 Welcome to Liquid Margin's and today's guest is Matt Salamone from Bridgewater State University and then our moderator today is Nate Angel. He's the Director of Marketing at Hypothesis. So with that, Matt, would you like to introduce yourself and tell us where you're from, what you're doing. No problem. So I'm Matt Salamone. I'm a mathematician, associate professor at Bridgewater State University. Bridgewater is kind of equidistant from Boston, Cape Cod, and Providence, Rhode Island. So we've kind of been southern Massachusetts. We're the largest of the state university system in Massachusetts. We have about ten and a half thousand students, mostly undergraduates, and one of the things that really drew me to BSU in the first place is that the mission of our institution is largely wrapped up in teacher training. So we're the largest producer of K-12 math and STEM in particular educators in the Commonwealth of Massachusetts. And, you know, we began as a normal school in the 1800s with that specific sort of mission and purpose. We were a horseman institution and that's still kind of in our DNA. And I've always had a passion for education and teaching and Bridgewater is the kind of place where we have a very large lever with which to influence the next generation of teaching and learning, both at the college level and also at the K-12 level. And so I'm coming into my eleventh year here in the math department at BSU. I've been the chair of our department for four years, which reduces my teaching but also increases my engagement with other kinds of work on campus and I'm getting to work with students in different ways and participating in conversations that are larger than the institution such as we're having today. So. Okay, great. Thank you. And Nate, would you like to say a few words and introduce yourself and then we'll just, I'll hand it over to you when we can get started. Great. Yeah, thanks Franny. And thanks for coming on the show Matt. It's really an honor to have you here. We've been, we've been really looking forward to this because a lot of the use of hypothesis and collaborative annotation has been, there's been a lot of noise made about its use in the humanities, people doing close readings, you know, and things, things like English and composition and history and disciplines like that. But we know that there's really vibrant possibilities and use in STEM fields and in math. And we know that you're one of the people who's really taken that the farthest. And so we're super glad to have you today. So I'm Nate Angel. I'm calling from sunny Portland, Oregon where it's actually not sunny today, oddly. But I work at hypothesis along with my colleagues who are here today on the call. And I, myself, I'm a humanities person. And so I don't necessarily have the, the background. So you really give this a full look. So I'm going to be relying on you Matt to kind of help steer us in math. But I did have a part of my career where I was focused on helping to produce open educational resources. And we produced a lot of math oriented resources, including even print textbooks, which is, boy, talk about a pesky problem getting math to print. As I'm sure you probably know, it's tricky. Absolutely. And so I definitely have had some professional experience with it. And, but before we get started on all that, I mean, we were talking a little bit before the show started, you know, this is probably, there's never probably been a start of a fall term that's been exactly like this in higher education in the United States or anywhere really. And I'm just, I, before we get started, I thought I might just check in, you know, how are you feeling Matt? Are you, are you feeling okay? Are you ready for this? Well, I don't think any of us can say that we're truly ready for the beginning of this fall. I think, you know, to the extent to which I've always been, I've been kind of a technology fan my entire life. And so I did a lot of things in my own teaching with technology that I've been doing for, you know, five, 10, in some cases, 15 years. And the extent to which I'd already integrated a lot of technological and online tools into my, even into my face-to-face teaching practices, that when we made the shift to remote learning in February of this past year, when we moved 100% online, it was an easy transition for me. And so I could really focus on helping out my colleagues in the department across campus, some of whom were not using any instructional technology to speak of with their teaching to try to help them to make the pivot. So I've been really more concerned lately with making sure that the people around me have everything that they need and access to the technology resources and training and the actual physical devices. There was such a run on webcams in February that we had trouble even just getting people to hardware that they needed. So we're looking for workarounds. So yeah, I think my own readiness is at a place right now or about a week out from the beginning of the semester. And my usual week out anxiety that's usually at about a five or six. Is it about a 15 or 16 right now? But that's just because of how much we don't know about what beginning of fall term like this is going to look like. Yeah, well, again, I really appreciate you coming on at a time when your anxiety levels at a 15. So this is a highlight of my week. Okay, all right. Well, let's make it as good as possible then. You know, you talked a little bit about Bridgewater's history and I didn't know that. I actually lived in Providence for a while. So I'm familiar with the area, but I didn't know Bridgewater's history. And it sounds like the teacher education is a big part of the school's DNA. Like you said, could you describe a little bit the kinds of classes that you do normally teach and what kind of students you have, what levels they're at? Sure. Me personally, I've kind of seen it all every corner of our curriculum at Bridgewater. When I first came to BSU, I came in as the coordinator for our developmental math program. So the math course is taken by students who we as an institution in one way or another assess. You might not be ready to thrive in a college level full-on, full-bore, you know, calculus or, you know, collegeable math class. And so from the beginning, you know, I was really working with that population of students who are coming in with a lot of what I like to think of as unhelpful habits and beliefs and attitudes about mathematics and their own role in their own agency within the subject. And really a lot of, you know, students come to college with a lot of trauma around mathematics that they're bringing in from experiences that they had as children coming up through the grade sometimes at home. And so, you know, my first priority when I got to my institution was, well, how can we do better by those students, both in the classroom with our feet on the ground, teaching them, you know, pedagogically, and then also kind of institutionally are other ways that we can be thinking differently about how to give students the support that they need to actually kind of find their voice in mathematics, understand that they have a voice within mathematics. Because to be honest, that's a voice that too often gets taken away from them at some point along their journey from early childhood into college. And so to the extent that we can try and undo some of that trauma in a semester or two of developmental courses, I've always tried to do that in one way or another. But even though that was my sort of main purview, my research as a PhD student was in sort of mathematics, straight up mathematics. I did research in celestial mechanics, the intersection of kind of physics and higher mathematical techniques. And so, you know, I always have also had an interest in just teaching anywhere across the curriculum. One of the places I've taught a lot in the past few years, is I've taught a lot of our abstract algebra course. So this is a course where math majors kind of open up the hood of the algebra that you learn in high school and figure out, well, what is really the essence of algebra? If we take the numbers away, what are the important questions that we can still ask and answer? What does that look like? And then sort of rebuild up to the algebra that was familiar in high school. So that's one of the courses where when I first started teaching it face to face, I taught it in a flipped classroom. And so I generated a lot of video content that I sort of unthinkingly really shared out online via YouTube. So I have these entire sort of course length playlists of 70 or 80 hours apiece that constitute this sort of semester's worth of 10 and 15 minute individual lectures that I used for my flipped classroom. Those materials have lived on past my teaching of those courses in ways that when the open education conversation came along, and I hope we'll get a chance to talk more about this, that I realized, oh, I'm kind of doing this already, sort of teaching with open educational resources that I created and open pedagogy. And that's been a really nice set of resources that not only can I reuse in my own teaching, but that I know that other folks out there are using particularly now during our remote learning environment. But so I kind of teach in all different segments of the curriculum. This semester coming up, I have a class of graduate students in our Master of Arts in Teaching program. So these are practicing high school math educators working on their final licensure master's degree. So I'll be teaching a course in the theory of algebra and knots, KNOTS, the actual strings tied up into various figures. There's a rich mathematical study around them that also connects into algebra, which fascinates me. And I'm also on the other hand teaching a course of business mathematics. So marketing and business management and accounting majors, taking their last required math course for their degree, which is also the one that scares them the most in their degree program. So I kind of get really two sides of the spectrum of math comfort and carried trauma this semester. That's really a lot. A lot was packed in while you were saying there. It's so much interesting stuff. I'm imagining now a class in the math of knots and sailing. That would be a good class for your area there. Absolutely. There's a great mathematician's name is Colin Adams. He teaches at Williams College. And he's a really performative guy. One of his routines that he does actually is a kind of a lesson on knot theory where he assumes the persona of a sailor, telling the audience, here's why I'm interested in knots. Here's how they work and here's some of the mathematics behind them. And he's one of those really gifted mathematicians that treads that line between scholar and entertainer and really has figured out how to speak to broader audiences about mathematical ideas that have a pretty low floor because everyone's tied a knot at some point in their life but they also have a very high ceiling. That there's a lot of active research going on in the math community around knots and what they can tell us about algebra, what they can tell us about topology and geometry, combinatorics. They have these connections into all different subfields of our discipline. So I'm imagining him talking like a pirate during those classes. He does indeed. Yes, it's a fact to follow. So one thing, since we could just talk about math all day, that's fascinating, but one thing that the purpose of the show Liquid Margins is to talk a little bit about how you might have related to collaborative annotation. And so maybe to get started, like how did you first learn about collaborative annotation and it might not have even been in the digital space, right? It may have been an analog practice or something. And how did you learn about hypothesis and get started with hypothesis? Yes. So I don't think that the concept of collaborative annotation was much on my radar screen as an entity until the open education conversation came along. So it was probably five or so years ago that our faculty development teaching and learning center on our campus brought in Robin DeRosa from Plymouth State to come and give a presentation on open educational resources and open pedagogy. And it was one of those presentations, Robin is also one of these really talented, almost evangelists, right, for open education and open pedagogy. And it was in that presentation that really sort of, it's one of these, what's it called, a threshold concept. Once you grab hold of that concept cognitively, your entire frame on things changes. And in the course of one hour, my frame kind of changed with, so I've been doing all of these different practices in my own teaching and I have these sort of tacit beliefs about education that I wasn't able to articulate. And then I kind of realized in that moment, hey, the way that I want to interact with my students and the way that I want myself and my students to interact with the broader community beyond our institution really is informed by the mission of public education as a public good and the mission of sort of open resources, because the information that my students are learning also is a public good. And particularly at a large public university, I understood in ways that I couldn't really put to words until then that the mission of our institution is to make information accessible and transparent both to our students but also to the community beyond our boards. And so she sort of is talking about all those things and showing us, impressing us with the various digital tools that she had used with her students to really sort of collaboratively author a compilation of works for an interdisciplinary studies course. But I remember that the social annotation piece of it was the piece that stuck out to me the most because one of the things that is almost a truism about math majors, which are primarily the students I was teaching at the time is sort of the inverse of what you hear from a lot of English majors. A lot of English majors will tell you, well, I chose the humanities because I like reading and writing and I don't like quantitative analysis, right math. But I hear the opposite thing from a lot of our math majors like, well, I didn't want to be an English major because I don't really like reading or writing. I just like crunching numbers. I like solving a problem that I know has a solution and I know that someone will tell me if that solution is right or not and I can move on to the next question. And so I always felt like both sides of that debate were missing something. And the thing that my math majors were, in my opinion, missing the most is I didn't see them engaging with text, both writing but also reading, right. And not being able to be there present while my students were trying to read about a math concept. I was missing out on a lot of that formative thought process, that wrestling match that we have with the new text that my students were going through or maybe they weren't. Maybe they were just blowing it up. Because I think what we find a lot in math is the way that students use and interact with text is very different than the way that folks in the humanities do because a math student will often come to their textbook because they have a problem set to work on, right. You have to do all the even numbers from two to 20. By tomorrow, so they open the textbook the night before they look first at the problems. And then it's only if they get stuck on those problems not knowing how to solve them that then they might go back and flip through the text and try and get into a paragraph or two maybe look for a formula and a big blue box on the page somewhere. Find me the magic tool that will help me to get this done. And that's not the way that I wanted my students to interact with text. But I wasn't there wasn't really a way for me to know how they were interacting with text unless we could create a social annotation context around it. So that's what sort of, you know, when I saw that there were tools like hypothesis allowing one to do this with open resources as the backbone. And Nate, as you mentioned, there's a lot of really high quality open educational resources and mathematics out there. But that was one of those aha moments for me that we could kind of have that experience in a virtual space of all sort of gathering around a text and kind of do the work of teaching math students how to read about math. Because math is also written about in ways that are not Congress with the way other subjects have written about. I mean, mathematical writing tends to prize brevity over clarity in a lot of ways. Like, well, why should I write a whole paragraph when I can just write a single algebraic formula and it should speak for itself? Well, they don't speak for themselves. Formulas and numbers don't have a voice until we give them one. And a lot of students don't arrive in college knowing how to do that with a math text. And so having the social context in which to do that with my students and watch their developing interactions with the text and the developing understanding really that for me is the niche that social annotation could fill in my teaching. Wow, that's really great stuff. I keep hearing that you talk about voice and agency and that's such a refreshing language to hear from folks focused on math. So not that I've done some sort of like great study of math teachers and whether or not they talk about that but I really love hearing those words. I want to actually even take a minute to get into the actual nitty gritty and so like it's really easy for people like me to imagine how one might collaboratively or socially annotate a poem, right? But when it comes to math, can you kind of show us around what some actual annotation looks like in the math context? Sure, so one of the exciting things for me is that there are a lot of different angles that you can take to annotating a math text because mathematical exposition has to take various forms, right? One of the things that we prize in math education is sort of the ability to communicate a concept graphically through, for example, using a graph or a diagram or something but also numerically using a table of data verbally using your written vernacular language and then also algebraically using formulas and symbols which is kind of the really unique language to my discipline. And so one of the things I'm going to share my screen with you here real quick, one of the things I try to use social annotation to do is to help students to translate between those different modes. So one of the prompts I'll sometimes give students is, you know, point out a formula somewhere in the text and try and put into words what that formula actually means, why somebody should care about that, right? Or if you had to explain this to somebody who's not currently taking our math class, what would that explanation look like? So this is an example of an open text called Active Prelude to Calculus. It's part of a sort of a three-volume open educational resource written by Matt Balkans at Grand Valley State University that I recommend anyone teaching in the calculus sequence has to check this out. It's designed not only as an open resource but also is aligned to kind of principles of inquiry-based learning. So it doesn't feed, it doesn't sort of spoon feed students the right answers and concepts upfront but sort of leads them through a series of activities that they can do with one another in which the important concepts emerge from those activities. So that too kind of jives with my own sort of values pedagogically. And so here's an example of sort of a sentence that has some mathematical notation in it. It's asking about a specific word. What is the domain of this function which is written as h equals g composed with d? What is its range? And so I asked my students to not only kind of address the question but also kind of tell us about the thinking that led them to their answer. So this student is probably not super visible but they're saying, well, for this domain here's the answer that I got for this. I was wondering if somebody can check my work, which to me that's much better than asking the sort of closed ended question, hey solve this question in an annotation, right? Instead, this person got an answer and then they're inviting further conversation for which then I have a couple of students who jump in and they verify, oh yeah, I also got this and here's how I did it. Oh, this person says it took me a couple of tries before I figured this out but I did get the same answer, right? So I think the first, one of the first context in which students will find it natural to kind of have conversations in the margins of the math textbook is just in the sort of problem solving context. Here's the answer that I got. Does anyone agree with this? Can you check me? What's your process like? But then the more that I have students working in this space, the more that I also invite them to contribute other resources to the formative process of understanding what's there. So this is from a pre-calculus course that I taught in this past spring semester and I invited students to, if you would like, if you find a helpful resource elsewhere on the web that helped you to understand this thing that's being written in the text, share it here. And the ability to embed, there's a lot of great YouTube content to explain math ideas. The ability to embed a YouTube video directly into the annotation is an option that students took that option a lot. If I look through this whole page, it's probably six or seven additional videos that they contributed in the annotation stream, in addition to the videos that are already embedded within the text as well. So making the annotation space a place where students can not only bring in their own ideas and their own experiences, but also, hey, here's some additional resources I found online that helped me to understand this concept. There's also an information literacy aspect to it because I want my students not to rely upon me or to rely on a single textbook for understanding what they're learning, but to build those skills to reach out and assess other sources as well and bring those into the conversation. Yeah, that's, that really makes sense. And can I just ask what, so it looks like you're in Canvas there. You're the learning management system for Bridgewater probably. And then embedded in that is this math text. Is that just a website or what? This is just a website. Yeah, so I guess the first thing to notice at Canvas is actually is not my university's main LMS. Ah, okay. We're actually a Blackboard campus. I'm using a Canvas free for teachers account in this. We won't tell. Yeah, right, no one will know. But one of the main reasons why I chose to use Canvas free for teacher rather than Blackboard is that I feel like it integrated tools like this in a much friendlier way for students. It had more flexibility for different pedagogical styles I found than Blackboard did. My grading system is a bit unorthodox. My grade based on a standard spaced grading system rather than points and percentages. So you clearly define learning targets and give students opportunities to revise their work and improve it over the semester. And their final grade reflects the number of those topics on which they were able to demonstrate mastery, full stop. And that includes all the revisions and resubmissions and those conversations that happen. So Canvas also was a place that was much more natural to embed that grading structure within that LMS. But yeah, so this particular open textbook, it just resides on the web. It's authored using a system called pretext, which is gaining a lot more purchase in the mathematical open textbook authoring community because it's got this really friendly structure to it with the sections and subsections and it has math type. As you were saying, typing up math in the first place is a challenge, but pretext is friendly to all of those things. And so it creates these really nice looking pages but also has the option to export as a PDF in a friendly fashion. So you create this central XML source code that then compiles into HTML that becomes a webpage that we're seeing here but also can compile into PDF and other eReader formats and stuff. So the same source material can get processed in different formats. So yeah, what we're seeing here in Canvas is just basically a hypothesis webpage embedded within a Canvas assignment. And I noticed, by the way, in the chat the question was the course that I used this with an online course or was it an in-person course? And actually, in my case this past spring, the answer is that it was both. I was teaching one section of face-to-face students and one section of 100% online students. And from the very beginning of the semester, my intention was to blend those classes together so that I created some small groups that were mixed. So three students from the web section, two students from the face-to-face section had an online discussion group that they worked with inside of the LMS. And so the course was a little bit of both. But then when we made the pivot to fully remote learning, my face-to-face students all became online students overnight, but they were already working with students that were in the web section to begin with and we were already doing a lot of interactions for the course in the online space, even with face-to-face students. So it was kind of one of those lucky choices that I made in January that turned out to make my life a lot easier in February. But I'll say that even before this past spring, I've used these tools with face-to-face students just as here is what your reading assignments for the course will look like. And you'll do these assignments virtually in this space outside of the classroom. Yeah, that's so that you're sort of... You got lucky, you say, but it sounds to me more like you were anticipating a pedagogical strategy that is actually a valuable one to carry forward, this idea of blending the face-to-face and online experience. I mean, even just to bring those two student populations together seems like a really valuable thing. Yeah, and I think particularly thinking about the challenges that 100% web students face. A lot of students opt for online courses, not for pedagogically sound reasons, but because of scheduling and expedience. And they don't always... The attrition rates for students taking web-only courses are much higher than face-to-face. And if nothing else, blending the two student populations, I had hoped would create a cohorting effect where they are sort of connected more to campus life because they're interacting with students in the face-to-face course. I don't know if it worked, but again, it seemed to have been a lucky choice. It seems like a really good idea that should be experimented with further, at least. Of course, we don't have the face-to-face aspect as much anymore right now. Fingers crossed, it will come back. So one thing I'm curious about, like I see a lot of annotation going on on the actual language around the equations and so forth. But two things I was interested to hear your thinking about was, are people annotating equations themselves? And then, as a follow-up, are your students making use of the ability to put equations into the annotations themselves? So the short answer is sometimes. The longer answer is that that I found depends a lot on my population of students. So pre-calculus is a course where most of my students were not majors in math or computer science. And so I didn't really... They generally did not make too much use out of typesetting their own equations within the annotation space. On the other hand, when I've used this as a tool with my math majors and my more advanced students who are already coming with a little bit more comfort typing equations, they don't always do a ton of it in their previous courses to mine, but they find it easier to pick up on, usually, to be able to write out the latex codes that actually create the math type within the annotation margin. But I do see it happening more with my students who are already in the math space and have already had some experience with doing that in the past. Yeah, it makes sense. I mean, typing latex is not like a common skill for every student. No. Yeah. You know, I wanted to get back to this idea that you kind of raised before about building people's sense of agency and addressing the trauma that people kind of experience in learning about math, especially in the context that you were saying at Bridgewater State being such an institution that focuses so much on teacher training. I'm curious, I bet you have, but have you worked in the idea of helping people that are themselves going to become teachers, bring that kind of set of practices to try to improve agency and lessen trauma into their work with presumably K-12 students that they'll go on to teach? Sure. So I haven't. So in our department at least for the courses that are mainly populated by future teachers, I guess they fall into two categories. So in Massachusetts, our future elementary teachers and early childhood teachers and our future middle and high school teachers have sort of different curricula and different expectations for getting their initial license and going forward. So for the most part, most of the future teachers that I see in my upper level courses in the math major, most of them, but not all of them, are planning on secondary education as their target middle or high school. Our future elementary teachers, in fact, every future elementary teacher at Bridgewater has to take three semesters of a math course for future elementary educators. And I haven't had too much experience teaching those students, but so they have a very different sort of, they come in because they haven't selected to be math majors in the way that our future middle and high school teachers have. And so I think, again, we're seeing the difference in the two populations. But definitely one thing that I do hope is happening with my math majors and also with my in-service teachers who are my graduate students that I've taught before. So I think it's important to model these practices with students who are pre-service educators and sort of be intentional and explain to them upfront, hey, here's a pedagogical choice that I'm making with our class right now, you, my pre-service teachers and me. Here's why I'm making that choice. Here's why I think that's important here, ways in which that's connected to sort of current research and how students learn and how they thrive in education. And just sort of talking through that with students and making them aware that this is a choice that I'm making that maybe your previous teachers made different choices. Why might they have made those different choices? And just sort of giving them the language to talk about such things. One of the things that the common core state standards do for states that are adopting those of which Massachusetts is one is in the math arena, they define not just what are the content skills that we expect students to be building in their math classrooms, but they also develop these standards of mathematical practice, they call them, which are kind of describe a set of habits, attitudes and beliefs that we want students to demonstrate about mathematics and about themselves that actually lead to this thing that we sometimes call productive persistence, right? We want students to be able to persist through the solving of a new and difficult problem by applying sort of strategies and developing cultivating a mindset and a habit of mind that makes that persistence productive. And so I think that, you know, when I talk to my future teachers and even my current in service teachers about this, that everything is kind of, I'm trying to point everything towards that goal of helping students to develop those skills that are not just about solving equations, but about knowing that they are mathematical themselves, that there's no such thing as a math person and that we can kind of, you know, unpack some of that trauma that they don't realize is trauma that they've developed coming up through through their early grades and just help them to become more comfortable talking about mathematics and mathematicizing questions and mathematicizing problems from their life and doing that in their own voice and in a community. Because what the practice of mathematics looks like not just at the teaching and learning level, but at the professional sort of research mathematics level is that mathematics is an inherently social enterprise. How we figure out whether a mathematical idea is even true in the first place is we don't submit it to some all-knowing all-seeing oracle in the sky somewhere, right? Is we submit it to one another and we engage in a conversation and we assess, you know, the success of a new argument or even the success of a very definition in community with one another. And that's what it looks like to sort of practice mathematics. I think one of the things that's telling about math is that other disciplines get different verbs out in front of them. We can practice in art. We can conduct a science experiment. We can investigate a question in social science. But what is the thing that gets attached to math? What's the verb that gets attached to math? We do math. And the reason that we do math is that it can be done and we can then do something else with the rest of our day, right? Just the choices of words that we use there. And it sort of glosses right over the fact that we don't do math as individuals most of the time. In math classes, that's what it looks like. In the field of math, we are sort of plagued with this myth of individual genius, right? We celebrate these individuals who have made contributions to our fields who are almost invariably young, white, and male. And so there's this folklore that builds up around math that that's what doing math for the sake of having it done looks like is one person that's brilliant, usually a white guy sitting in front of a piece of paper almost monastically and sort of elucidating it all on paper. But that's not what real math looks like. It's not what it looks like for professional research mathematicians. It shouldn't be what it looks like for first graders, learning basic facts in their classroom. And so even just the ability to come together and decide whether or not an argument is sufficient or decide whether or not even a definition of a term is the right definition. One of my favorite moments over the past couple of years has been the debate in math social media about whether a hot dog is a sandwich. This is something that caught on in not just the math sphere but I point the finger at John Warner's book Why They Can't Write as one of the places that popularized this as what he called sort of an unwinnable argument is a hot dog a sandwich. But to a mathematician, it's not that that argument is winnable or unwinnable. It's what does that argument tell us about the definitions that we care about? Why do I believe a hot dog is a sandwich where somebody else might not? It is one of our definitions better than the other. And so that's become an activity that I do at the beginning of the semester with all my math students. Tell me about whether you think a hot dog is a sandwich and why more importantly, and try and argue with a person next to you over why your definition is superior to theirs. Where does yours definition potentially break down? And those are the conversations that mathematicians have is the number one, a prime number. You can make an argument for a definition one way and a definition the other way. And the definition that prevails tends to be the one that makes the rest of the work that we do more comfortable and easy, right? It's not because it's right. It's because mathematicians have decided it's more convenient. So just getting students to understand that we all have that role to play in the development of mathematical knowledge is something that necessarily happens in a social context. Oh, that's really powerful stuff. And I love that hot dog practice. I've been in that with myself, plus the is zero and even our odd number question is always going to. I had a professor shout out to Arnie Langberg, well, a teacher back at high school who said that math was taught backwards and that we did all the boring stuff first. And so people had to slog through arithmetic and algebra and so forth until they got to the really interesting kind of math questions that happened in the more theoretical end. And he proposed the idea of reversing it and like, let's do the interesting stuff first even if people can't do the equation part of it. And then build toward, you know, the sort of more kind of specialized ability to actually work with complex equations and so forth later on when you have a reason to. Curious if that kind of resonates with your practices. That definitely, I think that resonates with most of our mathematical community in a lot of powerful ways because one of the things that the K-12 curriculum kind of does is that it trains us to believe that what math is is arithmetic, algebra, geometry, trigonometry, calculus. When in reality, that's a very narrow segment of what mathematicians study. There are all sorts of fields that have very little, if anything, to do with school mathematics that are yet still mathematical enterprises in the ways in which we construct knowledge and assess one another's work around them. And a lot of those topics do have very low floor. Talked about knot theory before. If you've tied a knot, you've done something mathematical, right? So just figuring out what that sentence even means, what it means to do something mathematical and then trying to unpack that into something that resembles a mathematical theory. That is something that students at a lot of different levels can do. And so there's this, one of the more famous sort of writings about this is a paper called Lockhart's Lament. It's this sort of article written by a mathematician looking at the K-12 curriculum and saying the thing that we're not doing well in K-12 is we're not allowing students to express their creativity in their math education. We're sort of stomping the creative juices out of our students in the process of their math education and standard curriculum. And a lot of mathematicians and math educators, I think, have taken that to heart over the years and said, well, how do we teach math in a way that we're teaching it to the whole person? Where they are able to bring their authentic selves and their voices and their creativity into the math classroom and instead of funneling them all to the one correct technique and answer, how do we help them to focus instead of to funnel and to do math as themselves instead of having to put on sort of an artificial mathematical formal persona in order to fit it in their classroom? So how do we teach to the whole student in the math classroom? I love that. And I kind of wanted to ask a follow-up. It sort of dovetails with that and also what you were just talking about earlier about voice and finding your voice. And so what does that look like when you have diversity in the classroom or even for your female students or your female students of color? Yes. A lot of us are told we're not good at math. So. Yes. That's an important point, right? Because to the extent that math anxiety is a social contagion, right? It's something that passes person to person, teacher to student, sometimes parent to student. Sean Bylock's research on math anxiety shows that the female young girls are more susceptible to those messages from other female authority figures in their lives, teachers and female parents. But that tends to be, they tend to catch that social contagion more readily through that route. And so you're right, first of all, that more students of color, students who are not white males are coming into college with the belief that I'm not a math person. I'm not good at math. This is not a field that's for me. I need to find something else to do, right? Because that's who I am somehow. And so I think that we do have to do that work of kind of not just helping each student to understand that they are mathematical, right? That they can magnetize a problem. They can be successful in mathematical reasoning. And it doesn't always have to look exactly like the mathematical reasoning that their high school math teacher showed them, right? But there are many different routes to an answer. And sometimes there are even many different answers, which makes a lot of math majors uncomfortable. But then we still do have to do that work to unpack the really the gendered and the racialized additional trauma that is picked up by students from sort of minoritized groups as they're coming up through their math education. And those students do disproportionately land in courses in college like remedial algebra, right? And so that's not just an individual mandate to help those students to come to grips with their own voice and agency in the math classroom. That's also kind of an artifact of an institutional bias that exists across higher education and that we're only now starting to figure out how to productively unpack. So you're right. And to the extent that any course in any discipline has to create safety productivity and sort of norms around participation in conversation. I think we have to do that in math as well, maybe even more so because of the additional weight that those students feel about having been stigmatized as not math people for such a long time. Yeah. And so like, and we should probably wrap up, but this is such an interesting thing to me. And so are there ways that you have found that you can use collaborative annotation to that end, to helping people find their voice or even being really transparent about all the biases that come in, that they're coming in with? Yeah. I don't know how much that's really intentional around that that I've done in the social annotation space. I mean, one thing that I do, the one thing that I do do, I'm going to actually share this on my screen again real quick is that I try, especially with math students, I find that you have to be really kind of directive to say, hey, here's some ideas for how to engage with this text. And sort of one of the important things, I think that has a disproportionate effect on students from minority backgrounds is just encouraging the asking of questions as the most important part of the enterprise of doing math and not worrying ahead of time about whether it's the right question, whether it's a smart question, whether other people might have this question, just setting the norm in the course that what I want you to do more than anything else is just to ask. And if we can lower the activation energy for doing that for all students, I think that it's one of those universal design principles that will have a disproportionately positive impact on students who historically have not felt the agency to ask a question, felt it wasn't their place, that asking a question exhibits ignorance or dumbness. That's that fixed mindset that I'm not a math person so if I ask a question, it's because I'm stupid. It sounds blunt and trite to say it that way but I really believe that that's what a lot of students are carrying with them out of high school. And so just to encourage that formulating of a question and then the sharing of resources around the answering of question, those are something that everybody can do and so creating the space and sort of walking students through the process of what that can look like. Without shutting down any of the discussion or saying, oh well, why didn't you know this to begin with? There's a lot of stigma that students feel coming into a math class and asking what they perceive to be an elementary question that they should already have known and just training out of ourselves as instructors and then training out of our syllabi in a lot of ways, that sort of belief that students should already know whatever the body of content might be and then opening up the floor to ask anything, I think is the most important thing that I do with annotations specifically. Wow, that's great. I really love that. I wish I was taking your math class now. I wish I could go back in time. But anyway, speaking of time, we're pretty well out of it but I want to thank you so much for being here today and for everyone who joined and Nate for all your really great questions. And I feel I'm going to walk away sort of a different person with less math anxiety myself just because I feel really inspired by your work. That's terrific. And I would like at this time to give you an opportunity to say goodbye and leave us with anything you want to leave us with. So thank you first of all for having me on today. I think that it's been, I don't know why but it surprises me that there aren't more math and science and engineering educators that have really sort of taken the leap into social annotation and really using text as a way to gather around with their students. So I'm hoping that that some of this conversation has been helpful to connect people with the ideas and the philosophies and the resources that can help other STEM educators to do that. And I would just encourage people if you have math and science educators on your on your campus to don't be shy to talk to them we're actually not we're not the prickly group we mathematicians that we have a reputation of being. And there is I'm sure a champion on your campus who would love to hear about how to do more sort of social active annotation activities with their math students. So don't be shy to talk to them or send them to me. You can find me on Twitter. You can find my website on Mathematics.com. So spell mathematics with a double t.com you'll find my website. I'm happy to help connect you with any other resources from there as well. So it was really a pleasure great conversation today. Thank you. Thank you so much.