Room to Grow Podcast

Season 6 Episode 11

00;00;01;29 - 00;00;30;01

Joanie 

In today's episode of Room to Grow, Curtis and I bring together some important ideas from previous episodes, exploring how it happen to student curiosity and sense making to deepen mathematical learning. We wrestle with the tension many educators feel to balance conceptual understanding with procedures, especially in relation to assessments and accountability. We hope you'll find a new way of thinking about teaching and learning math with us today.

So let's get growing.

00;00;32;04 - 00;00;52;01

Curtis

Well, Joanie, I am really excited for today. Today is a great day. We are recording the Room to Grow

podcast. This is our sixth season and it is our 11th episode. We are almost through with having done six seasons of this podcast.

00;00;52;11 - 00;01;25;07

Joanie

So fun. It's been such a joy to be able to have conversations month after month about learning and teaching mathematics. And the I just think reflecting back on all the great guests we've had and the way that my thinking has expanded and how much I've learned from all of our guests and from you, Kurt. So, yeah, let's let's dig into another conversation about some of the challenges of teaching math and places where we know we all have room to grow and see if we can, you know, uncover some different ways of thinking and ideas for folks today.

00;01;25;09 - 00;02;58;16

Curtis

Yeah. So today we're actually going to be looking at, kind of this idea that the dichotomy maybe of,

just kind of the thing that teachers wrestle with this idea of teaching with a curiosity mindset. So thinking about how do we develop and, and kind of use and maintain curiosity through the process of learning mathematics? And I hate to say verses because that immediately takes us to like, you know, this is a fight, but I, you know, and the alternative, or at least a different way of thinking is kind of this more direct. Very procedural and like, let's just get in and memorize some steps. Concept of teaching.

And in the idea that, at the end of the day, you know, retention matters. And, and this idea of assessment and, and really accountability even come into play. And so just thinking about, all of those things, I think it's important we just had this conversation.

couple of episodes ago, this idea of what is math? Really and and thinking about that. And so maybe that's a place that we can we can start. It's like, what are we even talking about when we're talking about accountability and assessment of of mathematics and curiosity mindset as you walk in

learning mathematics. And we may want to establish a little bit.

What do we mean when we are talking about this idea of mathematics?

00;02;58;19 - 00;06;42;28

Joanie

Yeah. And and I would just say, like. I appreciate your nod back to our September episode where we had the author team of decluttering mathematics on. And I think this conversation is really an extension of that one. Right. Like the the conversation that we had and some of the content in that book, which I've started to read.

And let me also just add one more lens on here that I hope doesn't complicate us too far.

Maybe we won't go too far down this path. But you know, the current state of technology in our world and thinking about the role of AI and, you know, kind of what how do we think about math instruction in a different way and paralleling that back to, you know, when graphing calculators came into the space, and I think of how disruptive that was to teaching and people saying, oh, now what?

You know, the calculator can do everything I'm trying to teach kids to do. What do we do now? It's going to ruin everything. And it actually made instruction better. And maybe we're having another similar moment with AI. Like, you know, what do we care about kids knowing if we know they're going out into a world where AI is going to be readily accessible to them, and they're going to have at their fingertips the ability to solve even more kinds of math problems than graphing calculators allowed us a few decades ago. So back to the question what is what is math really? And I think it is through that lens that I'm thinking about my response to that because, I really I really believe that math is not just about procedures and getting answers.

it's a it's a system. It's about understanding relationships between these mathematical ideas. And I, you know, for, for just a, I don't know, simple way of saying it, math is a way of describing the world.

learning math for me is really about, thinking and really about making connection and

relationships and underlying structures. And, you know, one of the things just to cycle back to the book a little bit. One of my favorite quotes very early on in the decluttering math book was, what are the things that like what math is, are the things you want kids to remember even after they've forgotten the procedures? Oh, man. With me. Like, yes, that is what math is.

Because this memorization approach of like, I, you know, I know when I was a kid, I would say I'm good at math. And what that meant was I was good at recognizing pattern, and I was good at memorizing procedures, and therefore I was good at school math for a long, long time.

And I it wasn't until I was in college as a math major, and I think my course was calc two, where I hit the wall where I was like, oh my gosh, I don't really know what I'm doing. I'm just doing procedures. And it was calc two, and I finally got to a point where I didn't know what procedure to do, where it just became like confusing and mysterious. And, you know, the things that we often hear kids who struggle or even adults who struggled with math, the way they describe math, like it's just a mystery to me, I don't know, how do you know what to do with any kind of problem? So moving past that idea of like, there's you're presented with a problem, and then there's a way to do it to get the answer. That's just been what math has been, what school math has been for so long that shifting to know math is a way of thinking. Math is a way of understanding. relationships can be really challenging.

00;06;43;17 - 00;08;59;10

Curtis

Yeah, I think I think that, well, calc two is also the time frame for me as well. The first and only,

math exam that I failed. Was my first. My first two. Exam. So that was a traumatic in, in college. The hitting that and going. Oh, no, I'm an engineering major. What is going to happen now?

It messed with your identity.And so it it really did for a minute. 

It really did for a minute. There was a there was some fear there. 

So, but no, it makes a ton of sense, like this idea. And I love that statement about, math is really the, the things that, are retained after the procedures have been long since forgotten.

And when when you said that and when that was stated in the in the podcast. In the book,

I immediately thought of my current level of management of calculus.

I have a I have a pretty solid understanding of kind of the relationships of the study of rate of change. But if you ask me. To carry out the procedure of computing a definite integral of some, even relatively benign statement like, I don't know, cosine x, right? Like, I gonna struggle. I'm gonna forget. Sine x, right? Or is it opposite sign? See, I can't even I don't even have it. It's one of those two. But,

anyway, I, I just, I this idea of of. Yeah, I can understand that those two are related by,

the rate of change in each one and integration being the sum and and finding area like I.

Those things make sense to me. I still remember that. But the procedures are long, long gone. And I think that's, I think that's a really important thing for us as we go into this. Okay. So what actually sticks? Conversation and what actually is applicable and what can students kind of like, move forward with?

I think it's important that we define mathematics and, and I think, the idea of ideas that stick beyond the procedures is a good one.

00;08;59;13 - 00;12;19;02

Joanie

Okay I want to just stop here for a second because the way that you described your own like longevity of understanding of calculus concepts, you know, as we are planning for this conversation, I was really specifically thinking about middle and high school math teachers. You know, that's typically the audience that we're our conversations are focused on. And I was thinking about teachers that I've worked with in recent years who describe being in a department where they are.

They are unusual in their belief that math is about thinking and, you know, their desire to not teach procedural, procedurally focused and give kids step by step ways of solving different kinds of problems. And I just think of, you know, wanting this episode to be about empowering that teacher to, you know, be true to what their belief is about what math is, and teaching that for their students, while not just going rogue and being disrespectful of their peers.

So all of that to say, the way you just described, like, okay, I might not remember all of my procedures from calculus, but I do remember big ideas and I do remember relationships. And even to your point about what's the definite integral of sign like you may not remember on the tip of your fingers what that is, but you would know

Sure.

could apply that because you have that foundational understanding.

So all of this to say, if I were to, you know, a teacher who's feeling alone in their department around this kind of approach like that might be a great conversation to start with their colleagues, like, what actually is algebra? What actually are the big ideas of algebra? And get getting deeper into, you know, the concept of, yeah, solving equations is one thing, but we could even go deeper, like, what does it mean to solve

Yeah. Right.

Right.

Oh, man, ideas of function? It's not. Here's the laundry list of functions that students are going to encounter in an algebra two textbook.

It's what's the idea of a function? And how do we want kids to think about families of functions? Why do we group different functions together as a family of functions? Why do we talk about quadratics for a whole chapter in our in our textbooks? Right. So I love that idea of what do you remember after the procedures. And I think that's even if if I started the school year or, you know, my, my next week's PLC meeting with other algebra two teachers thinking about what are the big ideas of the unit, we're about to start teaching and really get to what are the things we want kids to remember after they get the procedures? Like if that becomes my guidepost as I'm teaching that, that's a different approach to can kids just give me the answers to these kinds of problems at the end of this unit?

00;12;19;05 - 00;12;49;20

Curtis

I really think it is. So I, I think let's kind of think a little bit about this idea of why that, why that argument happens so much. I think there's this idea that, why this kind of conversation about, you know, curiosity and conceptual kind of teaching and thinking about, these big ideas and the retention of that and this idea, this fear, even this nervousness

00;12;49;23 - 00;12;51;10

Unknown

about the procedures themselves.

00;12;51;12 - 00;13;26;12

Unknown

I mean, those procedures are important. I don't want to I don't want to come in here. And I know we've said this time and time again on the podcast, but it's true understanding and being able to carry out procedures. And I think even in our conversation leading up to the podcast this morning. You know, thinking about solving a two step equation and just the, the idea of being able to identify an appropriate value of x, let's call it x, an appropriate value of a variable or an unknown that,

00;13;26;16 - 00;13;35;22

Unknown

makes the statement true, that sentence being able to say that and think about the solution that you come to, that's the important meet.

00;13;35;25 - 00;13;36;00

Unknown

But

00;13;36;04 - 00;13;39;15

Unknown

there is some level of procedure, whether it's

00;13;39;15 - 00;13;40;15

Unknown

reasoning.

00;13;40;18 - 00;13;42;19

Unknown

Whether it's,

00;13;42;19 - 00;13;47;12

Unknown

understanding, dealing with order of operations

00;13;47;15 - 00;13;55;14

Unknown

in, in sort of a reverse manner. Right. Like there are procedures and at least one of them,

00;13;55;16 - 00;14;06;08

Unknown

you know, a student needs to be able to manage that kind of of thinking. Right. Like this I this idea of being able to, to, produce solutions is an important piece.

00;14;06;08 - 00;14;16;10

Unknown

But I think the reasoning that goes into it is maybe more what we're what we're targeting. And I don't know, we think about this idea of,

00;14;16;18 - 00;14;27;18

Unknown

maybe getting to a, hey, all my kids are going to be assessed at the end of this. And they need to know how to do this problem or they need to know.

00;14;27;21 - 00;14;36;23

Unknown

And it it made me think about significance in statistics. Can I talk to you about this idea of significance in statistics?

00;14;37;00 - 00;14;58;08

Unknown

Well, and, you know, there's a lot of conversations about research. And of course, research always brings up this idea of statistical significance. And right now our listeners out there are probably getting hives. Because any conversation I ever have, even with even in a math audience and I bring up the idea of statistics, it, it seems to,

00;14;58;13 - 00;15;04;04

Unknown

seems to cause many folks a little bit of, of trouble, consternation, a little bit.

00;15;04;06 - 00;15;17;12

Unknown

Trauma, coming out of that. Right. But this idea of significance. Right. Like, I could, come in and and talk to my students about. Well,

00;15;17;15 - 00;15;21;10

Unknown

if you get a p value when you're carrying out an inference procedure,

00;15;21;13 - 00;15;33;00

Unknown

that's less than 5%. Well, then, you have statistical significance, and we could just memorize that, but, and I'm going to call out one of my math heroes mentors,

00;15;33;03 - 00;15;34;09

Unknown

Penny Meltzer.

00;15;34;12 - 00;15;42;04

Unknown

Penny showed me something long time ago. So, Penny, thank you for this, because a great number of my students.

00;15;42;04 - 00;15;44;13

Unknown

Benefited because you showed me this.

00;15;44;16 - 00;15;53;19

Unknown

And so what I would do when we got to this idea of thinking about significance, as I would get my most honest student,

00;15;53;22 - 00;15;55;12

Unknown

I actually did this a number of ways.

00;15;55;19 - 00;16;02;28

Unknown

So I took what Penny did, and I did it slightly differently. I took my most honest student, and I got him to the front of the room,

00;16;02;28 - 00;16;18;24

Unknown

and, I gave him a I gave him a coin, and I had him drop the coin inside of this box. And he would shake so that no one could see it and shake the box, and he'd tell us whether it was heads or tails.

00;16;18;25 - 00;16;34;06

Unknown

Right. And he would do this. And, in the bottom of the box, I had always say, heads, ignore the coin, always say heads.

00;16;34;08 - 00;17;08;07

Unknown

And so we would go and he would shake the box and he would say, heads, heads, heads. And it's amazing. So the probability of one head is 50%, right? The probability of two independent shakes together. Having heads in a row is 1 in 4, right? 25%. The probability of three is 1 in 8 12.5, and the probability of four is roughly 6.25%.

00;17;08;08 - 00;17;49;12

Unknown

Right. And these kids would absolutely blow up when he would say heads a fourth time in a row. Well, what is what is 6% awfully close to. Yeah. And so there was this gut feeling of, you're cheating. Something bad is happening. It's not what we expected.

00;17;49;15 - 00;18;24;00

Unknown

So. So then we line it up and we go. Did it down the row. Right. And it is amazing to me that either at four in a row or five in a row, every 100% of my students rejected the hypothesis that that coin was fair.

So then we got to have a conversation about what is statistical significance.

That's what we're talking about with this. And this is a perfect example of what we’re talking about.

End of Segment 1

00;18;24;00 - 00;18;34;00

Music break

Start of Segment 2

00;18;34;00 - 00;24;37;00

Joanie

okay I love I love, love love love that story. I'm totally going to steal that. I will give you an penny credit but I'm going to use that. I don't doubt I'm going to use that story again, but let's see if we can circle back to non statistics content.

Sure. It is, just by its very nature. Well in so many ways we can connect statistical understanding to our experiences with the real world because that's where statistics lives. So how could we think about a big idea in algebra in the same way where we're tapping into students ability to use their gut, use their curiosity, as a driver for understanding an algebraic concept.

So I want to go back even to where you started before you went into the statistics story of thinking about a two step equation,

Yeah.

there is something to the procedure.

I would argue maybe less so with a two step equation. But certainly the bigger idea there is, it's not only find the value of the variable that makes the equation true.

Like for sure that's the foundational understanding when we're solving equations, whether we're solving, you know, two x equals six or whether we're solving two x plus three equals nine, or whether we're solving a polynomial equation or a radical equation or logarithmic equation, like ultimately that's the idea, right? Find a value or the values of the variable that make this statement mathematically true.

But in that process, and so early on, even in middle school level solving, we're applying the ability to read the notation. What does two x even mean? Right. What does two x plus three mean. And and then how do we think about identifying the more numbers and math symbols that are surrounding that variable or multiple.

Instances of the variable within that equation? There might be variables on both sides. There might be multiple variables on one side of the equation, like thinking about all these kind of iterations of solving equations that we typically go through in eighth, seventh, eighth grade algebra one, getting at like, what is all of that math? And do we understand the notation?

And then how do we think about isolating the variables so that we can find its value like we would say isolate the variable. But that turns it into a procedure. If we stay grounded in, how do we find the value of the variable that makes this true? Well then we can start talking about, okay, all the x's on this side of the equation.

And the x is on the other side of the equation are all the same value. Right. Like this is one of the meanings of a variable that it's a single value. And we have to figure out what's that single value that makes this true. So then we we talk about what does two x plus three x even mean.

Right. So we're getting at. There's so many concepts to unpack there. And so the concept of what a variable is the concept of how you know like terms work like what can we combine and not combine. And how do we make sense of when we can combine and when we can't combine. And even, inverse operations like these are big, big ideas.

And I think that taking the time to unpack and let students reason about those things without the procedure attached to it, without the value of here's how you do it. Like now how do you think about this? Like that's a different question then how do you solve this? How do you think about this idea? Right. So focusing on their understanding of those bigger ideas of they're going to naturally go to just like you described, it took 4 or 5 of these fake coin flips to bring out that natural, like, wait, that's not right.

And then you could build on that and then you

And I think the same approach for solving an equation, right. If we allow kids to just reason their way through two x equals six, let's talk about what that even means mathematically. I would argue most kids are going to know x equals three.

Like it's just going to make sense to them. And then if we add, you know, two x plus three equals nine, they're still going to be able to reason with that without having to necessarily right steps down. They're going to be able to do that in their head. And taking that building on that sense making rather than starting from here's procedure and maybe you'll understand the underlying idea.

But my whole point here, too, is when we take the time to allow that curiosity to drive and allow them to make sense of the mathematics as we're going through, then their ability to generalize their thinking to a new situation is enhanced. So I would argue if I take kids through that process of just letting them reason, then when I add, the distributive property to a solving of equations, they can tie back into what they understand,

of inverse operations. What do I know about this symbol and what it means? And then how do I think through what to do about it? Instead of us saying, here's what you do when you see parentheses, right? We see that like if we lead with the procedure, it's really common. It wasn't my in my classroom to over generalize and apply that procedure when it’s not appropriate.

00;24;37;02 - 00;28;37;09 

Curtis

To all kinds of things.

Well and I, I'm going to take something you, you just said even more this I apply. That context to a new I, I want to build back. I'm, I'm feeling like we're building backwards towards something that often tries to get justified in the, the math world, and that is, I, I think in order for students to have, let's call it the engagement, the by in the personal meaning. Behind solving two x plus three equals nine and determining the value for x, that makes that statement a true one.

I genuinely think a context for which that makes sense opens the door for them to reason about what we're talking about.

Totally agree.

I would argue placed in the right context.

A, a relatively meaningful one for the students, whether it's basketball scores in the first quarter, whether it's, you know, statistics for their favorite player, or maybe it's, I don't know, they video playing video games or something like that.

Like it, it couched in the right context where they have some emotional by in just because of what we're talking about in the real world gives them a worldview to start thinking through.

And they, they do a lot of this mathematics in my head. I'm remembering,

a conversation that we had with Vanessa,

about this idea that we're doing a lot of math every day like we do it.

And we reason through things a ton just because we have to figure out how far it is to the grocery store, or how long it's going to take us to get to the airport or whatever it is. Right? We think through all of those things, and we do little computations in our head, and we know it's going to take us ten minutes longer than if then when we get here. And if it's twice as long from us for us to get here as it is to get there. And now all of a sudden, I've built myself a two step equation.

right.

For how to do this. And so I want to back us up even one more that like these tasks. When you hear about rich tasks and you hear about tasks that provide opportunity for students to to enter in

and think and do pattern recognition. And this is why this is the reason. What we're talking about right now is the reason why all of that stuff is being talked about. And we kind of are building backwards towards that, right? Like we're talking about, well, you know, we do think that procedures are important. We do think that coming up with responses and answers is, is an important thing.

But in order to do that, students really need to be able to have this conceptual or curiosity idea about the thing. And in order to develop that, you start with a real world context. You start with something that's meaningful for the student. And so we're kind of backing up to this idea of, you know, it really is important to start with a curiosity mindset.

It really is important to allow my students a moment or maybe three moments or five moments. At the beginning of my class period, to really have this thinking about what is we're doing here and how can I reason towards that? I think it's way putting. And what made me think of that was the statement I think a lot of students would come up with two x equals six.

I think they could come up with, you know, the value that makes that statement true. Well, maybe if we were talking about, you know, every one of my buddies has two cookies.

Right? Or how many buddies do I have? If there's six cookies in total or something crazy like that, I don't know, I was making that up off the fly, but, you know, like, like they, they think to that.

00;28;37;12 - 00;31;41;18

Joanie

it's true. And I just want to add a little bit of nuance to that because I think, you know, again, thinking back to my experience in the classroom and working with teachers, and there is some content that's important for students to know that it's really hard to come up with a real world example. And I think.

I think sometimes we overemphasize the need to connect it directly. One of the things that I talked with teachers about back in those days was, okay, what I think is more important than real world when it comes to providing kids an anchor to develop their curiosity and sense making is authentic. And I think you can have something that's authentic, that isn't real world, if that makes sense.

So like, real world might be a problem that they're actually going to encounter in their lives. And maybe like if I have six cookies and each of my buddies have two, how many buddies do I have? That's not very like, okay, I'm not really going to care about that in the moment. I'm not going to be thinking mathematically that way.

But the example that I would give that's authentic but not real world is, how many birthday cakes fit into Mile High Stadium? That's our Broncos stadium here in Colorado. No one's ever going to fill Mile High Stadium with birthday cakes. If they do, I want to make sure I'm there.

Yeah.

I brought up Vanessa and it made me think about how this curiosity led instruction and how this focus on giving students the time and space to reason and make sense, how that supports their identity and their feeling about math.

Like what I love about how many birthday cakes in my life Stadium is. First of all, there's not a clear right answer. There's not even a clear approach to how to solve that. And are they all the same size.

Students could think about that. Right. And the conversation that that opens up like, oh, no, I'm thinking about a, a three tiered, nine inch round birthday cake.

And somebody else is like, oh, no, I'm thinking about the sheet cake from Costco. Like your approach to solving that's going to be different based on the assumptions that you make and the way that you reason through and the that gives us the opportunity when we have those open ended, rich tasks, that gives us the opportunity to validate all the different ways of thinking that are meaningful and powerful.

And when every students reasoning can be validated, I mean, that's success as a teacher. Now, again, not saying that the procedures don't matter and the right answers don't matter, but they don't have to be our focus every day, in every minute of every class period.

00;31;42;00 - 00;32;39;16

Curtis

Well in the even in the birthday cakes in Mile High stadium conversation. Right. There's going to be a moment where a student is going to have to justify their response. And the student is going to have to carry out a procedure. And the student is going to have to, you know, do division. And arguably solve some one step equation.

00;32;09;00 - 00;32;18;22

Unknown

Right. At some point they're going to be doing that, in that. So I think you're exactly right. This idea of by in, in a, in a situation like that where,

00;32;18;28 - 00;32;23;04

Unknown

you know, the students really have an opportunity to connect and reason,

00;32;23;07 - 00;32;25;17

Unknown

gives us a place to have,

00;32;25;17 - 00;32;28;18

Unknown

have a kind of an anchor for this conversation.

That's good. That's really awesome.

End of Segment 2

00;32;28;18 - 00;32;41;25

Music break

Start of Segment 3

00;32;40;25 - 00;34;59;11

Curtis

So one of the things that comes up as a part of this conversation often, I think is also this idea that

assessment is a real thing. And it is we know that assessment and accountability are real. We acknowledge that, and they have real impact on my tendencies as a teacher.

I know that at the end of the day there's a level of accountability and somebody is looking at, how my students perform. Somebody is is measuring success based upon that? I'm not saying that. I'm necessarily saying that, but I, I, I know that that is a real thing.

And so really, I think the conversation now maybe shifts to, well, this is all this conceptual stuff is great, but like, they're at the end of the day, most of those assessments that we look at are pretty procedurally based.

We're not asking the students how many birthday cakes are fitting, into Mile High Stadium.

On the SAT, on any standardized exam, I'm going to say that it's probably not even on my end of course exam. In in any course ever. I, I well, that makes me think of one physics course that I took in college.

Where we actually went to the, we went to the dam, on the Klamath River, and we were supposed to be computing, the electricity, the electrical output from that dam based on dimensions that we measured and things of that nature. I mean, it was a very practical. Exam, but that's, that's pretty, that's pretty unique in  experience. That's probably not the way students do that, you know, do this most of the time. And so I'm now going to drive the question. Okay. If that assessment and accountability is in fact so real. And I do feel so much pressure, how do we how do we get to this idea of conceptual instruction? I get nervous right at me as a teacher, I get nervous.

Thinking about, well, if I don't do enough of these procedures, if I don't practice tying my shoes enough times, I may not know how to tie them.

00;34;59;14 - 00;36;11;24

Joanie

Right. A great point. And I think that's I love the shoe tying example. I know you just kind of said that off the cuff, but that that's an interesting analogy to thinking through this.

If you don't understand that the purpose of what the purpose of tying your shoes is,

right. You could you could execute the procedure. But if you run into, I don't know. I don't know if I'm able to follow this, that's all right. So I might bail here. But like, the purpose of tying my shoes is to get the laces out of the way so I don't trip over them, and to tighten the shoe enough that it stays on my foot while I'm walking. Okay, if that's my premise of of why I tie my shoes, yes. I absolutely need to know the steps to go through to do that there are also ways that I can get that outcome that might not follow a traditional step. Or what if I'm given shoes with really, really short laces and I can't go through the procedure that I know?

Well, if I understand, the ultimate goal is to keep my shoes on my feet and not trip over the laces, then I can adjust my approach, right? So I'm not owning that as a great analogy. I tried to build on that but…

00;36;11;24 - 00;37;37;12

Curtis

Yeah. And. Well. And that's okay. Now, I'll take it one step. I'll get it a little closer to. The idea of friction and the idea of knots. Right. I don't have to have a real understanding for my shoe tying event. If I don't, if I don't know how friction works, that may or may not be the thing. But I do know that the idea of a knot in that scenario is important, right? The idea of reasoning through, hey, I've got to have something that keeps this tight on my foot, right? I gotta have something that keeps these things out of the way so I don't trip. I there's a lot of ways I can accomplish that.

Yeah, exactly.

And just because I didn't do, you know, whatever the thing was with the rabbit that I never learned.

I never, ever understood that story. Just because I don't have that memorized, in that way, does not mean that the two times that I loop it through. On the bottom. Because I want to make sure that I have an increased friction. And then I do my loops, and I tie them in a hole like that.

That idea of getting a not tied. I learned a procedure that works for me, but it's more aboutholding tight. It's more about getting those out of the way. It's more about the shoe staying on my foot. It's more about traction as I walk and whatever. Right.

00;37;37;12 - 00;39;58;28

Joanie

Yeah. So generalizing that to math and let's go to our solving equations example. If I, if, if I'm in a class where the teacher has focused on procedure, has told me, here's how you solve a one step equation. Here's how you solve a two step equation. Here's how you solve an equation with variables on both sides. Here's how you solve an equation with the distributive property worked in where we've just gone through. This is a problem type. And here are the steps for solving it. And let's practice. And I'm not trying to diminish the importance of practice because that's important. But that opposed to if I took the time to say what is an equation even mean? What is that equal symbol all about? What does it mean to say two x? What does it mean to say two times the quantity x plus three? How are those things different? How do I how do inverse operations work? And how do I think about finding the value of a variable when it's got a lot of mathematical symbols and operations surrounding it? The difference for that student experience going into an assessment where they may be presented with the kind of problem that's different than anything they've seen in class, which do we which population of students, which experiences are preparing them for a problem? They haven't seen I can talk from my own experience. I've done it both ways. How many times did I have kids come up and say, you never showed us how to do this? And I'm like, okay, well, if you understood the concepts that all the other things I showed you, and that was an eye opening thing for me as a teacher.

Like, math is not about let me show you every possibility you might encounter on the exam. I really believe, and because I experienced it myself, because I've seen it with other teachers that I've worked with, when we help kids understand the underlying concepts and build on their natural curiosity and their natural sense making, they're going to go into that assessment with strategies for figuring out how to solve a problem they’ve never seen before. And if I don't have the concept of all I have is the procedures, I just think it's it's much less likely to happen. And there’s actually research to back that idea.

00;39;58;28 - 00;42;11;08

Curtis

Yeah. If I don't have the understanding of what the application of distributive property, what we're actually doing there. And really kind of how that connects to the solving or developing an expression or an equation. Like then I then I'm not going to be able to apply that when there's two sets of parentheses in a problem, I'm not going to be able to kind of see that. And I, I very vividly remember. Students coming and saying, well, you never showed us how to do this. And it was, you know, the, the equation in this case, I had, you know, XS and wise on both sides of the equation or something like that.

Right. And we've, we've never seen this before. And it's like, well I mean yeah, you actually have, you know, and and thinking about, the different kinds of examples and I love that, you know, we don't have to show and we shouldn't be showing every single example of every possible problem out there. There needs to be some level of extending. Because otherwise when they leave our classroom, we've created just, robots, you know, little math. Math robots. And and they're not thinking. Can do it. Yeah. Yeah. They're not having a reason. And do those kinds of things. And so I so I think we're, I think these are, these are all,

really good kind of ideas and thinking about approaches to my classroom. So I want to kind of wrap us up or at least have a quick conversation about,

Okay. So. So now what? Like, where do I start? If I'm a podcast listener who goes into this and I've got a teach on Monday. What what are the things that I can do that are maybe small shifts that I make? You know, we're in the middle of the school year now. We're getting close to, the holidays and and kind of a break are some things that I can maybe think about that I can do next week and maybe even next semester.

00;42;11;10 - 00;45;13;01

Joanie

I think maybe the most straightforward thing that wouldn't require a lot of, you know, throwing out my current curriculum and going out and finding deep, rich problems like that can feel overwhelming and daunting for a teacher who doesn't have hasn't been teaching that way. But I think shifting, my planning, like planning from that lens of what do I want kids to remember about this concept, even after they've forgotten the procedure and starting to build some of that level of question and building, and I think connecting that to students natural curiosities and abilities to make sense.

So again, back to I'm an algebra teacher. I'm in a school where all the other teachers have 20 more years of experience than me, and they very much want to just teach the step by step things. And those are the resources we're working with, and those are the tests that we're giving as common assessments. It doesn't mean I have to give in to that and teach procedurally.

I can still use those curricular resources to focus on what's the important mathematics here? What what is underlying, solving equations. Like what is what are the big ideas there? And getting into focusing on those and getting kids talking about those alongside talking about the step by step procedures. So I think just giving yourself the space and time and maybe you're just starting with a warm up, you know, maybe instead of putting two x plus three equals nine on the board and saying, okay, your warm up is solve for X, maybe your warmup is what's a what's an example in your life that this equation might represent.

Or what do you notice and what do you wonder? I mean, it can be that simple to allow kids to start to tap into their own curiosity and their own sense making. Yeah. I think the, the belief that we have to throw everything out to, get students to make sense and have, you know, an inquiry based curriculum that supports this. Like, I think that's that's a false narrative. That's a false dichotomy.

Yeah. Yeah.

But just shifting that thinking to tapping into their natural curiosity, getting at those underlying structures. And for me, thinking about what is the structure, what's the commonality here that I want kids to be able to walk away with? How about you, Curt? What would you what would you recommend a teacher do to start to make these shifts gradually or, or wrestle with and come to some satisfying resolution on the tensions that they might feel in that kind of a situation.

00;45;13;04 - 00;47;47;10

Curtis

So I, I love that you brought up.

Planning and thinking forward. And I think the thing that that came to mind for me, was what you, And I can't remember if you said this before we started the podcast or after we started recording, but the idea of of asking the question discussing with your colleagues that same question, what do we want the students to retain after they forget how to solve these problems? 

What do we want them to retain after they forget? This idea of isolating a variable or removing, undoing order of operations. Right. How many times did I say that we're doing order of operations in reverse? And and and, you know, and and I'm not saying that that statement is necessarily a bad one.

I'm saying that, maybe without context and without thinking that that statement might be a bad one. But why is it that we're doing order of operations in reverse? Why does that work? Why why is that in a, a, a secure way of kind of, working with these problems? And, so I, I want, I want to kind of think about that idea that it's okay to have a math conversation with a colleague and to ask a, a humble question, even a vulnerable question.

About why why do we spend so much time on this? What? What is it we're trying to have our students take away from this and and come together, have a have a real conversation. And so I think my my statement is maybe less immediately practical to what I'm going to do tomorrow.

And more about how do I maybe change my thinking about my approach.

Then I can think, once I've done that, once I've had a few of those conversations, I think it it naturally will go to. Okay. So next week we are,

we have this set of curriculum questions that we have this set of notes that we're going to be working through. What are the things, what are the questions we can ask?

What are the the stop and think things that we can say, to students to help maybe move more towards this idea of reasoning and curiosity and thinking. So mine is going to be ask some vulnerable questions with a colleague about why we do what we do.

00;47;50;10 - 00;48;10;02

Joanie

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!