[00:00:01.030] - Pam
Hey, fellow math-ers! Kim, think I can do it without reading?

[00:00:04.480] - Kim
No.

[00:00:06.340] - Pam
Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris, a former mimicker turned math-er.

[00:00:14.250] - Kim
And I'm Kim Montague, a reasoner who now knows how to share her thinking with others. At Math is Figure-Out-Able, we are on a mission to improve math teaching.

[00:00:22.230] - Pam
Y'all, we know that algorithms are amazing human achievements. But they're terrible teaching tools because... Oh, crap. I was good. I did have it till there. Because mimicking step-by-step procedures can actually trap students into using less sophisticated reasoning than the problems are intended to develop. Oh, so close.

[00:00:42.220] - Kim
In this podcast, we help you teach math-ing, building relationships with your students, and grappling with mathematical relationships.

[00:00:49.020] - Pam
We invite you to join us to make math more figure-out-able. Bam. Alright. I thought I'd try it.

[00:00:54.680] - Kim
How many times have we said that?

[00:00:56.370] - Pam
Well, a million. And so, one of the reasons I wanted to try it without reading it is because you and I have done a few co-presentations lately where we're in front of people and we obviously don't have notes in front of us. I mean, we have little notes. But we don't have, you know, whole introductions to the podcast. And I think we did pretty good in person, so I thought, "Well, shoot. I should be able to do it here." Maybe not.

[00:01:17.530] - Kim
It's terrible. It's terrible. When we got on, you were trying to decide if you were glasses on or glasses off.

[00:01:24.480] - Pam
Oh, thanks for telling the world.

[00:01:26.890] - Kim
Well, I think it's funny. It's a legit question because the headphones, man. Sometimes my ears get hot at the end of this.

[00:01:34.520] - Pam
I'm about to take my earrings out because it's...

[00:01:37.070] - Kim
That's what I was going to ask you is do you wear earrings? I know you love the earrings. Do you wear them during?

[00:01:42.140] - Pam
You know, I'm kind of an earring fan, so I almost always have earrings on during the day. I take them off at night. I slept in them for a couple of years. I don't know what that was all about. I don't do that anymore. But headphones. Yeah, it's not the most comfortable. So, we'll see. We'll see if I last through this whole episode. You might hear some clinking. If you hear some clinking, it's because I put my... I have a glass desk, so you can kind of hear stuff when it clinks. You don't care. See, here's my glasses. You probably can hear my glasses clink on my glass desk.

[00:02:10.600] - Kim
Oh, my gosh. Alright, let's get going.

[00:02:11.260] - Pam
I am glasses off. How about you, Kim? Are you glasses on or off?

[00:02:14.880] - Kim
I am... Yeah, I'm contacts, most of the time.

[00:02:17.110] - Pam
Ah, contacts. See, contacts are the wrong distance for my computer.

[00:02:20.880] - Kim
Well, mine are the weird multifocal.

[00:02:24.330] - Pam
Oh, same.

[00:02:24.980] - Kim
I've got old eyes.

[00:02:25.810] - Pam
No, same. But it's the wrong... Yeah. No one cares about our eyesight. Here we go.

[00:02:30.530] - Kim
Let's get started.

[00:02:31.410] - Pam
Kim, the other day, ironically, I was in a... That's not the right intro for that. Let me just think. The other day, not having to do with math education. You know, you and I both volunteer and do some things around the community. And I volunteer for my church. And I was in a training the other day, and they introduced me to this Results Pyramid. Now, maybe I'm the last person on the earth who has seen it. Everyone else is familiar with this Results Pyramid. But I was really interested in it because, man, it fits really nicely with how we think at Math is Figure-Out-Able, how we think about teaching and about professional learning.

[00:03:12.060] - Kim
Mmhm.

[00:03:12.220] - Pam
Okay, so I'm gonna describe the pyramid. I've told you about a little bit. You were like, "Yeah, yeah. It seems to fit." So, we thought we'd share it with everybody today. I'm going to make my desk go up. I don't know if you can hear that.

[00:03:22.250] - Kim
Uh, 100%. I can hear that.

[00:03:23.140] - Pam
Nothing like a good standing desk. The pyramid... So, you picture... I mean, it really looks like a triangle. I guess it's a pyramid, but whatever. On the bottom is the word "experiences". And then move up, so you have a little less space, right? Because it's like the triangle. It's the same size height, but because it's a triangle now, you have less area it's taking up, or if it really is a pyramid, less volume.

[00:03:45.540] - Kim
It's like a candy corn for all the different colors.

[00:03:48.990] - Pam
So, "experiences" on the bottom, then "beliefs", then "actions", then "results". Now, let me go down from the top. Results. Hey, who doesn't want results? Do we all want results? We all want results. Like, we're working out. We want results. We're... Help me. We're learning to cook. We want results. We're teaching. We want results. We are results-oriented people. We're looking for good results. Does it make sense then that actions produce those results? Okay, sure. So, it's results, then actions. Actions produce those results. And then underneath that, beliefs inform or what's... They...

[00:04:30.450] - Pam and Kim
Influence.

[00:04:33.510] - Pam
They... There's a better word I'm looking for. The beliefs that you have create the actions. It's not quite the word I want. They influence. They drive. They drive the actions. Inform! I think inform. Beliefs inform your actions, and your actions then produce these results.

[00:04:52.490] - Kim
Yeah.

[00:04:52.550] - Pam
And so, in that candy corn. Thanks, Kim. That's nice. In that candy corn, we've got these results on the top. Actions are creating those results. Beliefs inform those actions. They influence them. You do the actions based on your beliefs. If you don't believe in something, often you don't do that action, which then means you don't get that result.

[00:05:10.721] - Kim
Mmhm.

[00:05:11.270] - Pam
So, if you have these beliefs, you do these actions, these actions produce these results. So, remember the bottom one. If you can see my hands, I'm like up and down and up and down. The bottom one is "experiences". So, when we have experiences in our lives, then it creates these beliefs, and these beliefs influence our actions, and our actions produce these results.

[00:05:35.810] - Kim
Yeah.

[00:05:36.960] - Pam
That fits for me. What also fits for me, and this actually came out in the church training, was, "Hey volunteers. You might be tempted to say to people, we want these results. So, because we want these results, do these actions." And then you sort of walk away, and you kind of expect or hope that people will do the actions you suggested, so that we'll get the results that we're looking for.

[00:06:03.670] - Kim
Mmhm.

[00:06:03.670] - Pam
Kim, how often in education do we see where professional learning will say, "Hey, teachers. We want these results. We want better test scores. We want happier students. We want students who attend school. We want..." Let's be clear, better test scores. So, do these actions. When what we haven't done is give teachers experiences, so that they actually believe that those actions will then create those results. And so, they might do some of the actions for a little while. They might even get some results. Test scores might go up. We might see kids doing certain things. But it's not going to last because the beliefs didn't change. And if the beliefs don't change, people aren't going to continue to do actions they don't believe in or that aren't founded on their beliefs. Could this be the same with students? Maybe. But as you and I were talking about this and thinking about teaching and thinking about professional learning. So often we see, "Hey, we want these results better. I want kids to get the answers right. If I want kids to get the answers right, I'm going to tell them to do these actions. Hey, guys. Just mimic these steps of the algorithm. Bam, you're going to get correct answers. Oh, you don't have that right? Well, here, let's memorize better. Let's mimic. Let's learn some actions, some rhymes and mnemonics. And we might even rap to it. Or some stories that have nothing to do with math. But it's going to help you remember the actions. Do these actions." But we have students who don't believe that Math is Figure-Out-Able. They don't believe that they have the ability to do it. They don't maybe believe that it's worth their time and effort. How many boys do you have in your class? And there's probably some girls too. But I often would have this sort of louder, outspoken guy that was like, "Why are we doing this? What is this? Let's go play some football." Whatever. And in a way, if the actions we're asking them to do are based on, you know, doing rote things to get answers that really aren't related to their life and doesn't really interact their... You know, it doesn't affect them.

[00:08:07.970] - Kim
Yeah.

[00:08:08.780] - Pam
Then they might just refuse because their beliefs are, we can learn things in school that relate to me. We can do things that matter, that are... Golly, help me, Kim. We can believe that Math is Figure-Out-Able. I don't know if the kids know that right off the bat. But if we don't impact their beliefs, then the actions aren't going to... Yeah, keep going.

[00:08:36.280] - Kim
Well, like I'm sitting here thinking about how much of the time we talk about the results that we want. It's so prevalent in so many conversations. We talk about the results. And what you're describing here, when we focus on the experiences that impact the beliefs that inform actions, then we get the results. It feels like a more organic, more...

[00:09:02.240] - Pam
Experiential?

[00:09:03.450] - Kim
Yeah, I mean the results will come, but they come in such a way that we've changed the person. We've changed the subject. We've changed the appeal.

[00:09:14.980] - Pam
Nature of the subject, I think is what you mean, right? The nature. Yeah, math being figure-out-able.

[00:09:17.840] - Kim
Yeah, not math changes.

[00:09:17.840] - Pam
 Yeah. Yeah.

[00:09:20.880] - Kim
But I feel like we focus so much on the results that we're trying to get that we just keep banging our head against the wall to say, "Do this thing. Do this thing. Do this thing," but it's always somebody telling somebody else what to do.

[00:09:35.630] - Pam
Those actions. Instead of, and this is what we promote here at Math is Figure-Out-Able, what are the experiences that we can provide? Hey, how about podcast listeners? You'll notice we often do Problem Strings during the podcast. We ask each other math problems and talk about what we're thinking about. We believe in giving us, and our listeners, and people that we interact with, and the teachers in our coaching group, and the teachers in our workshop, experiences, so that that impacts their beliefs about the nature of mathematics. That's what I think what you just meant. The subject.

[00:10:09.060] - Kim
Yeah.

[00:10:09.320] - Pam
Yeah. And their beliefs in their self-efficacy.

[00:10:12.830] - Kim
Yeah.

[00:10:13.190] - Pam
That they can, and that it's worthwhile, and all of that stuff. That then leads to them going, "Huh. Well, I can do these mental actions. I can use what I know to reason and think about the math. I can do those actions." And bam, then we get results that aren't just test scores, but we get more kids in STEM. We get kids believing in themselves. We get kids with the doors open to them because Math is Figure-Out-Able, and they believe they can do it.

[00:10:40.530] - Kim
Can I tell you a story about something that happened this weekend that I'm thinking about right now?

[00:10:44.780] - Pam
I mean, is it completely unrelated? But yeah, go ahead.

[00:10:48.090] - Kim
Well...

[00:10:48.310] - Pam
I'm sure it'll be great.

[00:10:49.290] - Kim
So, you know that I love to run, and I do races when I can, and the weirder the race the better, and whatever. And my kids have known this growing up. And I have a son right now who is running cross-country. Which I absolutely love. But he has grown up knowing about these races. And he's like come and cheer me on and make the sign and whatever. And I did a race a couple of times that was... It was a long distance. So, it was a couple hundred miles. 6 people on the team. Blah, blah, blah. It was a relay. And the nature of the race...

[00:11:21.070] - Pam
You're in a van, right? And you switch off legs and all that.Yeah, uh-huh.

[00:11:24.290] - Kim
So, you do like a 10K-ish. Then the next person goes, next, next. And so after, you know, the round one, then you go back out. And the race itself ends up being a whole bunch of 10Ks in like 20 hours for each person. And Cooper knows this. My youngest. He knows that this is a thing that I've done. But it wasn't until cross-country that he had an appreciation for what 10K means. 10,000 kilometers, 6 miles. And...

[00:11:57.330] - Pam
10,000 meters? 10,000 kilometers?

[00:12:00.650] - Kim
10,000 kilometers. No, 10,000 meters.

[00:12:04.030] - Pam
10 kilometers.

[00:12:05.370] - Kim
10 kilometers. 10,000 meters.

[00:12:07.390] - Pam
10,000 meters. There we go. Okay. Okay, good, good.

[00:12:09.190] - Kim
6.2 miles. So, anyway, we went and did a long run this past weekend.

[00:12:14.870] - Pam
Oh, in the heat. Holy cow.

[00:12:16.190] - Kim
It's hot. We go really early.

[00:12:18.310] - Pam
Oof.

[00:12:18.610] - Kim
Anyway, so he's known about this. It's kind of in the back of his head. And, you know, he believes that I've done it, whatever. But when we were done, and we were sitting there stretching. I said, "Hey, remember that race that I did?" And he was like, "Yeah." And I said, "We just ran 8 miles. So, now that you have an appreciation for that, keep in mind that this race that I did that you said, 'yeah, I'd like to do that one day when I'm older,' is six or seven 10Ks over like a day and a half." And you should have seen his face, Pam. He was basically like, "I'm sorry, what?" And in the moment... This is related because he has an experience now of understanding what it means to run 6 to 8 miles.

[00:13:06.480] - Pam
Yeah.

[00:13:06.780] - Kim
He knows what that feels like. And before it was just like, "Yeah, yeah, yeah. Like, whatever. Anybody can do that." But like in that moment I was like, "I'm so cool to my kid right now because he gets it. He has an experience of these distances. And he totally understands now. And he has this belief about what can be possible. But he didn't understand it until he experienced it himself. And so, now when I say...

[00:13:34.270] - Pam
Not on that level for sure.

[00:13:35.490] - Kim
No. And so, now when I talk about like training or what you have to do, and like the actions you have to take to be able to get that kind of result, it's like it's more ingrained in him. Like, he finally understands. Yeah.

[00:13:49.990] - Pam
It means something different. Yeah, yeah. Sorry to interrupt. Yeah.

[00:13:51.450] - Kim
No, it's okay. I just... It was one of those moments where I was like, "This is a perfect," as you're talking about this pyramid. He now has an experience that helps him understand what it would take and the actions he would have to take.

[00:14:05.010] - Pam and Kim
Yeah.

[00:14:05.550] - Pam
Yeah, yeah.

[00:14:06.200] - Kim
Yeah.

[00:14:06.680] - Pam
To get those results. Nice. So, interesting. I'll tell a completely unrelated, related story. So, I have 3 daughters-in-law, so nobody will know which one this is. But I have a daughter-in-law who's recently been diagnosed as celiac. And Kim, like this is not just a preference. This is like, she gets really sick if  she eats gluten. And so, I have had to learn to cook differently. And, you know, I had a friend who was celiac, and she kind of told me about it, and I was kind of like, "Wow, that sounds really hard." La la la la la. But man, like having to do it. It's just been the experience of you have to rinse everything because if you'd ever put a measuring cup in the flour, you can't now use it, even though it looks clean. You got to rinse it out to make sure that there isn't any flour left on it before you put it in the... And you have to have a special container of sugar because if you ever put a measuring cup in flour, and then in sugar, then you've got flour in that sugar, so you can't use that sugar, even though it's sugar. That's, you know, obviously. And...

[00:15:08.200] - Kim
Yeah.

[00:15:08.820] - Pam
All the things. And then... So, you know, I've gotten fairly good at that. And the other day we were eating and I... Golly, what did I even do? I put my knife. No, I put my fork in the fruit bowl. Which is disgusting. I shouldn't have done that. But I was just, you know like. And my son goes, "Well, you've contaminated that." And I was like, "Okay, but I just touched that grape. Like, that one grape. I didn't..." He's like, "Don't put your, you know, your fork that's been in other..." Whatever. And then on a different level. So, we have family dinner together on Sundays, and they're most often there. And love it. Love family Sunday dinner. She came back after Sunday dinner and said that she'd gotten gluten. And was very polite about it, but she's like, "Can we think about, you know like, what happened, so it doesn't happen again?" And, oh, you know like, even more strong experience that it matters, you know, because she gets... It was kind of like in front of my face. I had this experience of like, "Oh, I just sort of poisoned this woman I love." You know?

[00:16:05.610] - Kim
Yeah.

[00:16:06.150] - Pam
Yeah. So, experience really does impact my belief about how important this is, which then impacts everything I do in my kitchen, which impacts the results that I want, which is a healthy, happy daughter-in-law. Yeah.

[00:16:20.850] - Kim
I mean, and until you have that experience, like you can think about it. You can like...

[00:16:25.310] - Pam
You can try your best.

[00:16:25.570] - Kim
"Oh, that's hard." Yeah, like you said with your friend. Like, "Oh, that's really hard." But you don't have to deal with that until you really do. Yeah.

[00:16:34.960] - Pam
It just doesn't really impact you the same way until you've had that experience. Yeah.

[00:16:39.300] - Kim
Huh. Okay, so let's talk about leaders and teachers.

[00:16:43.640] - Pam
Yeah, let's get back to math, math education. Good, good, good.

[00:16:46.770] - Kim
So, sometimes we see leaders who want to give teachers experiences, right? They want to do math with them. And that's great. Maybe often it comes from they've had an experience themselves, and they get a shift. And then like, "Oh, man. I want to provide this for my teachers. I've seen something different. I believe something different. I want that for my teachers." But often, they don't take the time to examine the beliefs as well. So, like they give their teachers this great math experience, but then that's it. It just like, you had the experience, and then that's it. And maybe they don't take the time to say, "Okay, when we've had this experience, what are you realizing now? What are you thinking? How will this look in your classroom?" Maybe they don't help people realize what the beliefs they have could be shifting, and then what actions that they can take that their new beliefs motivate them to do. So, an experience is great. But then helping your teachers then examine beliefs and what actions can come from that.

[00:17:47.050] - Pam
Almost juxtapose their beliefs about the nature of math and the nature of teaching math.

[00:17:51.450] - Kim
Yeah.

[00:17:51.850] - Pam
Like what we just did, how does that line up with the way you were taught math? And how does that line up with the way you want to teach math? And why?

[00:17:58.610] - Kim
Yeah.

[00:17:59.090] - Pam
And kind of helping ferret out. Sort of make visible. Or at least make in the ether. Get it out in the ether and help people gain clarity around their beliefs, so that they can then gain clarity about the actions they want to take, so that then they can get the results. Is that kind of what you're saying?

[00:18:15.040] - Kim
Yeah, absolutely.

[00:18:16.180] - Pam
So, that makes me think about manipulatives. Like, you and I like manipulatives. We like concrete examples. But we're also very picky about when and how to use manipulatives for reasons. And sometimes we see leaders toss out manipulatives. So, we're thinking of professional learning and they say, "Okay, solve this problem. And here's some manipulatives." And they say, Struggle to solve this problem because we need to get you to feel. This is experience, right? We want you to feel productive struggle. But then sometimes teachers look at... If the problem is such that it's too easy or too trivial or too forced to have to use the manipulative. If teachers can just sort of see the answer or do an algorithm real quick to get an answer, then they might just be like, "Why are you forcing me to play with these toys? Like, why are you forcing me to struggle in this kind of unnatural way? Yes, it is weird for me to try to shove my thinking into this sort of odd thing." I guess we're suggesting that the task matters. And in that case, if you sort of throw something out and kind of force people to use a manipulative when they kind of don't have to, when there's sometimes the answer's just there, or they can too easily solve it with an algorithm, then their beliefs about the nature of math and teaching math might actually get solidified that this is too hard with manipulatives, so why bother with manipulatives? Just tell the kids how to do it. They'll get answers. And it kind of backfires. So, you actually don't shift beliefs in that way. Or the teachers might now believe that they should force kids to count one by one with these manipulatives instead of helping them develop more sophisticated relationships like additive reasoning or multiplicative reasoning.

[00:20:06.159] - Kim
Yeah.

[00:20:06.690] - Pam
So, it's one of the reasons why we struggle to try to help give teachers and leaders the vision about the use of manipulatives.

[00:20:14.340] - Kim
So, here's another example is, you know, we love Problem Strings, obviously. And leaders will, very smartly, give people experiences with Problem Strings. But it might be like, you know, a leader is doing a Problem String with their kindergarten teachers, but they're doing a Problem String that's really too simple for the teachers, and then nothing shifts. Like, they didn't have to grapple at all. And maybe the emphasis was maybe only on the answer, and the teachers just get the answers, but there's no shift that really actually happens for the teachers. And so, I think it's important that we would recognize that it's not just any experience. Like, you're mentioning manipulatives. Don't just give them any experience. It's the right experience that is what's going to cause the change in belief. And that...

[00:21:00.710] - Pam
At least has the potential to influence their beliefs.

[00:21:03.940] - Pam and Kim
Yeah.

[00:21:04.640] - Pam
Yeah. What about student experiences, Kim?

[00:21:07.850] - Kim
Well, here's what I have been thinking about ever since we started talking is that we absolutely love hint cards or clue cards for multiplication facts. And I don't know that we're going to describe it all right now. But hint cards or clue cards are ways that we think that are personalized flashcards that help kids through experience work on owning their multiplication facts. Or addition facts if you use them for that. But what's so great about these is in order to create the clue card or the hint card, you're giving kids experiences where they're focused on relationships. So, when they have experiences with the relationships and they start to gain experience, and they start to get practice, and they start to learn...

[00:21:49.020] - Pam
Just to be clear, relationships among number, not like peer-to-peer relationships. Sorry. Relationships among numbers and like chunks of facts to build other facts. Yeah. Sorry, keep going.

[00:21:59.270] - Kim
Yeah. So, once they start having these experiences, they start to believe that facts are figure-out-able. So, then the actions are, "Well, I believe that facts are figure-out-able, so I want to engage in this more because the result I want is to know them."

[00:22:16.230] - Pam
Yeah. Oh, that's nicely said.

[00:22:16.810] - Kim
So, they get better in the end because they believe it's figure-out-able. They put the work in to work on their facts.

[00:22:24.170] - Pam
So, I can think of a higher... Oh, sorry.

[00:22:26.670] - Kim
Yeah, I was just going to say, so juxtapose that with I'm just drilling, and I'm not getting experience. I'm not changing my belief that this is doable. I'm cementing the belief that I'm not good at this, so I'm not motivated.

[00:22:40.070] - Pam
Or I'm a good memorizer, therefore I'm good at math.

[00:22:42.560] - Kim
Yeah.

[00:22:42.650] - Pam
And either of those beliefs are not. Like that held me back, right?

[00:22:46.910] - Kim
Yeah.

[00:22:47.490] - Pam
So I'm thinking of a higher math example. Sorry for interrupting a second ago.

[00:22:49.730] - Kim
No, you're good.

[00:22:50.440] - Pam
I'm thinking of the fact that I really emphasize the power of getting students to walk in front of motion detectors. So, high school teachers, even middle grades teachers, the idea of having students in front of motion detectors and seeing the results of their embodied motion really allows them to develop a sense of covariation that as... I'll say covariation. That means the variables are varying in a connected way. And we can describe that. As time passes, the distance from the motion detector is changing. And those two, time and distance, are connected because as time is varying, the distance is varying in a connected way, and we can describe that mathematically. And so, in a huge way, we're giving students an experience that helps them believe that their motion can be represented in this way, and that they have the power to then do a representation, so that represented as a graph. And then they can represent that graph with a function rule. And we get these results of more people math-ing, doing the mental actions of actually math-ing.

[00:23:59.470] - Kim
Yeah, and once they've had that experience of the motion that they are moving. Tell me more about their beliefs about those kinds of problems. Like, does it feel to them more like that there's not steps to do?

[00:24:14.880] - Pam
Definitely that. So, it's an example of here is a bunch of static information that's sitting on the page. What's a function? You pick an object from this set and an object from that set, and you pair them together and you've created a function. Like, that is a definition of a function. But in that, nothing's varying. And we need both definitions. We need variables to... Well, we can get into that. But we need variables to be sort of the static thing, but we also need them, especially, for kids to understand the general basic ideas underneath calculus, so that then they can be successful. And we can do this as early as like 8th grade, maybe even 7th grade. Where the kids really get good at this idea that the variables are varying. Time is changing. And as time is changing, the distance from the sensor is changing. So, there's this sense of motion, of fluidness. We have quotes from Leibniz and... Oh, golly, who's the other guy that did calculus? High school teachers all around the world are like, "Pam, it's Newton." Newton and Leibniz, who created calculus at the same time, both use these words about how variables that they use the words like flowing and fluid. And we talk about this in our 9-12 book. For the Developing Mathematical Reasoning series, where we want to help students believe that variables can vary, that have this sense and the feel for covariation. That means that when they are writing the equation of a line, they're not just looking at a formula and plugging stuff in and saying, "What did that... Why did I do that?" But they're actually like, "I can write a function that represents my motion. The motion I did, I can use actions to do that." And the results are we get more right answers, But more importantly, we get kids who are thinking covariationally, which builds towards so much success in all the high school math.

[00:26:02.960] - Kim
Yeah.

[00:26:04.080] - Pam
Alright. Kim, that was fun. The Results Pyramid, everybody. We'll probably talk a little about that a little bit more coming up. Y'all, thank you for tuning in and teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutable.com. Let's keep spreading the word that Math is Figure-Out-Able!