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Hello and thank you for listening to the teaching math teaching podcast. The teaching math teaching podcast is sponsored by the Association of mathematics teacher educators. The hosts are a with Anheuser me dusty Jones, and Joel Amidon. Today we're talking with Nick Wasserman. Nick is an associate professor of mathematics education at Teachers College, Columbia University. We're talking with him about his current interests and work in the area of secondary teachers advanced mathematical knowledge and development. Welcome, Nick, can you tell us a bit about yourself and your background? Sure, a dusty Hi, Eva, hydro. Hello. As dusty mentioned, I'm a faculty member in the program in mathematics education, Teachers College, Columbia University. I've been there for seven years. And I went through an undergraduate program at the University of Texas called the uteach. Program, which was where I figured out how much I enjoyed teaching and teaching mathematics and turn that into sort of interest in teacher education and the preparation of secondary mathematics teachers, what sort of things did you do between your time in the uteach program at University of Texas, and pursuing a PhD? Was it like one solid motion? Or did you do some? Did you teach high school? Yeah, I taught for several years in Austin at a large public school and the south of Austin, teaching a range of high school courses. And then started graduate school, where during graduate school, I also taught but I taught at a private school in New York City, and also taught secondary classes and taught pretty much the scope of math courses that are offered at the secondary level. Yeah, so what sort of thing made it the teaching of mathematics really interesting to you? When I was a kid, I wanted to be an architect. So when I went to school at the University of Texas, I actually was an architecture major. And I think I, in hindsight, I was an architecture major, because I sort of love math and was a creative kid. And that felt like the right kind of right mix. And during my undergraduate studies, I figured out that teaching math also combined these two things pretty clearly. And as it turned out, I was a mediocre architect, but thankfully, a better t shirt. Those are the things that sort of pushed me into teaching. And that was a positive experience for me, and the uteach program, one that I very much enjoyed. And it was really there that I started thinking about wanting to be a part of teacher preparation, wanting to be a part of training teachers to think about the issues that happen in classroom to think about mathematics and how it interacts with the dynamics of teaching. So in my work, I have undergraduate students who are math majors who are working to become certified to teach high school and I also work with some high school teachers in the area, high school mathematics teachers, and they have this question that I think both groups have this question of, what does a secondary math teacher like an algebra teacher gain from taking advanced math courses like abstract algebra? Like why the students in abstract algebra are saying, Why do I need this? The teachers are wondering, why did I have to take this? What's your answer to that question? I spent a lot of time thinking about this question, too, which is, I think, a large part of why we're talking today. And I think there's several different ways that you can respond to this question, I think the one that's most common or most often talked about, has to do with thinking about very content specific ideas. So in elementary school, teachers are required to or sort of tasked with teaching students about the base 10 number system, how we represent numbers, using a base, a sort of a place system. And one of the things that elementary teachers often are asked to think about in teacher preparation or other base systems, base two or base five, because it helps them think more deeply about what a base number system means. And that were specifically base 10, and how that informs some of the different procedures and things that we do that are very common in elementary school. And I think the same kinds of ideas are a first response to that which there are there are connections in in these courses that are intended to deepen teachers understanding of the secondary topics that they teach, I think, you know, one way to understand this, of course, like abstract algebra, by it's just in the title indicates that this is about abstract things. It's an abstraction. We're talking about algebra in an abstract way. And that what secondary teachers end up teaching is a particular it's an instance of these more abstract algebraic structures. And so understanding the secondary mathematics that one is teaching as an instance

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Have a broader, more abstract structure can be pretty helpful in the same way that an elementary school teacher understanding different base number systems might help them think about teaching students about base 10. I think this is a pretty common answer. And I think it's valuable and insightful for the kinds of things that that can be helpful. I will also note that I, you know, from the literature and and from teachers themselves, there's pushback against this. And I think we also need to listen to the pushback that's being given. And I think a lot of the ways that I've sort of pushed in my own work is trying to be much clearer about how it relates to the work, how some of these ideas relate to the work of teaching. And so I think an initial response to that question is the one that I gave that are very content specific ideas, but I think there are other kinds of connections that we can also talk about that help answer that question in a more well rounded way. So I gave an example of a kind of mathematical connection, that talks about abstract algebra as something more abstract. But I also think there's times when what we're learning in an abstract algebra course is, in fact, an example or instance of something that is more abstract that gets talked about in secondary mathematics. So just as a brief example, the notion of function gets used a lot. And that that idea gets talked about in high school mathematics. And some of the things that get talked about an abstract algebra course are examples of functions. So things like binary operations, these are examples of a topic that's in secondary math, that is a more abstract topic. And a binary operation is an example of a function in that context. And so thinking about other kinds of mathematical connections, whether they be things like I just mentioned, or whether they be more activity or process based, what does it mean to engage in doing mathematics and being very explicit about this, that abstract algebra, and a course of study actually models, what doing math can look and feel like, and the result of what doing math kind of is. And so I think there are ways to emphasize these not just content connections, but sort of process or the activity of doing mathematics kinds of connections, I've been thinking along these lines about, you know, where the activity of doing math relates to the activity of mathematics teaching, and how the, there might be some useful ways of thinking about the intersection of those kinds of activity or practices. And so I think for all of those reasons, there are ways to talk about what algebra teachers might gain from a course such as abstract algebra, I think a lot of my work has been trying to figure out ways that it connects to work that's done in teaching and how that translates, I'd like to jump in, Nick. And I was just thinking back to my abstract algebra course, which was many, many years ago now. Maybe almost 20. But thinking about some of the lessons I learned. And so you know, we've had different conversations about some of the stuff that you've talked about in the importance of this work in secondary math teaching. And so thinking about my own preparations, I was just thinking, like, what are the lessons that I learned that I still have with me that I still talk about with my own students now, my pre service teachers and thinking about, I remember my abstract algebra teacher, he was talking about the importance of definitions, right? The importance of definitions and knowing what things are and what things aren't. So if you could say, Is this a, I apologize, it's been a while since I've done a ring or a field or like, what if this operation has certain characteristics or things like that? And I can't I mean, I look back at my notebook, I can't make heads or tails of it, but I remembered you Steven, like, how important it was to think about is this thing, this definition or not, or, and then knowing that the importance of definitions and vocabulary and how like, sometimes in my own math preparation, it was just kind of skimmed over like that, is that like, I don't even know like, how would I define that? How would I put that in? And for someone that, you know, is is struggles in mathematics and like so is asking those why questions and like, and you're in this like, basic algebra class, and like, you're just skimming over some things like, well, what's the difference between an expression and an equation and a variable and a coefficient and all these sorts of things like, my, the importance for that, like in pointing those things out, and helping to find those things and helping students forget, like, came from my abstract algebra course. And like, I mean, I remember even him saying, like, if you know, all the definitions, you're going to get a C in this course. I'm like, you know, me being like, okay, you told me what to do. I'm going to do it. I'm going to memorize all these definitions and how that led to success in the course and just even thinking like, you know, then even going into the beauty of figuring out like, how do all these things fit together and like I was, I remember once like, just sitting

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I mean, like, wow, I'm actually doing abstract algebra like this, like, kind of pretty beautiful thing that was like I'm achieving something that very few people have. And it just made me like, I don't know, kind of embrace some of the beauty of mathematics too. And like just all those sorts of things came rushing back, as you were talking about some the importance of doing these things. I don't know if that's getting out which you're the kind of the emphasis of this work is or if that's slightly different, but yeah, no, that's a perfect illustration, a really good example of the way in which courses like abstract algebra, it sounds like the professor that you have cholas made some explicit acknowledgement of the role of definitions and the importance they play in mathematics. But this is kind of that engaging in mathematical practice, that these courses can force you to engage in doing math, and how that can help you understand sort of at a broader level, what mathematics is, what makes it tick, as a discipline, how it's different than other disciplines, what things are valued, and how they're valued. And I think that's super important. And I would say that you were lucky to have a professor that called your attention to some of that, because I think being explicit about those disciplinary practices can be really helpful for teachers in particular. So that sounds like one way that that math teacher educators can help connect, make connections for the students or helped us students see connections, I guess, students, I mean, future mathematics teachers, being explicit about those connections, what other recommendations do you have for people who are teaching math teachers? How can we help make these connections? And up? I'll give an answer. I'm not sure it's an exact answer to the question you posed. But I think math teacher educators, have a role to play exactly in this situation of connecting two worlds, which are the worlds of mathematicians and people who do in practice mathematics, and the world of teachers who are out in secondary classrooms teaching high school mathematics that the teacher educator plays a really important role in trying to find ways to bridge those two worlds and to help bring those two worlds together. And I think, you know, how do you help sort of people make some of these connections is, of course, a really hard question. But I think it takes an appropriate balancing of listening to and respecting both of these constituents, the sort of mathematical world and the very practical people who are in the classroom, out teaching mathematics world, and really not dismissing either end, but really honoring both of those groups and figuring out how to make points of connection across them. Because I think they can be really powerful opportunities, both when it's things like Joel mentioned about mathematical activity and practice being modeled in those courses, and being explicit about those. But when it's also specific kinds of content connections, I think that there are ways that teacher educators can help serve as a bridge between these two this sort of late It's, it's caused me a lot of a lot of my work in, in teacher education ends up being on the on the more mathy side of things, thinking specifically about the preparation of teachers through their mathematical coursework, rather than through their methods courses or pedagogical coursework that happens in an education department. It's pointed me to how divided we are in terms of teacher education, in terms of university structures, there are education departments and math departments, their education courses and math courses. And that, to me, some of the most powerful opportunities happen in the conversation between those worlds. How do we think about mathematics courses, as offering opportunity for pedagogical learning for secondary teachers is a really interesting question. And you know, the counter is how do we think about pedagogical courses as opportunities for learning mathematics. And those spaces to me are really interesting, I think, institutionally and structurally because of the divide between them. Those can be hard spaces to navigate, but increasingly, I find that, that it's important work to navigate those spaces, you know, from my own experiences, and some have conversations with others, there really is a lot of power. In a pure in a course that's devoted to learning mathematics that can translate into teachers reflecting on and learning lessons about pedagogy, and that those can be really powerful lessons when they happen within the context of a math course, as they're learning and experiencing mathematics, that those can be really powerful.

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Developing thoughts about how to teach how students learn thoughts about their own future work in the classroom, what sorts of tools are useful to you? I know I've looked at some of your work, I know you've done some things with abstract algebra I played around with I played around, I've used a lot of the stuff that you've developed for analysis for a couple different courses that I've taught. I know you've done some stuff with statistics, you kind of been around the, you know, the different content areas, what tools have you found useful as you work with secondary future secondary teachers, when you say tools that they can you give me a sense of what you're thinking about, for example, I know that you use GeoGebra, I learned a lot about GeoGebra, from just looking and seeing some of the stuff that you had made for the ultra analysis materials. And I was like, Oh, I didn't know that could even happen that way. So I thought my learning increased by just kind of playing around with the stuff that you put out there. So I know you use GeoGebra. That's one of the tools I'm talking about. Are there others, my immediate response was less about tools and more about resources. And okay, in that it was, initially my thought was, I think people end up being these resources, these sort of hand your own experiences, sort of engaging in math and engaging in teaching that become really powerful resources, for thinking about how to bridge this world, you know, bridge, talking about, you know, pure and more advanced and abstract mathematics and talking about teaching at the secondary level, in terms of tools that I found productive. Yeah, GeoGebra is an example of one. But I found various dynamic technologies to be productive tools at all levels of mathematics. So GeoGebra would be an example that is pretty versatile. But there's things with, you know, probability and statistics with simulations, other software that's similar, that would be in this vein of dynamic technology, allowing students not only to visualize mathematics, which I think is a really important piece that they bring to thinking about the mathematics, but also to interact with, you know, to physically feel like you're interacting with some of these mathematical objects, whether they be you know, sliders and graphs, or geometric shapes, or simulations of coin tosses, I find that those open up opportunities for learning mathematics period, I think they can be tailored, obviously, to talking about connections between Advanced Math and in secondary math, but I find them as useful tools for any, you know, math, math, teaching kind of situation, that visualization process, that ability to dynamically engage, to think about how changes in one thing might impact changes in the whole, that those are useful tools for learning. And you're not the first person dusty, who said, I downloaded that GeoGebra file, and I reverse engineered what you did, and that helped me learn GeoGebra better. So yeah, I think it's a good way to learn technology systems often, once upon a time, I think the first maybe the first semester I was using GeoGebra. I told a student Oh, yeah, GeoGebra doesn't do that. And then two months later, I realized, oh, GeoGebra does do that. I just didn't realize that. So now I just tell people, I don't know how to use GeoGebra to do that. The word bridge was mentioned earlier. So that's the transition I'm using. So Nick, I got a question for you about bridging. And so you're in a math, you're in the math department. Correct. I'm actually in a math education department, but a very unique one. I can talk more about that later. But it's more mathy. Yeah. Okay. All right. And so maybe, maybe you might have some suggestions for bridging, I'm just thinking about myself as a math teacher. So here's my situation, I'll just be totally selfish, math teacher educator in the School of Education. And these courses for secondary are taught across campus, which, you know, we can't access campus right now. But across campus, and so maybe some of these connections and why we want them to be taking these courses, maybe those things are not being realized in those courses. Or maybe just like that point is not being made like, Hey, this is why we want them taking modern or abstract algebra. What suggestions would you have are for making those connections or trying to, I mean, obviously, building a relationship with the people that teach them but anything else that you would suggest, I don't know if I have anything super concrete, besides its work, to establish connections when there's not only a physical divide distance between departments on campus, but just different goals, that it's work to figure out how to how to make those relationships happen. One of the things that I found productive in the approaching from a math education or teacher education side approaching mathematicians is trying to acknowledge the ways that mathematicians actually bringing something good

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Thinking about teaching, acknowledging that the ways that we, I think often we we try to in the teacher education, education lens, we try to clarify that our expertise has to do with education has to do with learning. And one of the results is that we often sort of downplay and say, Well, you know, to the mathematician or to someone who's just in their discipline, thinking about learning is our domain of expertise. And I think that's true. And we don't have mathematical expertise in the way that they do. But I think acknowledging the ways that they know mathematicians often bring thoughtful perspectives, about teaching, acknowledging that the ways in which that they think about pedagogy and in their own work, when they teach their own classes, it's important to, to actually acknowledge that those are valuable, and that this isn't some divide, where we're saying, you know, we know this and you don't, but rather than it needs to be a synergistic relationship where both sides value each other for what they bring, you know, one of the ways to do that is by trying to acknowledge how mathematicians can bring value to the conversations about teaching. And hopefully in the same way that it can be reciprocated in the ways that we bring value to, to the mathematical world, by the way, in which we think about the sort of depth of connections and learning processes that that actually helps inform ideas about mathematics, and even make an explicit like, Hey, we're kind of here for the same purpose, right to advance math, education, and like, building on those, you know, commonalities and, and like coming in with a, hey, I want to learn more about what happens here and trying to be in a posture of learning. Right? It's probably a helpful perspective to take. Yeah, I think I think that's exactly probably said better than what I just said. But yeah, that's exactly how you use good words. It's okay. So Nick, can you tell us about the recent book that's come out in the last couple of years that you edited, connecting abstract algebra to secondary mathematics for secondary mathematics teachers? How can a mathematics teacher educators use this book? Yeah, thanks, dusty. This was one project of mine in this world in thinking about abstract algebra courses, as you know, I've had in other conversations, and, you know, analysis and statistics and other things, but this was one that was trying to draw out trying to point out some of the difficulties that exist in this space difficulties that are just intrinsic to learning abstract concepts. But more specifically, difficulties that that happen is we try to point to connections between abstract structures and instances of those abstract structures, that, that that's actually difficult work. The work that we are asking of teachers often to do without any help is difficult work, it points to some of the challenges of learning the mathematics, but also learning the kinds of really deep mathematical connections that we would want teachers to have about these ideas that then become those can help shift their way of thinking and teaching some of these topics. So I think the book in some ways, wanted to point out their challenges in the space of abstract algebra and secondary math, and then to talk about some of the opportunities that are there. And these are opportunities that are in, you know, just connecting the mathematical ideas, you can sort of talk about this just in the space of connecting, you know, between the mathematical, on kind of a mathematical plane, so to speak, just pure the mathematical connections, but also opportunities, pedagogically ways to realize these in teacher education by thinking about how does the activity in abstract algebra maybe mirror the kinds of activity that happens in a secondary math course? Or how do we think about applying some of these ideas, not to the secondary mathematics, but to thinking about teaching? And how do these inform solutions are resolutions to problems that arise in teaching the work that teachers do in the classroom and how these mathematical ideas can help inform their responses. So I think the book was intended to try to kind of point out some of the challenges but also point to opportunities. It was meant to be both a practical resource, there are chapters that are really from practitioners who are offering sort of their experience in trying to navigate this space, and also chapters dedicated to studies and research that have been done in this space that have some empirical support, and maybe provide some insights that are that are useful. And so the hope is that the book as a whole is a good resource for teacher educators to think about navigating navigating these spaces between sort of tertiary and advanced mathematics and teaching secondary mathematics. Yeah, lots of good ideas and thinking I'm still in my head thinking about the example that you started with the alternate basis in the yellow

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Entry schools because that's my wheelhouse. And one of the reasons I do think that people use alternate bases or historical number systems or those kinds of structures is to push this idea of knowing what something is means knowing what it is not. So to really understand base 10, you have to kind of step outside of base 10. It's kind of like, when you're monolingual, it's really hard to understand about language and how languages are different, right. But once you start speaking a second language, you can start kind of seeing differences in so base 10 is similar, right? Because we grow up, like knowing how to say, you know, 10, before we have in serving Kids Count, they go 1-234-567-8910 1112. And there's no difference between nine. And for them, other than it's one more and I'm saying a different word. But structurally, it's an enormous difference, right? Because we're grouping. And so I think partially, we're using these alternate basis or historical numbers systems to recognize what is actually within base 10. And what is not. And I was trying to figure out, as you were speaking about why we're using those in the elementary system, whether that played a role or not. And I think I just got hung up on thinking about that throughout the conversation, whether website URL later. So this notion of variation, right, so this idea of to understand what something is, we need to kind of understand what it's not. So, for example, you could use time and use a standard algorithm of addition and subtraction in a time context. And it's not going to work because it's not base 10. And that's going to help you understand why base 10 works, because you go like, Oh, wait, there is this relationship of 10 to one. The other thought I had is that one of the reasons we really want to emphasize teaching base 10 in a meaningful way, and not an arithmetic way. I mean, there's many, many reasons. But one of the reasons looking at advanced mathematics is that if you actually understand base 10, solidly a lot of algebra falls into place. Right polynomials fall into place, I mean, there's, it lays the foundation for a lot of things. So and different argument would be we really need to know kind of what comes further up. So we know what we're preparing for. And we know what's important to teach. But that's a different argument and this argument to say let's step outside of what something is to kind of be able to see it, I've had several thoughts crossed my mind, we'll see how coherent my responses across them. But yes, absolutely, I think that's one of the powerful aspects about stepping outside of a concept to be able to help understand its sort of boundary. Right, you can, in some ways connected to what Joe was talking about in his course, about the importance of definitions, definitions define not only what's in that set, but also what's outside of it. And examples and non examples of definitions are, I think, are really important, because they help define the space of what you know, that object that's being defined really is and what it's not, and that that sort of scope or that boundary is, is a really important one to to navigate and to think about. So answer sort of immediate answer your questions, yes, some of those same ideas in having formed part of my thinking, I think one of the challenges, that exact thought is actually I have my own preference. In thinking about abstract algebra, there have been people who have talked about restructuring, abstract algebra for secondary teachers based on trying to align the study of rings and fields first, because rings and fields are more common in secondary mathematics and to do groups later. And so there's people who would advocate for an approach using rings in fields I'm, I'm somewhat hesitant about this in large part because groups become this point at which you can understand pretty clearly examples and non examples of what these algebraic sort of simple algebraic structures are. And I find the non examples to be really powerful for helping motivate, understanding what some of these things are talking about and what they're what's outside of those. So, you know, my own thoughts, I think align pretty closely with that. The last thing I'll say is, I think one of the ways that we often it's hard work

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To do a good job helping teachers understand what it is they're learning from engaging, you know, with different base number systems from looking at these more abstract ideas at non examples, I think it helping communicate clearly to teachers what it is that they're supposed to be gaining from these mathematical sort of learning opportunities that is specific to what they imagine themselves doing out in the classroom. And how that impacts what they do in a classroom is it is challenging, but it's really important. Some of the work that I've done is pointed out how teachers might approach like, sort of making connections. And often they end up being sort of on the superficial, trivial connections, because we haven't done a great job of saying, No, you're engaging in this because it, it accomplishes this in relation to the work you'll do in a classroom. I think we need to be better about being clear what those things are. Yeah, it's it's not enough to just be the Mr. Miyagi character and know that this motion is going to be useful in karate later on. So just go wax the car or sand the floor or paint the fence. But helping our students realize that we're talking about inverses here, or we're talking about functions in this context. And here's how it relates to what you did when you were in algebra one or what you will be doing what your students will be doing when they're taking algebra, algebra one was coming back to what we talked about, like the explosiveness right, Nick, like the

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that'll help them see when we make these connections will help make those connections. Right. Yeah, I mean, I think leaving things implicit, some people might see it, some people might not, but being explicit about it is really helpful. It It calls attention to things that you're wanting to call attention to, in a way that allows other people to start attending to those works in teaching parenting and in marriage all it's great. It's wonderful. Be explaining explicit, explicit, my gosh.

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Thanks so much, Nick, for talking with us. We really appreciate it. And I've got a lot of notes here. I'm looking forward to putting some of this into practice fairly soon, maybe at home with my wife and my children be explicit. Maybe more so with my future math teachers. Yeah. Thank you. This has been fun. Thanks again, for listening to the teaching math teaching podcast. Be sure to subscribe to the podcast. And we hope that you're able to implement something you just heard, and take an opportunity to interact with other math teacher educators. Hello, listeners of the tg math teaching podcast, we have some big news. Are you ready? The teaching math teaching podcast is starting a summer book club. What better way to grow as teachers and math teachers than to engage in professional learning together, and we would love for you to join us. In June we're going to be reading rough draft math revising to learn by Amanda Johnson. In July, we are reading high school mathematics lessons to explore, understand and respond to social injustice by Robert berry bazel Conway, Brian Lawler, john Staley and colleagues. The plan for the book club is to read the book throughout the month and host weekly interactions on Twitter and Instagram around the chapters for the week. At the end of each month, we will have a podcast that discusses what we learned from the book and how we can apply what we learned to improving how we teach math teachers. We also might be joined by some authors. In short, we're excited we hope you are as well. Follow us on social media at teach math teach on Twitter and at teaching math teaching on Instagram to stay up to date on how to participate in the teaching math teaching summer book club. Thanks again. As always, for listening to the teaching math teaching podcast, be sure to subscribe to the podcast follow us on Instagram and Twitter. And we hope that you're able to implement something that you hear in the podcast and take an opportunity like this summer book club to interact with other math teacher educators.