SPEAKER_01 0:00

The very nature of an assumption is oftentimes we don't know we're making it, right? So it's like how do you make yourself aware of something you're not aware of? That's a tricky question.

SPEAKER_00 0:09

Good meta question.

SPEAKER_01 0:10

Welcome to Room to Grow. I'm Curtis Brown. And I'm Joni Funderberg. We work together at Texas Instruments, and we're glad you're here.

SPEAKER_00 0:16

We're looking forward to continually improving our practice, and we understand that you are too.

SPEAKER_01 0:21

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. In this episode, Curtis and I get to geek out in a conversation about math content. We tap into elementary, middle, and high school math topics to illustrate the times that we as teachers make assumptions that might interfere with student learning. We talk about the implications of these assumptions and how we might be more intentional in our planning and instruction so that students get to deep levels of understanding of important mathematical ideas. We hope you enjoy the conversation as much as we did. So let's grow.

SPEAKER_00 1:01

Well, Joni, it's great to be in the TI offices here recording again with you. And today we've got a topic that I'm really actually very excited about, mostly because it's exposed some things in my own thinking that uh challenged me or at least caused me to have pause and think about, well, now wait a minute, what do I really say or what have I really said about that? Um and so I'm really excited to look at some of those assumptions that we bring to uh our math instruction about um student pre-knowledge, uh things that um maybe we just don't even realize that it's an assumption that we make. I know there are several of these that I was like, wait a minute, I didn't even think about that before. So I'm really excited to have this conversation today.

SPEAKER_01 1:51

Yeah, I'm excited for this one too. This is gonna be a great opportunity for us to really dive into some math, which you know, you and I love to geek out together in talking about math and our understanding of it. Um, so I'm excited for this too, and and really to think about how we make assumptions, not only about what students bring, but also the the mathematics we're teaching. And I think this ties back to, you know, it's really I know you can relate to this as a high school teacher. It's sometimes hard to teach kids a new math concept that you know and understand really well because it's hard to remember what it was like when you didn't know and understand it. So so um I'm gonna kind of dive in uh with a specific example, which is where our planning for this episode started. And that is thinking about the meaning of the equal sign. Yeah. So there's uh a really interesting article from teaching children mathematics that we'll link in the show notes. Um, and and this sort of resonated with an example that I saw and used when I used to work with teachers in my district years ago. And it was a way of checking if students understood the relational meaning of the equal sign. So it's one of those things that we assume, right? Like when you put an equal sign between two expressions, it's a it's expressing a relationship. These two expressions are equal. Or in the case of algebra where we may have variables, are these expressions equal?

SPEAKER_00 3:13

I was just gonna say, I will push back on that even just a little bit.

SPEAKER_01 3:16

It's like but it's a relationship, right?

SPEAKER_00 3:18

Right.

SPEAKER_01 3:19

But what we often see, especially for younger students in elementary school, is they view the equal sign as operational. The equal sign means find the answer to the stuff that came before it. So the article references, and I won't, I I will certainly get the numbers wrong because I don't have the article right up in front of me, but it it takes um an equation such as four plus three equals blank plus two. And the way that students might fill that blank in can give some indication to whether they're thinking simply operationally about the equal sign, where they would look at the four plus three that precedes it and go, oh, well, the number that goes in the equal sign must be seven, because four plus three equals seven. And the equal sign says, do the thing that came before it. Or if they're thinking about the relational meaning of the equal sign, they would see the plus two that comes after the blank and think what number needs to go in there so that blank plus two is equivalent to four plus three. So whether kids put a five or whether kids put a seven gives us a lot of information about how they're thinking.

SPEAKER_00 4:27

It really does. And I I it's so it's funny or even strange to have to admit sitting here thinking about that and having had you tell me that story, it still for me is hard to grasp hold of the concept that a student wouldn't see that. I I I'm trying to put my mindset of that that is not the way that everybody sees that equal sign. And and and the concept of that we're making a statement about these two expressions, four plus three and blank plus two. We're saying these two expressions are equal. Right. What is the value that fills in the blank that makes it a true statement? Becomes uh a little bit more of an algebraic question. Right. Right. It is like we're now solving or at least uh coming up with determining uh the value that makes that statement true. The initial statement is the statement four plus three equals blank plus two. Right. It's a statement. Right. We're saying these two things are equal. It could be true, it could be not true, right? Depending upon what I put in in that blank.

SPEAKER_01 5:37

Exactly. And I love how you said at the beginning of your statement right there, like I was having a hard time thinking that students wouldn't think about it this way. And I think that's exactly the point of the conversation we want to have today is how many times are we doing instruction and we're thinking about the mathematics that we're working with and that we're instructing students with, and we're making assumptions that they're thinking about it the same way we are. And um the danger comes. I mean, I I think there's it's impossible to avoid that altogether. Sure. I think we do that normally in our lives and in conversing as humans. But what we what we want to draw attention to today is the ways that making those assumptions may contribute to challenges with students really deeply understanding the mathematics that's happening later later down the road, right? Sure. Or even in in that particular grade or course. So I want to have us shift a little bit and think maybe we could think about um a secondary example of a topic where you know you and I have had lots of reflection on this. And as you said, to start us off, we both have acknowledged that we've done this a lot in our teaching. So could you bring up another example of where that assumption was made and you didn't slow down and make something explicit for students that would have contributed better to their conceptual?

SPEAKER_00 7:01

I can talk very recently, even uh, about uh an assumption that I have made for years and only in recent presenting of a session uh kind of come to realize oh man, this is a great aha moment for me. Yay, for those that I didn't even think about before. And that is in the idea of solving systems of equations. Okay. That we make this assumption when we uh begin teaching these procedures for solving these systems of equation, elimination, graphing, um, uh all the different substitution, the different methods that we kind of call out. Right. Um, there's a pre-step to that that um we sometimes use as the aha moment uh for students when they solve the system of equations. We go, oh, look at the graph, and that's where they cross, and this kind of thing, which is a cool expression. It's a cool neat thing to kind of wait and let that be the aha. But we make this assumption at the very beginning that the values of x and y make the values of each of those two uh equation sets equivalent. In other words, that the the value of x and y in the first statement makes that statement true.

unknown 8:23

Right.

SPEAKER_00 8:23

And simultaneously, which is why sometimes we talk about it as simultaneously equations, right? So simultaneously, that value of x and y makes the second one true as well. Like both of those two things happening um is an assumption that we uh often, at least I did, gloss over. And that comes, um that comes from, at least for me, jumping into we want to find the answer. For me, it was hey, I needed to teach you, I needed to show you how to solve a system of equations uh by graphing or by um by substitution or by elimination. I had this methodology that I wanted my students to understand, the methodology, the the the procedure. But I sometimes skipped over, and and then when we got to the graph or when we got to talking about it, there was an aha that oh wow, these X and Y's make both of these two statements true. But the converse of like making that assumption at the beginning, right, then kind of lets me know what it is that I'm trying to do and what I'm what I'm about to to do and why some of the things even work. Right. Um, because I was out of the classroom. I I'm full honesty here in front of a national audience or international audience or whoever's listening. Full honesty. Like I don't think I really grasp hold of that uh concept of both of these two things uh are true statements and this realization that um why I can subtract one of these equations from the other one, why that's an okay thing to do. Right. Like that concept was foreign to me while I was teaching it.

SPEAKER_01 10:19

Yeah, that's so interesting because you're we get caught up in the procedure. And I loved what you said earlier, like we almost rush to the procedures because they help us get the answers. Right. And the answers are so satisfying. Absolutely. Especially for us as teachers. Um but oftentimes that rush to the procedure takes away the meaning and the understanding for the students. So with your system of equations examples, there's so much I want to say. I've got to slow myself down here. But like I'm thinking about the textbooks that I taught out of, and they they often did, you know, solve this system of linear equations by graphing as the first method, right? That was taught. And then maybe the substitution method came next, and then maybe the elimination method came last. Um, and and I can see now reflecting back, especially in preparation for this episode, why they did that, right? The power of the graph and seeing the point of intersection. But there's almost like this prerequisite thing that I don't know I was always explicit with students about. And that is if I just look at one linear equation and draw a graph of that line, what am I doing? I'm actually drawing the representation of the solution set.

SPEAKER_00 11:36

Yes.

SPEAKER_01 11:37

Right. So then when we put two together, now we're talking about there are two solution sets to each of these individually. So now the meaning of what they have together, right? Like what's the solution to the system of these two together takes on a different level of understanding.

SPEAKER_00 11:56

I love how you worded that with solution set and relating that to what the graph of a relation is. Because I think sometimes even in our thinking about that, right? We get the pencil and paper out, we get our grid paper out, and we plot some points either off of a table or off our graphic calculator. We go to a process and we get a picture.

SPEAKER_01 12:18

Yeah.

SPEAKER_00 12:19

But the concept that I've actually really investigated the entire plane. Yeah. I I've thought about what is the entire plane, or at least some small representation of the entire plane, and I've found the values that make that statement, that relation, the equation of some sort, uh a true statement. It's all of the set of values that make that true. And then explaining that now we have a simultaneous linear equations or nonlinear uh set of equations, and we want to find the value or values for X and Y that make all of these statements that are a part of all of the solution sets. Right. I that's a whole different way of thinking about this concept. And so that is, I think that's an example, a prime example of one that at least for me was an aha moment of dang, I I never thought about it that way. Like I'll and I'll just tell you that the aha moment for me was I was um solving a system of linear equations and trying to think of all the variety of ways that students might do this. Um, and I took one equation set and subtracted it from the other, um, which is how you would do an elimination. And I took that xy equation that I got because they weren't equivalent coefficients for y or coefficients for x. So I got another equation with the difference is equal to x, whatever, whatever the two real coefficient sets were, um, and I graphed that.

SPEAKER_01 13:57

Nice the difference.

SPEAKER_00 13:59

And then I thought about well, I'm looking for when this difference is zero. Right because that means that's what that's where they're equal. And it and it only dawned on me after having seen that picture and realized the connect, like the values of X and Y were also the same value. Obviously, they were the same values. Right, right. It just was this, wait a minute. Oh yeah. Yeah. And I know I'm probably people are going, and I listen to this guy. Wait a minute, he doesn't know any math. Doug Curtis. The idea was, I mean, it was just a it was this, wait a minute. Yeah. Um I've been thinking about this for too long to not have seen that.

SPEAKER_01 14:38

I love that you shared that because I want to geek out on the math of that for a minute too. Like teaching that method, that elimination method, we sort of like, oh, your goal is to get, you know, the coefficients to be additive inverses so that you can add the equations together and cancel one of them out. Instead of saying, hey, in this, in this first equation of our system, we have two equivalent expressions, the part on the left side of the equal sign and the part on the right side. And in the second one, we have another set of equivalent expressions. And we know that one of the properties of equality is that we can add equivalent things to both sides of an equation and it stays equal, right? So we we I know I did. I'm not gonna say we, everybody does this, but this is gonna be like, why do we listen to Joni? Because I don't, I don't know that I like slowed down in my instruction to say this is why this is the process for elimination. Or when the coefficients are not additive inverses, we can multiply one or both equations, everything by a constant. That's a property. That's not a trick, that's not a procedure. That's actually math. Yes, right. So making the math take center stage over the steps of the procedure is actually helping kids better understand the concept rather than just rushing to getting the answers.

SPEAKER_00 16:04

Right. And I think that's I think you made a beautiful comment there, right? Just we're we're taking the math and putting it in the forefront, putting the spotlight on the mathematics and the understanding of the mathematics. And I I I made this comment before we started this conversation that I don't want to uh take away from the importance of procedure or say that teaching about procedures is bad. That is not the intent here. The intent is to say that while we're getting ready to teach about these procedures, we're thinking about why that's okay to do that. Right. Why why is it okay that you can multiply both sides of, I mean, we get that when we multiply both sides of an equation, we end up with an equivalent statement at the end, right? Right. But why it's okay to do that in the process of working to eliminate some uh value, some parameter is not one that I certainly didn't talk about. Um and so uh I think that's really cool that you brought that to our minds. Let's think a little bit. Um, I think we've also got another idea or another example of this um that might relate um in our elementary areas or even middle school. Uh sometimes this is a concept um that uh that can come up there. So can you talk a little bit about what we have planned ahead uh for that?

SPEAKER_01 17:35

Yeah, I mean, I think this ties to the conversation about um when, like, why do we have a standard algorithm and when do you teach the standard algorithm, right? Um, and I'm remembering back to uh a book that I studied uh in graduate school from John Vanderwall. Um, I'm blanking on the name of it, but I'm sure many of our listeners will know exactly what I'm talking about. But he suggests that, and this is maybe a little extreme, um, but he suggests that we don't ever teach algorithms, that we just focus our instruction on kids building understanding of what's happening, and then they will formulate their own algorithms.

SPEAKER_00 18:11

Wow, that is extreme.

SPEAKER_01 18:12

It is extreme. But but think about I'm almost like framing that through the idea of solving systems of equations, right? I want to go back and like experiment with my algebra one students if I help them really understand what the system meant and the whole idea around the graph is the solution sets, and can we find the X and Y value that make both true and let them come up with their own methods, like even your subtract the two, graph the difference equation, and find the intercepts. I want to go play with that now. Anyway, back to the point I'm trying to make. So working with my fifth grade nephew um in multiplication, multi-digit multiplication and going through the standard algorithm. And even just thinking about us as adults, think about your friends, like, okay, this times this, carry the one, bring this down, put a zero there. Like we mindlessly go through these steps um without really thinking about why we're putting numbers in the places that we're putting when we're carrying the one. What does that mean? It's actually not a one, right? Right. Or whatever the value is. Um, and then put a zero here. Well, why are we putting a zero? What does that zero mean? Um, and and the foundational concept here that I think I have seen in my experience becomes very problematic down the road for kids is that understanding of place value. Place value is one of those key, key concepts in the elementary grades. And when we go to that procedure without first building out the understanding of the place value of both the numbers within the multiplication problem and the sort of um, you know, interim operations that we're doing before we compute the answer, we're losing opportunity to really help reinforce what's happening with multiplication and how place value works.

SPEAKER_00 19:58

Yeah, that's a that's another brilliant one. I I talk about with my son all the time because of the language that we use. And I try to be careful with the language that I use, but ultimately I'm I end up being lazy and I say some of the things that I go, man, I really shouldn't say it like that because I don't know if if I'm saying it the right way for him to really grasp hold of what it is that he's doing. You know, if we're multiplying two digit values and I've got the the tens places and eight, uh I'll often say something like eight times instead of eighty times. Right. Um and I've really worked with him um in that algorithm to instead of uh writing ones and digits up at the top, just carry out that single multiplication and write it at the bottom. Right. And then let's add up the four values that we get instead of trying to create two values that we need to add up. That's I'm still gonna have to add. Right. So I might as well add things that I know the answers to, right? So 80 times two, I can write down my 160 down there in the in the bottom because that's what 80 times two is. Right. Instead of trying to write down 80 times two and write a one over here, which is really a 10 or a 100. And wait a minute, just I get I get lost in those kinds of things, certainly trying to do it in my head. And so the same kinds of concept for my 10-year-old who's working on these things, uh, I think that's worth it to kind of slow down and say, okay, wait, wait a minute. What are we really doing here? Um, so that he can grasp hold of that before we move to the efficient uh piece of that.

SPEAKER_01 21:32

Well, and I think that was exactly Van De Waal's point, right? Like keep doing it over. Like, I I don't know if I'm making this up or if this was actually in his book, but it was like, okay, if you if you made your son keep writing all four, you know, we're doing we're talking about a two digit number times a two digit number. And if he had to write all four products before adding them up, at some point he's gonna be like, ugh, and and he's gonna like skip steps, right? And when you skip steps and get to the most efficient, That's what we're talking about with the standard algorithm.

SPEAKER_00 22:02

That's exactly right.

SPEAKER_01 22:03

And that was Van De Waal's claim. Like if you just continue to reinforce what's happening conceptually, students will stumble into a more efficient algorithm when they're ready, when that understanding matches.

SPEAKER_00 22:16

Well, it's interesting. At the time of this recording, there's actually a really interesting discussion on the My Nctm blog about this very thing, about algorithms and should we be teaching algorithms and that kind of thing. And there's a variety of responses and lots of valid opinions and lots of different ways of viewing this. And so I would encourage folks, if you're not uh already following that, you should go check that out because it is a really interesting conversation. Um, it'll be in the archives by the time this goes out to our podcast. I'll see if we can link to it. Maybe we can link to it. Yeah. Joni, we've been talking uh about this and we've got a couple of ideas. We've put some things out there in terms of conceptually what are the things that we make assumptions about. And I know that there's more that I made that I was like, wait a minute. Uh what should I have made that assumption? Did my kids really get that? So there's plenty of ideas. Um, we brought up those two because they're pretty relatable, I think. Um, but what does this mean then? What can I do? Because I know for me, naturally, I just go in with those assumptions. Right. I need to what should I do differently? Or what can I do that I didn't do before?

SPEAKER_01 23:33

Right. I think it's so interesting because the whole, the very nature of an assumption is oftentimes we don't know we're making it, right? So it's like, how do you make yourself aware of something you're not aware of? That's a that's a tricky question.

SPEAKER_00 23:43

That's a good meta question right there.

SPEAKER_01 23:46

Um, I mean, I guess my first thought is for me, this is one of the most powerful components to um having PLCs or having collaborative teacher planning and conversation time. Because ensuring that I'm thinking about all the ways my students might think about something or uncovering, like you said, the things that wouldn't occur to me to think about is much easier to do when I'm in a group and talking with other people.

SPEAKER_00 24:13

For sure.

SPEAKER_01 24:13

Um, I mean, that's part of why this podcast is so fun, because it it it's a it's a different podcast than if it were just you talking by yourself or just me talking by myself. Right. So I think that planning together uh with other other teachers is one important part of it. And then I think the other maybe lesson learned here is to just be really intentional about developing conceptual understanding and not just rushing to procedure and answer getting. And I know again, I'll speak to my own experience. I was the most guilty of this when I was feeling the pressure of content to cover. Yeah. It felt like I have too much to do. I can't, I can't let them explore this idea today because I have to get through it. So I just have to teach them how to get the answer and move on. And and that's a very real temptation. And it's very real that especially in this day and age with the implications of COVID and and you know, the uh impact on students learning, it's it's really hard. That pressure of got to get them through, got to get them caught up, gotta at least, they have to at least see it. I think how many times I said that. But what we know both from personal experience and what research says is that just showing them how to do it is doesn't usually result in deep learning. So taking the time, especially for those big important mathematical ideas that we know are foundational for future, for future work.

SPEAKER_00 25:47

And to follow up on what you just said about taking the time for those foundational big ideas, um, I I I'll uh I'll call out one of my mentor teachers, uh Penny Smeltzer. And uh the way that we when I first kind of started teaching AP statistics and knowing that I was behind in terms of what I was talking about and what my students had seen, um, one of the best things that I began to understand that she helped me understand was this concept of, you know, you can spend time really deeply on one question or one topic. And your students knowing that one topic is so much better for them because they can learn or they can assume, they can make connections to other things if they know that one thing super deeply. And this brings up this when we were doing some of our pre-conversation about this this podcast, I was thinking about this. When I when I tended to rush to procedures, just like the substitution, elimination, and graphing methods, or uh when we're teaching about factoring uh trinomials or when we're teaching about factoring polynomials of any kind, um, this idea of looking for differences of squares and looking for when's the the leading coefficient greater than one, or whether there's a strategy I use for that, or there's a strategy I use for these things. Those can be discovered if I stick to what's the big idea of what I'm really trying to get to uh in this one concept. If I continue to just show different kinds of examples, and the big idea is I'm trying to factor this. And so, what are some things that ideas that you might have for that? And that this that kind of then lends itself to this concept of student ownership, right? Because if they discover the mathematics, even though it's existed for hundreds or maybe even thousands of years, like if they discover it for themselves, then they own it and they've really deeply uh brought it into their own minds, right?

SPEAKER_01 27:50

Yes, because they've made sense of it, right?

SPEAKER_00 27:53

Right.

SPEAKER_01 27:54

Right, yeah.

SPEAKER_00 27:55

And if I if I if I rush too quickly to the procedures, it becomes this kind of and I I pictured the one of those expanding balls, you know what I'm talking about. I don't even know what the name of them is. Um, but those ones that are kind of plastic and they link together and you pull it apart and then you can push it back together, and it's got all these little vertices on the edges of it, right? Right. But when you expand it, it becomes a smoother, like spherical, almost spherical, right? But when you push it all together, it's this whole bunch of little dots that are disconnected, right? On the surface, especially if I kind of wrap it in a sphere and I can only see those points where it's touching. And that's what it feels like. That's what's in my mind whenever I'm rushing to procedures. I've got a whole bunch of disconnected little things that uh if I can get underneath of the surface a little bit, I can see that they're all different connected procedures. Uh, but I have to slow down enough, or I have to at least eliminate my rush to procedures to be able to do that. And I have to eliminate my assumptions. Yeah because I I've made an assumption to rush to that procedure. And instead, if I can introduce the concept that makes whatever it is that I'm doing okay, or you know, that's gonna help me maybe make those connections.

SPEAKER_01 29:14

Yeah. I I love that you made that analogy of the little expanding ball because I'm thinking if I'm teaching in a way of how the ball looks when it's contracted, when it's all smooshed down, you're right. It looks like all these points are discrete things. And, you know, my husband is a person who struggled in learning math. He'll say he doesn't, he doesn't get math. And he thinks that, oh, each math problem is like one of those little points of intersection. And you have to like know, you have to be able to recognize what do I do with this point of intersection. But then when you think about that ball expanded out, what you realize is that actually all of those points connect to create this smooth, um, broader, bigger thing that you didn't see when they were all condensed. And I think that's such a great visual for thinking about teaching, like not teaching the steps and ways of getting answers in the condensed thing, but blowing up the idea and thinking, what's the big sphere? What's the big mathematical idea at work here? Um, and how do I teach to that so that students naturally see those connections rather than trying to retrofit the connections or have them have like an aha moment. Sure. Like unveiling that later on or assuming that they see it when we never show the experience. When you never got there.

SPEAKER_00 30:34

Yeah, that's uh that's exactly it. Now I I and I I would I just want to be uh sure that we say this one more time. We we certainly understand that the pressure of time and content and what are we doing here, right? And we understand that that's a difficult thing to to manage. And so I just want to make sure that we continue to reiterate that that you know sometimes the conversation uh needs to happen where we go, wait a minute, is it really like do I do I have to do all of that? Yeah, or can I uh do a few things really deeply and allow the students to make this?

SPEAKER_01 31:15

Yes, this isn't about everything you teach being those big ideas. And again, like you said, this is not this is not saying procedures aren't important, and this is not saying that getting right answers is not important, it is. Um, but but thinking bigger than that in ways that actually take students further. And I'll end just with a a quick analogy that um I used when I was working with teachers around preparing for the statewide summit of assessments that we took. And there was, you know, we'd get the released items and we'd see, you know, the things that kids missed. And I would always be like, oh, we didn't teach that, right? So next year we're gonna add that into, you know, the the area of a or the volume of a trapezoidal prism. There's the famous uh horse trough problem in the Colorado test years ago that was a trapezoidal-shaped prism, basically, and you had the kids had to find the volume of it. So we added trapezoidal prisms to our curriculum the next year, and guess what? It wasn't on the test the next year. So the analogy that I made was think about a dart board and your instruction is the darts. If your instruction is to teach the pinprick that a dart would make, then you can only teach so many pinpricks in the school year. And if kids get that exact pinprick on the assessment, they know what to do and they're good. But if their assessment question is a half an inch off where you threw the dart, they can't get it. Instead of trying to teach all the pinpricks that we anticipate, I mean, sometimes that's what curriculum feels like is just this massive list of things to teach. But if we can instead blow it up and think about, I'm gonna teach circles on the dart board, I'm gonna teach a whole section of things that would cover 50 pinpricks. And if kids can understand that big idea, then any of those 50 questions, they can get right without me having to had taught them exactly how to do that kind of a problem because they have the bigger understanding.

SPEAKER_00 33:18

I like that analogy. That's a really nice visualization.

SPEAKER_01 33:20

So teach for big circles and not for pinpricks. How's that? I love it. How's that for an ending?

SPEAKER_00 33:25

I love it. Well, that's great. Well, thank you, Joni. This was a really fun conversation. Um, looking forward to the next one.

SPEAKER_01 33:32

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. We would love to hear your feedback and your suggestions for future topics. And if you're enjoying learning with us, consider leaving a review to help others find us and share the podcast with a fellow math educator. See you next time.