Intro

Pam Harris 0:02

one of my favorite things to do is to ask how do you think about this problem? 99% of the time people will go, well, you know, that's how you think about it, but you're supposed to do the right way. And I'm like, what if we could teach all math the way you just logiced your way through that one. Algorithms amazing human achievements. They're just not very good teaching tools because, people like me can get trapped in them and believe that math is something that it isn't.

Vanessa Vakharia 0:26

Okay guys, it's me, Vanessa, and I could not be more excited for you to hear today's episode Okay, I know I say that every single episode is gonna change your life, but this one is seriously going to change your life. Because the conversation I'm sharing with you today is not just about how we can improve math education, but literally about how we can improve the way we all talk to each other as people. The first time I actually had a conversation with Pam Harris was literally an hour long argument because we were both guests on the Debate Math Podcast, so we were debating, we were actually discussing whether students should be able to use algorithms or tricks. See, Pam is the founder of Math is Figureoutable, and by that she means that students can make sense of math through reason and shouldn't be taught to rely on memorization or shortcuts. I, on the other hand, just don't like believe in black and white rules or thinking, and I believe that students should have access to whatever tools work for them because everyone's different. Now on social media, most arguments would just end there, right? Like, you're wrong. No, you're wrong. Okay, cool. The end. So I wanted to have Pam on to really dive into these big questions with nuance and truly get to the root of what we each believe. Spoiler, we have more in common than we initially thought. This convo is a must listen for anyone passionate about teaching and learning math. We explore not just where we differ, but also the shared goal that unites us, which is helping students truly think, reason, and understand, rather than just memorizing steps or like spitting out answers. I hope this episode inspires you to have deeper conversations with every single person in your life because it certainly inspired me.

Why are algorithms so controversial?

Vanessa Vakharia 2:06

I was trying to explain to David how I knew you and I was like, you know, I didn't really know Pam until Chris and Rob got us on their Debate Math Podcast. So our first real interaction was a debate, which isn't like the best way to form like a, a relationship with someone. You would think, especially because our debate was a little heated, which if you guys haven't listened to it, go to the Debate Math Podcast, listen to me and Pam's first conversation via debate, about algorithms, which we're obviously gonna talk about later. But my point is that like you would think after like a debate where two people are disagreeing, we wouldn't like become friends after necessarily,

Pam Harris 2:40

Sure enough,

Vanessa Vakharia 2:41

No, but we kind of did.

Pam Harris 2:43

Well, I I maybe we both went into it, I think with, the best intention, desire to have a conversation. We listened to each other and we spoke back to each other's points. And we didn't just have our, you know, our thing that we were gonna say, and I'm not listening to you, whatever. And I think it was a real conversation.

Vanessa Vakharia 3:03

For some reason, whatever it was, I didn't think the debate was going to be as like pleasant as it was.

Pam Harris 3:10

Huh?

Vanessa Vakharia 3:10

I, that's all I can say. Like it had nothing to do with you.

Pam Harris 3:13

in your face or something like,

Vanessa Vakharia 3:14

I mean, maybe just the term debate and also just because, I don't know. I've heard so many great things about you. I think maybe I was a bit intimidated, like maybe I was like, I don't know if like, your brand Math Is Figureoutable is really based on the idea that algorithms should not be used as teaching Tools and we can figure out math, right. Would you say

Pam Harris 3:35

those so

Vanessa Vakharia 3:35

Yeah. Flip it?

Pam Harris 3:36

emphasis is we can

Vanessa Vakharia 3:38

Yes.

Pam Harris 3:39

It's not rote memorizable, it's figureoutable. And then there's a sort of corollary that, a thing that follows. If we are about figuring out math, if, if, if it means that we can actually teach kids to logic their way through problems, which helps them create mathematical connections and relationships, which then helps them logic through new problems, which then helps them create new mathematical relationships, the algorithms can get in the way of that and it can actually trap students into not knowing that we can do that, that that's, that those are actually the mental actions of a mathematician. Algorithms aren't evil. They're amazing human achievements. They're just not very good teaching tools because, I'm pointing at me, people like me can get trapped in them and believe that math is something that it isn't.

Vanessa Vakharia 4:28

Okay. I wanna get into this so much, and I wanna say just as to wrap up the, the what I said before, so it doesn't seem like I thought you were some mean person, that's not it. I really admire you, I think there are so many cool things about you, and I think this whole philosophy is something I'd never really considered before. But more than that, I hadn't seen it in action. And I want people to get a little taste of what you're talking about. But first I was explaining this to David, okay. And he said to me, you and your guests keep talking about algorithms. So like, what is an algorithm, exactly?

Pam Harris 4:59

These are excellent questions because we as a mathematics education community have gotten muddy and we use words to mean different things, and I don't know that we know that. a true algorithm means I can you any numbers, throw 'em in, you follow the steps, all the steps every time, and you will come out with the correct answer. It took humankind until we had the, uh, Roman Arabic numerals to get to where we were in about seven 800 AD when Al-Khwarizmi created both algebra and he wrote a book that then, um, he created our, at least precursors to our digits focused algorithms. So first we needed the, the numer system that we have today, and then he said, Hey, we could actually align these numbers up and just focus on digits. You don't actually have to know what's going on at all. You just have to be able to do these steps in a certain order. Start with the smallest digits, and then the next digits, and then the next digits. Without even really understanding what's happening or the numbers involved, you can just do a bunch of steps, you'll always get the correct answer. Well, that was an amazing, historic step for mankind because before that, the only people who were able to do lots of digit computations people who had money. were in the sort of intelligentsia, they were the upper crust that could afford to send their kids to school to learn to, to work an Abacus. So when Al-Khwarizmi created the algorithms, it was almost like Gutenberg's printing press. Like Gutenberg's printing press is when he was able to print the Bible and gave the word to the common man. And all of a sudden the common man could read. And we had the Renaissance. Well, Al-Khwarizmi's algorithms were very similar. All of a sudden you didn't have to go to school to learn to work an Abacus. Could use an algorithm and all of a sudden the common man could compute. So hear me clearly, the algorithms are amazing, historic achievements. They are amazing that we came up with them. We can do better.

Vanessa Vakharia 7:00

Okay, but hold on. I need an example of an algorithm. Like give us an ex, like do you know what I mean? Like what's, give us a one that we all would kind of be,

Pam Harris 7:08

99 plus 67. The algorithm would say, line those numbers up. So 99 plus 67, I gotta even picture this in my head. So 99 67, that means there's a seven and a nine in the smallest. Those are smallest numbers, right? So we would add the seven and the nine together, we know that 16. We'd write down the, the six of the 16 part, carry the one or the 10 depending on how well you understand what you're doing. And then I would add that second column together. and then notice I've written the sum of the smallest numbers down and now the sum of the biggest numbers down. And then I would to read the answer, I have to go the other direction.

Vanessa Vakharia 7:43

So let me just make sure I understand. The algorithm part of it is the, can I call it a rule? The rule that when we add things, we line up the digits on top of each other in order from smallest to largest. We line up the ones, we line up by place value, and we add them vertically. That is an algorithm.

Pam Harris 8:02

Yeah. And then you have to regroup, then you have to carry

Vanessa Vakharia 8:04

Yeah. Like the, the whole, the whole thing of it would be like, that is

Pam Harris 8:07

our traditional addition algorithm. Notice I called it the traditional addition

Vanessa Vakharia 8:11

yep.

Pam Harris 8:13

because people will call things standard algorithms, and I'm gonna push on that hard because the way that my mom, who grew up in Switzerland, learned to multiply and divide, doesn't look anything like our traditional multiplication and division algorithms. I was just in Croatia, um, last winter. Their division algorithm looks nothing like our long division algorithm. Uh, many Latin and South American countries, we, they use a different subtraction algorithm than we do. So we have to be careful that we don't assume that our traditional algorithms are standard.

Vanessa Vakharia 8:44

I, It's funny 'cause David just held up a question to me, which he very rarely does, he said, why is she calling that an algorithm as opposed to just doing addition? And I think, so tell me if I've understood you right. I think it's called an algorithm because there's a specific rule about how you're supposed to do it and a pattern you're supposed to follow.

Pam Harris 9:04

Well, it's a pattern you have to follow every time, no, no matter what the numbers are. So let me ask David this question, the problem I used before was 99 plus 67. if you think the definition of addition is to follow that algorithm, you line it up, you add the numbers, and then you're like, duh, that's what you do. what if I were to say something like 99 Do you get an urge right now?

Vanessa Vakharia 9:26

Exactly, I, because I've, because I've seen Pam, I'd be like 99 plus 67. It's a hundred plus 67 minus one, right? It's 166. I can just round that 99 up to a hundred.

Pam Harris 9:37

face right now. Like, what is

Vanessa Vakharia 9:38

Oh my God,

Pam Harris 9:39

Yeah,

Vanessa Vakharia 9:39

I, here, David, he's like, no, he's thinking about it. Well, but why is that not an algorithm, but because that's, okay. So now he, here's his question. Why is that not an algorithm?

Pam Harris 9:49

Because I wouldn't want to do that strategy for something like 32 plus 67. I wouldn't wanna go 33 plus 67 minus one.

Is addition an algorithm?

Vanessa Vakharia 10:01

Can I try, can I chime in here and tell me if, well, if what I've learned from you is accurate, I think what Pam might say, even though I could just ask her, but I wanna see if I've learned, is the point is if you understand what addition is as a concept, you can, uh, change your strategy based on the numbers. 'cause you understand what adding means. So you could be like 32 plus 67, it makes more sense to do it this way. 99 plus 67, oh, I'm gonna do this because I understand the concept of addition and I can draw on all this prior knowledge instead of being like, no matter what the numbers are, I have to line them up. Am I right?

Pam Harris 10:36

Yes. And, and

Vanessa Vakharia 10:38

You're so kind.

Pam Harris 10:40

No, no, that was very well said. I just wanna add in a little bit of an additional thing. So a lot of really good people would say, if we really get some conceptual understanding down, then then good, you're, you're good to go. And I'm gonna say there's actually a small set of major relationships that lead to a small set of major strategies that we need kids to own. Or when we say, well, you understood addition, go ahead and add 'em any way you want. They're gonna be like, I don't know what to do. they don't own enough relationships. We need to help them own. So the, the strategy you just used was, I'm gonna start at 99, I'm gonna go to a hundred, add the 67 and then back up one. great strategy. But another kid might say, I'm just gonna give one from the 67 to the 99. So now I have that a hundred that you had, and I'm gonna add what's left over the 66. Those are slightly two different ways of thinking about the problem that will be important when the numbers get done as, as trivial. So 99 and 67 I use because, and we're, we're on a, we're on a podcast here, so we're trying to make sure people can,

Vanessa Vakharia 11:40

I was like, Pam, don't talk too much about math. It's been 10 minutes,

Pam Harris 11:45

so, so people might hear, well, of course you can do that. Pam, it's 99. Well, okay, but what about 98 plus 67?

Vanessa Vakharia 11:51

but wait, so why aren't, but now I'm kind of with David because now you've gone into, there's a bunch of different, like, kind of,

Pam Harris 11:58

a small set. It's a, it's a very small

Vanessa Vakharia 12:00

But how are those not algorithms?

Pam Harris 12:02

because again, I'm gonna let the numbers influence which of those small set of relationships I'm gonna use that,

Vanessa Vakharia 12:10

Hold on, let me, uh, let me think here.

Pam Harris 12:13

So I have a son who would actually agree with you right now. He would say, all of your strategies are algorithms, mom. It's just that for many they won't work well.

Vanessa Vakharia 12:22

Right.

Pam Harris 12:23

One of the relationships that I would develop in kids, I call the Over strategy. Um, I got this from my co-host Kim, uh, on our podcast Math is Figureoutable podcast, where if she was adding something like, and I'm gonna write this down 'cause I don't hold numbers in my head, all that great. So I'm writing down 23 and plus 49, um, 23 plus 49. I might think about, I could line those up and I could do the thing because I can do that for any problem. I could also think about what do I know about 23 and 49? Well, 49 is pretty close to 50. Well,

Vanessa Vakharia 12:55

50.

Pam Harris 12:56

I know what 23 and 50 are. So 23 and 50, that's 73. I didn't want 23 and 50, I wanted 23 and 49. So it's gotta be one less, it's gotta be 72. So that's an, that's an over strategy that works really well when you're adding 49. That's, that's a, it's a good strategy. What, what if you were adding 89? What if you were adding 149? What if you were adding 999? What if you're adding 3,496

Vanessa Vakharia 13:25

we have to stop. We have to stop.

Pam Harris 13:27

or, or even better, 3,996. But 3,996 is so close to 4,000. No, hang with that one. 3,996 is so close to four. Come on, you can do it. Vanessa.

Vanessa Vakharia 13:41

on.

Pam Harris 13:42

if I had some random four digit number

Vanessa Vakharia 13:45

I know, I'm sure. Yeah. Okay.

Pam Harris 13:47

was

Vanessa Vakharia 13:47

Okay.

Pam Harris 13:48

it, 4,996, what if we, 4,996? What if we

Vanessa Vakharia 13:52

I would add four. I would put four.

Pam Harris 13:55

So,

Vanessa Vakharia 13:56

then I would take four away from whatever the answer is.

Pam Harris 13:58

You're now thinking about, what do I know about 4,996. Ooh, it's really close to 5,000. I could use that. Notice that, that, that thought process is,

Vanessa Vakharia 14:09

Okay,

Pam Harris 14:09

think about the numbers. Let me actually consider the numbers here

Vanessa Vakharia 14:13

okay. Okay.

Pam Harris 14:14

I dive and choose what to do. That's the crux of the difference between a strategy and an algorithm.

Learning by thinking, not following

Vanessa Vakharia 14:22

I think you just, I think you just, something clicked, even though literally what happened is you threw a bunch of numbers into the ether and that some No, it was, it kind of worked because I think, I think something you said just clicked where you said the difference is that you're thinking about the numbers. There I think is it. Is that with a traditional algorithm of just line up the digits by place value and add, there is zero thinking. You are simply following. If however, you have a boatload of, not a boatload, but you've got a range of strategies to choose from that require a small set that require your prior knowledge and thinking and you get to be empowered to say, I'm gonna choose this because I know this, all of a sudden the student is in the seat of power, they're thinking and they're not just following and now it is no longer simply an algorithm. Did I get it?

Pam Harris 15:12

And they're using their agency. I'd throw that in. Yes.

Vanessa Vakharia 15:15

Their agency. I love it.

Pam Harris 15:16

have agency, they get to choose, they get to, they, they get to be

Vanessa Vakharia 15:19

I got to choose

Pam Harris 15:20

they get to be influenced by the numbers. And then say, for this, for this problem, I think I'm actually, how about No. Ooh, this, I have, I have, I have my, my oldest son, I was teaching high school math, knew what I was doing. Vanessa, I could tell you every, give me a high school problem, I knew the algorithm, I knew the steps well because if I didn't, I would have major imposter syndrome. So I knew that in order to be a high school math teacher, I needed to know all the things. But what I knew was to recognize which algorithm to do and then be able to do those steps. What I didn't ever do was let the problems influence how I would attack the problem. My young son was growing up and he would logic his way through problems. I literally went to him one day and I was, I just had a conversation with my kids' teachers who were saying, Pam, we're kind of concerned that our kids don't know facts anymore because we've been told not to give him time fact math and so we stopped 'cause we don't wanna give him math anxiety. So we stopped giving him time fact math tests 'cause we don't wanna give him math anxiety and now they don't know their facts anymore. Is that important? And I was like, that's important. Like kids need to know their facts. I went home to my own kid, but, but just like you, I don't wanna give kids math anxiety. So this is,

Vanessa Vakharia 16:31

Right.

Pam Harris 16:32

was fussing with this. I'm a high school math teacher. Think I know everything. I know it sounds so arrogant, but y'all, I had to be that almost to be able to survive in that, that place. Anyway, I went home to my own kid and I'm thinking in my head, do you know your multiplication facts? And I, what I asked him was, do you know your fives? I just picked one. And he said no. And I was like, I'm sorry, what? Like, dude, thou shalt know thy facts. Like this is I, I, as a math teacher, I panicked like my own personal kid has to know his F like he's got. So in that panicking, he goes, mom, mom, mom, you don't have to know your fives if you know your tens. And I said, what? And he goes, you don't have to know your fives if you know your tens. I was like, dude, I heard what you said. I don't know what you mean. Like Vanessa, at that point in my life I could not follow. Now maybe some of your listeners are like, duh Pam. Well you can look down on me all you want, but I am the product of the current system.

Vanessa Vakharia 17:26

Yeah. Yeah, yeah.

Pam Harris 17:27

I was a very successful high school math teacher who had no idea what he meant. Now many, your listeners are like, I don't either, Pam tell me. So I said to him, give me an example. said, okay, let's say that you don't know five times nine. Well do you know 10 times nine? That's 95 is half of 10. So half of 90 is 45. To which I said, good heavens, that is nine times five. Does that work every time? Give me another example. he said, okay, what about five times 23? Vanessa? I'm thinking about single digit facts and he's playing around with a pattern with relationships that can find five times 23. So in that moment I was like, wait, okay, 10 times 23 is 230 five is half of 10. So half of 230 is 115. And then I literally had to say to myself, is that five times 23? I did the algorithm to make

Vanessa Vakharia 18:17

Yeah, yeah,

Pam Harris 18:18

but then I thought if you have 10 groups of things and you cut it in half, you'll have five groups of things. And times 10 is so brilliant in our Base

Vanessa Vakharia 18:26

yeah, yeah, yeah.

Pam Harris 18:27

that it makes it a relationship worth knowing. Does that, does that track?

Vanessa Vakharia 18:32

It totally tracks. I love this story, by the way.

Pam Harris 18:35

I'm gonna suggest that that relationship, that five is half of 10, is one of the small set of major multiplication relationships that we need to build in kids. Like I could've thought that way, if I knew it was a thing, if I'd had a teacher say, Hey, it's sort of important that you know five is half of 10, and look how you can use it, would've used it. Like I could have let that influence how I solve things, but nobody ever shared

Is "natural ability" a thing?

Pam Harris 18:56

that with me. So Vanessa, my work is interviewing, studying naturally math people and going, wait, wait, wait, wait. How do you do that? Oh,

Vanessa Vakharia 19:05

What's a naturally mathy person?

Pam Harris 19:06

Ok i'm gonna suggest, and I know there are good math minds out there, math education, people who would disagree with me on this, and I would love to have the respectful conversation with them. I think we all have natural talents. you're

Vanessa Vakharia 19:19

Okay.

Pam Harris 19:20

the video, you'll notice there's a basketball behind me.

Vanessa Vakharia 19:23

Yeah.

Pam Harris 19:23

a baller. I played semi-pro basketball in Switzerland. the game. Um, I, I, I think I have some natural talent. To play that game. But I gotta be honest, there were people who were much better than me. One of the reasons why I was any good is I worked blooming hard at it. So I had some natural talent. I put a lot of effort into it. I think my personal kid, the story I just told, has some natural talent to run into a low dose of mathematical patterns and pull them together make new relationships in his head. And now that he owns those, pull those together, uh, he ran into the same low dose of patterns we all do. And for whatever reason, he had both the, the sort of talent and the interest. Like I think both of those go together. If you have the talent and the interest, I think pull out a favorite athlete of yours. Ooh, Alanis Morissette. Did I get a name? I think she has some natural talent.

Vanessa Vakharia 20:22

Dying

Pam Harris 20:22

She's

Vanessa Vakharia 20:23

fam. I love you an athlete,

Pam Harris 20:27

Well, okay, because, so, so we should probably tell everybody, you introduced me to Alanis Morissette. You were like comparing I I hadn't heard of her before.

Vanessa Vakharia 20:36

but you know who she is now, right? Do you know that song?

Pam Harris 20:38

no, not really. Sorry.

Vanessa Vakharia 20:40

like rain on my wedding day.

Pam Harris 20:43

to it. Now I know it.

Vanessa Vakharia 20:44

You've never heard that song?

Pam Harris 20:45

before then. I don't think so.

Vanessa Vakharia 20:47

Really? Okay. No problem. Go on.

Pam Harris 20:49

me

Vanessa Vakharia 20:49

It's fine.

Pam Harris 20:49

ignorant. Maybe that's probably

Vanessa Vakharia 20:51

No, no, no, it's fine. It's totally cool. Yeah. I'm not trying to shame you.

Pam Harris 20:54

has natural talent and drive and eth work ethic and, and when after interest she's interest in it. So I'm gonna suggest there are naturally mathy people who a, have maybe some natural talent or, or have enough natural interest that they go for it. And either one of those is enough for them to sort of take interest. Here's the rub, not the rub. Here's my point. We can all be mathy people, but we have to know what the actual mathy thing is. I

Vanessa Vakharia 21:25

Oh, boy. Okay. Yep.

Pam Harris 21:27

I want, I want we, everybody can do more real math than fake math. I know you hate those words. Sorry.

Vanessa Vakharia 21:34

Oh my God. Okay, so there's so much happening here. Okay, so first of all.

Pam Harris 21:38

take a

Vanessa Vakharia 21:38

Let's just, let's take a drink. We'll both take a drink. What water? Okay. Decaf latte for me. Okay. So, mm, but electrolytes are gonna be next, so I'm gonna need to hydrate for what's happening next. Okay. So first, I guess we probably don't have time to go into this. I will have to respectfully disagree about the term "naturally mathy", we, we likely don't have to, and I'll only say, I'm actually, I'm gonna respectfully disagree because that's cool. But my actual question I don't wanna tell, I wanna ask. No, I, I'm gonna respectfully disagree 'cause I'm wondering if this is an opinion you have or something that's corroborated by evidence. Because all of the evidence I've seen is that there is no. Um, you know though, no, because it's such a big conversation. 'cause are we talking about like at birth they pop out and they're naturally mathy? Or are we talking about culturally they become more naturally mathy because there's no actual scientific evidence that there is like a mathy gene in the brain. That's all. Something to add on is have you ever read, um, Outliers by Malcolm Gladwell?

Pam Harris 22:39

And, do I think that some people have a math gene and some people don't? Absolutely not. We all have the math gene. Do I think some people pay basketball better than others? Yes. Is that because they were in the gym a lot with their dad, who was the coach?

Vanessa Vakharia 22:53

Yeah.

Pam Harris 22:53

so as from a young kid, they had a ball in their hands. Like, so my high school coach, we, we called his kids gym rats. Like he, his

Vanessa Vakharia 23:00

Yeah.

Pam Harris 23:01

in the gym. Were they? And, and then they grew up to be, uh, basketball coaches as well. that because they were naturally basketball players or is it because they grew up in the gym with a

Vanessa Vakharia 23:10

Yeah. Yeah, yeah,

Pam Harris 23:11

So I think nature, nurture, there's a lot of, um, I, when I meet people who say, Ooh, I've always thought about numbers that way, and I just assumed everybody else did too. Are you

Vanessa Vakharia 23:22

yeah.

Pam Harris 23:22

that you didn't? And I'll smile and I go, I didn't. And

Vanessa Vakharia 23:24

Mm-hmm.

Pam Harris 23:25

at me like, really? if I ask that person, did you have some, so I would call that person kind of naturally math for

Vanessa Vakharia 23:33

Okay.

Pam Harris 23:34

As they were growing up, they played with relationships. I did not,

Vanessa Vakharia 23:39

Yeah. Okay.

Pam Harris 23:39

thought about using relationships and they assumed the rest of us did too, or they weren't sure why the rest of us weren't. did not. Many people that I meet, the majority of the math teachers I work with will say to me, I was like, you, Pam, believed that math was wrote, memorizing and mimicking. And that's what I did. I did not, I never played with numbers. Now, I, I don't know if the reason some people kind of did that on their own was because. They had a natural inclination, more talent, or it was because they had more interest or it was because they had a mom, a dad, an uncle, an aunt a a, some other significant person in their life who played with numbers with them. So often when I meet that person I'll say, did anybody play with numbers with you when you were a kid and they'd say oh my gosh, my granddad would put me on his knee and we played cards and we always had to figure out our score. And like often, often, so this was research I would love to do when I have a minute, be to like, like they have some influential person who early in their life helped them do the mental actions of math then that set them on the course of math. And sure, that could be, that could absolutely be true.

Vanessa Vakharia 24:51

Okay, so then we're kind of fine. Then. You know what, actually, I think you and I are so funny in this way because I am, so, like, I think it's, for us, most of our like quote unquote disagreements are simply in language.

Pam Harris 25:03

Yeah.

Vanessa Vakharia 25:04

I actually think we're saying the same thing. 'cause when you say "naturally mathy", it sounds like you're like, some people are just kind of born with this, like abil. But you're not saying that at all. You're you're not saying that at all. And that's exactly, and you know what, all the research actually corroborates what you are saying, which is what outliers is all about. Which is when we look at these people that are like the most skilled mathematicians or athletes or whatever it is usually nothing genetic. It is about this 10,000 hours of practice. Right. And whether you get that

Pam Harris 25:28

born. Glad. We'll also talk about right when the, when the hockey players in Canada were born,

Vanessa Vakharia 25:33

at different times of the year. Yeah,

Pam Harris 25:35

year. So if they were born so that they were the oldest kids in their

Vanessa Vakharia 25:38

exactly.

Pam Harris 25:39

magically they looked like they had more talent. So Yeah, absolutely. I, I completely can. I think you are right. I think often where you and I are have to, where we dive in to have a longer conversation is when there's some vocabulary that we're like,

Vanessa Vakharia 25:52

It is a vocab.

Pam Harris 25:53

do, what do we mean. Vanessa, I just wanna encourage anybody listening to this podcast, we in math education, need to have more conversations like this where we don't just nod at each other and

Vanessa Vakharia 26:04

Mm-hmm.

Pam Harris 26:04

yeah, you're good. So sure, I'm gonna agree. Like, we need to parse out what we're talking about because too many people think we're all on the same page when I don't know that we are

Vanessa Vakharia 26:14

Yes.

Pam Harris 26:15

the same page, when in fact we are.

Vanessa Vakharia 26:18

Which is actually, you know what's so ironic about this? Which by the way, Alanis Morissette's, uh, key song was called, isn't it Ironic? Um, what's ironic is that so far this whole podcast has actually, this whole interview has just been about language. What is an algorithm? What do we mean when we're saying algorithm? You know, it's funny, it's like when we use the term common sense, it bugs me 'cause I'm like, my common sense is not the same as someone else's common sense, right? So it's like there's no definition of common sense that we're all gonna be like, well it's common sense that blah, blah, blah. Because depending what you grew up, depending what your values are, it's not common. It's actually not common. And like when we use the term algorithm, we don't all mean the same thing. So we're having this conversation and we're actually saying completely different things because we have a different definition than we're going on. And actually as mathematicians isn't the number one rule to define our terms, isn't that literally what we're supposed to do before we get into any problem?

Pam Harris 27:07

It

Vanessa Vakharia 27:07

Okay. So,

Pam Harris 27:08

would be a good idea. Yeah. This is, this is tricky and super subtle and so let me

Vanessa Vakharia 27:11

Okay.

Pam Harris 27:12

what

Vanessa Vakharia 27:13

Hmm.

Pam Harris 27:13

You and I, which I love by the way. Thank you. We are not. these terms so that everybody has to use our definitions. We're defining these terms so that you and I can have a conversation

Vanessa Vakharia 27:25

Yes.

Pam Harris 27:26

we can understand what each other means. We're not commanding everybody now take on our definitions.

Vanessa Vakharia 27:32

Yeah. This isn't like going in the dictionary of Pam and Vanessa.

Pam Harris 27:35

Uh,

Vanessa Vakharia 27:35

No,

Pam Harris 27:36

uh, me trying to be arrogant to say I know best of what the

Vanessa Vakharia 27:39

no.

Pam Harris 27:40

and everybody

Vanessa Vakharia 27:40

Love it.

Pam Harris 27:41

I wanted to

Vanessa Vakharia 27:41

Thank you for clarifying that. Actually, no, I really, really liked that.

Pam Harris 27:44

Yeah.

Vanessa Vakharia 27:45

We're making sure we're speaking the same language so that we're actually,

Pam Harris 27:48

we can't have a

Vanessa Vakharia 27:48

or we can't have a conversation. Yeah, exactly.

How this all plays into math anxiety

Vanessa Vakharia 27:51

This is just, I mean, there's so much here. You know what I wanna ask you now?

Pam Harris 27:55

Yes.

Vanessa Vakharia 27:56

Just to, to bring it all together. You at one point said, our students deserve, to, learn these reasoning skills. You know, we don't need them to just pump out an answers anymore. They, we, they deserve this, and we are robbing them for that. If we don't um, facilitate that. How do you think learning in the way that you're proposing versus learning the standard algorithm actually shapes math identity? you know, obviously this is a podcast about math, trauma and math, anxiety. How do you think teaching one way versus the other plays into that?

Pam Harris 28:25

Huge, so if I'm in a math class and the teacher says, here is math. I'm gonna give you a problem. I'm gonna work it out. Now we're gonna do it together. Now you go mimic exactly what I just told you to do. Then the student has no agency and this, the math becomes this thing that I'm, uh, uh, I, what, what I do is, is mimic what you have told me to do. And then if I like that, if I'm, if I'm rewarded for that now, I've created, uh, an identity of, ooh, I'm good at, and, and it was exactly the identity I created. Like I, I was, I was a nerd. I didn't have friends, whatever, but hey, I, I won round the world, Arlene Wheeler. I wonder if she would ever listen to this.

Vanessa Vakharia 29:08

Who's that?

Pam Harris 29:09

she was my third grade nemesis. Well, third grade she won the award in middle

Vanessa Vakharia 29:13

Should we find her?

Pam Harris 29:14

two. Oh, I love that. We, it was the two of us that were, we were always like, like we were the quickest we around the world. It was the two of us that were always at the end.

Vanessa Vakharia 29:23

Mm-hmm.

Pam Harris 29:23

division first, like all the things. I got this identity that I was smart and I,

Vanessa Vakharia 29:29

Okay.

Pam Harris 29:29

math. I've heard you kinda talk about how, I like your take on that by the way, that we have this cultural sort of assumption that if you're good at math, you're smart.

Vanessa Vakharia 29:38

Yeah, it's gross.

Pam Harris 29:39

I'm just gonna say if you're quick at math. even good, but if

Vanessa Vakharia 29:44

Mm. Mm-hmm. Yeah.

Pam Harris 29:45

cause you gotta define good, right?

Vanessa Vakharia 29:47

Totally. Totally.

Pam Harris 29:48

my identity was based on the fact that I was quick. And then I became a high school math teacher, and I thought I was the god of math. Imagine my dismay when I realized wasn't doing the mental actions that other people were doing it all. I was memorizing and mimicking, and there was this whole other world. Now imagine my joy when I realized I could do that too. I just needed to know it was a thing. I bought into the myth and it, it sh completely shaped my identity. Now I realize I, I can do that too. I can be in that club and maybe better, because this is my life mission, I can invite everyone else into that club, like the real club.

Vanessa Vakharia 30:30

Okay. Okay. I was asking about, you know, how does, uh, teaching algorithmically and through mimicking affect, you know, one's math identity? And there's kind of two parts here. On the one hand, you and your pal, Arlene Wheeler, which if anyone knows where she is, please do locate her. You actually formed a very strong, confident math identity as a result of this procedure, uh, sorry, as, as a result of being taught this way because you were, for a while, because you were able to nail this. Okay. So on the one hand, we've got two kids here who form a very strong math identity. They're quote unquote good at it. They're quick at it. They can follow the steps. However, later in life, when that becomes challenged in a sense because they are presented with math in a different way, all of a sudden that identity kind of collapses. There's a bit of an identity crisis, and then there's this beautiful plot twist romcom ending where you discover math in a whole new light and you're like, I can do it to like beautiful, cool, some ups and downs. Now forget you and Arlene. Let's think about the rest of the class who are watching you and Arlene just kill it. And they're like, oh my God, we suck.

Pam Harris 31:33

Yeah.

Vanessa Vakharia 31:34

So now not only like, you know, we have this like kind of partial success story with you two, you and Arlene, that then later ends up in a bit of an existential crisis, but the rest of the class is not quote unquote good at math in this way, and they don't feel good. They aren't presented with any other options. They just leave the class thinking, well I guess I'm not a math person and I'm not good at it.

Pam Harris 31:55

Uh yes. And what a tragedy. Because

Vanessa Vakharia 31:57

and what a tragedy.

Pam Harris 31:59

in reality, they are a math person and they

Vanessa Vakharia 32:01

Mm-hmm.

Pam Harris 32:01

can math if they just need to know it's a thing, and, and

Vanessa Vakharia 32:05

Well, yeah,

Pam Harris 32:06

get experiences to help them build their brain to be able to math more and more.

Why we still teach in ways that don't work

Vanessa Vakharia 32:12

Well, and the crazy thing is everyone is at a disadvantage in this situation, including the two of you or whoever it was that formed a positive math identity. Because like my mind is now going somewhere else here where I have never thought about it in this way of thinking like. You know how there's all these dis, this, you know, hoop law now about people being like, well, why can't we just teach it this way? There are people who are good at it and those are the people who are good at math, whatever. You're kind of living proof of being like, well, hold on. And now this doesn't, this is not a general statement, this doesn't apply to everyone, but a lot of people who get through this system and feel good about it, it's not even happening for the reason that I think we'd all like, which is like we'd all, like everyone to deeply understand mathematics and numbers. I'm not saying that people don't, I really believe that a lot of people who are, who have been taught through this, uh, mimicking an algorithmic way, I being included, and many of us being included still have, you know, number sense and deep reasoning skills and all of this stuff. They've just gained it in a different way because they certainly didn't learn it at school kind of thing.

Pam Harris 33:15

Yeah. Yeah,

Vanessa Vakharia 33:16

Sorry, I'm saying a million things at once, but I guess I'm kind of like, where I'm, I'm kind of trying to get to is like, it's kind of crazy to think that we're trying so deeply to protect this way of teaching that actually didn't service most people. Even those who felt good at math were actually being robbed.

Pam Harris 33:37

Yeah. Agreed. Agreed. And can I add a third

Vanessa Vakharia 33:40

Oh, say that again. I liked it.

Pam Harris 33:42

agree with you.

Vanessa Vakharia 33:42

Okay. Yes. A third group.

Pam Harris 33:45

add a third group because

Vanessa Vakharia 33:46

Yeah.

Pam Harris 33:47

people listening who while they were being taught algorithmic mimicking, through it kinda like my kid and said, Ooh, I could logic my way through that. And they use relationships and connections and think that the way they were able to do these real math mental actions is because they were shown algorithms.

Vanessa Vakharia 34:08

Mm-hmm.

Pam Harris 34:08

Like they almost can't, they almost can't separate it out.

Vanessa Vakharia 34:11

Yep. Yep.

Pam Harris 34:12

this is the way we should teach because it worked for me. So does that make sense? In other words, there's me who,

Vanessa Vakharia 34:17

I wanna talk about this.

Pam Harris 34:18

it didn't actually work for me. And I'm clear on that. But there are people who are successfully math, they're doing the mental

Vanessa Vakharia 34:24

Yep.

Pam Harris 34:24

and they think it's because of the way they were taught. And I'm gonna respectfully invite the, that if you're listening, I respectfully invite you to consider if your teacher hadn't given you things to mimic? What if your teacher had actually helped you, knew, knew the things you were developing on your own, helped you do that intentionally. What if they had set up experiences for you so you could have developed those things you kind of did on your own, quicker, faster, better, because the teacher knew what they were and helped you intentionally build it. Think of how far and fast you could have gone if you hadn't had to do it on your own. Does that make sense? I don't think

Vanessa Vakharia 35:03

it makes, it makes so much sense.

Pam Harris 35:05

to try to parse out. How

Vanessa Vakharia 35:07

No, I.

Pam Harris 35:07

how did you become the person you are? Well, of course it was the experiences that that created me. I respectfully submit that when you were shown an algorithm for whatever reason, I don't know why you were able to see things through that the rest of us were not, if we were taught differently, all of us could have done what you did in spite of the way you were taught.

Vanessa Vakharia 35:27

Uh, you know what I think is, I think it's just a really hard ask. I, I don't think it's impossible. I think it's actually, this is what like, you know, most therapy is, is this idea of like, let's try and deeply reflect on what, um, you know, we weren't just quote unquote born this way or, you know, these things all played a role, but I actually think it's really hard to be like, okay, so imagine you were taught completely differently because like, it is so hard to parse those two things apart. Like even I'm trying to do it because obviously I've kind of fall into this camp of being like, yeah, I was taught the algorithm, but I still have these deep reasoning skills and I was able to build relationships. And I imagine, and I think this might be only because I've sat through a couple of your lessons and been like, oh my gosh. You know, like, I think, I think people need to experience it in order to, you know, there's this thing I read. Okay, I think you're gonna like this actually. So my a, a lot of my work is based around this phrase by Joe Dispenza, who's like. Let's call him a pseudo scientist. Um, he's like spiritual science, but I love what he says. He says, uh, his whole phrase is, believe, behave, become. It's like you have a belief. So maybe I believe that what you're saying could be true. You know, I believe that maybe what you're saying is could be true. If I was taught in this different way, I'd have a different relationship with math. I changed my behavior. So I go to one of your sessions, I actually see it in action and I'm like, oh my God, this actually has changed the way I think. And now I've become a person who's like, yep, you know what? I really believe that this can work and now I'm gonna teach this way. 'Cause when I sat in your problem strings lesson, like I've heard of it, I've read the book, whatever. I get it. I was like, this sounds cool, but I, my entire body was on fire. Like, I was like, oh my goodness. It's that. Aha. That's when we know something to be true.

Pam Harris 37:12

I love that. And, and you're not people will say to me, that's what you mean? Like, they'll sit in a session with their ma with their jaw open.

Vanessa Vakharia 37:21

Yeah.

Pam Harris 37:22

Yeah. And, and I'll be honest, I, I did too when I ran into Kathy Fasno, so she's an elementary math researcher. When I watched her, when I watched kids, it wasn't really watching the teacher at first. It was, it was like me watching learning happening live. I was like, I want that, I want that. I want, I want to. And I had to experience it. And then now my goal is to help us all experience it. Yeah.

Vanessa Vakharia 37:47

Well, and if you think about it when you're teaching a lesson, what you do, this is actually gonna be interesting 'cause we're gonna present together soon. I, I notice this, you guide us through the experience and then you explain what you did, right? So you're not intellectualizing it first. You're giving us the experience first, and then you're unpacking it anyways, k. Fine, fine. We really do have to wrap this up. But my point is, the whole thing is

Pam Harris 38:09

You have a point?

Vanessa Vakharia 38:10

well, I, I, I know it's, I don't have a point, but there is one thing I wanna say. I,

Pam Harris 38:13

Okay.

Vanessa Vakharia 38:14

I wanna make clear, I think, at least from my end, but you can definitely obviously chime in that if you're listening to this and you're a teacher who teaches through algorithms, okay. I personally wanna make it very, very clear,

Pam Harris 38:25

right? We're not

Vanessa Vakharia 38:26

which we're not, not only are we not blaming because like, you know, yeah.

Pam Harris 38:32

right. We're not. We're not. We're

Vanessa Vakharia 38:33

There's nothing bad happening.

Pam Harris 38:35

we're?

Vanessa Vakharia 38:36

I wanna make it so clear. I wanna make it so clear that it's like, Hey, first of all, we all learn to teach this way, so whatever you're up to. Totally cool. We all have the same mission. We all want kids to develop reasoning skills. if you are doing that, you are not doing something wrong. And this whole episode is meant to be like, oh my God, we're robbing kids like you are not robbing kids. Everything's cool. I'm sure they're learning, they're, you know, you're helping them feel confident. You're helping them build their identity. That is the most important thing. And if you're interested in perhaps shifting a little and exploring how you might teach in a different way, maybe you'll like it, there's an opportunity. Would you agree?

Pam Harris 39:12

Absolutely. Yes. Yes. Very positive. we know better, we can do better. I'm suggesting there might be something worth investigating, worth experiencing

Vanessa Vakharia 39:21

Yeah.

Pam Harris 39:22

that you might find, um, will help you shift you're teaching to reach more students in less time.

Vanessa Vakharia 39:30

Yeah, in less time, that's who doesn't wanna do that? And also, I also would imagine that there are some teachers that are like, but I don't understand the math fully enough to be doing all this stuff. Like I learned through algorithms, just so everyone knows that is me. Like I'm a trained high school math teacher. That's Pam. She's pointing at herself. That's her. So. I would personally invite you two to be like, just see what happens. Like, take one of these workshops, like put, you know, whatever, if you can see Pam present, like go do it. Um, explore it just for your, I actually think for you, the listener, and I'm talking to myself too, it's gonna build your confidence. Because what happened is I sat in one of these sessions with Pam and I was like, oh my God, by the way, there's nothing that intimidates me more than elementary school math. Okay. Like the worst. Like, I'm just like, oh God. So I was very intimidated and yeah, right. I found myself having so many ahas, like sitting quietly in the back being like, oh my, oh my God, oh my God. And I was like, oh my God, I feel like I understand more than I gave myself credit for. So even if you just go as an explor, uh, exploratory, session for yourself, just, I want all teachers to have the confidence themselves. And you know, I think so many teachers have walked away with so much math trauma from the way we were taught, I just want you to have a good experience. You know, because yeah, just to see that there's hope. We can all be learning and growing and changing and there's magic in math and we can discover that every day.

How to connect with Pam

Vanessa Vakharia 40:53

Tell people where to go before I wrap up with the final two questions.

Pam Harris 40:56

Excellent. So the first thing I would do is, uh, suggest, wait. I have a free ebook. That would be a great place for you to start. So you can go to mathisfigureoutable.com/big B. 'cause I think it's big, it's a big, it's a big add to the world. That would be a great place for you to start. We have the math is Figureoutable podcast where Kim, we've talked about a little bit, uh, where she And I talk about all things, Math is Figureoutable from super young all the way through high school. Um, another thing that you could do is join me on social media at the hashtag Math Strat Chat where I threw out a problem on Wednesday evenings to people from around the world chat about their strategies and comment on each other's, uh, thinking. So that's totally fun. Those would be some super good ways to sort of kick off. Uh, joining me to make math more figureoutable.

Vanessa Vakharia 41:41

Oh my God. I can't believe we have to end this. We didn't even talk about the fake math thing, but I mean, whatever.

Pam Harris 41:45

We could send 'em to the Debate Math Podcast if they wanted to hear a little bit

Vanessa Vakharia 41:48

You can, but I feel like both of us have changed since then.

Pam Harris 41:51

well, okay.

Vanessa Vakharia 41:52

You know, it's not important. Um, it's not important because I feel like it's old news and I feel like we actually covered it in a way by talking about all these other things.

Pam Harris 42:00

Well, it's

Vanessa Vakharia 42:01

Do you?

Pam Harris 42:01

mean. Everything we talked about today Is the important stuff.

Vanessa Vakharia 42:04

Is what I mean. David's doing this, by the way. He is doing the wrap up. I love telling people that. It's so fun.

Q1

Vanessa Vakharia 42:09

Okay, so final two questions. Quick fire. I ask every guest.

Pam Harris 42:13

I'm ready.

Vanessa Vakharia 42:13

Number one, if there's one thing you could change about the way math is taught in school is, what would it be?

Pam Harris 42:21

Holy. I would help teachers experience what it means to do the mental actions of math thing, because that would shift everything they do.

Vanessa Vakharia 42:31

I love it.

Q2

Vanessa Vakharia 42:32

Okay. So what if a teacher's listening to this and they're like, you know, this is so awesome. It sounds really great, but I just, I'm not a math person and that's just it. I'm just not a math person. What would you say?

Pam Harris 42:42

I would say you're, you're not a math person yet, or you actually are a math person. Um, and, and one of my favorite things to do is to ask them, how do you think about this problem? And, and then, and 99% of the time people will go, well, you know, that's how you think about it, but you're supposed to do the right way. And I'm like, what if we could teach all math the way you just logiced your way through that one.

Vanessa Vakharia 43:03

Oh, I love that. Well, that's all she wrote. I mean, it's not, I have like 20 more questions, but that's all. That's all we have time for it on today's segment of Pam and Vanessa, figure out what words and math mean and truly understand each other. Thank you so much. This was so much fun.

Pam Harris 43:25

You're a gift.

Vanessa Vakharia 43:26

Oh my God, you're a gift. This is so sweet. We should tell Chris and Rob that you know they really were the start of our love story. Thanks guys. The matchmaker, the match makers. Okay, until next time.

Pam Harris 43:37

Woo.

Outro

Vanessa Vakharia 43:39

​Okay, guys, be honest. When we started that convo, did you think we'd be besties by the end of it? I loved this talk with Pam. In a world where our differences and disagreements are amplified. I just found it so refreshing to have time for actually nuanced debate about some math questions that people seem to lose their minds about online, but that really shouldn't be so controversial. Especially when we all have the same goal, which is just helping kids learn and feel confident. So, I have some homework for you this week. Maybe my biggest challenge for you yet. I want you to find someone you disagree with about math or like anything, and sit down for an hour, okay, that's, that's a long time. But like, sit down for like 20 minutes and really talk about it. See each other's side and truly try to understand where they're coming from and where you actually might share common ground, one piece of common ground. And, ahem, a good way to break the ice with this person might just be to send them this episode. And whether you try this challenge out or just wanna share your thoughts on the episode, please text the podcast, DM me on Instagram at the Math Guru, or email me at vanessa@themathguru.ca. Links to do all of those things are in the show notes. Also, my audio book is officially out and I cannot believe how many of you guys have already bought a copy and are listening to me narrate Math Therapy in my voice, in your ears, that's actually wild. You can grab it wherever you buy your audio books, including Audible and actually Spotify where it just got added like today. Finally, my Math Therapy merch line has dropped and I'm dying for the cute pics you guys are sending me with you in your MT swag, please keep sending me those, and you can shop the drop at maththerapy.com/merch. Math Therapy is hosted by me, Vanessa Vakharia. It's produced and edited by David Kochberg, and our theme music is by our band, Goodnight Sunrise. Okay, that's it. Go forth. Trust yourself. And remember, your inner mathematician is already in there. You just have to give them the mic.

Outtake - Britney Spears

Pam Harris 45:41

They're logical relationships, but we're not logicing our, see, I've made it my own word. They're logicing. We're

Vanessa Vakharia 45:46

I like it.

Pam Harris 45:46

logicing our way. What is on your shirt?

Vanessa Vakharia 45:50

Brittany Spears, do you know who that is?

Pam Harris 45:51

do, but I, there's two people on your shirt that, so they can't be.

Vanessa Vakharia 45:57

They're, they're both just Brittany. It's like both, just two different,

Pam Harris 45:59

right now. There's

Vanessa Vakharia 46:00

no,

Pam Harris 46:01

on your shirt. Sorry. Okay. Pop, pop culture ignorant here.

Vanessa Vakharia 46:07

I mean, no, you're not ignorant. You're, you know who Britney Spears is.

Pam Harris 46:10

mean, ish.

Vanessa Vakharia 46:11

Yeah, it's just a very poorly designed shirt.

Pam Harris 46:15

We won't tell anyone that you said that.

Vanessa Vakharia 46:18

Okay.

Pam Harris 46:18

We won't put that out publicly or anything.

Vanessa Vakharia 46:22

Oh my God. Okay.