00;00;02;10 - 00;00;33;12

Joanie

In this month's episode of Room to Grow, Curtis and I had a great conversation with Pam Harris about her latest book, Developing Mathematical Reasoning Avoiding the Trap of Algorithms. Pam shares what she's learned about how teachers and students think about math. The ways that algorithms can get in the way of developing understanding, and her foundational belief that math is figure out of all. We enjoyed this long and rich conversation and hope you will get lots of new ideas from it too. So let's get going!

00;00;36;10 - 00;00;58;18

Curtis

Well, Joanie, I'm super excited to be recording the Room to Grow podcast again with you this year. We've been tackling some big things. We went through the Principles to Action, all of the, the teaching practices outlined in there. And now we get to kind of follow up with one podcast conversation I am so excited to have. We have a great guest with us today. 

00;00;59;20 - 00;01;30;11 

Joanie

I know I'm so excited too. And, you know, because we've gone through the Principles to Action series while we did the eight teaching practices and then kind of our summary episode, it's been a long time since we've had a guest, so I'm really excited for us to be shifting to more than just our voices in the conversation, and we could not have picked a better guest to bring us back into the realm of guests.

We are so excited for our listeners to be able to hear some great thoughts from Pam Harris today. So, Pam, welcome. We're so excited to have you.

00;01;30;11 - 00;01;34;00

Pam

Oh my gosh, you guys, I'm so excited to be on. Thank you. I'm honored. I'm honored to be here today. 

00;01;34;11 - 00;01;52;11

Joanie

It's great. So Pam, let's start. We kind of a standard question we use for guests that join us on the podcast is tell us about your personal and professional experiences that have led you to your current work and your current passions and I know you've got some great stories, so let's give a little background for folks who might not be familiar with you and your work. 

00;01;52;11 - 00;03;12;28

Pam

Yeah, I love it. Often people will assume a different origin story, so I would love to. Yeah. Share a little bit. I was, what you would call a good math student.  I played school well, I played the game, I did the thing, I wrote, memorized, and mimicked. Super good. Got really good grades. Went on to major in math at  Brigham Young University, hit linear algebra. And, that was, I began to wonder. It was like, brick wall time math teacher. Yeah. Like all of a sudden I couldn't memorize my way through the proofs, and then I hit abstract algebra and really struggled. I made it through all of that, kind of had a bit of a little crisis, but I was like, you know what? I'm never gonna have to teach that stuff. So it's okay. It's it's kind of how I. Then I had a brilliant opportunity, my cooperating teacher, I walked into her room and I swear the heavens sang and said, this is where you need to be.

So her name was Pam Giles. Pam, if you ever listen to this episode. Oh my gosh, I have so much gratitude for the experience that I had. So she was one of the original teachers who met with the king of the godfathers of graphing calculator, Frank De mana and Bert Waites, when they were just coming out with, we think that the power of visualization could help us teach math better.

00;03;13;00 - 00;03;34;25

Pam / Joanie / Pam

She was one of the originals that was in that very first group, then came back and was using their paper bound textbook to teach precalculus. She had the TI 81’s and the old potty chair thing. That was like the way you displayed that 81 on an overhead projector. I mean, I'm sounding so old right now. Oh, I'm right there with you early, early days.

00;03;34;25 - 00;06;26;10

Pam

And for the very first time, I began to play with mathematics. I could I had I had no idea what a parent function was. I had no idea what transformations were that was not part of any of my background. And I could type stuff in and see what happened, and then change and see what happened. And all of a sudden I had this action result kind of thing that I know to now does a whole lot. There was a brilliant moment for me to go, this is not only something that I can have fun with because I like teaching and I like kids, but I'm actually having fun with the mathematics. It was it was a whole new vision. Then I got hired for my first gig. But wonderful. Maybe the best teacher I've ever taught with Scott Hendrickson, who, said to me, what, what, what your ideal classroom will look like and I was like? Well, I mean, that's easy. It would have technology and it would have like, I just went on and he he wanted that spark of someone that was bringing tech, and he brought the wisdom of the ages knowing, because I would say, oh, we don't need to learn that calculators can do it.

And he's like, but well, in calculus we're going to need…

And so he brought sort of the maturity and that perspective taught high school math loving it, differently than I was taught best I could. And then I had my own kids. The best way that I can describe how that affected me was one day I was driving home and I was really fussing about a conversation I just had with my kids teachers, and I was fussing. That's a Texas thing, I think, because they had just said to me “Pam, we're getting the message.” Elementary teachers. “We're getting the message loud and clear that we don't want to give kids math anxiety. So don't give them time for math tests. Well, we don't want to get kids math anxiety. So we stopped giving them time to fact math tests. But now we're noticing kids don't know their facts anymore. Is is that important?” Oh well, as a High School math teacher, I was like, yeah, that's important, but but just like everyone listening today, we don't want to give kids math anxiety. So I'm fussin as I'm driving home, you know, like, how do I we don't want to throw the baby out with the bathwater, but what does that mean?

I drive up to my driveway, walk into my house, smell a double fudge chocolate pop tart. That means my kids home. And I was like, okay. Hey, you. You're like, in fourth or fifth grade. This is describes my kids so well. I said to myself, do you know your multiplication facts? Hey, dude. And I just picked one at random.

I said, do you do you know your fives? He goes, no. I was like, panic, math teacher panic. If my kid doesn't know his multiplication facts and maybe none of the kids know their multiplication facts. And I was ramping up and he could tell he was like, “Mom, Mom, Mom, you don't have to know your fives if you know your tens.”

00;06;26;10 - 00;07;24;16

Pam

I said “What?”

He said “You don't have to know your fives if you know your tens.”

I said, “Dude, I heard what you said. I don't know what you mean. Like, give me an example.” 

So this is me, a very successful high school math teacher. At this point, I'm already doing Professional Development workshops for TI, teaching people how to use graphic calculators.

I'm like, “What do you mean you don't have know your fives if you know your tens?” 

And he said, “Well, okay, an example. Let's say you don't know five times nine. Well ten times nine is 90 and five is half of ten. So half of 90 is 45.” 

And I was like, “Good heavens, that is nine times five. Does that work every time?”

Like I this first time I ever thought of using a relationship. That right. I said, give me another example. And he goes, oh, okay. How about five times 23? Like, like I know what the example I'm thinking of, like, right then I'm fussing with single digit facts and he's playing around with a pattern.

00;07;24;16 - 00;08;00;26

Pam / Curtis & Joanie / Pam

So do it with me. If I was so ten times 23 is 230, but five is half of ten, so half of 230 is one of 115. Yeah. Is is that five times 23? Like people probably want to check real quick like I did. Yeah. Yeah. So in that moment I was super aware that the mental actions that my kid was using to logic his way through problems were not the mental actions that I had done as a student, and still too often I was doing as a high school math teacher, that my high school students would kind of wait me out.

00;08;00;26 - 00;08;41;12

Pam

We would do some exploration and kind of understanding stuff, but they knew I would. Eventually, I was going to give them a rule and some steps to mimic, and then they were going to go off and, and get answers. And that was sort of the goal of class. So in that moment, I wondered, could we create intentionally the kind of mental actions that my kid was using? Could we create those in my high school students? But Joanie and Curtis, let me be honest. Honest right here. Now, I could admit I actually wondered, could I create what he was doing in me? Yeah, I wanted that kind of mental action happening in my brain. And so I dove into research. I read everything I could get my hands on.

00;08;41;12 - 00;09;00;21

Pam / Joanie and Curtis / Pam

I read journal articles and research studies, and at the same time, I dove into my kid's classrooms. I literally went to my kid's elementary teachers and I said, May I enrich your students? Which was totally code for can I experiment? Can I sit in on your classroom and do well? I experimented like I went in, I experiment with them.

00;09;00;24 - 00;10;19;05

Pam

I tried all the things that I was reading and brilliantly built my numeracy well, I built theirs. And so this kind of grew up a few years later, the state of Texas. I was doing some graduate work and they had me do they turn my attention back? So first I wrote Building Powerful Numeracy for Middle and High School students, because I took all of the work that I'd been doing, building numeracy and young kids.

And in my head and myself and my psyche and I, and I said, we can use this to help our middle and high school students be more successful in the math that we teach. If we know how to build numeracy into the content that we teach at the secondary level, then with some work with the state of Texas and some other things, I turn my attention back to secondary math. And the work that I've been doing lately is really K-12. What does it look like K-12 to have kids doing the mental actions that I now know mathy people actually do, not what I had done as a student, which was really rote, memorize and mimic. And so I'm on a mission to like, spread this world, this work, this work to the world that, that it's not what we many of us are. Maybe most of us have most as I have a difference. Yeah. And so, I'm. Yeah, I'm I'm on a I'm on a mission. Let's let's, let's bring it to everybody. 

00;10;19;05 - 00;10;35;00

Curtis

I'm so excited to have you on mission and on our podcast talking about this. This is, I'm just listening to you getting all excited about all the wonderful things I can go now and talk to my boys about, right here and curse on them.

00;10;35;00 - 00;10;50;09

Pam

Lead with this lead with, “Did you know math is actually figure-out-able?” Yes, that's my tagline. Now that math is not rote memorization, math is figure out. And the good news is, we can teach it that way. So yeah, love it.

00;10;50;09 - 00;11;57;02

Curtis

That's, that's absolutely so true. That's so good. That's so good. Thank you so much for that story.

You know as we we're talking now. We're talking now. The you've just released your, your next book, which is really exciting. Just not very not very long ago, you just you just released it. And so we're here to kind of chatting a little bit about that. And in the preface of your book, which is developed, developing mathematical reasoning and really, it's the, the, the avoiding the traps of algorithms.

Right. And I love that, that, that tagline, they're avoiding the trap, of algorithms. And we'll maybe, get at what that, tagline is about here in just a minute. But in the preface of the book, you really you talk a little bit about, you call them distortions. There's three distortions that we as math teachers can kind of kind of bring to the math classroom and affect the way that we teach. And I'd, I'd like to hear a little bit more, about that and kind of maybe explain what you mean, because maybe those distort distortion sounds kind of negative and scary. And so I want to know, maybe a little bit about, about that. 

00;11;57;05 - 00;17;28;17

Pam

Yeah. I love it. Thanks. So, yes, the book came out in February of 25.Hit hit do best release and bestseller when it came out. So in our small niche of math education, that was pretty cool. Excited. My son actually helped me write it, and it's the son I tell the story about because it was literally him that set me on this journey. And he's a writer. And so brilliantly he, he, it's kind of fun because he adds, just one more, one more. I can say it. One more, uptick of snark. And so, like, I'm pretty snarky and he's just one more level of snark. And so it kind of makes that, I don't know, book a little bit more, fun and readable and kind of has this perspective where, you know, he, he has never had one of these three distortions.

And one of the things that he helped me think about is, I actually did a, an ignite, talk that's out on the internet where I called it three perspectives. And he pushed back on that because he said, if you call it a perspective, it kind of make you sounds like you're making it, valid. And what we're actually suggesting is, is that these three distortions were distorted ways of viewing math as a student, right then, can impact the way you teach math as a teacher. So, for example, I kind of just told you about my distortion. My distortion was I played school well when the teacher said, this is what it means to multiply. You line the numbers up and you do all the single digit multiplication, then you do all the single digit addition. And if there were decimals involved and you butt cheek the decimal at the end. And I'll just start out with the yeah, I mean, well we can see what you do remember, but you guys can see it shaking. Okay. There we go. You know, we got that butt-cheeking from a brilliant seventh grade student, who? Just butt-cheeked every problem twice. But she would only there's only there's only two butt-cheeks and.

Well, we said but but why do you ever do it twice? She's like, what? Because there's, Yeah. So, you know, tricks and, these rules that expire, things like that, they they, they seem helpful in the moment, but they can get overused and and and really the I think the biggest damages they create, kids like me who had the distortion that to do math means to wait until someone shows you how to do it.

You memorize that and then you mimic those steps and and you are doing math because you're getting answers using somebody else's procedure. Yeah. And I'm just inviting the world to consider that's a distorted view of what mathematics of doing mathematics actually is. And, and and I was seeing through that distorted, distorted lens. I am now actually using the mental actions that mathematicians actually use. So I think there's a couple of other distortions. One could be you could have been more like my kid. For whatever reason, he went through life running into the same low dose of mathematical patterns that all of us do. And for whatever reason, he kind of had a natural inclination or proclivity or interest in grabbing those together and creating patterns in relationships. And that created those relationships and connections in his head that then when he ran into other lotus patterns, he could pull those together. And, and Mom, Mom, Mom, you don't have to know your fives if you know your tens, because he was sort of doing that. And he was he was logic his way through problems. And as he did that, his brain grew, it got stronger, and he had more connections that he could build on and pull from. And, and all of that, could. So let's say that you're a kid like that, and your teacher is showing you a bunch of steps to do, and you are using relationships, and you might then think that it was the teacher showing you steps to mimic that helped you create those relationships. So become a teacher. Now you're a teacher. You liked math. You want to help other kids actually do the mental actions you were doing. And so you do what your teacher did. You show them the steps, and you expect that they will naturally be able to pick up on all that stuff you actually did on your own, not because of the teacher showing you a bunch of steps.

You you had this for whatever reason. And so now you say to your kids, okay, ready everybody here. And then you go, oh, crumb, there's a link to it. Or they're actually getting what I'm doing and they're oh, okay. Sucks to be the rest of you. You're just going to have to mimic these steps because, that's why my teacher did it. Because that's how. Okay, the rest of you, you just, like, do this, mimic this stuff. I don't know why you're not getting it. Bummer. Wish you did. Because it's really cool to be where I am. Yeah, but. But the only way to teach is. Is the way that we've kind of been shown that we've experienced. So it impacts the way you teach because it's all you've got. Like, even those are. We all have good hearts. If we're teachers, we're trying our best. Right? Nobody goes into it. Today I'm going to mess up kids. Everybody goes in saying, you know, I'm going to do the best I can. But if you thought that the reason, the way, the method that you created all those mental relationships and used them, if you thought it was because someone showed you steps of an algorithm to mimic, I respectfully invite you to consider not so. Actually, you were kind of doing that on your own. Think how far and how fast you could have gone deeper. You could have gone if teachers actually helped you do the things that you were kind of doing on your own. The good news is we now know how to do that. We now know how to actively help everybody do what a small segment of people were doing naturally on their own.

00;17;28;19 - 00;17;38;20

Music break

End of Segment 1

Start of Segment 2

00;17;38;20 - 00;41;56;19

Pam / Curtis / Joanie

Are you ready for a third or do you want a comment? I'm ready to know, like I can. I mean, I have comments about like, oh, go, yeah, I was I was okay comments too. But I'm going to say I so identify with your number one, frame, kind of like telling me. Yeah. All right. All three of us number.

00;17;55;28 - 00;18;21;25

Curtis / Pam

Yes. My number two kid is definitely number two. In the idea, he sees me. Interesting. But I don't know that he necessarily, like. I don't know that anybody besides maybe me myself is out there, like, helping him to see those and seeing why he thinks those and clarify them and put words to them. Give talking about your brand and how get language around it.

00;18;21;25 - 00;21;40;09

Pam

Just helps your brain solidify the relationships. And so a lot of people that are kind of like my kid was only go so far because they can only do that on their own. So. Right. And then, so Kim, the co-host on my podcast (in-audible) podcast was a lot like my son, and she'll admit she got to precalculus.

Boys were around. She's like, I just had other things to do. It was too hard to do on my own. I just started mimicking the teacher. Yeah, like she made it that far. And a lot of teachers will, kind of, So I'm not I'm not suggesting any of these three distortions are cut and dry. You're in a silo. You're stuck forever kind of head. Yeah. It's just a helpful. I'm offering it as a helpful way for us to maybe reflect. Is it possible that the way I saw math as a kid is influencing the way that I'm now teaching? And if we can honestly reflect, that gives us a chance to now choose maybe a different path, maybe, oh, that's why some people are suggesting these other ways of thinking about teaching. Yeah. So the the third distortion, the I would say that the first one came from my son, the, the, the first one was me, the second one was my son, the third one was my daughter. So it's good thing that God gave me her last because I needed all that. She just was sort of different from me. I had three boys and then the girl. And so for the, for the moment she popped out. She was a throw me off balance a little bit. And one of the ways that that was true is that when the teacher would put up that ruler, that procedure to mimic and memorize, she would say, yeah, how does that work? And the teacher would say, yeah, you do this step first. So she's like, no, no, no, I don't I don't want you to tell me how to do it. I want you to tell me, like the reason I understand what's going on. I'm not going to be able to memorize your stuff. I don't memorize well, whatever. But if I understand it, like, I'll do it like, like, get it. And she would come home so frustrated, she walked.

I remember the day she walked in, she's like, oh my gosh, “Ours is not to reason why. Just invert and multiply. Mom is division fractions figure-outable?”

I was like “Yes, yes, yes it is.” 

She's like “My teachers. They think they just told me, just do it. Think they won't tell me why.”

I'm like, sweetheart, they don't know why. Like because they come home. She just was. It was funny because she just, like, refused to quietly sit through class and then come home and figure it out. She would, like push up against those teachers in front of everybody. I, I have a great text from her in algebra two. I get a text from her that says, Mom logarithms are stupid. 

And I was like, sorry, sorry what? She goes, why do you subtract when you divide? Add when you multiply. Logs are dumb. And I was like, oh, okay. Right. She's like and then she sends me a picture of the board with the rules. And she said logarithms. You know, she said, I asked like, tell me why. And they said, just do it. She's like, that's not math. And I was like, I come home, we'll talk all things math when you get home. And funny, I actually show this sometimes in presentations. The the comment underneath it was, when you have passed your driving test at the DMV because we were literally going to take her driving test that day. We'll talk all things logs. Will we get there? She pass pass her test? I'll just say first time when the boys took more than that. Yeah. So she was a little proud of that. And then all I had to say to her was, hey, you know, those exponent relationships that we worked on last week, think about some of the and she literally I said almost that much.

00;21;40;09 - 00;22;05;09

Pam / Joanie / Curtis

And she goes, oh, because like when you have like bases that you add the ex. Oh that. And so that. Yeah. Okay. Log logs are cool. Like we've built enough exponent relationships. So at that point she could sort of like she can, she can math.you know, with the best of them. the best connection between exponents and logarithms. Yeah absolutely. She can math. She can math with the best of them when she when she understands it.

00;22;05;09 - 00;23;54;00

Pam

Sometimes I think people will hear me talk about this third distortion as none of these are negative. These are just always that we were sort of growing up right, right. That, for her, they will hear and maybe here's the distortion that they'll hear, oh she couldn't do it. So she needs these alternative whatever. No, no, no she could math with the best of them. She just refused to memorize your stuff. It's like we had her on. I do a challenge three times a year. We invite your listeners if they want to join our free challenge, where I just teach math for a few nights. It's super great. And she was my special guest when one people were like, we want to meet your kids. I was like, really? Okay, so my then in college, daughter got on and she said, I wonder. I had no idea she was going to say that. So it was a little bit like, but it worked out well. She said, you teachers, when you show kids to rote memorizing mimic steps, just consider to me that's like you're showing me an inkblot. She said, I can memorize your inkblot, but then tomorrow you change it just ever so. And then you want me to memorize a new inkblot? She's like, yo, just teach me the thing. And then I don't have to memorize any inkblots, I can just logic my way through the problems. Yeah, super confident she was that she was everybody's tutor in her, university, accounting classes. She's now a landscape architect. Go, Abby go. Math did not keep her from doing what she wanted to do in her life. She could absolutely math when she understood what was going on. She just refused to memorize. It was just dumb. Wonder how many of our kids out there are like that where they're like, this is dumb. You just want me to do these steps? You're telling me not to reason right now? I'm never going to use this. This is nonsensical. I, I opt out. What if that's not mathing? So it's a distortion to think that math is an ink plot we actually could all understand. There you go. 

00;23;54;20 - 00;26;55;00

Joanie

I just want to give the flip side of that distortion three, because that's my husband as well.

Except he wasn't the out there, bold personality that you describe your daughter. He internalized it and said, well, I must be dumb. I guess I'm not a person I am I'm not a math person because when the teacher, the teacher does the example and then tells us to go do a different example and I have to raise my hand right away or go up to the desk, and the teacher's response to me would be, oh, it's just like the last one. And he's like, but it's not just like the last one. And then at work or the teacher would say, he would go up and say, I don't understand this. And she'd say, oh, that's easy. And then he'd go, oh, I can't even get the easy ones, right? So the message that he was getting was, he was the problem.

Not that the approach to the learning was the problem. And I found the same thing. You know, we've been married for 35 years. He's married to a math person who, you know, very much loved and appreciate, who doesn't what you ask for math a very a very similar, a very similar journey to you of starting out as a memorize or in pattern recognizer and then discovering and learning math as an adult and just falling in love with it. And now that I have that perspective and he and I can have math light conversations, he has a beautiful mind for mathematics. He just spent so many years having it shut down and having him told he was wrong for the way he was thinking. So yeah, how many kids out there are? It's dumb for you to make me follow your steps. And how many more are not saying anything because we've just completely shut them off from thinking math is ever something that's meaningful or enjoyable in their lives. So I so appreciate these distortions. And and I really do appreciate the frame that you put around them that, you know, there's nothing right or wrong about any of these, but understanding and recognizing the ways that our experiences as learners, as children, even into our adulthood, those things impact how we interact with the students in our classroom. And when we can have conscious awareness of those and direct those in ways that we know, help kids get to the outcomes that we want for them. You know, we all want kids to be successful. And, I know lots of times and I've had lots of math colleagues, math teacher colleagues who are frustrated when they can't reach a kid. So I think this is a great frame to start to think about, you know, what kind of what kind of breaks down in that. So awesome. I think that was a great place to start. I want to shift us to this. The subtitle of your book, Avoiding the Trap of Algorithms. Nice. Very powerful. So can you elaborate on two of the words in there? One, I'd love for you to talk a little bit more about what what even is an algorithm, because I think that's a word that gets tossed around and everybody assumes we're all talking about the same thing, and maybe we're not. So what do you mean by an algorithm? And then the other word that jumps out as trap, why do you why what is your proposition around algorithms being traps. So tell us more about that.

00;26;55;15 - 00;32;08;12

Pam

Yeah. So I'm going to invite everyone to consider that we have gotten a little muddy in math education of using terms. Now, terms are social knowledge. We kind of it's by convention we agree what things mean over time. But we have in in math education specifically, we have used the word algorithm, especially lately, different than most people use algorithms. So if you go look up the definition of algorithm, if you ask a scientist if you as a statistician, if you ask, an engineer, they're going to have a very specific definition that an algorithm is a series of steps that you must follow all of them to solve a problem in a certain class of problems, a certain type of problems. We've gotten a little muddy in math education lately where some people will say student generated algorithms. I'm just going to suggest that's really rare that students actually generate an algorithm. Students will often generate strategies which work for particular numbers, particular structures, you know, like why bring a bulldozer if you can just use a spoon? And, and and those strategies, while they are kind of generalizable, they're not they're not a general procedure that you could put into a computer, that a computer could follow all the steps every time in fact, a computer must follow all the steps every time. A computer isn't judicious, it doesn't look and let the numbers or the structure influence how they solve the problem. They just computer follows the steps. So algorithms are amazing human achievements, but they're terrible teaching tools. Yeah, we need algorithms for computers now. In the past we have needed algorithms for bookkeepers and shop owners and people that needed to to keep the finances. We have needed algorithms because we didn't have technology. We didn't have the ability to do a lot of sums with a lot of numbers and a lot of digits. And so really, to be efficient, we needed ready. We needed a way for people who didn't actually even know what was going on, to be able to mindlessly follow a bunch of steps and get the correct responses. I offered to do that when those shopkeepers were doing that kind of math. They weren't mapping, they were calculating arithmetic, following steps, mimicking steps. And if they did that, well, they were getting correct answers. Did we need that? Absolutely. We needed that until we had technology. I give you a similar comparison. I know I'm talking to math people, but go with me a little bit. If you think about Gutenberg's printing press. So the world did not have the written word until Gutenberg came along and he had his printing press, and all of a sudden the common man had the written word, and they could they were sort of set free. And now all of a sudden, the knowledge wasn't held by just those that were able to sort of be wealthy and go to the universities and were all that stuff. The common man had the written word. Similarly, we had a, a shift in our in mathematics, where around 700 A.D., a gentleman I'm going to massacre his name, but it was something like Al-Khwarizmi came up with algorithms. So we get algorithms from his name and we get algebra from his name, both because he invented, at least a form of most of our traditional arithmetic algorithms.

And in a huge way, those arithmetic algorithms set the common man free. All of a sudden, you didn't have to be the rich person who was sent to the university to learn how to man a university, to school, to learn how to run an abacus. So prior to these algorithms, you had to use an abacus in order to calculate.

After that, the common man could use those algorithms, and all of a sudden I could keep my own books in my shop. And it's it's part of the Renaissance. Like all of a sudden now the common man was set free to do more math. I'm inviting everybody to consider. We now have technology. We have a similar revolution. We have a similar place where we don't need anyone anymore to do columns of addition or any kind of arithmetic at all. What we need are people who are actually doing the mental actions of mathing the what, what are the the, the way to logic your way through problems and in the process building more and more math connections. It's not just fuzzy thinking better. Sometimes people will look at the title of the book Developing Mathematical Reasoning, and they'll say, oh, you're just going to make us better thinkers. I'm air quoting here like some fuzzy, you think better? No, no, no, no, we're going to I'm suggesting we need to develop mathematical reasoning, which means content like I'm owning. I'm still all the standards that are out there. We're still meeting those standards, but we're doing it in such a way, the way that mathy people were doing it. Not the way rote memorize or mimic or robots.

00;31;43;13 - 00;32;00;21

Pam / Joanie / Curtis

We're doing it now, remember, if you're like, I can't believe you just call people rote, memorize or mimic a robot. That's me. That was me. I was totally, you know, like, so it's not I'm not. I'm not being ugly. Guilty as charged. Yeah, I'm being real. Like we were fed a line from good teachers who meant well because that's what they were fed. Yeah. Now we know different. Now we know different. We give everybody an opportunity to go, hey, you don't have to know your fives. If you know your tens and a whole lot more.

00;32;08;12 - 00;32;08;12

Joanie

I want to just pause you because you said something that just really, really struck me. And, you know, in some places I'm going to put on my Pam snark over here in some places, well-meaning policymakers criticize math standards.And, you know, one of the one of the criticisms that I heard a lot, I was working for a nonprofit in the days of Common Core being released. And one of the criticisms I heard a lot is like, why do we need to teach kids 17 different ways to multiply two, two digit numbers? And, you know, really push back on that and, and even I even saw some unfortunate generalizations of that idea in the classroom where it's like, who can do it a different way, who can do it a different way? And it's like the goal here is not to enumerate every possible way to get to an answer here. The goal is and I and I wrote down the language, you said I might have a little off, so fix me if I did.  But it's to logic your way through in order to understand the math better, right? The the outcome is not the the goal is not get the answer.

73 is not the goal. No at all. At all. It's the thinking that you do and the deeper understanding of what's happening mathematically. That's important about having access to these different strategies and different ways of thinking that are often cut off from us when all we do is focus on algorithm. 

00;33;33;20 - 00;34;06;17

Pam

Yeah, it's literally building kids brains. It's it's building all of our brains to be able to think more sophisticatedly and I use that word cautiously. Like when I say sophisticated, I don't ever refer to a person like, as in posh, like, oh, you're sorry. Not that. It's it's it's a level of complexity and being able to think about more, in, more levels at a time, more things simultaneously, more complex reasoning. And sometimes people will say, why don't you just use the word complex?

00;34;06;19 - 00;34;40;03

Pam / Joanie / Pam

Because complex kind of has a connotation of just complex for like like, yeah. What did you say, Joanie? I said harder. Yeah. It's like it's like harder. It's just, more, more decimal places. More. Yeah. It's not like it's not really about good reason. Yeah. Yes, exactly. I want, I want higher level reasoning. I want more deep connections. I want more understanding so that it all is making sense because I'm mapping like I'm actually doing the mental actions that a mathematician does when they math. I'm not rote memorizing mimicking. Yeah. 

00;34;40;03 - 00;35;14;17

Curtis

Okay. That's a more robust, that's a more robust position to be in. Right. If, if, if now that I, now that we've done this and I can logic my way, the way the mathematicians are doing the things that they're doing right, if I'm now in the position to, to handle that kind of thinking and I'm there, I now have a more robust ability to manage these all these other different things. It's applicable here. It's applicable there. I can take different applications, I can take different scenarios. And now I understand what's going on significantly better

00;35;14;17 - 00;35;14;17

Pam / Curtis / Pam

or at least at least you can dive in. It's all like right. If we have time for a quick other story. Oh we do please. So remember I told you I turned my attention back to secondary math and I was really then working on okay, how do we logic our way through high school math problems?

00;35;29;03 - 00;35;50;10

Pam

Like what do we need to build? What comes next? You know, like, again, I still want the outcome of kids knowing all the standards, but I want them, like, more than that, more than just being able to answer questions. So, I came home one night and my, third son, my third son was in physics and calculus, and he said, hey, mom, I have a physics question.

00;35;50;12 - 00;36;06;16

Unknown

And I said, what about? And he said, angular velocity and momentum. And I said, oh, goody, because I did that in high school and I've never taught it. And I didn't get it when I did it in high school. And it was a little bit, it was a little enough on the test that I could just kind of not worry about it too much.

00;36;06;16 - 00;36;23;14

Unknown

And I still got my A, and so I didn't worry about it. That's my that's the feeling that came over me when he's like, can I have help on that? And so I said, sure, after dinner hoping let's learn together. Well, no, no, no, I was hoping that life would get busy and he would forget. And he'd go, tomorrow when you get help from his teacher because I was really clear, like, I mean, I could dig into it.

00;36;23;14 - 00;36;40;04

Unknown

I had confidence that we could figure it out. But. So anyway, dinner ended, he came. He said, hey, remember, I need that help. And I was like, oh, crud. Okay, sure. This is still relatively early in my journey. Not totally, but it's certainly early in my secondary journey. Yeah. And I said, okay, we'll go get your textbook.

00;36;40;04 - 00;36;58;23

Unknown

And he goes, what? And I was like, go get your physics textbook. So this is not pre-internet, but you're still not looking up a lot on the internet at this point. So I get your get your textbook and he goes, mom, we don't need a textbook. Physics is figure-outable. Well, it did. You just throw my own words back at me you little snot-nose brat.

00;36;58;26 - 00;37;11;08

Unknown

And I was like, dude, I haven't, I haven't, I've never taught this. I did it in high school. I didn't do well in high school. Like, seriously. And he's like, mom, I'll tell you what, just help me figure out what the problem's asking. I was like, well, all right, I can do that. So we sat down, he goes, okay, this.

00;37;11;08 - 00;37;26;12

Unknown

And I was like, so then that, yeah, it is, it can do this. Okay, then that and then oh well then it would be this. Hey, we just solved the problem. Oh my gosh. Physics is figure out of all like it was that moment. It was totally that moment where I was like, okay, do we know enough to dive in?

00;37;26;15 - 00;37;51;19

Unknown

Like, there might be some social things that we might need to, you know, look like you've never heard of something. You might have a social convention that somebody might need to fill you in on, but then from there, you use what, you know, if anything else. In developing mathematical reasoning, avoiding the trap of algorithms, my main point is to say the mental actions of mathematicians amassing is actually where you say to yourself, what do I know?

00;37;51;21 - 00;38;04;02

Pam / Joanie

And how could I use what I know to logic my way through as far as I can continue? And yeah, that's what dimensional think? Yeah, yeah. 

00;38;04;02 - 00;38;04;02

Curtis / Pam / Curtis

So I went so I came to your session, this summer at CAMT, and I was sitting in the room and, you were it was the secondary version. Did I know that? Right? Did I see you there? You did. You saw me. I think you saw me afterwards. Okay, okay. You probably didn't know during the session, but afterwards you saw me. So, he was texting me during my mission, so, yes, I was, I was, and I took a bunch of notes, and I'm actually looking at my notes right now.

00;38;31;08 - 00;39;50;00

Curtis

And I just, So the reason I bring this up is because it goes exactly with what you were just saying. All right, so you were surprised by the fact that there wasn't a there was not a kindergarten teacher in the room. Okay. And, and and what happened was you had that you had the, and you probably know the example that I'm talking about where you have, we had, a list of numbers.

I think it was, 127 254 so we had this relationship, right? And we come up with the equation, the linear equation that models this, and then you went back and you said, well, what if we change the numbers and we just took all of the, the ones in the Y column and we subtract one. Right. What's the new what's the equation of the line. Right. And in my head I'm like trying to figure this out. Right. Because I'm an algebra person. I'm just like now I'm putting together my slope. You know, all these things are a thing. And looking to find the slope. Put it in this guy. Right, right. Yes. Yep yep yep. Okay, so here's my note, okay. Because you were looking for the kindergarten teacher. Here's my note. This is literally the words I said. Dude, she had 127 254 then we worked out the equation line and then she sets us up with 126 253 whoa. The transformation of functions. I totally just started seeing rate changing, but I didn't I didn't see that the rate did. Why am I so tripped up by this?

00;39;50;00 - 00;40;15;17

Curtis / Joanie / Curtis

The difference in the Y's was the only thing we did. I was so frustrated with myself going, how did I not see this transformation of functions thing happening? Right? So ingrained. You're so ingrained. You said it's at the most people look at that and they go up. Well, you just shifted the, but us math teachers, especially our high school math teachers who've been like, trained with like, this is what you do when you're finding the equation of a line.

00;40;15;20 - 00;40;37;11

Curtis / Pam

That's where I went. That's what I did. And you're so not alone. You're so not alone. And so one of my messages to the world is, let's be kind to each other and let's realize if and ourselves and our and ourselves. Absolutely. Yes, yes. You know, I just had an interesting conversation on LinkedIn the other day where, a very nice, I don't even remember.

00;40;37;14 - 00;41;35;11

Pam

He's not a math teacher, but he's like a physics engineer or something, and he, I said something about, Golly, let me remember. Oh, I said leaders of math education. You could use a prompt like this to help your teachers consider what do they think math means? What are the mental actions of math? And then I had, a proportional reasoning problem. So a good middle school problem. It was written kind of interesting kind of think about it. And this very nice person said, well, if they don't know that these are the mental actions of math, they shouldn't be teaching math. And I was like, then you're just like kicked out 90% of math teachers that I work with because I work K-12. So, you know, k-eight, most of them are broke, members are bad workers. And I'm not trying to be ugly. It's just we we created them to be that way. If that's high school math teachers, this is part of my call is let's all just, like, chill a little and go, okay, wait, what are the mental actions of mathing and how are the ways that we can help kids do that?

00;41;35;11 - 00;41;54;23

Pam / Joanie

How can we now send the correct message? And let's just all like, let's be kind. Let's be kind to everybody. We're not trying to we're trying to give the world this new view, their true view. And then now we can move forward. Yeah, yeah. And yeah, when we know better, we do better. And now we know better. So we're trying to do better.

00;41;54;25 - 00;41;56;19

Pam

Absolutely.

00;41;56;19 - 00;42;06;21

Music break

End of Segment 2 

Start of Segment 3

00;42;06;21 - 00;42;24;05

Pam / Joanie / Curtis

I'm glad you like that one Curtis. That was, that was oh that was that was earth shattering. Yeah. He's just told me that story at least twice. Yeah. Well, it was earth shattering to me because I could totally see why if I wasn't caught up in the the the high school math, what to do right in the what to do.

00;42;24;05 - 00;43;13;26

Curtis

If I hadn't, like, been in that mode, I could totally see why someone would go. Would you just moved? Like you just shifted up one or down one? Whatever you want. Yeah. And it's like, oh. Absolutely. You know, I just just that that. Well, yeah. Like you just changed it by this much. So of course the line just went like this. What what it was such a, it was such a a powerful moment for me, especially because I was sitting next to you, a former student of mine who is now an elementary teacher. She was actually up for a presidential award last year. Oh. Congratulations. She was. She was a nominee. She she didn't end up getting the Texas one, but she was still super like.

00;43;13;26 - 00;43;35;00

Pam / Curtis 

That's a huge deal, right? I'm so proud of you. But I was sitting with Lauren, and, and one of her coworkers, and we were we were chatting, who's a middle school teacher. And so we were kind of the three of us chatting through this thing and just the the mind blowing moment of like, what? Was just it was so powerful for me because it was just cool, so thank you.

00;43;35;00 - 00;46;44;26

Pam

Yeah. You're welcome. So maybe just a one other fine point that I'll put on that is so we have algorithms. We also have formulas, especially secondary teachers. We think about formulas. I'm going to I'm going to differentiate those. And you might you might disagree with my definitions and I'm okay with that. If we could just agree that this is what I mean, then we can kind of have a conversation.

So if an algorithm is a series of steps to solve any problem of a class, and you have to do all the steps all the time, you might then go, well then what's a formula? Well, a formula is a general representation of relationships. So when I say that like y equals mx plus b isn't going away. But what I'm suggesting is it doesn't initiate a to do sequence. So I'm quoting Courtney Pierce who's works on my team. And she said, who one day she's like a formula does not initiate a to do sequence. It's a general statement of a relationship. So, we still need and I usually do y equals b plus x because I like to develop it in kids as the outcome is equal to where you began, “B” plus how you move, “M” times the input, variable x.

And I develop that. I don't just tell kids that we develop it, they I end up pulling that out of them as we develop like. So it's not that that general, statement of relationship goes away, but what it is, is like. So Curtis, you saw that table values. I said find the equation of a line. You do this. We initiated a to do sequence. What we're suggesting is, is that to do sequence that we want kids to initiate is to use what they know to logic their way, not to reach back for a procedure, an algorithm. Yeah, yeah. So one of the things you asked me talk about is the trap of algorithms. So one trap of algorithms is that we all three of us, you just said, and many other teachers that I work with get trapped into thinking that the mental actions of math is to rote memorization, mimic instead of the mental actions of mathing, being, using logic, using what you know to logic your way through and build more relationships in the

process. That's one. But there's also another set of traps that are really important. So if I can do that really quickly. But, I have a great graphic in the book that's I don't know if we can do this verbally, but I'll try audibly here. There's a series of, nested ovals, and the smallest one says counting strategies. And then we build on that that encompasses counting strategies is additive reasoning. And then build on that to encompass those two is multiplicative reasoning building on that to accomplish those is proportional reasoning. And then encompassing all of them is functional reasoning. That means that if, as a high school teacher, I want kids reasoning about functions and relations and graphs and tables and transformations, then I'm building on all of those other reasonings. So, here's the biggest trap of our traditional algorithms. They can trap kids into looking like they're using one of those kinds of reasons I just mentioned, but they're actually because they're getting correct answers, mimicking a procedure. But actually, as they're doing that procedure, they're using reasoning in the level before. So an example would be that if I was doing you're listeners are usually secondary.

00;46;44;26 - 00;47;01;21

Pam / Joanie

So let me think about a secondary example. Let me do a quick elementary because it'll help me do a secondary to an elementary. We have lots of elementary listeners. You're good. They're yay elementary listeners. Do I have time to tell you real quick? Do you know my favorite teachers talk with her K-5 teachers? Yeah, elementary. You know what?

00;47;01;26 - 00;50;09;15

Pam

Yeah. Oh, yeah. Because if I can convince them, they will actually do it. Yeah. Middle school is a crapshoot. 50-50 maybe, high school, nah. High School teachers will look me in the eye and they're like, that's really cool. And I'll say, are you going to do it? No. Like, come on now. I will say high school teachers, you have staying power. So when you do finally move, then high school teachers, stay like they, they stay the course. Elementary teachers are a bit more fly in the wind. They're like squirrel shiny. So one of the reasons I do these nested, ovals is to help everybody, K-12 realize what's our goal. Our goal is develop these kinds of reasoning. So then, no matter what tick, tick, trip or tip, I can't say those two words together. Tip or trick that you try to throw at me. I'm going to weigh that up. Is this a way for kids to get answers that could allow them to be using less sophisticated reasoning than I should be developing and using? Then I'm going to say, and or is this something that's actually helping develop the kind of reasoning that we're getting at, than that, that I'm going to use as a teaching tool?

Oh, I like that. So, yeah. All right. So let me demonstrate what I mean. If we have a kid doing a multiplication problem, like, and I'm just going to do something like, I don't know, 27 times 99, just random numbers, then, in the traditional algorithm, I'm going to do that seven times nine first. Right, right. So consider that I've now asked kids to think about twenty-seven 99’s or ninety-nine 27’s.

But I have said don't think that way. Actually. Just think about seven times nine. Think about the smallest numbers in the problem. So a trap is I instantly turn a problem into a bunch of digits. I break it all up and I focus you on the smallest, most inconsequential parts of the problem. That's a trap. Second trap that I want to bring out is this least sophisticated reasoning trap. So now I'm thinking about nine sevens or seven nines. So a kid says, oh, I need to do that. Ready? Nine plus nine is 18, plus nine is 27. And they literally skip count a bunch of nines. Or maybe they skip count a bunch of sevens. So what kind of reasoning. We're doing a multiplication problem. So that's going to get rid of any right.

No that's additive reasoning. Nice journey. So I could do every single one of those single digit multiplications and then do all of the single digit additions, never using an out of an applicative reasoning at all. I never multiplying ones. Yeah, yeah. Now you might say, Pam, my kids, you might be a fourth fifth grade teacher. You're like my kids are using multiplicative reasoning.

They're thinking about nine times seven is ten sevens. And if I have ten sevens then I could think about nine sevens. Is one less seven. Ten 770 entity one less seven. Okay. Seven from 76 to 63. It takes way more time to talk about than it would for kids to just sort of think about. And you're like, Pam, don't we want kids just know that? Well, actually, I want kids to have ways of reasoning about nine times seven as ten times seven, in the same way that they'll reason about nine times 83 as ten times 83 and get rid of an 83. How about nine times 6752 as ten of those minus one of them, like, I want kids to develop that kind of logic and reasoning and relationships.

00;50;09;20 - 00;50;28;28

Pam / Joanie / Pam

To do any of that, I'm going to have rounding and estimation and reasonable is all involved as I do that kind of strategy anyway. So and it's I would also think about 27 times 99 is 27 times 100 -27. What you just stole my thunder because I was going there a second. Yeah, it's all right. Way to go. 

00;50;29;00 - 00;50;50;15

Pam

Let me let me finish a little bit so we could we got sorry, I sorry I got. No, no, it is brilliant, I love it, I love that you saw that. Right. Like, because you're like, well, what could what could I think about? If a kid is using a traditional algorithm, they could be at best using single digit multiplicative reasoning in that algorithm, never thinking about twenty-seven 99’s or ninety-nine 27’s.

00;50;50;15 - 00;51;12;16

Pam / Joanie

And Joanie, you just said, how come you didn't think about twenty-seven 99’s? Why did you think about ninety-nine 27’s? Because it's way easier to think about 100 of something than it is to think about 99 of something. You let the numbers influence how you attack the problem. Those are math. That's math. Behavior mapping behavior is to think. Let the numbers in the structure influence how you solve the problem.

00;51;12;16 - 00;51;35;16

Pam

And you did it right there. You thought about ninety-nine 27’s by saying, well, 127 that's just 2700. I just gotta get rid of a 27. And now I might need to know a little bit about getting rid of that 27. And so we play a lovely routine called I have you need that Kim, my co-host on the podcast made up where we, help kids think about what's the partner of 100 to 27 and what's they like?

00;51;35;16 - 00;51;57;21

Unknown

What's 73? Well, bam, I'm at 2673. And that's thinking multiplicatively. And with good additive reasoning in the middle I've built my multiplicative reasoning. Bam. All right. So for your secondary teachers, all too often we get into proportional reasoning. And we say to kids, to solve this is over of whatever we teach them. Some way to set up a proportion.

00;51;57;23 - 00;52;16;06

Unknown

And then we say like memorize, memorize this. And then we say cross, multiply and divide. Notice that we are in the oval where we should be using a developing proportional reasoning. But how are we going to do it. We're going to cross multiply and divide. That's the reasoning in the level before the inside. That's the inside one.

00;52;16;06 - 00;52;45;19

Unknown

So at best they're using multiplicative reasoning. But notice if they're using the algorithms to do the multiplication and division. They could be using additive reasoning. So we're supposed to be building proportional reasoning. They're getting correct answers. They look like they're proportional reasoners. But they're actually using less sophisticated reasoning that we could be using and developing. So we don't want to trap kids into looking like they're successful when we're actually giving them a way to use less sophisticated reasoning.

00;52;45;21 - 00;53;01;26

Unknown

But, Pam, it's so much easier to just give kids this method. They do it, they get the right answers. So then I say back to you, okay, so let me just be clear. You're if you're you're saying to me your goal is for it to be easy for kids not to mess it up for them to get exact correct answers.

00;53;01;26 - 00;53;23;13

Unknown

That's your goal. They don't really have to think at all. That's your goal. Then let me hand you technology. Punch the buttons, ask Siri, ask ChatGPT like if your goal isn't to build thinkers reasoners mathematical understanding and connections, well then don't even like why force the memorization of these nonsensical steps right now? Somebody's going to say they're not nonsensical.

00;53;23;13 - 00;53;53;10

Pam / Curtis / Pam

Okay, I'll say of these opaque steps because they're not transparent, right? Most algorithms are not transparent at all. Think about synthetic division for just a minute. Oh, Lord. Gosh, like what are the least transparent, like nobody is like, oh I clearly that's what we're doing it now. Teachers will go, oh, I know exactly what it means. Know what you mean? What you know exactly is how to do it. You're not thinking about the connections in a relationship anyway. So just there's, there's a very opaque example of, of an algorithm. 

00;53;53;10 - 00;56;51;26

Curtis

I, I love that. And while you were telling the story, particularly, about this, I was I was thinking about my son, Teagan and this is very, I know we're probably up against time, but I'm going to tell a story anyway, and we can we can drop it if we need to, but I, no, seriously, I want to hear it. I want to hear it. I was wrestling with him the other day, not physically wrestling with him, but because he's he's at least as strong as me and he's 14. He's going to beat me up one of these days. But that's another story. No, he, he and I were talking a lot about the…So he's he's an algebra one class, and he's, he's doing, they were solving some, some, equations, and they were having equations of different forms, and many of them were had, you know, a, a, a number multiplying a multiplied by, some binomial, factor. Right. So, like, you know, three times the quantity x minus four equals 27 or whatever the number was, Four. And I made those up, but, he and so he had it in his head. He was just like, pounding down, like, distributive property and doing that, like he was just doing this thing. And I'm like, Teagan, let's stop for just a second. What three times a number is 27. What does that number have to be?

And it took us like, I don't know, 12 different examples. Like he was doing these things and he was just like, no, dad, that's not how I'm supposed. I'm supposed to be multiplying these things. And finally I just I stopped, we had other things to do, and he had to go off and finish his homework, and I had to go somewhere else and do something else. I came back the next day and we were still working on one of these problems. And he says, I said he he did the problem. He says, and I just divided by two, to start, I said, wait, why did you just divide by two? Like he started doing it and I was just I got goosebumps whenever he was doing this. I was like, dude, you're seeing the structure. You're you're getting like, you're thinking this through because he's very much I think he's still kind of in the same boat that that the three of us were in. Yeah. He's, he's pretty good at, like, you tell me what to do and I can do it, I can execute. I'm just like playing school.

00;56;13;18 - 00;56;32;24

Unknown

Right. He's got expectations. And and he's pretty good. He's got he really has grabbed grasp hold of the numeracy piece. Like he's starting to really see numbers and being able to recognize relationships. But this was the next step, right? And I was like, dude. And so when he came back the next day and was doing this, I said, you're doing it. You're like, really logic going through this thing is, is this it was the happiest thing. The good thing is, is that he's still willing to follow me. Like, he's still willing to do those things. Right? So when we talk this through, he's he doesn't he he is willing. We do have to wrestle from time to time though but like the parts being structure.

00;56;52;01 - 00;57;07;05

Pam / Curtis / Pam

Yeah. So the next, string I would do with your son would be where I would have an equation purposely set up where you would want to distribute, and one where you would not want to distribute. Right. And one by one where you could do either. And I would want to that would be a good problem String to go;

00;57;07;05 - 00;58;02;19

Pam

When do you want to which strategy? Now this is an example of strategy we've got we've got an equation. And you could just do the same steps every time. Or you could let structure influence how you do it. And, and this is this is one of the parts of my work that I'll just invite your listeners to know.

What I've done is once I realized there were only a small set of major numeracy strategies. So, Johnny, when you said earlier that it was it was mis it was a misunderstanding. And I agree with you. It was a miss. When teachers are like, let's have 49 million ways of doing a multiplication problem. I put them all on the board and they're all equal. That set up this idea that that anything you do goes, whether it's counting, additive, multiplicative, anything you do, well that's yours. You own it. That's your I don't know that's that's a necessary starting point. That's how you're now thinking about it. Let's build on that. If you can see me I'm building those okay. Let's build it. Yes, I can build that.

00;58;02;20 - 00;58;18;20

Pam / Curtis / Pam

The ovals. Hi. You're watching my hands all day. So that's why I would want to take your son. And I would go, okay, now let's sharpen that. When, which which. What is it about the structure that's nudging you here? What's it? That structure that's nudging you there? And then I give him one. And this is hard for you to create.

00;58;18;28 - 00;58;55;21

Pam / Curtis / Pam

Create one where doesn't matter which way you you know, where you contribute first, you divide first and then have him create one. Bam. Now, now he's focused on structure, not just on answers, but boy, he's really gotten. Yeah. That's a there's an actual example. I'm really good. I love it I love it. Now the three and 27 was just happenstance. That actually worked the way that it was supposed to. But well, kind of if you look at any 1 to 29 odd textbook, you're going to have both kinds. Oh yes. You for sure. Well, absolutely. And it would be a great that that would be worth everybody's time to spend some, some time looking at structure and an example that there.

00;58;55;21 - 00;59;11;20

Joanie

Yeah. There's such a great way to make use of those kill-n-drill worksheets. Right. Just just a question. It really is a question for short. These sort these where, you know, how are they alike and why are they alike and why do you sort them that way, and why do you sort them that way. Yeah, that was a great strategy.

00;59;11;27 - 01;00;38;02

Pam

Can you give you another idea to take that? So I love your sort. You can also say solve your favorite five and nine. That's nice. Now they're looking for structure to find the easy ones. Now circle the five you don't want to solve. Yeah. And then maybe don't even make them solve it. But talk about why. So again looking at structure what is it about now all the things that we who were willing to play school and just do one through 29, all 29 times, as we did that, we came up with little things.

We looked at, ooh, solving systems of equations. Golly, if the coefficients are same, let's do elimination, not substitution. Like we had these little things that sort of occurred to us. The good news is we can actually teach those explicitly, not by telling kids, but by putting it up in front of kids and creating the conversation about it. So here's what my what I think I add to the mathematics world is.

First, this progression and not getting caught in in the traps. But secondly, there are really only a small set of major strategies that are that are built on a small set of major relationships. There's a lot of minor, but we really need to focus on the major ones. And if we focus on those major ones, we got kids that are successful, but they're mathing, and I'm going to suggest they're more successful than ever before, and they're doing the mental actions of the mathematician, which means they now have the world open to them. Math is no longer their gateway to closed down. They can now, do whatever they want to do because math doesn't keep them from doing it. 

01;00;38;02 - 01;01;27;15

Curtis

What I love about what you just said, too is that the, the one through 29 odds worksheet thing like that, there's a real use for it, not just go through and execute all of those things, but the explorations that both of you guys just put up there aren't…I mean, we're solving these problems like, these are things we should be able to do. These are standards. You mentioned that earlier. Like we're we're still teaching. Like we still need to be able to do those things. It's not just the wheels are off and we're all going out and doing our own willy nilly thing, but we actually are executing some things, and you're pointing out the ways that we could utilize some of those, those things that even exist right now.

It's not like we have to throw out the baby with the bathwater, out everything we've ever done before. But it's easier now looking for great ways to even make use of the things that we have now. 

01;01;27;15 - 01;01;27;Curtis / Pam / Joanie

I love. Yeah, absolutely. Nice. We could go another hour. I know we could. I'm having so much fun. We're all talking fast.

01;01;35;26 - 01;01;53;25

Joanie

We're all like our body positions, I love it. I know we're all enjoying this conversation so much, but, we are under time constraints, so, Pam, maybe you can wrap us up just share a little bit of information, because what you've shared with us today is incredible. And it's only the tip of the iceberg of what you have to offer the math ed community.

01;01;53;25 - 01;02;53;20

Pam

So where can folks learn more about you and your work? Math is figureoutable.com is a great place for lots of good free stuff where you can also get Ahold of my book, Developing Mathematical Reasoning Avoiding the Trap of Algorithms. I have AK2 version that just came out actually the week we're recording this podcast. Oh, so there's a companion. It's a companion series. So we have the K 12 that's out K2 just came out. Three five will come out in February of 26. Then a  68 will come out six months later, and then a 912 version will come out six months after that. So those companion books, in a big way, avoiding the trap of algorithms, is kind of what not to do with with quite a bit of what to do.

But then I really drill down into those great bands and, and really what to do. How do we develop the levels of reasoning that we need at those grade levels? So I invite everybody to check those out. Also the math to figure out what podcast is a great place to hear Kim and me talk about, all things math is figure outable. K12 yeah, those are those would be great ways to connect.

01;02;53;20 - 01;03;12;05

Joanie / Curtis / Pam 

Sounds amazing. Well, I don't doubt that you're going to be a repeat guest. And boy, I'm thinking book study all because, I mean, yes, this is going to be amazing. Oh, so you might you might be interested. We have a book study, website that's a companion website to the book that would help out in two ways.

01;03;12;07 - 01;03;53;15

Pam

Hey, there's videos and other things from the book, but then also a book study guide on how to, you know, questions and things to do in a book study. So perfect. Yeah, yeah. Grab those for anybody that's, running a bookstore. It's a it is a, I think, a unique K-12 opportunity to talk math. We have a lot of K-12 books out there that are kind of pedagogy. Pedagogy. Yeah, very few that are math. And do you know, Corwin said to me, there's a lot of math in this book. I was like, it's a math teaching book. Yeah. Look, we had to hire a math editor and all the things. It was great. And they were totally willing to do it. But yeah, I mean, and you can have a K-12 mathematics conversation. So I think it's a little bit of a unique opportunity to do that. Yeah. Thanks so much for having me on.

01;03;53;15 - 01;03;56;15

 Joanie / Curtis

So thank you for coming. 

01;03;58;25 - 01;04;16;23

Joanie Outro

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!