Pam  00:00

Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris, a former mimicker turned mather.

Kim  00:09

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.

Pam  00:16

We know that algorithms are amazing human achievements, but they are not good teaching tools because mimicking step-by-step procedures can actually trap students into using less sophisticated reasoning than the problems are intended to develop. I did not do that in one breath. So embarrassed. Look out more, Kim.

Kim  00:36

I don't think anybody is judging you about how long you can talk.

Pam  00:39

I'm judging me! 

Kim  00:42

In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships. 

Pam  00:49

You have to wonder how many people are thinking right now, "Good gosh. If you say one more thing, one more time about saying that sentence all in one breath." So, I'll stop. I'll try to stop. 

Kim  00:58

Okay.

Pam  00:59

We invite you to join us to make math more figure-out-able. We hope you guys just enjoy the fact that we re record this every time. So, to make it interesting for us, you get our real. This is real. It's the way we are really. Anyway, Alright, Kim, be real.

Kim  01:16

We're going to be real today. (unclear). But before we do, we actually got a review a long time ago. And Mrs. Seaver, way back in July, said, "I love listening to the podcast and learning about math and teaching from Pam and Kim. The host keep the conversation informal, and they both bring different perspectives while sharing their common belief that Math is Figure-Out-Able. This podcast is professional development in my car and has helped me improve my own math skills and my teaching." 

Pam  01:50

Aww! That's awesome.

Kim  01:52

You know, I got to tell you. I totally saved... You know, I only have a few more left, but I kind of scour every once in a while and grab a few. But I saved this one on purpose for a time where we could like duke it out a little.

Pam  02:07

Oh, because it says that we bring different perspectives.

Kim  02:09

Yeah. And I don't mean "duke it out" like in a mean way. But, you know, there are lots of conversations that you and I have where... 

Pam  02:15

Are you saying we don't agree on everything? 

Kim  02:17

Oh, yeah. Yeah. I mean, we agree on the important things, right? But we're different people.

Pam  02:22

Yeah.

Kim  02:22

And so...

Pam  02:24

And we have different experiences.

Kim  02:25

Yeah.

Pam  02:25

(unclear) different perspectives.

Kim  02:26

Absolutely.

Pam  02:26

(unclear) some things. Different life.

Pam and Kim  02:29

Yeah.

Kim  02:30

And if it's something really meaty. Like, one of the things that we've always loved is we take the time to like really beat things out. I know there's other partnerships out there or groups where they have... You know like, one of our Journey members, Cathy, one of our JourneyLEADERs. She's got a tight knit group of people that she works with, and they ask really good questions of each other, and they beat stuff out. And, you know, there's some really nice groups that that we interact with who have a person that they have heavy math conversations with. So, I mean, we do that. And so, on the outside, it probably looks like, you know, it's a completely 100% everything's the same. But we get there over time. 

Pam  03:13

Yeah. And there's maybe still a few things that we're still working on, working out, like for the best words and the best way to say things. And I think we're going to do that today. However, I do want to mention that she also said that it's "an informal conversation", which I like that as well. 

Kim  03:28

Yeah. 

Pam  03:29

Alright. So, Kim, poke away. 

Kim  03:32

So, not long ago, actually, in our work messaging system, I sent a message really aimed at you but including some other people.

Pam  03:44

I didn't know that. You tagged more than just me, but okay. 

Kim  03:46

I did, I did.

Pam  03:47

Alright, alright.

Kim  03:48

An audience for our banter. I said, "I don't know where this goes. Real math and fake math has always bothered me as phrases because it's not the math itself that is real or not. It's the way we tackle solving problems that's good or not. Mathing is probably not my favorite, but I can hear mathing versus step-by-step procedures. Is there a world where we leave behind real and fake? You're either mathing or you're not." And I say that because "real math" and "fake math", we both said it. It's been a part of the conversation and in a million presentations. And I appreciate you let me poke you publicly about that. 

Pam  04:28

Yeah, so just to be really clear. For quite a while, we've been talking about math being figure-out-able and not being rote memorizable. And no metaphor is perfect. But I've used the the terms. We really want to do real math and not fake math. And you're poking a little bit saying that... And for a while you've said it. You don't really like to talk about real math and fake math because it's math. And our our bent is to try to help the world realize what mathing means.

Kim  05:01

Yeah.

Pam  05:01

What mathematicians actually... The mental actions mathematicians actually do when they math.

Kim  05:06

And I think the words are important here. And I'll let you kind of share kind of a history, maybe, if you want to about how you settled on those words. But I think the words here actually matter more than we've really considered because I can hear where really fine educators and mathematicians hear "real math" and "fake math", and in their heads go, "What are you talking about? Math is the same. Like, this is where you guys do all this stuff, and like it's all about, you know, like whatever kids want, however they want to do it." And that is not at all what we would propose. 

Pam  05:42

In other words, you're saying people might hear me say "real math" and "fake math", and they might think, "Oh, you're proposing that it's fuzzy and kids don't actually have to learn math, and that we're just going to let them kind of feel it, do whatever comes naturally to them, and that they can solve it however they want." And it becomes this... Golly, where we're rejecting the mathematical traditions.

Kim  06:07

And we're watering down heavy duty, complicated, complex, solid math properties, and (unclear). 

Pam  06:17

We're being accused of that. We're being accused or we could be (unclear).

Kim  06:20

Well, it we could be, yeah, yeah.

Pam  06:21

We could be accused of doing that.

Kim  06:22

Yeah. 

Pam  06:22

And we would suggest that we actually do quite a bit of really sophisticated content. That when we talk about developing mathematical reasoning, it's not some fuzzy, just think better. No, it's reasoning mathematically. Which means you more than own content. You're able to use it and reason logically in an internal, consistent system. Yeah. So, I completely hear that a mathematician, somebody uses math, an engineer might say "Real math and fake math. I'm looking at a student mimicking an algorithm. That algorithm is real math. You can't call that fake math." Is that kind of what you're saying? 

Kim  07:05

Yeah. That it's not the math itself. We're all talking about the same mathematical ideas.

Pam  07:13

We're not calling an algorithm fake. Or at least...

Kim  07:16

Well, we have been. 

Pam  07:19

Okay. And I think maybe that's your problem.

Kim  07:21

Yeah. And I think that we have not said algorithms are fake. We really are saying "real math", "fake math" but what we're talking about actually is the teaching, the fake understanding, the lack of numeracy, the mimicking that's happening. It's not the math.

Pam  07:43

Yeah, depending on how you define "math". Let me tell you why I start. Or at least one place. When you say, "Can we stop using those phrases in like our marketing and the stuff that we put out?" Here's the one place that I'd love for you to help me find better words because I really... This happens to me very often. I run into people often who say, "What do you do?" I got a massage the other day. I was on the plane the other day. Where else was I... You know, I'm randomly talking to somebody at church, and someone says, "What do you do?" And I say, "I'm revolutionizing the way we teach math." And they say, "Oh, math." And then they try to, you know like... Not everybody, but a lot of people kind of scoot aside a little bit, or they look kind of sad. You can almost see some people's adrenaline kick up when, you know, kind of math anxiety kicks in or something. And when people say to me, "Math. I was never good at math. I like to be able to say, "You were never good at... Like, when you were a student?" And they're like, "Oh yeah. It's terrible. I hated it. Whatever." I like to be able to say, "You know what? I bet you were actually bad at fake math. I bet that thing that you're thinking about that was probably fake math. I bet you're actually pretty good at real math." I like to be able to say that. Now...

Kim  08:55

Ugh. 

Pam  08:56

...you hate that. 

Kim  08:57

I don't like it.

Pam  08:57

You don't like it. I tell you what, people perk up, and they go, "What do you mean by that?"

Kim  09:03

Yeah.

Pam  09:03

So, I've thought about that a little bit. And I don't know that it would have the same punch, but I suppose I could say something like, "Ah, I wonder if your experience was that you were actually fake mathing..." See, I just don't know if that would play. "...and that you're actually pretty good at real mathing." So, whether that would play well or not, would you buy that? Would that be better? 

Kim  09:27

I think that you are either mathing or you're mimicking procedures. And there may be some gray in between. But I think if you are really mathing, you're cognitively involved deeply in the math. "Mathing" is not my favorite either. I'm not sure what the word is that would super satisfy me. And maybe there's not a word. Maybe we'll never settle on a word because it's an action that we're looking for. 

Pam  09:55

It's the mental actions that mathematicians do when they do what they do.

Kim  09:59

I think that what people don't love is the math that happened in school. I think that...

Pam  10:07

The math that happened in school or the mathing that happened (unclear). 

Kim  10:10

Well, I don't think they were mathing. I think that they were... I don't think... When I grew up, I don't think... Well, I was, so that's weird to say. But I was mathing in my head. I was mathing in my personal life. But in school, I was mimicking. I think if you're truly mathing, you are involved in it. You're not mimicking someone else's procedures. I don't think there's real mathing and fake mathing. I think if you're mathing, it is like you embody, you think, you're using relationships. Mathing, to me, is the word that we use when we... Like, that's the goal. I want people mathing

Pam  10:50

And not just on their own, but like we're deliberately trying to develop. 

Kim  10:54

I think we want to foster mathing in school.

Pam  10:56

And foster more sophisticated mathing as we grow and use more sophisticated relationships, build our brains to think.

Kim  11:04

I think that's the best that I can mess with right now. There might be a better word than "mathing". It's hard to describe. You know, I know you're looking for a quick thing because you got just a few seconds with somebody before, you know, they (unclear) recoil from the math conversation. But I think that what might be nice to say to people on the plane is, "What you did in school is not what math is really about. It's not... That's not the beauty of mathematics, and I'm sorry you never got to experience that because of the way you were taught in school."

Pam  11:45

I'm thinking. Yeah, deep breath.

Kim  11:48

The beautiful thing is not only can we not settle on something because we promised we wouldn't.

Pam  11:55

We told the team we wouldn't walk away with any hard and fast rules that we were going to follow from here. 

Kim  11:58

No. But, you know, this is a meaty conversation because I think when you are in the public eye, people want to understand what you're saying. And I understand the pull that you feel to like have a phrase that really embodies what we're talking about. And I think it's hard to wrap everything you're trying to say into a couple phrases. So, I understand why real math, fake. You know, it's short phrases that capture people's attention. I just... I hesitate on attaching it to like mathematics. 

Pam  12:34

Yeah, and just a couple things I'll say. I've been traveling quite a bit and working with people who speak other languages as their first language. And "mathing" is also tricky in other languages because there are other languages where you can't take "mathematics" and shorten it and then make it into a verb. 

Kim  12:54

Yeah.

Pam  12:54

It kind of doesn't work the way that it does here. So, that's one thing I'll mention. Another part of this conversation for me is the idea, or at least it's correlated for me, is the pushback that we've gotten from mathematicians. It's very polite. I've had some fantastic conversations with mathematicians who say, "Hey, why are you dogging algorithms?" And then we have a great conversation about I'm not dogging algorithms as an algorithm, as good mathematics, as a generalization that's phenomenal. We even say in the intro that it's an amazing human achievement.

Kim  13:27

Right.

Pam  13:28

What I'm doing is I'm suggesting that they are not good teaching tools.

Kim  13:32

Right.

Pam  13:32

And that depends on what our our purpose of math class is. If the purpose of math class is to get kids mimicking, rote memorizing, mimicking, so they can get answers, then, okay, use algorithms. But if the purpose of math class is to develop students who are mathematically reasoning, reasoning mathematically...and maybe we call that "mathing"...and developing their brains to reason more and more sophisticatedly, then algorithms are terrible tools to get that to happen.

Kim  14:00

Right. 

Pam  14:01

I think a tricky thing, just staying on that vein for just a second, is if you are a mathy person, and when you saw a teacher present an algorithm, the steps of the algorithm, there's a good chance that you saw relationships and connections, you noticed patterns, and you mathed. You did the the mental actions that mathematicians do while that presentation of the algorithm is happening.

Kim  14:28

Yes. 

Pam  14:28

And so you then equate that those two things that you you did the mathing you did because the teacher was showing you those steps. 

Kim  14:38

(unclear).

Pam  14:39

Yeah, and I've talked to quite a few people where...

Kim  14:40

No. Okay.

Pam  14:41

This is absolutely true. That then it becomes this very tricky conversation to have because people are saying, "But that's how I learned to math. Like I do math. Pam, I do those things that you talk about that are the mental actions mathematicians do, and I learned those while the teacher was showing me, because the teacher was showing me the steps." And I'm asking respectfully, to see if you can parse those things out and wonder if you actually could have learned even more faster if we would have put the patterns that you kind of naturally found on your own that were sort of hidden behind the scenes, if you could have developed those faster, and better, and more sophisticated, and stronger if we would have actually just done those patterns and relationships not make you find them in the midst of the algorithm?

Kim  15:32

I actually wonder... And you can tell me more about it later. 

Pam  15:36

Yeah.

Kim  15:37

I actually wonder if people who say they did mathy things while they watched their teacher present the steps, I wonder if they couldn't follow, didn't understand why, and so they kind of like set that aside while they watched. And thought. 

Pam  16:00

I think it's highly probable that some people did that. I think it's also highly probable that others literally were like, "Oh, wow. Look, that's a 10, and so it goes over there, and it became a 10." Like, they saw more than what... 

Kim  16:12

What was happening. Okay, alright.

Pam  16:13

What might have been being presented. 

Kim  16:15

Yeah.

Pam  16:15

Or maybe they had a teacher who actually presented the algorithm, you know, with a lot of conceptual understanding. Which most students wouldn't have been able to follow.

Kim  16:22

Right, right, right. 

Pam  16:23

But they had enough already. For whatever reason. Maybe they're just naturally had made connections. Or they'd had a lot of experiences before that. That then when the presentation was being made, they were able to see, make connections more than 98% of the other kids that were in the room who just bought that what math means is to rote memorize and mimic these procedures. 

Kim  16:48

Yeah.

Pam  16:49

Yeah, so when we talk about the mental actions that mathematicians do, I wish we had some phrases for that. What do we use here? Reasoning mathematically. And then I'm very careful to say which includes content because it's not just this fuzzy reasoning better, thinking better. It literally is I'm reasoning mathematically. That means I own the multiplication facts. But I don't just have them, rote memorized. I own them deeply. I have connections between multiplication facts. So much that there isn't almost a category of single-digit multiplication facts in my head because I'm reasoning multiplicatively about single, multi-digit, double-digit, like fractions. Like, so it's almost an insignificant or a not consequential distinction to talk about single-digit multiplication facts because I'm reasoning multiplicatively, so that includes all kinds of numbers. 

Kim  17:44

Yeah.

Pam  17:46

Yeah. 

Kim  17:46

Well, I think... You know, you and I have talked before about how much language tends to get in the way of what people mean. You know, we've talked, asked each other. You've said that before. And, you know, I think that a lot of people are saying some really nice things, but using different words, and so it feels like different messages. And I think sometimes people, like us too, who have used some phrases that maybe make people question. And so, I think if we're all educators, all people who are talking about math and teaching, are willing to think about the words that they're saying, and how it can be heard, and the message that it might be sending. You know, sometimes we don't get a chance to really... Not everybody's got a podcast, doesn't have a chance to clarify, to get questions, to get pushback, to say, "Well, this is what I really mean." I think the more work that we can do to continue to clarify, and I love the people that ask questions so that we can, then I think we're moving forward to making real progress. And I'm still hung up on "real math", "fake math".

Pam  18:56

I'm going to add one tiny thing that I think is actually a major component that would help us continue that clarification. 

Kim  19:04

Yeah.

Pam  19:05

Is that so many of the people that are kind of pushing that message that you just talked about. And we support that. I would invite you to bring the mathematics into your conversations.

Kim  19:18

Mmhm. 

Pam  19:19

I think there's a lot of people out there that are using words, like we just said, are tricky, and have different meanings, and different connotations. And so, we need to have the mathematical examples. We need to actually do the mathematics we're talking about, so that then we can refer to those as the mental actions that mathematicians do. And now, we have a common experience that we're working from as we're defining those terms. 

Kim  19:45

Yeah, yeah. 

Pam  19:46

And I think that could be really important. Hey, I want to go back to "real math", "fake math" for just a minute. 

Kim  19:49

Do we have to?

Pam  19:50

Yes. 

Kim  19:51

I'm just kidding, haha!

Pam  19:53

So, when I run into somebody who's like, "Oh, yeah. I hated math. I was never good at math. I couldn't do math." Blah, blah, blah. Could I say something like, "I wonder if that thing that you're talking about right now that you weren't really good at, I wonder if that was presented to you in a way that isn't actually what mathematicians actually do, but that we could have done with you. We could have helped you learn what mathematicians actually do. And I bet you're actually pretty good at that." And maybe I'll follow that with... And I didn't say this earlier, but I usually then follow that with, "Like, how do you think about something like 99 plus 37.

Kim  20:28

Yeah, you do that (unclear). 

Pam  20:29

Yeah, so I'm going to follow the advice I just gave, which is I don't stop the conversation by just saying those words. I then say, "You know like, how do you think about 99 and 37?" And I tell you what, Kim, a ton of the time, people go, "Well, you know, it's just 136. But, Pam, those are like tricks I learned because I couldn't do the real way. And, you know, those are just like..." Or they'll say that, and they feel bad about those quote, unquote "tricks" that they just came up with themselves. Or they'll say, "Well, you would line up the 99 and 37, and you would do the..." And I was like, "Oh, yeah." Okay, so yeah, you you bought into the myth that math is a disconnected set of facts to memorize and rules and procedures to mimic. It's actually not what mathematicians do. It's not the mental actions that mathematicians do. And good news! You can learn to do that." Would that fit better for you if I had that slightly longer conversation?

Kim  21:18

Well, I'll certainly think about it, you know, because I'm not sure if...I almost said "run of the mill people"...everyday people care about what mathematicians do if they don't already know what you mean by that. But I think...

Pam  21:32

Yeah.

Kim  21:32

You know like my mom, I'm thinking about. Like, my mom does a lot of math in her life. Like, every day, does things with math and probably wouldn't consider that math that she's messing with because it wasn't what she was taught in school. 

Pam  21:46

Doesn't look like a bunch of x's, and lining them up, and... 

Kim  21:49

Mmhm. Or, yeah, yeah, steps to do. So, we're going to keep talking about it, though, right? 

Pam  21:53

Well, alright, let's keep talking about it. 

Kim  21:54

Okay, so we're super excited because we've got the Math is Figure-Out-Able challenge coming up. 

Pam  22:00

Dun, dun, dun!

Kim  22:01

And last week we announced that when the challenge was going to be challenge week, we shared that Jo Boaler is going to be our very special guest. And we are super excited about that. 

Pam  22:11

Super excited that she's willing to come on and be our guest, talk all about her Math-ish book. Yeah, we're super excited. Thank you. Thank you, Jo. And can't wait for that to happen. 

Kim  22:20

Yeah.

Pam  22:20

And so, we also have another announcement today that we're going to have a second special guest.

Kim  22:27

Yay! 

Pam  22:27

Woo-hoo at Deborah Peart is going to join us. We're super excited. Deborah Peart could have been in on this conversation because she says "Readers, read. Writers, write. And mathers, math." She even has t-shirts that says that. Yeah, so we're super excited. Her site is My Mathematical Mind. She's one of the authors of the Illustrative Mathematics K-5 curriculum.

Kim  22:51

Yeah. 

Pam  22:52

We're super happy to have her come on and share her mather perspective on our challenge. Ya'll, if you have not joined a Math is Figure-Out-Able challenge, check out the next one that's coming up. We do them several times a year. We would love to have you as a  guest in our free Math is Figure-Out-Able challenge. Those are super, super fun. Alright, Kim. Well, maybe someday we'll decide how we're going to get our language just right.

Kim  23:18

(unclear). We'd love to hear from other people about this too. 

Pam  23:21

Yeah, ya'll, communicate with us. Especially in the Math is Figure-Out-Able teacher Facebook group. We're on any social media channel. We'd love to hear what you're thinking about, words to describe what we are all heading for, which is helping students reason mathematically. Alright, thank you for tuning in and reasoning more... 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. And keep spreading the word that Math is Figure-Out-Able!