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Joanie Funderburk

It's part of the fun of taking the time to have a lighter topic like this is just like, give us all time to reflect. Like, Oh, am I doing something without intention that might be leading to confusion?

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Curtis Brown

Welcome to Room to Grow. I'm Curtis.

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Joanie Funderburk

Brown. And I'm Joanie

 Funderburk. We work together at Texas Instruments and we're glad you're here.

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Curtis Brown

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

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Joanie Funderburk

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 share our hot sports opinions, answering the question What is the most important math topic students need to learn? Although we recognize that there are lots of right answers to this question.

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Joanie Funderburk

We had a lot of fun sharing our thoughts and reasoning. We hope you enjoy listening and be sure to share your thoughts with us to. So let's get going.

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Curtis Brown

Well, Johnny, it's really exciting to have you in the office today. We are recording in the same room, which we don't get to do very often.

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Joanie Funderburk

I know it's really nice to see you in three dimensions instead of two, and I like your bodily movements instead of just your face. So it's great to be in the room with you.

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Curtis Brown

And there's no, like, lag in my voice, in my face, facial expressions.

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Joanie Funderburk

Gotta love it...gotta love it.

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Curtis Brown

So today we've we've chosen a topic that I think is going to be a lot of fun to talk about. We've been having a little bit of a discussion before this session and actually a couple of weeks ago when we started to plan this session, we were going to ask the question, what is the most important mathematics that we teach?

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Joanie Funderburk

Yeah, I love this question partly because there's no right answer there. There are certainly a lot of a lot of topics that one could argue are of utmost importance. I think we're going to have a hard time like landing on the most important math concept, but I think it's going to be fun to kind of chat about what we think are some of the most important.

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Joanie Funderburk

So for sure, I know that we have a little bit of insight on to what each other want to talk about, but not a lot. So I would, I would love for you to get us started. So, Curtis, what do you think is the most important mathematics topic that we teach and students learn?

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Curtis Brown

So it's an interesting question. I think this is a great question to think about. And as we consider math topics, I jumped immediately. My first reaction was one that I'm not going to talk about. All right? I just want to I want to put that out there that my gut reaction was probably, I don't know, influenced a little bit by things that I've been thinking a lot about at work there.

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Curtis Brown

And so my my first reaction was this idea of data literacy. And and while I think that is incredibly important and I think it's incredibly important from a both a a mathematical idea, but also just as a as a civic idea. Right. As part of being a citizen and and a person who lives in the world. And incredibly, this world that we live in are so data driven.

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Curtis Brown

That's an incredibly important piece of the puzzle. But when I stepped back to think about what is an important math topic, right, that we talk about, data literacy didn't fit in a space that I could say this is a math topic related to all these different spaces in.

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Joanie Funderburk

Math, right?

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Curtis Brown

So the one I'm going to talk about is the idea of equals. And I could really focus in on the equals sign itself. But I really want to think more broadly about this concept of two things being equal and what are we saying when we say two things are equal, right? They have equal value. They have equal place.

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Curtis Brown

What is what does it mean when we start thinking about equal? And so what that led me to think about was even in our early grades in the in the kindergarten, first grade, we're learning to add and subtract. Sure. We're learning. We're learning these algorithms four ways to do that. We're learning. What does it mean to put things together, to take them apart, take some away.

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Joanie Funderburk

Right.

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Curtis Brown

And to make these statements that are mathematical phrases? Sure. Right, Right. And we might say things like three plus one equals and we might ask a student, we might put a question mark on the other side of that. And we say three plus one equals. We put a question mark there. In that case, the student interprets that often as an evaluation sign that equal sign is interpreted as an evaluation sign.

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Joanie Funderburk

Sure do the thing that comes before it.

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Curtis Brown

Before.

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Joanie Funderburk

It, and then. Right. What your result is after it, right? Yeah.

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Curtis Brown

But wouldn't it be okay if a student responded three plus one equals two plus two. Mm. Could we say that. Would that be an acceptable thing. And is that a concept that we begin that we can begin to kind of start talking about. Right. And think about this idea that equal all is a statement, right? It's it's a declaration about the state of two different expressions.

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Joanie Funderburk

Right.

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Curtis Brown

And this idea goes all the way through all of our mathematics. We use this idea of equals all the way through at one place. We talked about it in a podcast a while back, this idea of of equals and systems of equations. And when we when we talk about solutions to systems of equations and what does that actually mean, right?

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Curtis Brown

When we're finding the solution to a system of equations. And it's really that that set of x y coordinates or that that point that makes both of these two equations true statements. Right, Right. Because we can say things like two to the third power equals seven.

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Joanie Funderburk

Right. Say that. Well, and this is where I think we're differentiating between the idea of equivalence, right. Like you're three plus one equals sign two plus two. Right. Is a statement of equivalence.

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Curtis Brown

Exactly.

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Joanie Funderburk

But oftentimes and I remember this when I was in the classroom and trying to help kids understand sometimes that equal sign is a question mark. Right. Like does two to the third power equal seven? Just because the equal sign is there is that, you know, we could be evaluating that statement for whether it's true or false.

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Curtis Brown

True. And that's exactly it. Right. The question might be, like I said, two to the third equals seven.

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Joanie Funderburk

Is a false mathematical statement.

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Curtis Brown

Right. But it's an it's an acceptable thing to try to say, I could write it down. And that is an equation. It's just an equation. That's not true. Right. Right. And so this idea of equals, I think, really goes into so much of our other mathematics that we that we think about when we work in algebra and we and we work at solving equations and then we look at this systems of equations piece, and then we work up into, you know, even calculus or pre-calculus or look at some of the evaluation things that we do.

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Joanie Funderburk

Oh, I was just talking with my niece yesterday, who's a sophomore in high school, and she's doing trig identities. And I'm like, Oh, there you go. Trig identities. Is these just series of statements about equality, right?

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Curtis Brown

Yeah, that's totally a thing to think about, right? It's this idea. It's the equals is so, so important as it as it moves up through. Because everything we do when we're finding solutions or finding these values that make a certain thing true, we're using that idea of equality, right? To make these mathematical rules right that we do. When students add the same thing to both sides, they're using this idea of equality, right?

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Joanie Funderburk

They're using a property of equality.

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Curtis Brown

Exactly right. Exactly, exactly. So that's kind of my hot sports opinion. The most important mathematical topic or idea is this idea of equality among phrases or among. Yeah, yeah. Expressions.

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Joanie Funderburk

I'm going to put you on the spot a little bit. What do you think are some of the common ways that students might be misled or how maybe misled isn't the right word, but what are what are the some of the things that we might inadvertently do as a teacher that doesn't contribute to a student's robust understanding of equivalence?

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Curtis Brown

Well, the very first thing I can think of is that example that I started with things like three plus one equals four plus seven equals seven minus eight equals. And I write a whole worksheet or a whole set of things where every problem looks exactly the same. All of the expressions are on the left hand side, and then I have equals.

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Curtis Brown

And then. Question mark, question mark, question mark. All the way down. That equal sign is now looking like just evaluate, write, evaluate whatever is on the left and put the answer on the right, right. If we do things and if we do things like write it where the question mark is on the left hand side. Yeah. And the equals sign.

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Curtis Brown

And then I write three plus seven.

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Joanie Funderburk

Yeah.

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Curtis Brown

What equals three plus seven. Totally different idea. Right.

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Joanie Funderburk

Right.

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Curtis Brown

In that it lets me know that what I'm really looking for is something that's equivalent. Yes. Not necessarily the evaluated value of those of that numerical expression. Right.

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Joanie Funderburk

Yeah. And I would say even take that a step further and do like, you know, two plus blank equals.

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Curtis Brown

So that's the next one.

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Joanie Funderburk

Yeah. Right, Right. So it's like really building out that idea that this is just a statement of equivalence of the two expressions. And I'm thinking about, you know, the generalization up to the high school level. And you know, I taught algebra one for years and years and how kids are sort of ingrained to always get there solving, you know, equations with a variable in them there.

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Joanie Funderburk

They're like, put the variable on the left.

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Curtis Brown

X is always on the right.

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Joanie Funderburk

They always do that. And I'm like, doesn't have to be. And I can remember doing examples on the, on the whiteboard and they'd be like, Wait, what? Why wait? Your X is on the wrong side. And it's like, the whole point is to get X alone, not to get X on the left.

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Curtis Brown

So I had someone that I knew that and they shall remain nameless, who shall remain nameless. You can decide whether you thought it was me or not. Okay. All right.

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Joanie Funderburk

I'm asking for a friend, wink, wink.

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Curtis Brown

Asking for a friend here. So that drew a line when we had these equations equals signs. And I drew a line down the equals signs.

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Joanie Funderburk

You did or your friend did.

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Curtis Brown

I? My friend. Sorry, my friend did. And. And wrote T X up. Up at the top. Yeah. On one side. Yeah. And then everything else on the other side. Yeah. And then we sang the song All my, all my exes live in Texas.

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Joanie Funderburk

Oh, my God, that's hilarious. Okay, we're going to like, I'm going to have Curtis do a separate recording of this of his song. Well, those are the Lincoln notes. I mean, okay, that was pretty funny.

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Curtis Brown

So. So, I mean, I knew. I knew that I knew someone I knew someone had.

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Joanie Funderburk

Happened.

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Curtis Brown

To that may have.

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Joanie Funderburk

Friends among us.

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Curtis Brown

So I'm just saying, like, but that's a thing that really continues to go along with this idea of mis understanding or maybe only partially. Yeah. Developing the concept of equality.

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Joanie Funderburk

Sure. And doing that inadvertently, because surely as a teacher you understood that concept of equality and you and you weren't intentionally, you know, believing that, that, you know, it has to always be this way. We just fall into those habits and so on. So I think I mean, it's part of the fun of taking the time to have a lighter topic like this is just like, give us all time to reflect like, Oh my, doing something without intention that might be leading to confusion, or have I ever stopped to think about my own understanding of the equal sign and the meaning of equality?

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Curtis Brown

Yeah, And and it's not necessarily even having to do with confusion, because I think my students really got the concept of what we were trying to accomplish, right? We were trying to to gather like terms and all of these things, but what I think ended up happening there was we we didn't lead to a full understanding, a full development of this concept of equivalent expressions.

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Curtis Brown

Right. Right. And being able to to deal with what we were looking for was a value of X that actually made those statements.

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Joanie Funderburk

The original statement. Yeah, that.

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Curtis Brown

Expression and any subsequent one right to be true.

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Joanie Funderburk

Right. And working through those. And again, we touched on this on a previous podcast, like as you worked through those steps of isolating the variable to determine what the value of the variable was to make the entire thing true, you were repeatedly applying properties of equality.

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Curtis Brown

Exactly.

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Joanie Funderburk

It's it's such a huge and robust topic. I, I love that you picked that one. I think that I No, seriously, I really do, because I think there are those there's application across all grade levels and courses and it's a big deal.

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Curtis Brown

It is a big deal. Yeah. So, Johnny, I'm not going to put you on the spot and say, so do you agree with me now that you have. No, I don't.

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Joanie Funderburk

Get to disagree with you. That's an important one, but it's not the same one I picked.

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Curtis Brown

All right. So, Johnny, what is the most important mathematics topic?

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Joanie Funderburk

Well, in my hot sports opinion, I really think one of the most important concepts that students learn is the idea of proportional relationships. Yes. And I think I'm I'm certainly biased by my experience and my expertise being in the at secondary level, middle and high school for sure. And I would argue that it is maybe the most important concept of middle school.

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Joanie Funderburk

MM. I can't, I can't think of something else that tends to fall in the middle school curriculum. I mean and I'm thinking, you know, ratio goes alongside that because you can't really talk about proportional relationships without talking about ratio because a proportion is defined as a set of equivalent ratios. Sure. So it's together that concept of a ratio and then a proportional relationship.

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Joanie Funderburk

And and the reason I thought about this one, and I do think it does kind of back down into the elementary grades a little bit and push back on me if I have my math wrong here, maybe go out on a limb. But I think about the idea of multiplication and when we teach multiplication as repeated addition, we're missing this whole other side of multiplication.

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Joanie Funderburk

And that's the idea of multiplication is scaling scale. Yeah. So when you think about multiplication of a fraction, times a fraction, for instance, repeated addition makes no sense in that context. Sure, but scaling does. Yeah, right. And that and that's this sort of multiplicative reasoning. And I think back to a lot of the professional learning that I did with with teachers when I was a district math coordinator at the middle school level, was understanding that idea of scale and thinking about, okay, we have a picture frame that is for a four by six photo and we want to make it for any by ten photos.

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Joanie Funderburk

So how do we scale up? And I'm messing up the numbers, but the point is, when we scale like there is, it's our natural instinct as humans, a lot of even adults think additively. They don't think multiplicative for sure. So I feel like this is a really important concept because that ability to think about multiplication as scaling and then that idea of scaling up with multiplication or scaling down with multiplication, it is the foundation of that idea of equivalent ratios and proportional relationships.

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Curtis Brown

Yeah, I think so.

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Joanie Funderburk

And then being able to understand that and how broadly that applies across the rest of the math that you learn, you know, from the time you learn the ideas about ratio and proportional relationships all the way up to the calculus and beyond, you know, calculus.

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Curtis Brown

Is study of change. This idea of rate, the.

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Joanie Funderburk

Whole idea of rate of change. In fact, rate of change was a topic that I considered as one of the most important. And I just kind of like took it back to scale back.

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Curtis Brown

But it just is just that proportional reasoning.

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Joanie Funderburk

Exactly. Exactly.

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Curtis Brown

Expanded a little.

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Joanie Funderburk

And then as I went and tried to, like, gather some evidence for my argument about why I think this goes in some of the similar.

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Curtis Brown

Notes, by the way, we're going to put some of those now. Now, it's for.

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Joanie Funderburk

Sure it's a little ridiculous, but as you and many of our listeners know, I used to work at Student Achievement Partners, and one of the most amazing resources created by Student Achievement Partners that lives on the website Achieve the Course is the coherence map, right? For those who might not be familiar, the coherence map is this amazing interactive tool that aligns to the standards, and you can go in and search by a grade level, by a domain, you know, all the way down to the standard level.

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Joanie Funderburk

And what the coherence map does is show how these mathematical ideas are related in terms of the standards language. So it's it's a visual where there's little boxes that have the standards and the standards language written in them, and then they're arrows that are either solid line arrows or dashed line arrows that connect that standard to any other standard.

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Joanie Funderburk

So if there are arrows coming in, those are prerequisite mathematical ideas that if students have understanding of those in place, then they are more likely to be successful in writing this one. Right. And then arrows going out for which this standard is prerequisite for these other standards going out. So I was admittedly did not do a full analysis of the coherence, but when I pulled up the seventh grade standard on proportional relationships, if you're a Common Core state or familiar with the Common Core notation, it's seven RPA two is the standard, right?

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Joanie Funderburk

That one had nine arrows going out of it, meaning that standard is prerequisite for nine other standards. And some of those were middle school and some of those were extended into high school summit, seventh grade, some in eighth grade and well beyond. And on a quick glance, like I said, I did not do a thorough analysis, I admit, but on a quick glance, there aren't other standards that have that many arrows going out that that show that are so foundational for so many other concepts.

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Joanie Funderburk

So you think about in seventh grade understanding this idea of, you know, how do I think about a set of equivalent ratios? How do I generate that set? How do I analyze whether a set of of ratios is in a proportional relationship? How do I recognize it graphically? You know, all of that leads into understanding around function outside and rate of change, like you said, like there's just so much that that leads into.

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Joanie Funderburk

So that foundational understanding I think is pretty darn important.

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Curtis Brown

I think that's an exceptionally important piece of the puzzle. And I love how you kept talking about equivalent.

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Joanie Funderburk

Right.

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Curtis Brown

Ratio.

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Joanie Funderburk

You know, I was thinking that too, as I you know, I see the smug look on your face, right?

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Curtis Brown

I mean, I don't I don't.

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Joanie Funderburk

Know how to talk about Chinese topic without talking about stuff like.

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Curtis Brown

I don't I don't know how else I can mic drop without breaking our we're not breaking our listeners eardrums but that just seemed was pretty good thinking that's pretty good to me as we were chatting about this. Now this is really I agree with you. I think that's an incredibly foundational piece of the puzzle. And we start thinking about where are we going once we leave middle school and once we begin to kind of get into the high school curriculum, everything I everything, everything is so dependent upon this understanding of proportional relationships and the idea of two things varying together.

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Curtis Brown

Yes, Right. That's right. Even if it's the idea that two things vary in a non proportional way. Right, right, right. The just the establishment of the fact that two things can vary together. And we talk about that in a proportional relationship sort of status.

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Joanie Funderburk

That's it's parallel to the equal sign is not always a statement of factual equivalence. It could be a question about potential equivalence.

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Curtis Brown

Oh, you right, right, right. What does make this a truce? Exactly right. Because we're asking when is this a true statement? What values of these variables make this a true statement? So, yeah, yes. This is a this is a fun topic. A lot going on here for sure. And I'm sure that there are plenty of other hot sports opinions out there that we have not thought about.

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Joanie Funderburk

No doubt about it.

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Curtis Brown

Both of our arguments or maybe even coming up with another topic that is also extremely important. We'd love to hear from you guys at our at our email address.

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Joanie Funderburk

And special bonus points if you can come up with one that does not require you thinking about equivalence.

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Curtis Brown

So special bonus points if you haven't, if you can work.

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Joanie Funderburk

On it, help me get that.

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Curtis Brown

That will help Joni get back at me on that one room to girl math at gmail.com. And it's an email address. I'd love to hear from you guys. Joni This is a lot of fun.

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Joanie Funderburk

It was great. Kurt Always, always loved love these conversations, and now we've had more smiles and laughs in this one in a while.

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Curtis Brown

So yes, this is good. Well, thank you very much for hanging out with me. We'll see you next time.

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Joanie Funderburk

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.

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Joanie Funderburk

See you next time.