In this episode of Room to Grow, Joanie and Curtis host special guest Dr. Katey Arrington to discuss the importance of teacher content knowledge.
Dr. Arrington is the Associate Director of the UTeach Institute at the University of Texas at Austin. She is also serving in the Presidential line for NCSM: Leadership in Mathematics Education from 2022-2026. Previously, she led the K-12 system services work at the Charles A. Dana Center, served as a mathematics coordinator for a growing, diverse district in Texas, instructional coach, and taught in both K-12 and community college systems. Katey earned a Ph.D. in Educational Policy and Planning, Master of Arts in Mathematics Education, and Bachelor of Science in Mathematics. She has extensive experience in leading networks for growing impact and designing and implementing system-level change for increasing equitable outcomes.
Early on in the conversation, it is agreed that caring about students in a non-negotiable and important component of effective teaching, but caring without math content knowledge is not likely to result in student learning. Our hosts and their guest explore the ideas of pedagogical content knowledge, math content for teaching, and approaching mathematics as ways of thinking, not just ways of getting answers. Expanding learning is presented as a group endeavor, but also something a teacher can pursue on one’s own.
We encourage you to explore the resources below, referenced in this episode:
Did you enjoy this episode of Room to Grow? Please leave a review and share the episode with others. Share your feedback, comments, and suggestions for future episode topics by emailing firstname.lastname@example.org. Be sure to connect with your hosts on Twitter and Instagram: @JoanieFun and @cbmathguy.
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Joanie & Curtis
I learned math deeply when I started teaching, and I can distinctly remember sitting in my at my kitchen table in my apartment on a Saturday morning planning my pre-calculus class and having these aha moments of like, Oh, that connects to this. And like how I totally understood that. Welcome to Room to Grow. I'm Curtis Brown. And I'm Joanie Funderburk.
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Joanie & Curtis
We work together at Texas Instruments and we're glad you're here. We're looking forward to continually improving our practice. And we understand that you are, too. We hope that you'll find this podcast as a room for you to grow along with us as we wrestle with and explore ideas about teaching math even better.
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In today's episode of Room to Grow, we talk with Katie Arrington about the importance of teacher content knowledge for teaching math.
Katie brings an amazing background with experiences as a math teacher, coach, district coordinator, professional learning provider, and with the You Teach program supporting the development of high quality STEM educators for each and every student. Katie Curtis And I agree that although it is important and required that teachers care about their students. Caring alone is not enough to ensure learning.
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As teachers, we can be most effective when we have a deep understanding of the mathematics we are teaching and also the math students have learned before and will learn after their time in our classrooms. There's a lot to unpack here, so let's get growing.
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Well good morning, Joanie. I am so excited to be having this conversation with you today, recording The Room to Grow podcast today.
We've got a really great opportunity to have Katie Arrington with us, and I'm really excited to talk about really a topic that the three of us are all passionate about and really this idea of teacher content knowledge and its importance in the classroom, both in instruction and making connections. This started with a conversation. Joanie, you and I were having around an end CTAM thread, a mayan CTAM thread that was going on, and I watched it and I thought, Oh my goodness, we've really got to chat about this.
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Talking about is it necessary for teachers to understand math in order to teach it? And so we thought, Oh my goodness, who can we have come and chat with us about this? And Katie Arrington was the person we thought of first. And so, Katie, we are so excited to have you with us today. Katie, can you just do a real quick introduction of who you are and what you do and get us started?
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Sure. Thanks so much for having me. I'm excited to be here to talk to you guys. I am. I live in Austin, Texas. I'm a UT girl all the way. I work at U Teach, which is a secondary STEM teacher preparation program. We work really hard to make sure all students have really well prepared STEM teachers in their lives.
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They kind of produce more and more of them to make sure all students have access to them. I'm also currently the president elect of NCSM, the premier national organization that supports leaders in mathematics education. And I've been a classroom teacher in both the K-12 and higher ed world. I have been kind of a curriculum coordinator and played that role for a very diverse, quickly growing district in Texas and just really love this idea that we're talking about.
How important is teacher content knowledge and what does it do for the teacher when they're working with students? I'm super excited to be discussing that with you guys today.
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Yay, Well, Katie, I just echo Kurt's excitement. I'm super excited that you're on the podcast with us. I know our listeners are going to learn a ton from you, and just your brief introduction is the perfect explanation as to why you were the first person that popped into our minds.
I mean, I just think your background of having been a classroom teacher, having supported classroom teachers, your presidency to be at NCSU, supporting leaders who support teachers, and then your work that you teach and understanding teacher preparation program just gives you such a well-rounded point of view on this. So why don't we just let you start us off on saying, What are we even talking about when we say teacher content knowledge?
And then what role does that play in a teacher's work in the classroom, and how is that separate from the other important skills and knowledge that teachers need to be effective?
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Yeah, this is such a great question and I think that I multi-specialty or on a regular basis is something that I really think a lot about. When you said that we were going to talk about teacher content knowledge and how important teacher content knowledge is, I immediately thought of Bob.
Bob was my first mentor teacher. My first year teaching. I was assigned as my mentor. He was the department chair and he was an amazing human being. I appreciate all of the things that I've learned from him, but he came through a very different path and I did right. I went to the university, I got a degree in mathematics and got a secondary math teaching degree at the same time through U teach, I had to throw that in there, and I feel like I was very well prepared in content knowledge, but also some of those other things that we'll get into that are really important when teaching.
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Bob was a career changer. He came into teaching much later in life. He did not have a math degree and one of the first conversations that we had, he told me, I feel like content knowledge is really not that high on the list of things that are important to being a great teacher. What really matters the most is that you're here, that you're putting your best forward, but you really care about the kids and whether or not they're learning.
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And I think about that a lot because he put those two things as caring about the kids is so much more important than content knowledge. And I wholeheartedly agree with him that that is incredibly important. It doesn't matter how much you know, if you are not really caring about what the kids are learning and you're communicating with them.
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And so content knowledge is not sufficient, right? But I do feel like it's necessary, you know, after years and years of being in the education system. And so actually last night I was thinking about another mentor of mine, Ernie Triesman. So before I worked at U teach, I worked with the data center and worked with Ernie for almost ten years, more than ten years actually.
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And he used to talk about how we can't tell kids that they're capable of doing rich, rigorous mathematics and then not giving them rich, rigorous methods to do something that's,
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Curtis & Jaonie
Oh, wow, it seems so obvious, but yeah,
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it's kind of pulling the rug out from under them and they feel Gaslighted Right. But how do you give them rich, rigorous mathematics without knowing that yourself, without as a teacher, having experienced that and having learned from that and really thinking about how to build those experiences for kids out of those experiences that you've had.
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So I feel like Bob was right. You've got to really care about the kids and really care about what they're learning. I think there's also a whole category of kind of allegorical content knowledge, like how do you express what you know to others? How do you bring them into your world and have them think with you? But all of that lays on the foundation of really a deeper content understanding. I think, the way I think about it.
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Well, I appreciate that response. And there's there are so much in what you were just saying in terms of your conversation with Ernie's work that it is about knowing the math yourself, right. So that you can provide those deep, rich experiences that you're telling kids that they have the capability to do.
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But you also have rattled off all these other really important things, right? Like how do you then communicate that? And I think so often people perceive the teaching career as just telling, right? You just you just tell kids and then they learn. And we all know that's absurd. None of us learns by being told something. So it's not about like, does the teacher have the magic words to say the thing in the way that kids understand?
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It's much more nuanced than that. Right? So you talked about like, how do you create the experiences for kids? It's an easy thing to say. It's not just like, Oh, well, I write a lesson plan or I design a classroom activity that they're going to do in small groups like there is there is an art to that. There's an art to figuring out how do I ask the questions that first tap into the previous content knowledge that students are bringing and then help guide them to this new understanding that I want to connect to that previous content knowledge.
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And I might even want to intentionally uncover some common misconceptions preconceptions, as Curtis and I like to say, that help deepen and enrich their understanding of that. So all of those things I am 100% with you and you both know, and I don't think it's a surprise to our listeners either, that this idea that teachers need to understand the math content very, very deeply and this is a soapbox issue for me, this is one of my most passionate things to talk about.
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I don't see how an educator can guide a student through the kinds of experiences we're talking about that lead to deep rich understanding without having that themselves and a whole lot more.
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Yeah, I agree. I think that's that's the crux of it actually is the intermingling of the content knowledge that we have. And Kati used a great term.
I love that this idea of pedagogical content knowledge, right? It's not just that I know the connections, but it's the way in which I plan to communicate those connections, or even better, to have the students discover those connections through the course of interaction, through the course of some lesson, through the course of some hands on development, whatever it happens to be.
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There's a process that I have to plan to make sure that that happens. And I can't do it if I don't both have the content, knowledge and the pedagogy and put them together in this pedagogical content knowledge. I love that term. That's such a good term.
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Yeah, I, I think too there is like and doing some research for this episode and actually even before that, in my work supporting educators as a district math coordinator, I came across this concept of content knowledge for teaching, right?
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So it's not only you have to know the content, but you have to know the nuance of the content that allows you to teach it. And I think for me, this came up a lot in the when Colorado adopted the Common Core State standards and in the elementary grades, for instance. Now there's like all these different ways of exploring the operations right?
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There's a whole bunch of standards in the early grades about additions and understanding additions, and it all leads to the addition algorithm. But if you get the algorithm without an understanding a place value, that doesn't become something that you can build upon.
So I would love Katie. If you can think of a content example, I know I'm totally putting you on spot right now, so if you don't think of what off the top of your head, that's totally fine.
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But like, could we talk about either with a piece of example content or without like what's the consequence? What if a teacher can do math? Like, I also got a math degree, what have you. You know, the math that you don't know the math for teaching. Like, can you help us like bridge what that actually means and what it looks like if that's not there.
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Yeah, well, so I think, you know, a lot of people think about math as you're just answering questions, right? You're kind of filling in the blanks or you're getting an answer. And I think the biggest thing about really having a deep content knowledge is that you understand that it's, you know, yes, we do find answers, but it's about leveraging structures and it's about leveraging patterns and it's about our number systems working.
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And all of those things come into play so that you can understand what's happening. I guess I'll give an example. I had this really kind of maddening discussion with one of the teachers that I stay in touch with, and we are still friends, but one of the first times that I talked with him, we were talking about factoring in graphics and how we kind of put a unit together around developing student understanding of these things.
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And I suggested maybe we should start with graphing, right? Maybe we should start with what does this function look like and what does it do and how does it act and what are the interesting points about it? And he insisted that we could not graph before we factored equations, but did the simple manipulation of finding the zeros so that we could put those on a graph.
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And I think the more he delved into it, the more he thought, Oh, really? We're building an understanding of this whole concept and that it all kind of flows together and we can choose to go through that learning in different ways. But really, if we're thinking about how our students are going to come to understand what these things are, I mean, we're going to talk about zeros, we're going to talk about roots.
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We're going to tell you just all kinds of different things. And together we were able to come to a good understanding of how the student might see this and he was just dead set on. We have to manipulate the symbols so that we can find the zero, so that we can put those on the graph, because that's kind of the process that he had been taught.
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And when he stood back and looked at it from a bigger picture, if we're thinking about this conceptually, there's lots of ways we can graph and if we can see a picture of this and we can think together about what's happening, then maybe we understand more when we start to factor what it is that we're even looking for.
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That's a really great example and it's one that I can I can totally relate to as a secondary teacher myself, right? Having gone through some of those same sorts of experiences the other day, we had a planning call with you and we were chatting a little bit about just some of our experiences with this kind of thinking and really thinking about what it means to both know the math deeply.
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Because unlike you too, I did not get a math degree. Actually, I have an engineering degree, which I suppose is an applied mathematics degree. I don't know. But it's still this idea that we have an opportunity to know this content at a deep way and then going back and learning about how do I communicate that to students. Now, we chatted the other day about, you know, all three of us have a secondary background.
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We kind of have this expected kind of experience in that space. And you chatted about your job as a K-12 coordinator and when you kind of walked into that space and thinking about looking at the elementary mathematics, you had an experience. And I just want to give an opportunity for you to tell that story. I think that was a really important one.
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Yeah. So coming in with my mathematics degree, you know, I've been really talking about calculus and AP statistics and all of these high level math courses for a long time. I taught pretty much everything at the high school level from algebra one to AP statistics. And I took a job as a as a coach first and then as a math co-ordinator supporting K through 12.
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And I had an inkling and a feeling that I had some learning to do about let me a little self-reflection there that I as a secondary math teacher, if you need to grab something or talk about functions and translations and all kinds of things, I can do it in a heartbeat. You know, I was thinking to myself in a student doesn't know how to count, I have no idea what to do.
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And I, I had a feeling that there was a lot to be learned. And so I really took it upon myself. When I took this job supporting K through 12 educators to do some deep learning because I knew I didn't know. I mean, obviously I know how to count, but I don't know how to teach a kid to count.
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Right? And all of those things that it's secondary. You know, we complain them not knowing their multiplication tables quickly, but if they know how to count, I don't know what anybody would do. So I got a lot of experiences. I really made an effort to try to learn elementary math. And one of the first things that I did and I'm so thankful for is I went to this professional learning experience that was about week long.
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I think they still do it. I think the abbreviation is DMI and I think it was developing mathematical ideas. I think that's what it stands for. And I just remember after the first day we did a lot of modeling and talking about addition and subtraction. And I just remember thinking I had no idea how complicated subtraction could be. I mean, how many different ways there are.
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Think about it, you know, different models, different ways of understanding what you're doing when you subtracting. And I feel like people don't have that appreciation, especially in elementary, but continuing into secondary really, math is about ways of thinking, right? So I found it interesting. One of their objectives was to have the educators in the room discover how to continue learning about both children in mathematics.
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That was one of their objectives was for us to think that how to continue learning. And I think that's exactly what I look for in a teacher. And that knowledge is important, but it's about that like curiosity and desire. Yes. To discover new things about both our students and about mathematics. I think that's that's super important. So it was really important for me to maintain that curiosity and to continue learning.
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And I now elementary mathematics is my favorite thing to do. And moving into middle school, I love talking about fractions and ratios and all of those kinds of things that are similar and different. It just it like that. My life when I get to talk about this, you show how I love that.
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I love that and I love how you made that connection too.
And I just want to say when I kind of again pull out some threads, the idea around using our teacher content knowledge to create good learning experiences for kids is picking up some words you said that I actually wrote down. We think about how might the students see this right and plan from the students perspective and that ties so nicely into your power to continue learning.
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Comment Continuing to learn about our students has also continue to learn about how students are thinking about the mathematics and then how do I think about the mathematics and then how do I support students in developing the mathematical ideas to get to the outcomes we want? And I sort of had a little bit of an aha moment and maybe this is obvious to everybody else in the room and those listening.
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But like I think part of this, part of this lack of understanding that math is about ways of thinking is the ways of measuring mathematical success are rarely about ways of thinking. They're about they're about getting answers and they're about getting the right answer. And in some realms, it's also about getting the right answer quickly. And I just think that there's such a disconnect between that and what we're talking about for sure.
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So it's something that I know all of us as teachers have to wrestle with, right, because that accountability measures are often about procedures and getting to an answer. And that's important. But the idea of building deep understanding lends itself to those assessments because it allows kids to get better at getting the right answer.
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Now, that's a good connection, Joni, because I agree with you that one of the key difficulties is that accountability and the testing measures that have been put in place and those are important, right?
We need measurements. But thinking about the structure of those measurements and what are we actually measuring? It's easy to fall into the rut of trying to prepare students for that measurement, and it's a very difficult thing to sit back and trust the process of teaching for conceptual understanding, teaching for a deep learning experience, and realizing that as a byproduct of doing that, the procedural understandings and the ability to perform on those set assessments also occurs.
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And it's a very big challenge because we get nervous. Oh, I haven't shown my students exactly how to do this particular problem, and if it's presented this particular way, here's the most efficient steps or whatever. Rather than realizing that a student who understands this math at a deep level can see a problem in a way that they've never been presented it before and determine them in a good approach for tackling that project and that problem.
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And I don't know if I can say this, so we may end up having to take it out of here. But one of the big things that I've always long felt is that we are going to have to change how we assess if we really are going to have to change that, because that's that's such a big piece of how we end up in these ruts of teaching those procedural kinds of things.
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Yeah, And what's interesting about how you just described that Kurt is a classroom teacher, has a lot of control over how they assess in their own classroom. They may not have control over the station or or district mandated assessments, but certainly they can align their assessment to more assessing for thinking and sort of test that theory that we all suggest that if you teach for deep learning procedures and answers, take care of themselves.
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I'm reflecting back and Katie, I'm going to shoot this back to you, but I'm reflecting back on my own teaching. And, you know, I often tell people like I was the kid who was a quote unquote good math student because I could memorize really easily and I could recognize patterns really easily. And I my brain worked really quickly in terms of computation.
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So I got this reputation of being good at math. And as often happens, by the time I hit, I think my wall was calc two in college as a math major, I was like, Oh my gosh, I've been faking my way for 14 years. I don't really understand what I'm doing here. I just can get answers. And I often say, like, I learned math deeply when I started teaching for sure.
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And I can distinctly remember sitting in my at my kitchen table in my apartment on a Saturday morning planning my pre-calculus class and having these aha moments of like, Oh, that connects to this. And like how I totally understood that. But I would have been at a loss for knowing how to fill in the gaps of that lack of understanding or thinking about helping my kids develop deep understanding.
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If, if all my understanding was just procedures and how to get answers. So for our listeners who may recognize that there are some math concepts that they don't know as deeply as they would like to know, to be able to support their kids, maybe they have had the experience of a student asking a question in class that they didn't know how to answer, and the sort of internal freeze and insecure moment that comes up as a teacher like, Oh my gosh, I should know the answer to this and how to respond.
00;24;53;01 - 00;25;11;20
Or I also think that's where we often veer away from student creativity because the teacher isn't sure if what the kid's doing is right and should I let them go down this path if I don't know if they're on the right way? So I just would love some of your thoughts about if I'm coming from the perspective that I can always get better and I want to continue learning.
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What are some ways I can think about doing that? How do I get more content knowledge like, do I have to go back to college and take calculus again? Like what? What might that look like?
00;25;22;11 - 00;25;56;21
Well, that's that is one route you can take know. I think there's tons of professional learning opportunities, both kind of in-person and virtual that are available these days.
So I would say don't hesitate to look around and see if you can find something that you feel like might suit your needs. Obviously, we have conferences and all kinds of things that you can attend, but I think really at the heart of it is about embracing that curiosity, right? Yes. Finding maybe some other folks who want to be curious with you.
00;25;56;23 - 00;26;15;15
You know, I mentioned earlier that my favorite things to talk about fractions and ratios and all those kinds of things. And when I was at the Dana Center, I will always remember this one day we were doing some professional learning around these ideas, right? Middle school math fractions, ratios, all these kinds of things. And we were like, How do we talk about this?
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How do we make this really accessible for people? Because I feel like fractions are definitely very things people don't want to talk about. And we literally just sat there or I'm not kidding you guys, I think I'm going crazy for hours talking about the different words and what they mean and how we can use them and how we can picture them and how they're the same and how they're different.
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And just really delving into it. We drew models of operations when we got to that place, right? Like we tried modeling just multiplication. And then how would we do multiplication of a whole number of times, a fraction? How would we do a fraction? Times of fraction? And we didn't get it all right the first time. Yeah. Yeah. And different people have different ideas, but just being in the room and talking about it together really does so much for us to share ideas.
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We look things up, we write down other ideas from resources and just really engaged in that curiosity, right? How do we do this and do it well? And so I would encourage people to form math and what can we call them? Math, math discoveries.
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I mean, it's my math discover, math geek out. I mean, it's yeah, that's exactly it. Let’s play together
00;27;33;26 - 00;27;52;17
Absolutely. And just if you want your students to be curious and engaged I feel like this is a great thing for you to engage in and model and talk about it and just really let your geek flags rise what they say, right?
00;27;53;20 - 00;28;26;09
Yeah, I think there's there's great wisdom in what you're saying right there. The bulk of math teachers, especially at the secondary level, we were, I don't know, quote unquote, math students. Right. We were we were math geeks. We had some connection to it. We enjoyed it. And I think that we we can and should lean into that process and that experience on our own and kind of model curiosity for our students thinking vertically.
00;28;26;09 - 00;28;57;11
Right. How does that idea of fraction move among the grade levels and what does it look like and how how can I model that differently? What are the things that it sets me up for as I go into the higher grades? Or how does it move backwards? Like, what are the things that set up a student for understanding the idea of fraction and thinking about placement on a number line and the idea that all of the things that we talk about and we geek out about, that's good.
00;28;57;11 - 00;29;19;21
That's what we should be doing. That's what we should be celebrating. I'll add one other thing. Just because I am a real geek, I fly the flag high all around this kind of stuff. I also think that it's worth it to go and explore mathematics that are beyond the current understanding that I have.
I am a number file geek.
00;29;19;24 - 00;29;57;09
I love that YouTube channel. There's actually multiple ones. They have number, file and number file two, and each one of the contributors has their own YouTube channels and I follow several of those as well. But I just realizing that there's enough of that out there in the world. Yeah. To go and watch, you don't have to understand all of it, but it creates curiosity about, ooh, I didn't know that there is this thing out there that you know and you can kind of follow it and you go, Oh man, I don't know if I even understand that, but there's a level of just getting curious that translates into passion and passion for what I'm teaching
00;29;57;14 - 00;30;16;25
translates into students having, at minimum, a joy coming to class. And as soon as we have a joy coming to class, we're free to kind of start learning the things that we're doing in class, even if we initially weren't interested in it. So, yes, curiosity become curious.
00;30;18;00 - 00;30;30;10
Absolutely. And I will say I want to acknowledge that. I feel like I have been very privileged to be able to be surrounded by people who are very curious and who want to delve into this with me.
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And I've been giving the time and the space to be able to do that, and not everybody has, right? So I know there's probably lots of teachers out there that are saying, I didn't get to graduate week long developing ideas, things. So like, that's what I think. Any time you have the opportunity to engage in something that's more formal, like discovering mathematical ideas, but also you can do some of that on your own to just dive in and find there's all kinds of I love the visualizations of like, I'm not sure how much more (?) you completing this work can be if you're just doing the simple manipulation.
00;31;05;28 - 00;31;14;20
But there's tons of visualizations of how that works and what it means and that are so engaging and you just have to go look for it, right? There's tons of resources out there.
00;31;14;25 - 00;31;46;07
Yeah, I love that. Liz Squares regression line. Like why is it called that? There's some cool visualizations of that in videos around that. So awesome. Well, Katie, thank you so much.
It has been such a pleasure exploring these ideas with you and I think we have such a nice balance in our conversation of high level thinking and ideas around this and down to the practical of how can I actually think about getting better at my own content knowledge. So grateful for the conversation for both of you and Katie. We look forward to talking to you again soon.
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Thank you so much. Any time you want to talk about math, I'm down.
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We could math geek out together, all three of us.
00;31;53;13 - 00;31;54;13
00;31;57;20 - 00;32;33;16
Well, that's it for this time. Be sure to check the show notes for the resources we mentioned and others you might want to explore.
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