Christina Tondevold

Christina Tondevold 0:03

at five years old We aren't expecting kids to come to baseball, being able to hit the ball into the outfield, We aren't expecting every kid to win, every kid to be perfect. We are laying a foundation of we're all here, we're learning. Unfortunately for some kids, they end up getting coaches who are all about winning, and all about getting it right, and then guess what happens? They don't love the sport anymore, and they stop doing it. And that's exactly what we're doing in mathematics, it becomes only about the winning, only about being perfect. Flexibility is the piece that gets everybody feeling like they can do this, that they have a part. And yet we have to balance it a little with, accuracy and efficiency do matter, but if we put too much focus on that, that becomes an issue, and then kids are performing out of fear,

Vanessa Vakharia 0:56

Hello, and welcome to another amazing episode of Math Therapy, the podcast that unpacks math trauma and helps you build a better relationship with math. I'm your host, Vanessa Vakharia, and I'm here today to get you pumped up about going back to school. And yeah, I know, what a buzzkill when it's still summer and, like, 10,000 degrees out, but it's coming, so why not get fired up about it? A new school year brings so many possibilities, one of which is the chance to redefine our terms in the classroom so they empower us, and today's word is one of the most vulgar F words in all of math: fluency. We keep hearing that we can't, like, teach students to be fluent in mathematics while also honoring their progress and thinking, but today's guest, Christina Tondevold, joins me to tell you that that is total BS, and yes, you can have both. Christina is a self-proclaimed recovering traditionalist who initially taught mathematics by expecting students to parrot back to her what she taught them. But she now works with teachers in developing students' mathematical understandings in a far more balanced way, developing conceptual understanding while working on procedural fluency. She does this through her site buildmathminds.com, where you can get PD literally at your fingertips. She's a powerhouse. She is, like, the person I think about when someone says fluency, and that's why I decided she was the person I had to talk to about reclaiming this F-bomb. If you think that timing things in math will ruin a kid's life, and if you've ever been like, "Well, yes, I want my students to feel good about math and their thinking and themselves, but I also need them to be fluent in math, and it feels impossible to do both," or if you've previously thought, just like me, that fluency equals speed at the expense of all else, this episode is for you. And if you're like, "Sorry, I don't even know what fluency means, period, why it's important, or why I should give a fuck," well, this episode is definitely for you. All right, enough from me. Here's my convo with the amazing Christina Tondevold.

What is a "recovering traditionalist" teacher?

Vanessa Vakharia 2:52

Christina, welcome to the podcast

Christina Tondevold 2:55

Thank you for having me. I am excited to be able to chat with you.

Vanessa Vakharia 2:59

I love how your, the first thing on your website says that you are a recovering traditionalist, and I kinda wanna just start there. I wanna know a bit more about that. What does that mean?

Christina Tondevold 3:09

When I started sharing things online, I decided to call my blog that, The Recovering Traditionalist, because it really is a journey, and I feel like I'm never fully done. You know? Like I'm, i'm not just a thinker now. It is really hard to get out of our traditional ways of teaching mathematics because if, assume that, you know, people my age, uh, I'm nearing mid-life, that we all kind of grew up with very traditional style of teaching of here's the procedure, learn it, memorize it, follow the steps, and you'll get to the correct answer, and that was how everything was graded in math, was did you follow the steps? Did you get the right answer? That was the way that I taught, was still very traditional. I stood up there and gave students the notes and taught them the procedure, and then expected them to practice it as a whole group. And then when they got to the independent practice time, it was like hands were going up, and I did not have another way to help them. I just would repeat the rules and steps, 'cause that's all I knew.

Vanessa Vakharia 4:19

I wanna make sure I'm tracking here that when you say traditionalist, you are referring to, um, a, a historic way of teaching math that relies on procedure. Is that correct? Like, we're relying more on procedure, we're on... Okay,

Christina Tondevold 4:36

Yeah, and, and for years though, I think it's, you know, as everything in education, the pendulum swings. And so the pendulum swinging away from the traditional style of teaching into more letting kids think and play around with mathematics, um, w- kind of went a little too far to where it was, let's just let kids play and discover. And but we have to remember it's guided instruction still. It's not just letting them play. It is letting them play and explore, but we still have to use that information to then pull them, push them along their trajectory of learning. It's not just let them play and flounder, right? We need to learn how they're thinking and then use that to guide our instruction to move them through mathematical understandings.

Vanessa Vakharia 5:30

Yeah, it does feel like the guided piece is what, when we hear about this in the media, gets missed. And I, I, I'm actually curious, like, did we ever actually just, like, let kids play and flounder, or was there just, like, an idea that we... Like, were we ever like, "Oh, we're actually not teaching anything"? Like, just, I don't know. Like, I know that there are some schools or, like, communities where that happens, but I feel like that was never, like, the instruction to teachers was never like, "Let them just, like, vibe," and, like, the end, right? Like, it was like, "Let's figure out, like, how they're learning, what draws their interest, like, toward what their tendencies are," and then exactly like you said, "Let's use that to guide how we're gonna teach a certain concept. Like, let's prov-" Right?

Christina Tondevold 6:12

I could say even from, like, personal experience, but also stories, like my own children's experience

Vanessa Vakharia 6:19

Yep

Christina Tondevold 6:20

Um, I have four kids ranging from 18 right now to 12. and there have been times where, I mean, they've had a whole range of types of teachers, especially around mathematics. but there have been a few have taken things that they've learned and, like, wanting kids, you know, what we talk about of letting kids solve the problems, uh, not telling them how to solve it, kind of to an extreme

Vanessa Vakharia 6:52

Okay

Christina Tondevold 6:53

"Here's the problem," and they, and then, like, if they are floundering and they aren't even sure where to start or anything, th- then the teacher's like, "Well, figure it out. What can you

Vanessa Vakharia 7:03

Okay

Christina Tondevold 7:04

you know, a little, like, not just play and discover, but even as we're open, giving kids an open task, and

Vanessa Vakharia 7:11

open

Christina Tondevold 7:12

yeah, too open, and we just, like, let them flounder a little too much sometimes.

Vanessa Vakharia 7:17

I would love to actually talk about this because I think that this is one of the biggest issues that I'm hearing right now from educators, is like they know that the move isn't to, like, be so prescriptive, right? They're like, "Okay, I get that I'm not supposed to give the kid a problem and then, like, a step-by-step formula on how to solve it. I understand that. But at the same token, I know I'm not supposed to just be like, 'Okay, well figure it out. Like, see ya. I'm gonna go eat a peanut butter sandwich.'" Like, do you know what I mean? Like, that's not the move.

Finding balance between teaching methods

Vanessa Vakharia 7:43

So it's like, but I also do feel like teachers are constantly being, like, wrist-slapped for doing too little or too much. This feels like a Goldilocks situation, actually, right? Like, it's like

Christina Tondevold 7:55

Yes

Vanessa Vakharia 7:56

too cold. What was it? It was too hot or too cold, the porridge, I think, right? And then there was, like, the middle. What are some tips you can give on, like, what you might ask yourself in terms of what the right balance might be? Like, what is that, that not too hot, not too cold bowl of porridge? Like, what should they do?

Christina Tondevold 8:14

In general what I try to keep in mind Is I don't want to start by telling. Actually, okay, so w- in our talk that we had a couple days ago, we mentioned our friend Raj, right?

Vanessa Vakharia 8:28

Oh my God, Raj, what a vibe.

Christina Tondevold 8:30

Raj is so fun. So in I got to chat with Raj for a thing that we have upcoming that Vanessa's a part of, and one of the things he mentioned to me that still is- is on my brain is that we can always go back to direct instruction. But our default has been, let me first tell you how to solve it, and then you work on parroting back and mimicking what I've just told you. So instead, like, to just get started and, give kids a problem. Just give them a problem, and see what they think. And you should always have in mind, though, where you're wanting to go with them. Like, what is the ultimate goal? And, a lot of times we think our goals are the standards of students will be able to blah blah blah. It's not about what they're able to do. It's about what they understand. And so we need to keep in mind is it that I want them to understand about this?

Vanessa Vakharia 9:34

I love this 'cause, like, uh, what I'm hearing, and this makes so much sense, it's like d- direct instruction or just telling, okay, let's just call it just telling 'cause I hate all the jargon. Just telling someone is in your back pocket that you can use any time, right? But asking somebody, like asking somebody to notice for the first time can only happen in the beginning. You can't tell them and then ask them to notice for the first time, right? Whereas, like, the telling could happen at any time, so it kinda makes sense to front-load your instruction with the stuff that can only happen at the beginning as like an experiment, right? Like, to be like, "Oh," like, "what would you do if you saw this with no information?" 'Cause once they have the information, they can't like imagine they had no information and like, right? Like you can't like you--

Christina Tondevold 10:19

something

Vanessa Vakharia 10:20

to unsee

Christina Tondevold 10:21

pretend

Vanessa Vakharia 10:21

and you can only be...

Christina Tondevold 10:22

Yeah

Vanessa Vakharia 10:23

Exactly. So I kind of love that of being like, fine, like especially for the teacher who's like, "Oh my God," but like, "What if I have to..." It's like, great, you can tell them. I almost wonder if like a nice little therapeutic way to like acclimatize yourself to the thinking model or, or getting farther and farther away from telling is to even set a timer and be like, "I'm not gonna tell them anything for two minutes." And then the next day be like, "I'm gonna set my timer for three minutes," like I'm not allowed to tell anything for th- you know what I mean? Like you can build your tolerance from uncertainty this way.

Christina Tondevold 10:56

Here's one thing I would recommend. If your lesson is on doubles plus one, right? And they aren't getting it in the first 10 minutes, uh, might be some deeper issues at, in play. Because mathematics is really a web of understanding, and we try to teach it in isolation. Like, here's your doubles, here's your doubles plus one, here's your make tens. Or with multiplication, I don't know how you learned them, but I learned them times ones, then times twos, then times threes. Like, total isolation.

Vanessa Vakharia 11:28

Yep

Christina Tondevold 11:29

and now we know, like, okay, well, if you know your tens, those are connected to your fives. And twos are c- can help you with your fours and your eights. Like,

Vanessa Vakharia 11:37

We never did that, did we?

Christina Tondevold 11:39

No. No, never. Never. And, and so I could, I could envision, like, you know, I, I don't want people to take away that, oh, okay, today we're learning our times twos multiplication facts. Um, and if they don't have it by the end, I'm just going to give them a bunch of flashcards to practice it. Like, there, there are really big things in mathematics that just take time. They take time to build. And one of the things that really bugs me in our standards, actually, is we do a really great job with addition and subtraction facts because it starts in kinder, then they, every state is technically different. In, in Canada, yours might look a little different as well. But typically in the United States, um, it's second grade that it talks about being fluent with addition and subtraction within 20, which is your, your quote, unquote facts. So they have kindergarten, they have first, they have second grade that they are building an understanding of what addition and subtraction are. And then by the end of second grade, they're supposed to have become fluent with it. Well, then, here in the United States, then third grade comes along, and all of a sudden they're supposed to master multiplication and division facts all in one grade level. Like, that's, I mean, that is an unrealistic thing. Um, and so I'm just saying, like, sometimes our standards are great in giving us time to build these things for kids, and sometimes it's just thrown at us and then we're like, "I've gotta get kids fluent with these all in one year?" And so we get the pressure of, oh, I just gotta tell you, I've just gotta tell you. But there's a lot underneath that we can build that will help them be able to see a lot of connections, and the more connections we can make, the easier it is to recall info.

Show students how connected math is

Vanessa Vakharia 13:35

I'm gonna be honest here. I'm listening to this, and I'm stressed. Like, I'm listening to it, and I'm like, "Oh my God," there are so many variables here. Again, no pun intended. But there's so much going on, right? Like, you're kinda like, you're kinda like, l- like, there's a lot of messages. And this is interesting. I remember talking to another educator earlier this week, and she was like, "You know, sometimes teachers just want, like, what are the questions I should ask? What are the rules I should follow?" And it's like, y- this is where your teacher's intuition comes in, right? Like, as educators, we are healers. We have an intuition. We have experience. We know that certain students need this. We can feel the energy of a class when things are being understood and not. And I feel like we have to, like, pause and be like, "Listen, we're saying a lot of things here, and none of these are prescriptive." You kinda just have to feel it out and trust that you're doing the right thing. That being said, it's kind of hard to hear, you know, that in grade three we have this basically, like, impossible standard, and it's gonna be really, really hard, but also, you gotta just, like, know what they need to understand about the standard, and it's not about the standard. It's about them. Right? Like, it's like, oh, my God. And now-- And, like, I, I get that almost in a way this is supposed to chill people out, to be like, "Don't worry so much about the standard. Just, like, make sure they're understanding some stuff that's gonna be important." But what if... Can we take a step back and, like, forget the content, forget the actual things, and, like, what is something that a teacher, especially a beginning teacher who's hearing all this, what's something that might give them a bit of hope and relief to-- if they're l- kind of like, "Yeah, man, like, this feels hard. It feels like some kids aren't gonna be, like, getting all this content. It feels like I've gotta do too much." Like, what is your, like, go-to, like, just remember this, this, this thing. Like, j- what-- Like, do you have something?

Christina Tondevold 15:19

Yep.

Vanessa Vakharia 15:20

Oh, okay, great.

Christina Tondevold 15:21

Is not a set of isolated things. It's all connected. And so it does seem overwhelming when you, like, you open up even your standards and you're like, "Oh my gosh, I have to get all of this in this grade?" Especially when w- I have students maybe coming with some gaps or two back. and when I saw mathematics is so connected, opened up a whole world for me. Like, I was middle school, so I w- at one point I was doing sixth, seventh, eighth grade. Um, in the, in the sixth grade textbook, there's a chapter on decimals, there's a chapter on fractions, there's a chapter on percents Those are all the same thing.

Vanessa Vakharia 16:12

Yeah.

Christina Tondevold 16:13

Tho-

Vanessa Vakharia 16:13

I love it

Christina Tondevold 16:14

the same types of numbers, yet we teach them in isolation without connecting them together. Fractions are, you know, like a part of a whole. I mean, that's a simplified version of it, right? It's

Vanessa Vakharia 16:27

I loved it. It was cute. Yeah

Christina Tondevold 16:30

decimals are just special fractions because they have denominators that are a power of 10. They're a 10th, 1000th, right? And then percents are even specialer because they are out of 100. So it's a fraction, but always, you know, percents are basi- based out of what it is out of 100 of something. And yet those are not connected together. I didn't learn them as connected together, and I didn't teach them as connected together, and then I had this group of kids in front of me, uh, when we get done with it, and they don't see those connections. Like, once I realized how connected mathematics was, it made me really start to look at the textbook, the standards, and say, "Okay, hold on. So these are all listed individually, but what actually connects together?" And there is great, a great document out there, um, by Achieve the Core, and it talks about the... I mean, they're not the power standards. That's not the phrase that they use. I think they call them content emphases. but it's like what's the major work at each grade level? And then they also talk about what ones are connected, they support that major work. And then it also shows which ones are just additional. Like ex- for example, kindergarten, basically everything is connected to the s- the work. The major work in kindergarten is getting kids to count, understand how numbers work, you know, like the, the, the number system of how things work, and a little bit of addition and subtraction. Well, everything you're doing, if you're working on geometry stuff, of recognizing shapes, and then you ask a kid, "So how many circles do you have?" You're really also working on counting. Like, it's all

Vanessa Vakharia 18:21

It's all co- Well, I also feel like it's like this is, like, a message of hope to be like anytime you're working on anything, you're working on something else, too. Like, you're getting more bang for your buck than you think. So it's like I really like that example of being like fractions, percentages, and decimals. You're looking at it being like, "I've got three chapters to teach," right? When really it's like what if we look at it as, like, you have one chapter with some tentacles. You know what I m- oh, I hated that I said that. I'm sorry. That grossed me out. But do you know what I mean? Like, you have, like, one chapter with, like, w- like, if you teach fractions, you've now taught, you've taught a third of percentages. You know, if you teach fractions and decimals, you've taught two-thirds of percentages. Once you close it off by being, like, percentages, now you've taught the entire thing. Oh my God, I just turned the chapters into a fraction. Did you see what I did there? I think that worked, right? Like, right, right, right? I think, as we know, time is, like, one of the biggest struggles for teachers, and it's always like, "I'm running out of time, I'm running out of time." So if you can feel like you're doing more, like, th- whatever work you're doing now is investing in all the future chapters. Like, if they're understanding fractions, they've already understood a third of percentages. You're getting there. Like, you're filling that. I do really like the way of looking at- I Can I switch gears for a sec?

Christina Tondevold 19:37

absolutely

What is fluency?

Vanessa Vakharia 19:38

Okay, you've said a word several times, and it is what I think of when I think of you. And it's a word I feel we throw around a lot in math ed and don't define, and that word is fluency. You've said fluency, kids being fluent, and I think when people hear fluency, what they think of is speed, and I know that we're missing the mark on that, and I feel like you're, like, one of the fluency experts in my world, and I would love to hear from you what fluency means.

Christina Tondevold 20:07

So I will say everything I learned is from other people. Again, I, once I start seeing connections, I, I learn something over here from somebody and something here, and I bring it together to make it make sense for me.

Vanessa Vakharia 20:20

Yeah

Christina Tondevold 20:20

so I, I really love Susan Jo Russell's definition of fluency. She had a article back in 2000 called Computational Fluency. It was published by NCTM, National Council of Teachers of Mathematics. And she talked about three things that make up fluency, one is efficiency. We do need to be fast. I mean, that is part of And I feel like fast has kind of gotten a bad rap

Vanessa Vakharia 20:46

Yeah, agree

Christina Tondevold 20:47

of all the trauma that comes with it, that the way that we have used speed and timed activities in the past has been harmful. But the, the need to be efficient and fast still is a part of our world. We do want to be able to get things done quickly. Um, and, and this isn't just for mathematics. This, this version of fluency, uh, for those that are listening, think of anything you feel like you are fluent at or trying to become fluent at.

Vanessa Vakharia 21:19

Like a language obviously we'd think of

Christina Tondevold 21:20

A language. Yeah, yeah. If you're fluent at driving your city that you live in, right? You are efficient. You know the fastest way to get somewhere.

Vanessa Vakharia 21:30

we've used speed and timed in a way that was harmful, can you say a bit more? Like, how have we used it?

Christina Tondevold 21:36

The classic of the timed tests of you have to get this many in a minute, and then you, uh, switch papers and somebody grades you. You know, and so everybody's seeing it. And then, um, if you don't get 80 problems in a minute, I'm totally exaggerating, but 80 problems in a minute, you have to stay in at recess, and you don't get your name up on the wall with a star that you have achieved, uh, this thing or whatever it was. I mean, f- there for a while, the multiplication sundae, I don't know who started that, uh, but it was, like, everywhere that you would get a scoop for every, you know, you learned your

Vanessa Vakharia 22:18

Like actual ice cream?

Christina Tondevold 22:20

Like, they would make a sundae. And then when you got your sundae complete, you could have, like, you got to join in on the ice cream sundae. Like, I

Vanessa Vakharia 22:30

Wait,

Christina Tondevold 22:30

was

Vanessa Vakharia 22:31

what?

Christina Tondevold 22:31

years ago.

Vanessa Vakharia 22:32

didn't have this. I would like an ice cream sundae. Okay

Christina Tondevold 22:36

right? But everybody who, who was good at it loved that. But those kids who struggled and never achieved that 80 in a minute or whatever it might have been, uh, never got their scoop on the wall, never got to participate in the, the sundae party that everybody else got to. So the timed activities, those timed tests, the mad minutes, all of that was used as punitive. It was as a punishment, not as some- And we thought it would motivate kids. We thought if they got to see their name with a star up there, that would motivate them.

Vanessa Vakharia 23:13

and that's the thing, it's like, I mean, this is a whole other conversation of how do we define motivation, because like, okay, I guess it did motivate some kids, but like out of fear, right? Like, it's like, yeah, like that's like not really motivation exact- Like it is motivation, but it's like

Christina Tondevold 23:30

true.

Vanessa Vakharia 23:31

of motivation that we like want Let's just like for argument's sake say we are gonna time you to do like 20 questions in a minute, or 10 questions in a minute, whatever. There's gonna be no sundae, there's gonna be no star, there's gonna be no nothing, but you're still gonna be timed. Would that be okay? Is it the way we treated the entire thing as... Yeah? Okay.

Christina Tondevold 23:51

Yeah, yeah. So actually, the research around- so there's, uh, a What Works Clearinghouse and the Institute for Educational Sciences, IES. Um, two groups look at all the research that's out there because we can say, "Oh, well, research says this, research says that." That might be one study, right? What Works Clearinghouse and the IES puts together these practice guides. And, um, back in 2009, they did the first one that I ever saw, maybe there was one before it, but the first one I ever saw for what works with students who are struggling with mathematics. And then in 2021, they just released, an updated one. I said just released. That was, like, five years ago. Uh,

Vanessa Vakharia 24:36

it

Christina Tondevold 24:37

very many people know about it. We,

Vanessa Vakharia 24:39

Yeah

Christina Tondevold 24:40

So this document has six recommendations that have strong research evidence to back them up for what works. One of them is timed activities. So

Vanessa Vakharia 24:53

what works for fluency

Christina Tondevold 24:55

f- for, no, just mathematics in general.

Vanessa Vakharia 24:58

Oh, okay

Christina Tondevold 24:59

So it's, um, what works for students who are struggling with mathematics. It's actually a guide for t- for intervention, what intervention should look like around mathematics. So there's things like, um, vocabulary, like getting clear on vocabulary helps kids, being precise in your language, using word problems, using lots of representations in mathematics, so models. Um, everybody seems to be, to love the first five recommendations, and then the sixth recommendation is to use timed activities. And people are kind of a little like, hmm, I don't konw about this."

3 pillars: efficiency, accuracy, and flexibility

Vanessa Vakharia 25:37

I'm so into this because this is juicy, 'cause it's kind of like I'm just, I don't know, I'm processing this with you right now, but it's like you're right. We have our backs up with speed. As soon as someone says speed, I'm like Like as soon as a kid is like, "I'm not fast," I'm like, "Oh my God." Like, that's not the thing. But you're kind of right. Like, there's nothing inherently wrong with wanting to be efficient with something, let's say, but there's so much baggage that has come with it that I now think we're almost doing an over-correct. Like, the pendulum is swinging where we're like, oh my God, like, no, like, speed doesn't matter at all. But, like, doesn't make s- what if it took an hour for someone to add two plus two? Like, we would- like, that, that's too long. That's not good

Christina Tondevold 26:21

Yeah. So we, again, it's a thing of balance, right? Like, we shouldn't be only focusing on efficiency. That is only, I'm gonna use a fraction again, one third,

Vanessa Vakharia 26:32

Okay, tell us the other things. Yeah. Okay, I love it, I love it. Wow, we are doing the whole unit. Yeah, 30.3.333. Oh my God, we're, we just did it. We just converted them all. We just taught, we just taught three chapters of math in one second.

Christina Tondevold 26:47

There we go.

Vanessa Vakharia 26:48

Oh my God. Okay

Christina Tondevold 26:49

efficiency is one piece. Let me get all three of them out so then we can

Vanessa Vakharia 26:52

Okay.

Christina Tondevold 26:53

it,

Vanessa Vakharia 26:53

Yes, yes, yes

Christina Tondevold 26:54

all together. So efficiency is the speed part. And then accuracy. Correct answers do matter. And again, the pendulum swung away from

Vanessa Vakharia 27:04

ChatGPT, yes. Oh my God

Christina Tondevold 27:06

You know, it's been like, oh, it doesn't matter if they get it

Vanessa Vakharia 27:09

Hmm.

Christina Tondevold 27:09

or wrong.

Vanessa Vakharia 27:10

Oh my God

Christina Tondevold 27:11

thinking. I'm sorry. Correct answers do still matter in

Vanessa Vakharia 27:15

It matters

Christina Tondevold 27:16

They need right answers. Accuracy is important. yeah, it is important to look at their thinking, but they also need to be told when it's wrong, like,

Vanessa Vakharia 27:29

Oh my God. Yep

Christina Tondevold 27:30

That matters. And then the third one is flexibility.

Vanessa Vakharia 27:35

What's that?

Christina Tondevold 27:36

that's been my focus is because for the longest... I mean, I've been doing this a long time. I started teaching, I graduated college in 2000. So I have been in the education space for a long time, and focusing on building number sense and, and flexibility for more than 15 years of that time. and so I have pushed heavily on getting teachers to focus on flexibility.

Vanessa Vakharia 28:07

What is it? What is that?

Christina Tondevold 28:10

about, uh, that in a

Vanessa Vakharia 28:10

Doing math when on, in a handstand while doing the splits

Christina Tondevold 28:14

yeah, yeah. Let's see how, how good your downward dog is or whatever, you know. Um, but the hard part is that that put the focus away from the first two of accuracy and efficiency. So I'm gonna talk about flexibility, but do not take away that that is the only thing we need to build.

Vanessa Vakharia 28:33

We're just defining it

Christina Tondevold 28:35

And I've, I've had to, like, say that now because I've put so much emphasis on flexibility and people have, have gone all the way to the

Vanessa Vakharia 28:43

Okay

Christina Tondevold 28:43

the other side. So flexibility is just having alternate routes. So let's go back to, you know, if you're, if you feel like you're fluent driver in your city, what happens if there's an accident and

Vanessa Vakharia 28:56

Oh, God. Oof.

Christina Tondevold 28:58

Do

Vanessa Vakharia 28:58

I'm stuck.

Christina Tondevold 28:59

to get there without

Vanessa Vakharia 29:00

No, absolutely not.

Christina Tondevold 29:03

You're not

Vanessa Vakharia 29:04

I am not.

Christina Tondevold 29:05

your... right?

Vanessa Vakharia 29:06

I'm not

Christina Tondevold 29:07

That's, and that's essentially what our kids are. So they might have one way to do it, and that's also been an, an issue is, like, once all these different strategies came around, it's like, well, if kids have one way to do it, that's fine. They don't need to know other ways. what if that one way doesn't work on that problem?

Vanessa Vakharia 29:25

okay? I'm gonna do a little recap for the listeners, and for myself, quite frankly. Okay, there are three aspects to fluency. We have efficiency, which has to do with speed. We have accuracy, which has to do with getting the right answer. And we have flexibility, which has to do with being able to, to do things in different ways depending on the circumstances. Context dependent. Having to... Okay, so now...

Christina Tondevold 29:47

love that. Yeah.

Vanessa Vakharia 29:48

Okay. Oh my God, thanks. Okay, so now I'm like, oh my God, you are so right that we have g- we have completely over-corrected. It's almost like, because flexibility wasn't a thing when, you know, like 20 years ago, we didn't care about it. We only cared about efficiency and accuracy. Now we're like, "Oh my God, flexibility," like, "We left you out." Now we're gonna, like, focus on flex- it also feels like, to me, flexibility is like the feel-good one of the gang. It's like, and you know what I mean? It's like, oh, you'll feel good. Don't worry about getting it right. Don't worry about being fast.

Christina Tondevold 30:22

Yeah, it's like everybody wins kind of

Vanessa Vakharia 30:25

Everybody wa- so it's giving participation ribbon. Oh my God. what?

Christina Tondevold 30:30

I mean, we need that. We need everybody to feel like they have a starting point, like they have a way to contribute, like they have a way to participate in mathematics. That's, we do want that. I'm not saying it's a bad thing,

"But the kids need to get the right answer!"

Vanessa Vakharia 30:43

I'm having a real epiphany. I'm like, "Oh my God." I do think, I do think... You know all the haters on the internet that are like, "Oh my God, they're treating kids like snowflakes"? I don't know if I'm using that right, but do you know what I mean? Like, participation ribbon, da, da, da, da, da, da. It's like they're not actually entirely wrong, okay? So, like, let me just say that, right? It's like I get where they're coming with this. Like, for sure, if we really were like, "Don't worry, take two hours to add two plus two together. Don't worry if two plus two equals five or six or seven, but, like, as long as you're doing something, you get an A+," that would be a problem. I agree, that is a problem. But I think what's actually happening, and tell me if I'm wrong here, I feel like flexibility is not only useful, but it is the inroad for the math-anxious student into doing what is needed to get to an efficient, accurate place. The flexibility is the thing we can be like, "Look, do it whatever way. We can reward you because everyone has their own way of doing things." Now you feel more confident, and because you're feeling more confident, you're not gonna freak out when you get something wrong or you're too slow. You're gonna keep at it because you have some con... Like, am I onto something? You know what I'm saying?

Christina Tondevold 31:52

Absolutely. And, and as, as an, um, a sports mom, you

Vanessa Vakharia 31:57

Okay

Christina Tondevold 31:58

I can see the analogy too with the participation trophies and the, and the entry points in, right? Like, if, if at five years old we're expecting kids to come to baseball, being able to hit the ball into the outfield, and know which- where to throw it, and know which way to go. Because, uh, if you've ever seen, like, play T-ball, they hit the ball and then they'll run to, like, third base, not first ba- Like, you know, they're learning

Vanessa Vakharia 32:23

Oh, cute. Right

Christina Tondevold 32:25

not They don't have the accuracy and, uh, all the, that stuff down. And, and so we do that build that and, and allow for all kids to have an entry point, right? We aren't expecting every kid to win, every kid to be perfect. We are laying a foundation of we're all here, we're learning, right? And then unfortunately sometimes, uh, for some kids, they end up getting coaches who are all about winning, and all about getting it right, and will yell, and all of that kind of stuff. And then guess what happens? They don't love the sport anymore, and they stop doing it. And that's exactly what we're doing in mathematics, is that we might be having these entry points, you know. Like, again, we've, we've gone to the wayside where we're just letting everybody come in and have this entry point piece, but we also need to balance that with, hey, they Uh, like, we do need to get to the efficiency and accuracy part, but even then it's still flexibility. The flexibility still matters. It's not like flexibility goes away. that's unfortunately sometimes we see in sports, is it becomes only about the winning, only about being perfect. And then kids are playing out of fear because they're, uh, s- scared their coach is gonna yell at them or bench them because they did something wrong, right? Like, there, there's different styles of coaching, for sure. I've done Uh, this is part of a keynote that I do. Um, so it's something I love talking about. But the same is true for us as educators, that we- that flexibility is the piece that gets everybody feeling like they can do this, that they have a part. And yet we have to balance it a little with, you know, accuracy and efficiency do matter, but if we put too much focus on that, that becomes an issue, and then kids are performing out of fear, and they're motivated only because those external things that they're afraid of.

What's our ultimate goal in teaching math?

Vanessa Vakharia 34:21

I wanna bring this all home for us here, which is gonna be a bit hard, and that kind of is a baseball metaphor that I'm sort of using, I think.

Christina Tondevold 34:29

Yeah,

Vanessa Vakharia 34:30

right? Uh, is that where it comes from? Is it like ra- is it-- Oh, my God, Christina, wow, we are basically sports commentators. Okay, this is gonna be hard to track 'cause I have about 30 thoughts in my mind, but here are some of them. The first is, I think we have to recognize, for anyone listening, I don't think anyone who thinks this listens to my podcast, but if you are listening and you're like, "I hate this flexibility stuff. It's all about speed and accuracy," I'm so glad you're listening. Actually, thank you for taking a chance on the podcast. Thank you for being here, and here's what I wanna say. If you're like, "Well, exactly, like, we are losing those things at the expense of flexibility," here is, here is my thought on that. I actually think we have to recognize that historically math has been such an anxiety-inducing subject that so few students are willing to walk in the door. And because of that, it behooves us, I don't know what behooves means, but it feels like it fits in the sentence.

Christina Tondevold 35:23

Yes

Vanessa Vakharia 35:23

It behooves us to actually put an emphasis on flexible thinking as a starting point because it allows the most flexibility in terms of who can walk in the door. Because, you know, there are not many ways to think incorrectly. If you're thinking, if you're doing something in any way, you are doing something, and that is what flexibility is all about, right? So, like, it kind of makes sense for us to be really emphasizing that so we can get all the people in the door. Once they're in the door, we are, we are dedicated as educators to helping them solve the problem and get the correct answer and to do so efficiently. But, like, let us, let us emphasize flexibility for a bit, because the lack of flexibility is what has led to so much math anxiety, hence so few people walking in the door. What do we think so far?

Christina Tondevold 36:07

Mm-hmm. Absolutely.

Vanessa Vakharia 36:08

Okay, so great. So that's nice. But now you're making me think too, we've now really centered on the concept of fluency, right? But it kinda, I know that fluency isn't the only goal in math, right?

Christina Tondevold 36:21

well, so that could

Vanessa Vakharia 36:23

It feels like,

Christina Tondevold 36:24

right? I

Vanessa Vakharia 36:25

it feels pretty good. It feels like it covers a lot of bases. Oh my God!

Christina Tondevold 36:30

We're just gonna keep at it.

Vanessa Vakharia 36:32

Okay

Christina Tondevold 36:33

um, I've had to really rethink my own understanding of the purpose of mathematics,

Vanessa Vakharia 36:39

Okay

Christina Tondevold 36:41

My traditional view of mathematics was I use mathematics and solve problems, like, I'm just doing math. However, to me, and, and I got this from, we in the United States call it Singapore math, Ban Har says, "It's just math, but we do it in Singapore." And their focus is

Vanessa Vakharia 37:07

Hmm.

Christina Tondevold 37:08

We want kids to be problem-solvers, and we do that with mathematics. Um, and so fluency, uh, like even in the United States, though, even with this flexibility piece, we still do have this push of get them fluent, fluent, fluent, fluent. When, is mathematics, like, being fluent in mathematics, like, the end-all be-all? Or is it that we want kids to be capable thinkers and problem-solvers, and we

Vanessa Vakharia 37:39

it sounds...

Christina Tondevold 37:40

through mathematics

Vanessa Vakharia 37:41

But hold on, fluency sounds, the tenets of fluency, efficiency, accuracy, and flexibility, is that not what it takes to solve a problem?

Christina Tondevold 37:52

It is, but it's, it's, it's flipping how we're looking at it. Like, the end-all be-all... 'Cause okay, so, um, let me see if I can reframe this too. Like, I always thought, like, even on, when I first started becoming more of a recovering traditionalist, I still thought my job is to teach the mathematical skills, to help them understand the mathematical skills, like the operations,

Vanessa Vakharia 38:15

Yes

Christina Tondevold 38:16

that they can solve problems, right?

Vanessa Vakharia 38:20

it

Christina Tondevold 38:20

can solve problems. When I heard Dr. Bonhar talk, he said, "Our goal is to help kids be problem-solvers, and the byproduct of that is they learn some math skills along the way." And so it's the rever- Like, just

Vanessa Vakharia 38:39

Oh,

Christina Tondevold 38:40

is our

Vanessa Vakharia 38:41

yep, yep, yep, yep, yep

Christina Tondevold 38:42

wa- I was still focusing on help my kids learn the math skills, help my kids learn the math skills, and if they become great problem-solvers, that's a great byproduct. But instead

Vanessa Vakharia 38:53

hear you.

Christina Tondevold 38:53

let's put

Vanessa Vakharia 38:54

I hear you

Christina Tondevold 38:54

And so even if we're like, "Okay, fluency will hit all of that stuff," it's still we're putting the focus on fluency, fluency, fluency, and then if they become thinkers, eh, a great byproduct of that.

Vanessa Vakharia 39:08

Oh, wait, 'cause it kind of seems like fluency, if we think about the w- you remember, you were, like, fluent in driving, fluent in cooking. It almost feels like fluency is dependent on the thing you're fluent in. You can't just be, like, fluent in life. Whereas you can be a problem-solver in every domain, fluency is domain-specific, and right now we're talking about fluency in the domain of mathematics. Oh my God. Oh my God. Okay.

Where do we go from here?

Vanessa Vakharia 39:32

And I think, we're gonna wrap here, but it's okay that we don't have a conclusion. I actually think it is for the educator listening or whoever listening to be like, "Huh, what is my approach going to be?" Right? Like, what is it? Like, is it, is it going to be fluency first, problem-solving second, or is it gonna be pro- Like, and again, I actually don't know that there is a generic answer. I think it can depend on your situation, on the student. Maybe a student's really interested in some mathematical concept, and through that they develop fluency, and through that they learn how to solve problems. Maybe a student's... Right? Like, do, do you think, do you think there's a right answer to this?

Christina Tondevold 40:11

well, I would encourage, I guess, um, maybe it's the recovering traditionalist in me that we start opening our mind to maybe it is more about the problem-solving si- piece of it, and that when we have that as the center goal, it makes all the other things seem easier it all connects back to that one thing. like fluency with math facts still connects to that because we do... If, like, if we're out solving the, like, some problem comes to mind when we're out there, we do need to be quick that so that we can figure something out in the moment, right? Like, I, know. I- it's a good conversation to keep going and a, a, a good one to really think and grapple with.

Vanessa Vakharia 41:01

Yeah, think about it

Christina Tondevold 41:02

the center focus of this,

Vanessa Vakharia 41:04

we're ta- w- you know, a lot of people listening are, teach math. And I'm like,

Christina Tondevold 41:08

Mm-hmm.

Vanessa Vakharia 41:09

Have math as a gateway. Like, I, I mean, I love math, but I don't particularly, I'm not like, "Math is the most important thing." I'm like, math is just the most beautiful gateway to so many things. And one of the gateways that it opens up is the ability to problem solve, right? Y- yes, you're learning content and you're learning skills, and you're becoming fluent in this and that, but at the end of the day, the, the broader skills you're learning is how to solve a problem, right? So I guess I'm kind of thinking, like, because at your disposal, what you have is math content that you have to teach.

Christina Tondevold 41:41

Mm-hmm. Mm-hmm.

Vanessa Vakharia 41:43

that's kinda what you, you're supposed to be doing anyways. I don't know, I'm stuck at that end of that sentence. But I'm kind of like, so if you, if through teaching a child to be fluent in math facts, that was your way to teach them to be a problem solver, that almost seems like starting with the trees instead of the forest.

Christina Tondevold 42:07

Yeah, it's an interesting conversation of just, um, trying to grapple with, you know, all these things that come at us. You know, there's, oh, well, oh, now we should be doing which one doesn't belong, and now we should be doing

Vanessa Vakharia 42:20

Cat

Christina Tondevold 42:21

word prob- Like, there's so many different, like, things that we can be doing in the math classroom, and now, like, high-quality instructional materials and, and all of those pieces that get thrown at educators. And so whether you have been teaching for 30 years or you've been teaching for three years, you've probably already noticed this pendulum swing and bandwagon. You're getting on the bandwagon of this new next thing, and then guess what? We're gonna hop off that here in six months, a year, maybe two years from now. And so it's not about the things that come along. we really do need to think about the bigger picture what our goal is when we are helping students learn mathematics. Um, and I think that's something we all need to grapple with a little.

Vanessa Vakharia 43:14

Well, this is perfect, 'cause I was gonna say, can you leave th- us with something? It is summer. I want you to leave us with a, like, what's something we can do right now so that when we get into the classroom, you know, when school starts again, we feel more centered? And I think that's the thing, is to think about what you feel your purpose is up there in front of the classroom. What is the point of what you're doing? What's the point, and then let your answer to that question guide how you teach.

Christina Tondevold 43:41

Let your own answer to that, and then what you're hearing from your students where they are in their learning guide the instruction moving forward.

Vanessa Vakharia 43:51

Ugh. Okay, this was so fun. I have two questions to ask you that I ask every podcast guest, and you have 30 seconds to answer both of them. But I do have to tell you,

Christina Tondevold 44:01

Speed.

Vanessa Vakharia 44:02

this is...

Christina Tondevold 44:02

I

Vanessa Vakharia 44:03

Right? You gotta be fluent. You gotta be fluent.

Q1

Vanessa Vakharia 44:06

Okay. Someone's listening to this podcast, they're like, "I love everything Christina said. Honestly, that's so interesting," and like, "Yeah, for sure, maybe I have some math trauma," like maybe that's not why I'm good at math, but honestly the truth is I'm just not a math person. How do you respond?

Christina Tondevold 44:19

Mm. you have a wrong version of math because your version is probably like mine. I th- I had the opposite. I thought I was good at math until I had to go teach it, I learned that I was good at arithmetic, mathematics. And so you may see yourself as not good, but you've been not good at the arithmetic part of mathematics, of getting correct answers, and there's so much more to math beyond just correct answers. It is about a problem solver, and so, and everybody's good at that.

Vanessa Vakharia 45:01

It's so hard right now because I know I don't have time for what I'm about to say, but I'm gonna have to say it.

Christina Tondevold 45:08

Okay

Vanessa Vakharia 45:08

I really don't like... This is not your fault, okay? And I love your response, but I don't like that in the response you put yourself down. And the reason I don't like that you put yourself down is, is I don't like this new trend in general, and I'm not saying you're, like, sorry, but I don't like this trend of us in, in order to include more people in math, we now have to exclude people who were included. I don't like this 'cause I feel like it's like arithmetic is a part of math, and you were good at arithmetic. It's not like it wasn't real. Sorry, Christina. I shouldn't be yelling at you

Christina Tondevold 45:42

No, that's a good point because I say it a lot, um, and you're making me reflect on that part of it that I, I was good at a certain part of mathematics.

Vanessa Vakharia 45:53

great. No one's good at every part of mathematics, even like prolific mathematicians. It's not like they're... There's like game theory experts that don't know shit about like calculus. It doesn't mean that they're like, "I was never actually g- Sorry, this is like I shouldn't be taking this out on you, especially within the time confines that I gave you. Okay.

Christina Tondevold 46:12

We're all on this journey. That's why I call myself a recovering traditionalist. Like, there are times that somebody needs to hold a mirror to me and be like, "No, I don't... You're n- you're not saying that correctly," or, or, "Rethink about how you think

Q2

Christina Tondevold 46:25

about that."

Vanessa Vakharia 46:25

Um, the n- final question, I swear to God. final question is, if you could pick one thing to change about the way math is taught in school, what would it be? One thing

Christina Tondevold 46:34

Hmm, balance.

Vanessa Vakharia 46:36

Hmm. I love that

Christina Tondevold 46:37

of balance becau- and it goes with our theme of the pendulum. I wish I could tie baseball into this somehow, uh, but I don't see that. Uh, that we, we swing one way, we swing the other, and, and we realize that's not working, and so we'll swing all the way to the other way. And we need some of both all the time. We, we do need some timed activities. We need kids to be able to play and discover and guide their instruction. We need, um, sometimes we need a lot of practice. You know, when textbooks have moved one way and it's like a lot of contextual problems and discovery, and then those textbooks are lacking in practice, and we do need t- time to practice things. So we, if I could change anything, it would be to find a way to have some balance of both sides of these pendulums that we find in math education.

Vanessa Vakharia 47:34

I love this. And honestly, to really take us home, David has texted me, I think your landing metaphor may be what you were looking for. Not sure if it really is it. He said, "Gotta play both offense and defense to win a ball game

Christina Tondevold 47:50

Yeah.

Vanessa Vakharia 47:51

Honestly, this was such a treat. I've had so many epiphanies. It was so great talking to you. I can't believe how many metaphors we wove into one another. Like, I, I feel it was truly prolific. Um, thank you so, so much for

Christina Tondevold 48:06

for asking me, and it's fun to get to chat with you, and hope to get to meet you in person one of these days at a conference. So

Vanessa Vakharia 48:14

me too, and I'm really excited for the World Series when the Blue Jays hopefully make it again, because this time when I jump on the bandwagon, I know that I have a baseball girly to talk to about all the things.

Christina Tondevold 48:24

terminology, I'm

Vanessa Vakharia 48:26

I do.

Christina Tondevold 48:27

Yeah.

Vanessa Vakharia 48:28

See you on the field. Is that what they say? No?

Christina Tondevold 48:32

Play ball.

Outro

Vanessa Vakharia 48:36

Okay, I'm dying to hear what you guys think because honestly I feel like my entire relationship with fluency has changed, especially when I start thinking about using that word in contexts other than math, right? Like, the word is not polarized when it comes to languages. Why has it been so polarized in math? And I'm really curious if Christina shared anything that changed your mind and might even change your practice this upcoming school year, or if she said anything that made you double down on what you already think and what you're already doing. Either way, we wanna hear from you. Text the podcast, hit the link in your show notes. It literally says, "Text us," so text us. Also, if you want more of me and Christina, I'll be speaking at her free virtual thinking summit August 10th, 11th, and 12th. You should totally come. It's completely free, and there are such awesome speakers like Pam Harris, Zak Champagne, Deborah Peart, Juliana Tapper, Mike Flynn. I could go o- Brittany Hege. Like, there are so many people that you know and love that are gonna be speaking at this thing, and it's totally free. So head to the show notes. The link is in there to register for free, so come. As always, thank you guys for listening. I would love for you to do all the things I always ask you to do, like share this episode with a friend, rate and review the podcast, that stuff makes such a big difference to a small little operation like ours that is growing thanks to you and the amazing work you're doing out there. But also, I know we're busy. I know you are booked and busy. I know the last days of summer are approaching, and you have a million other things to do, so if you don't wanna do any of that stuff, that's cool. Just have the best day ever, and I will catch you back here next week

Outtake - baseball

Vanessa Vakharia 50:13

uh, David, I have a producer's note from about half an hour ago when we were trying to find that fairy tale. He's now said, "You've run out of pitchers in your bullpen and putting position players on the mound at this point", I don't know what that means. Okay, here are the two questions. You have 30 seconds for each, okay?

Christina Tondevold 50:29

Toss me a softball question here.

Vanessa Vakharia 50:31

Oh my God. All right.

Christina Tondevold 50:33

that

Vanessa Vakharia 50:33

I,

Christina Tondevold 50:34

to hit out of the park?

Vanessa Vakharia 50:35

I do know that, from when I watched the Blue Jays World Series. A soft toss. So here are two soft tosses that you have 30 seconds to answer. Okay.