Don’t Lose the Mathematics

00;00;00;00 - 00;00;18;23

Joanie

I used to joke with my students, like your friends who have the Algebra one teacher down the hall are probably learning something today called foil. And you know, they might be talking about that on the bus home tonight. And here's what that is and here's why you're not going to learn that. You know, so kind of this joke of empowering them with the math.

00;00;18;25 - 00;00;39;26

Curtis and Joanie

Welcome to Room to Grow. I'm Curtis Brown. And I'm Joanie Funderburk. We work together at Texas Instruments, and we're glad you're here. We're looking forward to continually improving our practice. And we understand that you are, too. We hope that you'll find this podcast as a room for you to grow along with us as we wrestle with and explore ideas about teaching math even better.

00;00;39;28 - 00;01;03;13

Joanie                

In today's episode, Curtis and I talk about teaching tricks, mnemonic devices and ways of getting answers. It's not uncommon to see and hear criticism for things like foil the Butterfly method. Or please excuse my dear Aunt Sally. We wonder whether it's always bad for teachers or students to use tricks and whether we can flip these ideas to empower students to create their own ways of remembering.

00;01;03;16 - 00;01;12;28

Joanie

This isn't a black and white issue, but we hope it sparks some thinking and conversation. So let's get growing. 

00;01;13;16 - 00;01;28;28

Curtis

All right, Joanie, I am so excited that you're still in the office today. And we are recording another episode of The Room to Grow podcast. This is our sixth episode of Season three, which is really exciting. And today we're going to be talking about this idea of don't lose the math.

00;01;29;00 - 00;02;05;06

Curtis

There's a lot of conversation out there around the idea of nixing the tricks. It's a book that was written, or 13 Rules that Expire, another book that was written around this idea of what's happening when we teach these students ways of remembering math things that either apply nicely to mathematics or perhaps they're just kind of made up little procedures that students memorize and they don't necessarily teach mathematics or they don't really relate to mathematics.

00;02;05;06 - 00;02;16;20

Curtis

And so what we're really talking about here today is this idea of not losing the mathematics, of really focusing in on what's our job as teachers and what's our responsibility to our students. 

00;02;17;06 - 00;02;16;20

Joanie

Right, Right. I love that you took the time to kind of spell that out because you know, we went back and forth a little bit about what to title this episode and thinking we wanted to have a conversation about tricks of teaching how to do mathematical procedures that actually can sometimes be more harmful than helpful for students and not wanting to focus on the tricks as the topic of what we're talking about, but really not losing the math.

00;02;41;21 - 00;03;08;27

Joanie

So I'm excited to unpack this a little bit. And I just want to start with like a story from my own teaching days where I did this all the time. Like as a teacher, I would teach kids ways of getting answers because I thought the concept was really challenging. I felt crunched for time, like I don't have the time to actually build out their understanding of this idea because I'm supposed to be giving my chapter test in two days or whatever.

00;03;08;29 - 00;03;25;28

Joanie

And it was really satisfying for me to be able to say, okay, hey, students do this, then this, then this and this. Here are your five steps and you'll get the right answer. And they would get it. And by the end of the class period, they felt good. I felt good. It was like, Oh, we're all being successful.

00;03;26;01 - 00;03;49;23

Joanie

A week later, after they had taken the test on that, our quiz on that, you know, whatever procedure it was, if they had never learned it because they had no conceptual understanding to back it up. So I certainly can relate. I did those things. I even created some little gimmicks and tricks for teaching different mathematical concepts to myself to differentiate me from other teachers, to make a relationship with my kids.

00;03;49;24 - 00;04;06;21

Joanie

So all of those things are very understandable about what we're getting at with this title is really our ultimate jobs. And why we all went into this is to teach kids math and we want them to to actually learn and actually know the math after we're done. 

00;04;06;24 - 00;04;09;21

Curtis

I think that's a very important thing for us to be distinguish, right?

00;04;09;22 - 00;04;31;23

Curtis & Joanie

If I did the same thing, I in fact, I think in our last episode, I may have admitted something that kind of sounded a lot like some of these ideas. All my ex’s, live in Texas. And so you can listen to the last episode if you missed that. And there wasn't there. Songbird is back and well, okay, yeah, there there was.

00;04;31;25 - 00;05;01;17

Curtis

And there is a great, a great country song about that. So anyway, so what we really wanted to kind of look at today is, is this idea of, of these tricks and, and, and really focusing in on let's not lose the math. And I think that's the reason why. Right. That's the reason why these books are written. That's the reason why folks like Phil Dero are out talking about these things, is that we really need to and we really need to concentrate on our students having a conceptual understanding.

00;05;01;17 - 00;05;20;15

Curtis & Joanie

But I'm willing to recognize that that's hard. Yeah, for sure. And that it takes a lot of time. It can be intimidating. Yeah, for a lot of things. Right? My students may not be prepared. They may have holes in their coming in. Right? And so we need to try to figure out how we can fill those holes and how do we how do we answer these questions?

00;05;20;15 - 00;05;37;26

Curtis

How do we help our students get beyond this? That's right. And and the fact of the matter is, is sometimes it just takes time. 

00;05;37;16 - 00;05;46;26

Joanie

I want to take a minute here because I think you've made that. That's a really good point. And I think, you know, a lot of our our planning conversation was focused around, is it is it a trick?

00;05;46;29 - 00;06;07;25

Joanie

How do we know if it's a trick? Is that just a remembering device? Are all remembering devices bad? Are all true, you know, quote unquote tricks bad. So let's unpack that idea a little bit. Like when we say nix the tricks or don't teach to tricks that don't support conceptual understanding, like, let's take a minute and kind of unpack what we think that means.

00;06;07;25 - 00;06;28;14

Joanie

Like, what are some of the characteristics that would say, Oh, yeah, no, that definitely falls in trick category versus something that's like, well, that could be a trick. If it were presented at a certain time during a student's developmental understanding of this idea or in a certain way. What are some of your initial thoughts about that?

00;06;28;15 - 00;06;52;20

Curtis

So I think an example probably is the best way to to do this.

And I'm like, I'm going to beg, beg, borrow and steal, please, from the introduction to the next the tricks book and this this idea of I taught them a concept, right? It's just to help them to remember. So things that may fall in that category. And in fact, the example that they give at the beginning of the book is so Katoa.

00;06;52;25 - 00;07;16;05

Curtis

Mm hmm. Right. So it's a mnemonic device, right? It's a mnemonic device. There's no underlying reason. I'm just quoting directly from the book. There's no underlying reason why the sign is the ratio of the opposite to the hypotenuse is it's a definition, right? Right. That's part of it. And this pneumonic device, this abbreviation, helps me remember those things, those definitions.

00;07;16;05 - 00;07;46;18

Curtis

Right. Those definitions that are there versus a trick that that people pick on all the time. It's on the front of the book foil. Right. Right. This this first outside, inside last, this idea that that is the way we multiplied to by no meals is that trick for ordering the those multiplications that we need to get done it loses a bit of of the flexibility of the distributive property of cross multiplication.

00;07;46;18 - 00;08;11;18

Curtis

Ah sorry of multiplication across division Ah. Addition. Well I can't even say that that's such a mouthful again after we come up with things like foil. Exactly. It really is. And maybe that is the, that's the thing, right? We're trying to help students remember those things, but we had this conversation in the beginning of our planning. We can teach those students that concept of distributive property, right?

00;08;11;21 - 00;08;41;16

Curtis

If the student comes up with a way to remember method for remembering, yes, then, then they're okay. I mean, that's the magic. That's the magic that has to happen. And so this I can't teach them foil. I can teach them. I mean, even the suggestion here, the phrase each by each is more powerful because at least then I'm remembering all this one's got to multiply by each one of those values.

00;08;41;16 - 00;09;11;06

Curtis & Joanie

And the other part of my expression. Right, Right. And this one's got to be multiplied by each of those other values in the expression. So I have each by each that at least gets me the idea of distributed of property. Yes. Yes. Expandable. Right. Right. I now could do a trinomial ten times a trinomial. This idea of each by each then is at least helping me remember all of those pieces where foil is applicable to one application.

00;09;11;08 - 00;09;32;14

Joanie

Right. Okay. So let's see if we can kind of crystallize what you were just trying to the point you were trying to make with those two examples. So so katoa as a mnemonic device helps me just to remember a definition. There's really not even a concept there. So Katoa is not about a concept, it's about definition. It's the definition, right?

00;09;32;14 - 00;10;08;28

Joanie & Curtis

Like which which relational sides in a right triangle are related to the angle. So there's no like danger in Sokoto. I use it all the time. I do too, but something like FOIL, which is also a mnemonic device, but it's not a mnemonic device for remembering a definition or a fact. That's always true. It's a mnemonic device for remembering a procedure, procedure that is a very specific, limited example of the bigger concept of the distributive property and multiplying polynomials for sure.

00;10;08;28 - 00;10;38;09

Joanie

So what makes that problematic is it's limited in scope. It's not grounded in the mathematical it there's nothing mathematical about foil. Like you said. That's a it's a randomness. Linemen man of letters that could be, you know, life. Right right so there's yeah where so katoa like I guess we could say tolka so but like we couldn't randomly rearrange those letters, right?

00;10;38;11 - 00;11;03;22

Joanie

It's a, it's attached to that definition, right. So the groundedness in the mathematics I think is really important concept. And the fact that it generalizes beyond a specific use case like foil is very specific to binomial times. Binomial was it? This was always my issue with it. I used to joke with my students, like your friends who have the algebra one teacher down the hall are probably learning something today called foil.

00;11;03;25 - 00;11;26;18

Joanie

And you know, they might be talking about that on the bus home tonight. And here's what that is and here's why you're not going to learn that. You know, so kind of this joke of empowering them with the math. So that might be one filter for teachers to think about as we reflect on our own practice and some of these things that we use with the intention of helping students, you know, am I actually doing damage with this remembering device?

00;11;26;20 - 00;11;44;19

Curtis

Yeah, And I think there's a really great even a really nice little tree diagram in the next the tricks book on the very next page from the example that I was just showcasing and talking about this idea of is it a trick and being able to kind of walk through and ask myself the question, is it a method or a definition?

00;11;44;21 - 00;12;10;26

Curtis

If it's a method and the students can explain the shortcut, then it's really not a trick, right? If it's a method and there's not really an explanation that goes with it. Instead, it's just because this trick says so. That's what I do next. And it's a trick for remember, ring those steps and there is no there's nothing with it right then it's a trick and you need to be very cautious about that.

00;12;10;27 - 00;12;31;05

Curtis

Yeah, in the same way, if it's a definition and it's a new term, you have this mnemonic that goes with it. And we're going to talk about a mnemonic here in just a second that I think we can discuss this a little bit versus based on previous terms versus that trick is based upon things that they should have known or previous things before.

00;12;31;11 - 00;13;00;04

Curtis

Again, a trick, right? So this is a nice little tree diagram to kind of think about some of the things that we're doing and introducing in our classrooms right now. I think about it for me, another piece of the puzzle and maybe it's subtly in here is the expansion or the the ability to apply, said math, pneumonic, whatever I call the thing thing to another or an expanded application of that piece.

00;13;00;04 - 00;13;25;10

Curtis

And I think of I think about the box method. And for those of you who are not familiar with the idea of the box method, but drawing a box, a square or a rectangle separating it into for say, multiplying a two digit number by a two digit number or a binomial times a binomial into four squares and associating the square one set of squares with one of the values on the side.

00;13;25;10 - 00;13;49;08

Curtis

So maybe it's, you know, 78 and I write a seven and an eight on down in a vertical column and it's 37 across the top. And I write a three in A and a seven associated with each of those four squares. And now I'm multiplying and I can see the 30, I could write that as 30 and 730 plus seven, and I can write that as 70 plus eight and write those there.

00;13;49;08 - 00;14;18;00

Curtis & Joanie

And now I'm multiplying and I can see my place values, things happen. Yeah. And I could see you even using you could even use a scale in how you draw your boxes to actually make connections to the area models or multiplication as well. The same thing could totally be done, but that's expandable. Right? Right. If I'm, if I'm multiplying a binomial times, a trinomial and I write my, you know, my your terms in each one of these columns and my linear terms in each of these columns are my constant terms in another column.

00;14;18;00 - 00;14;44;21

Curtis

And then I've got my binomial and I've got my linear term and my and my constant term here, and I can multiply each of those things and now I can see all the parts and pieces and I can now collect. Yes. My like terms, right. I can see that hey, this grouping makes squared terms or cubed terms and this grouping makes squared terms and oh my gosh, as a constant times a square ends up in the same kind of term as a linear times.

00;14;44;22 - 00;15;11;20

Joanie

A linear. Yeah. Like each of those pieces. Yeah. So what you're describing there is I'm trying to find better words to describe the different things that we're talking about. Right. So that's more an organizational tool is what it sounds like, right? It helps me keep track of the individual products and it helps me to be able to think about organize the term so I can think about proper combining of those terms.

00;15;11;22 - 00;15;47;17

Curtis

Right? That's what I'm thinking about. Yeah. When I'm thinking about these things and trying to help me as a teacher differentiate from that versus some other trick sort of thing that might not necessarily fit into that space. And so that's really what we're talking about today. That's what we're really trying to think about. And really maybe it's worth looking at another couple of those examples, thinking about a couple of other different examples of things that are out there that, depending on how they're applied, could be a trick.

00;15;47;19 - 00;15;53;19

Curtis

Yeah, or could be an actual conception. Could actually have. 

00;16;02;19 - 00;16;18;19

Joanie

One of the things that I would like to bring up is pandas. And I'll just tell a quick story here because I like to tell stories. I have a nephew. Well, is he my nephew? I guess he's my cousin, my cousin's son. When he was in the fifth grade, he was doing his math homework.

00;16;18;19 - 00;16;37;19

Joanie

And I got a call from my cousin and her husband and the fifth grade son on the phone at the same time. And parents were completely exasperated at the fifth grade homework, which was on order of operations and my cousin said to me, it's please excuse my dear Aunt Sally, like that's what he's supposed to be doing here, Right?

00;16;37;19 - 00;17;04;04

Joanie

So we did. Please excuse my dear Aunt Sally, and we got this answer. But the answer key says, the answer is this other number. And as we unpacked the problem, what I realized is they were doing when it got to the m d you know, my dear and and Sally, my cousin, the adult in the situation was interpreting this as multiplication comes before division and addition comes before subtraction.

00;17;04;07 - 00;17;33;04

Joanie

And of course, our mathematical listeners know that multiplication and division are equivalent operations in terms of the hierarchy of order of operations, and same with addition and subtraction. They're equivalent. So they should be applied in order from left to right in a mathematical expression. So that was the problem that caused tears in my cousin's home. And that's my like, I struggle with that a little bit with Pam Dos because I think of it more as more like our Sokoto example.

00;17;33;04 - 00;18;10;23

Joanie

Right? Right. It's a way of remembering what is a defined hierarchy of operations, an agreed upon way to evaluate a mathematical expression. But by trying to put a letter or a word to each operation, we lose that equivalence between multiplication and division. I'm sure our listeners have plenty of examples out there that can address that problem. Sure. But I think this is a great opportunity to say in structurally, wouldn't this be a great chance to allow students to create their own ways of remembering?

00;18;10;23 - 00;18;31;29

Joanie 

Right, Right. Our job as educators is to build their understanding, to be sure they understand the nuance, like, hey, let's hit that concept dead on of does multiplication come before division? Like, I would be sure to give my kids an example that would lead them to do that so that we could talk about it and be like, Oh, actually this is how that works.

00;18;31;29 - 00;18;52;00

Joanie

And why are multiplication and division equivalent in the hierarchy? That would be a really fun conversation to have with my students. But after we've gone through the teaching of all of that and gotten to that solid mathematics now, hey, why don't you make up a way that you're going to use to remember? How great is that and empowering them?

00;18;52;02 - 00;19;01;23

Joanie

They're going to be much more creative than me every day of the week. And it's it's helping them to own their learning. And we know that that leads to better outcomes. 

00;19;02;02 - 00;19;01;23

Curtis

That's awesome because, Joani, you're actually leading us right into what the next question I had for you, which would have been but now you've sort of answered it when, when, if ever is a trick.

00;19;13;23 - 00;19;35;06

Curtis& Joanie

Okay. And what if a student creates their own trick? Should we discourage them from using it? Should we encourage them to create their own tricks? Like how does that even work? Should we be doing that? Should that be part of what we're exploring? Yeah, and I'll turn that question back around on you. Thanks. I'll take a moment to add some nuance, too, to my thoughts.

00;19;35;06 - 00;20;05;14

Joanie

And I think, you know, one of the things there about, well, what if a student creates their own and in my in my Pam dos example, like, okay, I'm going to I'm going to unpack this mathematical concept. I'm going to be sure that students have an understanding of the nuance and the ways that confusion can arise. And then I'm going to invite them to create their own way of remembering if in a different situation, a student says, Hey, Miss Funderburk, we've been doing for thousands of these kinds of problems, and I noticed this shortcut, can I use it?

00;20;05;17 - 00;20;29;21

Joanie

You know, that might be something that happens. And and you think about my own experience and thinking about educators that I've worked with over the years. That's kind of scary, right? Yeah, because, well, first of all, do I actually understand what the kid is doing when they're explaining their shortcut to me? And second of all, I have to have a lot of faith in my own advocacy around is it mathematically sound?

00;20;29;21 - 00;20;54;10

Joanie

Is what they're doing actually based in the mathematics. And that can be a really scary place to go as a teacher. None of us wants to be wrong as the math teacher up at the front of the room. So I just want to acknowledge that there is some like, Oh, a little trepidation that can come around that particularly when a student comes up with a with an invention or a shortcut, I might not be able to evaluate.

00;20;54;10 - 00;21;14;14

Joanie

Well, so I just want to acknowledge that and say if I found myself in that situation as a classroom teacher, I would encourage the student. I would praise the student for their recognition and and then I would do everything I could do to determine if it was mathematically sound. Yeah. So that's how I would answer that. Like, should we encourage them?

00;21;14;17 - 00;21;37;29

Joanie & Curtis

Yes. Yes, mostly. But also we need to be careful and ensure that the things that students are coming up with are actually accurate and mathematically based. Yeah, and you hit it on that. You hit the nail on the head with the idea that it is incumbent upon me as a math teacher to have my feet grounded in that concept.

00;21;37;29 - 00;22;09;04

Curtis

Yeah. Such that I can recognize when things that students bring to me are applicable, are expandable, are things that apply beyond the scope of this one series of problems or set of things. And just as a side note, I need to be careful about the practice examples and the things that I set my students up with, right? That they're varied enough that patterns that my students recognize are generalizable.

00;22;09;04 - 00;22;31;25

Curtis

Generalizable for sure. Because I think that's I think that's one of the things that can happen to us if we set up way too many examples where the leading coefficient is one. Right. And we miss an opportunity there, right When the leading coefficient is not one two, we could potentially set them up for recognizing something that isn't totally expandable.

00;22;31;25 - 00;22;55;18

Curtis

Yeah, right. Yeah. When we think about those kinds of things. And so I agree. I think that there's encouragement and there's high praise for a student who does this. My son, when we were working with adding fractions of unlike denominators and we were doing some kind of area representations of this, this concept. Yeah. And, and trying to figure out how we could piece these things together.

00;22;55;18 - 00;23;21;11

Curtis

And, and we worked on a ton of examples of doing this and repeated over and over and over again this, this area piece and having him recognize these things and he came up with on his own this need for common denominators and a method for coming up with it on his own. That was a completely appropriate method for him to to do that.

00;23;21;11 - 00;23;42;09

Curtis & Joanie

Sure, he discovered it and he came up with it and now talks about it in that way. But I had to be aware of the possibility of him coming up with something like that and to recognize recognize what he was doing. Exactly. Exactly. And you also sort of hinted at something. I want to take a moment to unpack to that.

00;23;42;11 - 00;24;12;13

Joanie

Because of his repeated exposure to the concept and his own pattern recognition, he was able to come up with this method, this shortcut, this trick, whatever you want to call it, for doing that mathematical procedure. I would argue that when students have a deep understanding, they're just recalling what they know, right? And their own personalized selection of way to recall what they know.

00;24;12;15 - 00;24;33;10

Joanie

I was telling you also another story before we started recording of a friend of mine whose daughter was doing Long Division. And, you know, you know, I kind of looked over her shoulder and said, oh, what what are you working on? Oh, well, I'm doing Dad, mom, sister, brother. It's like you Wait, you're doing what? And she didn't know she was doing division.

00;24;33;10 - 00;25;02;29

Joanie

Like she couldn't even say I'm doing division, right. My argument there is like, that is a trick because, like, she's not even understanding what operation that she's doing. And for me, if you understand the process of division, you don't need a mnemonic device for the standard algorithm for a long division. You just need to understand what division means and how division works and how do I think about partial products and, you know, tracking place value and those and those kinds of things.

00;25;02;29 - 00;25;31;09

Joanie

So right when you have that strong conceptual understanding, you can build the algorithm, right? Based on your understanding. I think I've mentioned this on a podcast before. John van der Waals book says we shouldn't ever teach algorithms for operations with fractions. We should teach the concepts of what it is we're trying to do and do lots of practice with manipulatives and concrete examples and then let kids discover how does that work?

00;25;31;09 - 00;26;02;19

Joanie

When do I need it? When do I need the denominators to be the same? And when do I not need the denominators to be the same? And how do I work with these numbers to get an answer so that I'm kind of anchored in that? Like it should always be based on the understanding of the math. And if there's a way that kids gravitate towards that helps them tap into their understanding of the math, that's a great trick.

00;26;02;22 - 00;26;24;28

Curtis

I think that's I think you hit it again right on the head. I have watched a math hero of mine on YouTube quite a bit. Number five and well, no, this a number five reference string. So sorry. I know. Sorry. Made the assumption, please. That's very rude of me for interrupting. Only another one. Mother, your problem Now you're going.

00;26;24;28 - 00;26;58;02

Curtis

Okay now. So James Tanev and I watched this this episode of his of his work where he develops the idea of not trigonometry, but circle symmetry because of if you go from the way that we have learned these relationships over time and you look at the history, the development of those concepts, it's very sense making. It just makes sense.

00;26;58;04 - 00;27;26;17

Curtis

And I think one of the things that you're hitting on this multiple examples of multiplying or adding fractions or working with fractions and not teaching students these methods for those things, but rather instead having them discover the patterns on their own. If, if that's it, if they can do that and in, in the same way, this idea of circle imagery turning into trigonometry because we were interested mostly in studying circles, right?

00;27;26;17 - 00;27;57;27

Curtis

And then we recognize these things in triangles, and that's kind of the process by which we got to where we were with the circle pieces. So the same thing applies in all of these mathematics that we're doing is having students come up with and recognize those patterns and these repeated pieces over and over. If they can do that and they've created their own method, just like we talked about a minute ago, it's learning.

00;27;58;01 - 00;28;15;25

Curtis & Joanie

It's learning, but it's they're not upon me to say, okay, now, is that expandable? Is that something that we're going to write? We're going to be able to move forward with? Or have you kind of noticed a niche thing? And that applies when these conditions are true, Right. And either way, that's a great conversation. It's a great conversation.

00;28;15;29 - 00;28;35;05

Joanie

And I just want to add one more caveat on that. Like, I appreciate how you've said it's our responsibility as a teacher to be firmly grounded in the content and to be able to help navigate students through that. And at the same time, it's also perfectly realistic that we're not going to know everything there is to know about every topic we're teaching.

00;28;35;08 - 00;29;00;18

Joanie

And there's always room to grow, right? Like, I mean, there's nothing wrong with reaching out to my colleagues, to the Internet, reaching out and learning all the time more to help understand and even just the classroom having a discussion as a class around. Wow. What do you think about this? When if we've ever seen a really gifted teacher lead a number talk, oftentimes they'll do that.

00;29;00;18 - 00;29;23;07

Joanie & Curtis

They'll say, Oh, Curtis's method was da da da. And Joanie's method was not an either method, more like Curtis method, or Joanie's meant like they're giving that ownership to student thinking is just is super powerful. I think that's super powerful and it's very important, right, to set the students up for success, future future in later on for sure.

00;29;23;09 - 00;29;40;00

Joanie

So as we wrap up, Curtis, what's what's your big takeaway for our listeners? If I'm a classroom teacher and yeah, you know, as you and I both admitted, we do some we do some teaching of things to help kids get answers more quickly or help them remember what's, what's your big takeaway for our listeners. 

00;29;40;01 - 00;29;40;00

Curtis

So I think my, my big takeaway, even from the research we did and the reading and the talking, the pre talking that we had about this, my takeaway is focus in on the mathematical concept and then allow students the opportunity to develop their own method for remembering.

00;30;02;07 - 00;30;29;26

Curtis

I'll call it that. I don't want to call track their own concepts so their own method for remembering and then be prepared to be humble and have to go ask. Yeah, or do a little research. And does this make sense? Concentrate for a minute. And because I know I keep saying it's incumbent upon me as a teacher to know what it is, but it's important out don't if I don't know right away.

00;30;29;26 - 00;30;51;28

Curtis & Joanie

Yeah. Like I don't have to know right now. Yeah. And I think that's, that's my big takeaway. 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 us.

00;30;51;28 - 00;31;06;13

Joanie

Fellow Math educator. See you next time.