was a I think I'd finished even finish my, my second year of graduate school, and they said, Well, you're gonna have to teach your own class. Now you can't be a grader anymore. Because we had a lot of international students in the program. And they said, Well, you speak great English, or you need your own class, we need to put you teaching and I really fought it and hesitated because I was very shy. And just standing in front of class was the last thing I wanted to do. But they forced me to it.

Amy Langville  4:28  
And I had a great time. I was shocked. I was shocked. You know, I really didn't expect that at all. When I look back, it kind of makes sense. Again, I've been educated in different settings. And and that was it from there. I said, I think one of the classes they had me do was a precalculus. And I had students that didn't want to be there and had some discipline issues and I was not a disciplinarian and so I realized quickly I wasn't going to go on to teach lower level math classes. So then I was like, Okay, well what do I have to do to do these upper ones like well, you should be a professor then so I just stayed in it I make my pass.

Derek Bruff  5:01  
There you go. Thank you for that. I like hearing how people come to education roles. Let's talk about the Deconstruct Calculus project. And let's start as concrete as possible. So if I were a student in one of your courses, using a deconstruct calculus, I call it a textbook, but I know you guys don't call it a textbook. What would I find in that in that book? What would it look like? And what what could I do with it?

Amy Langville  5:34  
I'll let you start Kathryn.

Kathryn Pedings-Behling  5:34  
Okay, so we, unlike many, many classes that you see today, and especially post COVID, we actually require our students to purchase a physical textbook, they are required to purchase a physical textbook. And at first at the beginning of the semester, we have a lot of requests for ebooks, can I just buy the ebook? No, we want you to have this in hand. And I think by the end of the first chapter, they get it, they understand why. You know, one of the activities in the first chapter literally says, rip out the next page and make a paper airplane.

Derek Bruff  6:14  
Which it's hard to do on an ebook. 

Kathryn Pedings-Behling  6:15  
Yeah, yes. And so different things like that, right, we have flip books on it, they can flip the pages, and they see the animation of secant lines turning into tangent lines, of Riemann Sums becoming smooth function, you know, quote, unquote, smooth functions, that you can integrate. Those types of things that you just can't get from an e book, and are difficult to construct yourself in an online setting. So our students learn very quickly that it's a readable textbook, which many are not. And they're very, I think they get very excited about that. But it is readable. It does make sense. And just loads of activities and applications throughout each chapter.

Derek Bruff  7:05  
Say more about why you think it's readable, what makes it readable?

Kathryn Pedings-Behling  7:09  
So I have to, I might throw that over to Amy, because she really started the project. And really, one of her focuses was that, you know, you pick up another large calculus textbook, you know, the god of calculus textbooks, who shall not be named. In it's just,

Derek Bruff  7:29  
I know who you mean, and some of our listeners know who you mean. 

Kathryn Pedings-Behling  7:32  
It's jargon, right? It's heavy, it's difficult. And we wanted well in I don't want to speak for Amy, but the goal is for it to feel like a conversation, right? We're in class, we're sitting down together, tutoring, anything like that. And we're having a conversation about this, you know, what is rate of change? How can we talk about that? Instead of using epsilon and deltas, right, like, how can we how can we talk about it, have a conversation about it? And you don't know if you want to add anything on to that?

Amy Langville  8:03  
Sure. I'll jump in. So I think many textbooks are written. They're written for the instructor, and they're written for their colleagues, and you're worried, did I say this precisely enough, and is one of my colleagues gonna come in and nitpick about this or that. And we wrote for the students, we really wanted, as Kathryn said, that they would have a feel that they were having a conversation and one of our with a deconstruct and constructive learning, we want to get that in the title. We want students to feel like they're not just sitting and we think many books are printed form of a very passive lecture. And we wanted this to be active learning. And one of our slogans is that like, it's kind of like playing with like Legos in the sense that you are very much interacting with the book. As Kathryn said, every few pages, almost every page actually, we're asking you to hit do a checkbox do something manipulative with with the books and physical. So we were hitting some embodied learning. And that was very important to us, the student centered exposition. 

Amy Langville  9:08  
We also found as Kathryn said, in the first week, well, we'll get some students saying, couldn't I just have the ebook this is easier. And and then by the second week, there we have some assignments that they've already completed. And then we started getting a lot of feedback. Like yeah, this is so cool. I'm showing my friends and my family this, which we know we wouldn't expect a math but this is the coolest book and it's a math book that they're showing folks so we love hearing that.

Derek Bruff  9:33  
I've never heard that from my math students. Yes. Hey, look, mom my textbook is awesome. Yeah. So and and so it's part of it is the way that you've written that part of it are some of these physical activities that students will do with the the object of the book right the paper itself. It's but they're meant to write in it right? Cuz you guys call it a journal. What were some of your inspirations for this format, because I think I've seen a few of these at at the bookstore.

Amy Langville  10:05  
So years ago, one of my students I had probably done 10 years ago, we were talking about this idea that we wanted student centered. And we really talked about concepts as much as calculations. And he handed me a Wreck This Journal. And so if anyone's familiar with that, that's more for the art side. Keri Smith is the author of those, you work in them. And then many people submit through a wiki and show what they've done. And it's it's very art centric, it's "to destroy is to create" is the slogan of that volume, so it's very, and so that title "wreck" had this verb in well, we wanted to deconstruct calculus ideas and have them be assembled. So but we say it's, it's, if anyone's familiar with that, it's like a "Wreck This Journal" meets a textbook. So we kind of mash them together, you got the result?

Derek Bruff  10:53  
Gotcha. And so students are, you have multiple choice questions that they're supposed to check off answers, right, and they can check themselves at the bottom of the page, what are the other kind of smaller interactions that students do on a page by page basis with the book?

Kathryn Pedings-Behling  11:09  
So a lot of times we want we we want them to reflect not only after they've learned something, but even before they learned something, right? Think about in your head, envision an increasing function. What are list things that you you would know about that increasing function. Right, before we even define that? You know, and we have them really think about before they approach a topic what, in their own words, the mathematics they may be about to learn? Right? You know, because they like to use words like, we're going uphill, right? We're, you know, they like to use words like that, in general. And we want them writing about that, we want them thinking about that before we say a function is increasing. If between x equals a equals b, f prime of x is positive, right? Don't get me wrong, like we get there. We are not opposed to precise mathematical language. But we need them to really understand it intuitively, before we start talking about two points A and B, and what's happening between those points from a mathematical point of view.

Amy Langville  12:27  
And some other activities, I think, along those lines, so there's places for them to write or draw. So give their sort of ideas. There's often even study skills sort of reflections, Kathryn added a good bit that here and there, there are what we call faded examples. There's places where instead of doing an entire example, we might have a little box for them to fill in easy part of the derivative. And so they can see, like it follows with example, with actual answer is, so we want them participating again, instead of as if they were in class. And it's not all just presented to them immediately. 

Kathryn Pedings-Behling  13:04  
One more thing, I will say that that we have added in the past year or so. On any of the activities that are that are matching, we have incorporated something called find the error. So the answer key at the bottom of the page, one of them is incorrect. And so the students have to figure out and justify why that one is incorrect. And we found that to be very helpful and improving just the thought process behind the matching instead of just like a very passive like, Okay, this is f this is B, you know, whatever. So that's been really an interesting addition to the books as well. 

Derek Bruff  13:45  
Yeah, yeah. All right. I like that move. I've done something kind of like that in one of my classes where I would have students generate examples of something. And I added my own example to the list without them knowing that and my example was wrong. And so then I asked them, you know, now that we have this list, which one is wrong? 

Amy Langville  14:05  
They love that it's sort of a detective it puzzling nature, for somehow some reason that totally changes the tone of the project. Like Kathryn said, it's very straightforward and dull in some way. And then suddenly, oh, it's just this little twist, right? Intrigue.

Derek Bruff  14:20  
There's a bit of metacognition there, too, because you're asking students to kind of assess themselves, how confident Am I in my own answers, right? The Answer Key says it's this, but I think that's the wrong one. Right? I think actually, my answer is better. That's fascinating.

Kathryn Pedings-Behling  14:35  
It actually came from an accident. We do a lot of iterations. And we would make errors, right? So occasionally, there would be errors. And I would have students emailing me saying, I really think this is supposed to be F like the books has B, but I really think it's F and I was like, Oh, you're right. It is if we just we wrote the wrong letter. And so then we realized there's really something to that to them sort of fighting for that answer. Yeah,

Derek Bruff  15:02  
Well, and now remembering I took a, I think it was a ninth grade physical science class in high school. And the book was actually riddled with errors. I don't know what textbook we were using, but it had a lot of problems. And it became this kind of like fun scavenger hunt we did with our teacher. And we would all rejoice when we got to say the book is wrong about this. Right. That's fascinating. 

Derek Bruff  15:24  
Let me ask one more question about the kind of the physical book and then I want to talk more about the teaching context that you're in, because I think that's also super relevant. In terms of the the the visual, the use of visuals in the book, right, my understanding, I've seen a few of these pages, and you guys are pretty intentional with what things look like, right? Graph paper versus blank paper versus what are some of the visual cues you have in the book to help students work their way through it.

Amy Langville  15:50  
Yeah, thanks for asking that there. Because that that is a lot of work went into that. And as Kathryn said, we've had many iterations, this has been going on for about 12 years, we started this project, and every semester, we have feedback, and we refine. And one of the things we did was learn a lot about graphic design, because we do think that's important how students approach it, and we think it can be useful in an instructional way. And so we have certain types of pages with different backgrounds. So Kathryn's mentioned, several of these activity pages, they're all set on a background of graph paper and kind of have that, you know, we're gonna get to work feel. And then traditional lined notebook pages is when you're doing your sort of usual calculation problems, like, you know, here's the function, find the derivative, here's a function by the drip those kinds of what are in your typical textbook, and they actually right there in those pages, and then our information pages have the usual call outs on the sides, some nice boxes. And then we also have included these checkboxes and multiple choice questions. And then we have several example pages. But again, to make those more interactive, and less like a typical textbook, we added in the past couple of years, these faded example boxes where students have some things they fill in.

Amy Langville  17:03  
And then there are also challenge problems at the end of every chapter, that are meant to be low floor, high ceiling problems that students at many different levels can be captivated and interested in. And those are on a background a grid background of dots. So just to distinguish, and I think students do find that, at least the main two that they see that they have in their homework assignments are, do three of these five activities, activity pages, and they could quickly find those and do we have usually have less choice on the calculation pages, but we'll say do two of these four calculation problems on this page. And so they can quickly look for the line notebook pages versus the graphics pages.

Derek Bruff  17:47  
Yeah. Well, and that I know, that may sound subtle to some, but students taking calculus often don't know what they're supposed to be doing. Yes, right. It's, it's, it's all it's new material, it's hard material. Often, they don't come in with the study skills they need for this type of subject. This is something we talked about on the previous episode of the podcast that often when they hit college, students will have these kinds of generic study skills that worked well in high school. But now they need to develop more discipline specific study skills. And so learning calculus requires certain types of doing of things, right. And what you're doing is you're cueing them to the things they should be doing. And I really love that. So let's talk more about the teaching context here. So this is done in an online asynchronous class, am I right?

Kathryn Pedings-Behling  18:34  
That's correct. Yes.

Derek Bruff  18:36  
So how does that work? So what else is going on in this class? And how does the use of the journal kind of feed into the overall structure of the class?

Kathryn Pedings-Behling  18:45  
So like all campuses, at this point in time, we have an online LMS. And for each of the chapters, we have recorded instructional videos, so it's Amy or I just doing some of the problems, talking through the concepts. And then the students will work we one of the things you're required to do every week is read the textbook. So we say that specifically, like we want you to read the textbook, and they learn quickly that they will gain a lot from that. After reading the textbook, we asked them to choose a certain number of activities and a certain number of calculations to work on for homework that week. They then scan them and upload them to us so that we can see their thought process. We can see where they may be struggling.

Kathryn Pedings-Behling  19:35  
So the largest thing that we do in our asynchronous class is group work. Group work is actually the largest percentage of their grade in our course, which as you can maybe imagine for college students that you're not really excited about it first. So we randomly put them in groups at the beginning of the semester and And they work every two weeks when we're doing a unit on those challenge problems at the end of the chapter, they are allowed to choose the challenge problem that they want to work on. They are required to do level one, but they can do any of the level ups that they feel like they can do as a group together. So again, that low floor high ceiling. 

Kathryn Pedings-Behling  20:19  
We require them to do a planning document, where they we walk them through the process, what it means to plan as a team to do this challenge problem. And then they have to write up an official report on the challenge problem for us, and submit that to us. And by the end by even by the middle of the semester, that you just have realized how valuable their groups are, and how important it is to do math with other people. And that's very important for us. And we do, I'll say one more thing, our students it is a 100 level math course. And what we have found over the years is that they do need quite a bit of support towards the beginning of what does it mean to work in a team? What does that mean? And so we provide that support at the beginning. And we are getting better and better every semester. 

Amy Langville  21:19  
Sure, sure. That's great. So we do realize as though there were 100 students that they need some nudges. And we've had many discussions and debates ourselves as instructors about what that might mean. And some things is it can be as simple as inserting in their weekly quiz, one of the 10 questions they have is, is it nudge? Have you texted or contacted your teammates? Or have you been a good teammate this week. And then other things that are more extensive are in that challenge planning document that Kathryn talked about every week we might have whatever those are, do, we'll have another little paragraph about what it means and in different, different topics of what it means to be a good team member. We do feel like they need that coaching. 

Amy Langville  22:06  
And we've realized we've gotten a little better every semester about this. And students are starting to say things like, oh, you know, I still contact that teammate, and I'm now taking history class with them, I really tried to sync up our schedules or, you know, we decided to do aerobics together. So it's, that's the kind of stuff that we like to hear to that they are they've, they've made these friendships. And in person class, it was very important to us that you have to discuss, you have to talk about maths concepts. And it's not just a solitary activity, we really wanted them discussing them. And trying to figure out how to do that in an online set, an online asynchronous setting, you know that it's gotten a bit of trial and error, because it is I think, from your book, we know to like this is a new topic. And there isn't a whole lot of research on this yet. And so we have tried some trial and error and read whatever is available, and try these techniques to get a little bit better with that, making the collaborative part of an asynchronous class work.

Derek Bruff  23:07  
Sure, yeah, establishing that social presence among students is totally doable in an online course. But it just takes a different set of tools than what we would do. And I think more intentionality, I think some of that happens naturally, when students walk in the classroom and sit next to each other. And you say, Turn to your neighbor and talk about a problem. And, you know, there's there's kind of natural opportunity opportunities for those social connections. And in an online course, we just have to create those. And it sounds like your team structure is doing that really well. That's, that's really great.

Amy Langville  23:38  
I want to follow up on one of those comments. Derek, is that something Kathryn I recently sort of come to is that, as you said, it's not maybe natural. It's more natural, but part of that Naturalness is the fact that we all we were schooled in live classes, we grew up and had decades of that. So this asynchronous is newer to us as instructors, we're just not we haven't seen many examples of doing it well and doing it bad. Many of us haven't even taken an asynchronous class ever in our lives. And then we're teaching one. And so I think that intentionality is like, yes, what, you have to think differently. You can't just port things that worked in a live setting or even the synchronous setting to an asynchronous, it takes some it's doable, but it takes some thought, and it's just something we're not used to. 

Derek Bruff  24:25  
And so it's gonna take some time to figure that out. Absolutely. And, Amy, I think you told me, speaking of the kind of shifting modalities, you guys were using these journals in the spring of 2020, when all of college teaching went remote, and did you find that having the journals was a useful tool in that transition?

Amy Langville  24:48  
Yeah, so that was that first semester just thinking back and it seems longer than that? Wow. It was a it was a wild time, you know, and I had been using these journals in my at all my classes prior to COVID, were in person in a traditional setting. And I'd been using these for years and refining them. And for all the same reasons students enjoy them and interact interacting with them. But there was that first several weeks of the semester that we were in person and I quickly we shifted, and I noticed that a lot of my colleagues were really "this stinks" and struggling and how are you Is anybody learning anything and and mine were just seem to be rolling along smoothly, because they had what Kathryn I talked about those journals is becomes very important, I think in an online setting to have a physical touchstone, something you're going back to because so much is on the screen, so much is delivered, your quizzes, your exams, so much is screen time. And then to have this thing that is a complement to that it's so different. It's kind of can be like a security blanket for them. And, and, and ours are trying to be completely self contained. So it just worked out.

Amy Langville  25:56  
I had, Kathryn had been in distance education. You know, as a pioneer of like, 10 years ago, 15 years ago, when all this originally started, and I was still the college was still in a live setting. I was still in that and and probably never envisioned myself doing this. And I've found it very fascinating. And exciting and got me really, if you have any over course of a career any sort of like, oh, I'm board with with this, like it totally changed all that, was really fascinating to switch to this new setting. 

Derek Bruff  26:27  
Because you, before 2020, Amy, you had not taught online asynchronous, right?

Amy Langville  26:33  
 I had not taught online in any way.

Derek Bruff  26:35  
 And now you are volunteering for that modality.

Amy Langville  26:38  
Yeah, right. And I hadn't even I didn't even really use the learning management system. A lot of people that were in the live class would use that. And I really wanted them there. I had this philosophy that they, you know, if I put things on, you might not come to class. And so I was hesitant, I was resistant to it. And now I do see the possibilities with it. And something else Kathryn talked about, you know, this has changed the world of education and work and and I think there's we need to train our students to be prepared for this new world. Yeah.

Derek Bruff  27:11  
And that means how do you work together in a team when you're not in the same place at the same time, right? Well, let's, let's kind of step up a couple of levels of abstraction here. What do you think some of the ingredients are of the Deconstruct Calculus project that help it work well, for students, especially things that might be kind of translated to other disciplines that aren't calculus?

Amy Langville  27:35  
Okay, I'll jump in on one of these. And I know Kahtryn has several. But things that Kathryn mentioned earlier, we're really big on choice. And I think that applies to any, doesn't doesn't have to be STEM, it could be any discipline, is that we want to provide students with choice. And so we say Kathryn mentioned they can pick will have three challenges at the end of the chapter, they can pick which challenge they want to work on, discuss it with their team members. What's interesting in that as we started to find students, and this was a little bit of nudging with mentioning it. But we said fine, why don't you choose one for a topic you're not good at? Don't pick the three, one of the three challenges. Often you find at the beginning, they chose it because we knew how to do two derivatives really well. So we're gonna do choose challenge number two. Later there say we didn't know how to do integrals very way. Well, so we chose challenge number four. And again, a little bit of nudging there. And then choice, we think that choice gives them empowers them in a weigh in and they're not as stressed about, you know, even on the calculation problems, choose two out of four on this page, we don't need to over check what they're doing. And, and they seem to enjoy that. So that's what we call deconstruct principle that we think is important for teaching in any setting. And I know Kathryn will probably mentioned a couple of other Deconstruct principles.

Kathryn Pedings-Behling  28:59  
I just want it Yeah, I want to go back to those activities. When we're talking about deconstruct with people from other other disciplines, we often bring up, James Lang's "Small Teaching" work. And we really believe in the power of the short activities, the sub five minute activities, and that the majority of the activities, not the challenge problem, but the activities in the book are less than five minutes, but we think it makes a powerful difference in the way that their brain is processing these mathematical concepts that they're learning. And that's something the short, you know, I feel like teachers hear the word activity and then they're like, Oh, my God, how much class time I lose to this. I don't have time for that. I don't have time to do that. You know. And, and we really want to change the idea behind that, like the word activity doesn't need to be you think lost a class period, that we're talking about five minutes here, five minutes there, right? Really incorporating that. 

Kathryn Pedings-Behling  30:10  
And so something else I want to mention that we haven't mentioned yet, but it talking to other people from other disciplines, they seem to enjoy this idea throughout the book. Um, we don't believe our students know necessarily what it means to be a mathematician, right? 

Derek Bruff  30:28  
Sure. Yeah. 

Kathryn Pedings-Behling  30:28  
I'm gonna say that person's a mathematician, what are some of the things that they're able to do well, right, or what are some of the principles that they use in their job? Or in their life?

Derek Bruff  30:38  
I will often hear, so do you invent numbers?

Kathryn Pedings-Behling  30:41  
There we go.

Derek Bruff  30:42  
I do not do that.

Kathryn Pedings-Behling  30:43  
I do not do that. Thank you for asking. So we, throughout the book, we have what are called math morals, and they're very short. But the idea is that they're snippets into the brain of a mathematician, right? Mathematicians look for patterns, right? We have that when they're learning how to do power rule for derivatives, right? That's how we can take you know, that's how we can use the limit definition, isn't it? Oh, okay. I get it, I get how to take the derivative of a polynomial. And so we encourage people from other disciplines, what is your what are your chemistry morals? Right? What are your engineering morals? What are your physics morals, right? And really thinking about helping our students understand what it means to be a professional in that field, and what skill sets those professionals have and that they are building in that course.

Amy Langville  31:45  
And those math morals appear throughout the journals as little like post it tags when one of them is used to solve a problem, their little post it. So for instance, another math moral is "approximate, refine, limit." That's what a tool that's used often in calculus so there's just three words. And it was neat for me when I taught a calc three from these Deconstruct journals. First time it was asynchronous during COVID. Worried how this was gonna go, I hadn't taught calc three ever and. And students started, students started using that in their assignments. So to solve a challenge problem, they'd say, Oh, just like we've done before, you know, we approximated refined and then took a limit. And it was their own clever solution to this challenge problem that I hadn't thought of doing it in that way. But they would, they would repeat these math morals. 

Amy Langville  32:31  
So we do find that they're helpful for students. And we do think we take for granted some of these things as the instructor and as someone that's so in the field. But as as Kathryn said, we've done a couple of webinars and we got some really great answers from the instructors in chemistry or engineering and, you know, art history, that things that they find that there are morals, well, let's be don't be subtle about that. Let's be explicit, yeah, and keep hitting it.

Derek Bruff  32:58  
Well, and I'm also reminded of, for my old podcast Leading Lines, I interviewed Remi Kalir who does a lot of work around annotation. And he takes a pretty broad view of annotation. But that can be anything from the library book that you check out that has someone else's notes in the margin, right, to a tool like Perusall where you have students annotate a text document collaboratively online. But one of the things that he talks about as being powerful about annotation is that the thinking about the thing is anchored to the thing, right, it's, it's the note in the margin, and a little arrow drawn to the comment that you're making a note about, right. And that anchoring is really powerful to help you understand someone else's idea, or even work through your own ideas. And that's what I hear about your math morals, it's the post it note in the margin next to the example of the thing, right, if you if you tried to communicate those morals in the abstract wouldn't be nearly as helpful as highlighting it when it's happening as the students are working through these activities.

Amy Langville  34:01  
Right. And the thing I like about that annotation that you've mentioned, too, I remember that from your book, I was really fascinated and how you've used it in your math classes, is it's also the memory. I think that's one of the things that we were hoping and we've heard this from students, especially my calc three students that they were shocked at how little they had to study for the final because they just knew everything. And these are intentionally designed that you do an activity often it is something embodied or physical, you're going to tie the, tie a loop around the journal and lasso overhead and throw it and what's gonna go in the direction of the tangent. Well, you never forget,  you've physically done it. You understand the tangent later. And many of the things are just thought experience. Imagine you're doing something you don't have to physically do it, you have the same benefit. And if these math morals and these annotations those are our mental cues that that sort of emblazoned in in the brain right then and there in a way that requires a lot Is it is a repetition is a great repetition in a way that requires less later on down the line. 

Derek Bruff  35:05  
So what's next for deconstruct calculus?

Amy Langville  35:10  
So we're not quite through that. So it's a series of books that we want to cover the entire calculus curriculum. So we've got the business calculus on the one end, or calculus for non stem a liberal arts or humanities majors. And then we've got a series journals, we'd like to course test and journals are complete and ready for Calc One. So derivatives and integrals. We're now working on the missing piece of the puzzle is Calc Two, because we do have the calc three ready. I had to teach that during COVID. So we're trying to fill in that gap.

Derek Bruff  35:41  
Well, that's really exciting. If other math faculty want to teach with some of these journals, is that an option?

Kathryn Pedings-Behling  35:47  
Yes, we would love for you to all you can find all of our information, we have deconstructcalc.com. Because here's all the information about all of these journals, what's available, and how to get them and how to access us how to how to get in touch with us. Because we love to provide support. Wherever, wherever it's needed.

Amy Langville  36:10  
And give feed feedback. 

Derek Bruff  36:11  
Yeah, right. And keep iterating right, making it better and better. 

Amy Langville  36:14  
Exactly. Approximate, refine, limit. We're still in the Refine phase.

Derek Bruff  36:20  
I love it. Well, thank you both. This has been really great. I've enjoyed this conversation. And I really love thinking about teaching a very familiar subject like calculus through some very new and innovative ways. So thank you for sharing with the podcast. Thank you so much. 

Amy Langville  36:33  
Thanks for having us. 

Derek Bruff  36:39  
That was Amy Langville, professor of mathematics. And Kathryn Pedings-Behling, adjunct instructor of mathematics, both from the College of Charleston. Thanks to Amy and Catherine for coming on the podcast to share about deconstruct calculus. I was blown away when I learned about this project. And I hope you can hear that excitement in the interview. Visual thinking, embodied learning, active learning through a printed book, so much good stuff. I hope you got some inspiration for ways to engage your students, whether that's in class or online. And if you know of similar projects in other disciplines, please let me know about them. 

Derek Bruff  37:14  
See the show notes for a link to the deconstruct calculus website for more information about the project and how you can be involved. And I've got a few links to other resources mentioned in the episode. And my Patreon supporters can find a little bonus clip from the interview where we follow a rabbit trail on embodied learning. 

Derek Bruff  37:31  
This episode of intentional teaching was produced and edited by me Derek Bruff. See the show notes for links to my website. The signup form for the intentional teaching newsletter which goes out most Thursdays, and my Patreon which helps support the show. For just a few bucks a month you get access to the occasional bonus episode, Patreon only teaching resources, the archive of past newsletters and a community of intentional educators to chat with. As always, thanks for listening

Transcribed by https://otter.ai