Pam  0:01  
Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam, a former mimicker turned mather.

Kim  0:09  
And I'm Kim, a reasoner who now knows how to share her thinking with others. At Math is Figure-Out-Able, we are on a mission to improve math teaching.

Pam  0:17  
We know that algorithms are super cool human achievements. But ya'll, they're terrible teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop.

Kim  0:31  
In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships.

Pam  0:38  
Ya'll, thanks for joining us to make math more figure-out-able!

Kim  0:42  
My very favorite is when you say, "Oh, shoot!" and then you start talking. 

Pam  0:45  
Did I say, "Oh, shoot"? 

Kim  0:45  
You did. Like, right before you started doing the intro, and I didn't know if you like dropped

Kim  0:45  
a pen or what happened?

Pam  0:46  
I don't even know what I did. I think I need to take drink.

Pam  0:53  
I'm going to take a drink. Of water. Of water. 

Kim  0:55  
Oh, good. Good, good, good. Well, I found a podcast review that's actually kind of recent. And this is from DVLCATRNG. When I see T-R- N-G, I'm like, "Training?

Kim  1:12  
What's training?"

Pam  1:13  
I think it's training. It's developed. It developed CA training, 

Kim  1:16  
California training? See, California? 

Pam  1:18  
Maybe.

Kim  1:19  
I don't know.

Pam  1:19  
Cat Training. develop call training, develop (unclear).

Kim  1:24  
I don't know.

Pam  1:25  
What's another CA? California. Is California the only CA?

Kim  1:28  
I don't know.

Pam  1:28  
There's got to be a country CA. Canada! Develop Canada training.

Kim  1:32  
Maybe. Okay, well, you have to let us know if this is you. This person said "Just in time learning. I was not a podcast person until this year." Can I be honest? I have (unclear)

Pam  1:33  
Are you still not?

Kim  1:39  
(unclear) listen to podcasts. 

Pam  1:44  
You're still not a podcast person?

Kim  1:47  
Mostly because I like quiet in the car and when I'm running, I need music. Otherwise, I'm not listening to much. 

Pam  1:53  
Oh, it

Pam  1:53  
just depends on the state of my brain. There are times where I'm like, "I need something else happening right now or I'm going to spin out." 

Kim  1:58  
Yeah.

Pam  1:59  
And so, yeah then I have to be 1.5 because it's... 

Kim  2:02  
Oh,

Kim  2:02  
for sure, for sure. Yeah, I try really hard to listen to. There's some great math podcasts. But anywho. "...not a podcast person until this year. I'm a high school math teacher for 27 years who moved from being a mimicker to wanting my students to reason and explain their thinking without knowing that was what I was doing. This podcast helps me continue my Journey to help more students think and reason. Thank you, Kim and Pam. I look forward to binging this podcast."

Pam  2:28  
Oh, that's amazing! Especially high school teacher. I love it. I love it. I love it.

Kim  2:32  
Yeah, that's what I was going to say. I love hearing people who have taught 27 years at the high school level being open to learning more. I think it's fantastic.

Pam  2:38  
That is amazing. Thank you so much DVLCATRNG. 

Kim  2:45  
Yeah.

Kim  2:47  
Okay. Hey, today is a super fun episode. Like, picture... 

Pam  2:53  
They're

Pam  2:53  
all super fun, Kim. Come on. 

Kim  2:55  
Okay, well, yeah.

Kim  2:57  
This one's... This one is...

Pam  2:59  
Should we vote someday which ones are not your favorite?

Kim  3:02  
Oh. Probably, yeah. We should have a top five for each of us, and like a, "I wish that hadn't happened."

Pam  3:12  
You really have, "I wish I didn't happen" episodes. 

Kim  3:18  
No, gosh. 

Pam  3:18  
I'm going to be thinking about that all day today. What were the wish? What were the ones Kim wished wouldn't have happened? Hmm.

Kim  3:19  
I mean, honestly, we just put stuff out to the world. Honestly, like, we put videos out of us teaching. We talk about all the things. Like, the shame and the embarrassment and the like joy. Like, whatever. It's gone. I don't care.

Pam  3:32  
(unclear). Do you know, I have to tell you. So, really, really quick. What is her name? Oh, it'll come to me. Mmm, maybe. She's out of Houston. She's... I saw her present forever ago. Juanita Copley. I got it! Juanita Copley.

Kim  3:46  
(unclear). 

Pam  3:46  
I'm pretty sure now that I've said her name, this is a positive. Yeah, this is a great thing. It totally is. I had to think for a second because I was like talking before I thought. So, I never do that. She gave a presentation forever ago. I was super young, and she showed video of her teaching young children. And I remember she said something about... I'm not going to quote her correctly, but she said something like "Thinner Juanita and fluffy. You have to get over that. You have to get over the fact that you're (unclear)."

Kim  4:14  
For sure.

Pam  4:15  
And gol, Kim, sometimes in our coaching group. I'll look at the videos that we've posted, and it's like, Pam with pink.

Kim  4:21  
They all have a thumbnail of you.

Pam  4:23  
I got all these thumbnails of mine. Pam with hair down. Pam with straight hair, Pam with wavy hair. It's like, oh! It's just like.

Kim  4:23  
I know. 

Pam  4:23  
So, yeah, anyway. You have to give that up. Okay, moving on.

Kim  4:23  
Yeah.

Pam  4:24  
Let's dive into today's episode. 

Kim  4:31  
Okay. 

Pam  4:32  
What in the world are we talking

Pam  4:35  
about? 

Kim  4:35  
Well, so today we're going to talk about word problems. Which I think is a great topic. I think a lot of people deal with either loving or hating word problems. And as I'm laughing a little bit inside because as we said, "Okay, let's talk word problems. I actually loved problems. Word problems growing up. 

Pam  4:53  
You loved them. 

Kim  4:53  
I did. 

Pam  4:54  
Alright, yeah. 

Kim  4:54  
And I thought... You know, I'm picturing like a paper 1-29 Odd. And then there was always like the four problems at the bottom. 

Pam  5:09  
Four star problems, yeah. 

Kim  5:09  
Maybe two, maybe four. But when I.

Pam  5:09  
They were the word problems, right.

Kim  5:09  
Yeah, (unclear). 

Pam  5:09  
The story problems.

Kim  5:09  
Mmhm.

Pam  5:10  
Mmhm, yep.

Kim  5:11  
And so, I never had an issue with them because I found them like a little bit more intriguing and interesting than the boring 1-29 at the top of the page. It was like, "Oh, finally. You're going to give me the story. I don't have to make one up about the problems that I'm doing."

Pam  5:28  
Like, wait, wait, wait, wait, wait. You're saying you were making up stories about 1-29 Odd, the naked problems? 

Kim  5:34  
I don't think all of them, but.

Pam  5:36  
Just to be really clear. Most people didn't do that, Kim.

Kim  5:39  
Well.

Kim  5:39  
I mean, I was thinking about the problems, and so like it helped me to think about what was going on. Then I would like create a story. But at the bottom of the page, like that was done for me. So, I would just like read the story, and I'm like picturing what's going on, and I'm doing the thing.

Pam  5:54  
Oh,

Pam  5:55  
that we could create more of you. 

Kim  5:56  
You didn't do that?

Pam  5:57  
Heavens, no. Now, that's not to say I didn't enjoy word problems. I enjoyed them for a different reason. And I'm not proud of this. But I was one of the only people who was able to figure them out, and so it was a self esteem boost for me. 

Kim  6:10  
Oh, like,

Kim  6:11  
they hard for kids. 

Kim  6:11  
I never... 

Pam  6:12  
Mmhm. So,

Pam  6:13  
it was less about these are cool, and it was more about I'm cool. Again, I'm not super proud of that because now I realize I was more fake mathing. 

Kim  6:20  
Yeah.

Pam  6:21  
But I'm also going to suggest that if you're rote memorizing and mimicking stuff, word problems are completely different than the naked problems. 

Kim  6:28  
Yeah, it

Kim  6:29  
wasn't until I was a teacher that I realized that. It wasn't until I was a teacher that I was like.

Pam  6:31  
Because they weren't for you. They weren't completely different for you.

Kim  6:34  
They weren't for me. And I could instantly go, "Okay, well, if you..." Like, if the reading the language was difficult, then that made sense to me why those were more challenging. But as far as like the mathematics behind them, like I didn't. Until it was a teacher, I was like, "Ah. These. That's why these are more challenging for kids." (unclear).

Pam  6:56  
So, let's start this episode by suggesting. Start? Are we in the middle? By suggesting that we actually... Our goal is for naked problems and word problems to kind of be the same. 

Kim  7:08  
Yeah.

Pam  7:09  
That that's our goal for students because I'm going to just put out there. If we teach reasoning, if we really help kids develop their brains to reason mathematically.

Kim  7:22  
Mmhm. 

Pam  7:22  
Then they are reasoning in a context list naked problem and in a context word problem, and the reasoning is the same. So, I get it. I get it that for most of us word problems were completely different because you memorize these rules and did these procedures, and then all of a sudden you came to words, and you were like, "Ah! Which one do I do where? Are these..." Kim, when I was very first teaching is actually right before I got my first teaching gig. I did a long term sub gig for a couple of weeks in alternative high school. And when I got in, I taught... I don't remember if I was teaching, but the part I remember was the gal that came back to me for help, and she said... So, high school student. And she goes, "Ma'am, miss, were these the ones we did on Tuesday or the ones we did on Wednesday?" And I said, "What?" And she goes, "Well, if they're the ones we did on Tuesday, then I know you do this. But if they're the ones we do on Wednesday, then you do that thing." 

Kim  8:13  
Right, right. 

Pam  8:15  
I was like, "Oh, my gosh. You're putting so much mental effort into like categorizing on the day you did it and what the rules were. You can actually just think about the problem. Like, if you just like actually tried to make sense of what was going on. So, that's why, if they are completely foreign, it's now you have to like sort of guess. That's why teachers dive in and and try to have kids memorize keywords, and why we better practice all the sample questions from the high stakes test because if they haven't seen this exact, you know, question worded this way, they're going to get thrown off. They better see that a lot, so they can sort of memorize their way through that question. Does that make sense, Kim?

Kim  8:54  
Mmhm.

Pam  8:54  
Like, if it's they're completely separate and word problems are this different entity, and I have to guess and flip a coin about which algorithm I'm going to do when. 

Kim  9:02  
Yeah.

Pam  9:03  
Then we better have all this other stuff happen. And I'm just going to mention that I appreciate Dan Meyers work, where he says, "You know, remove all the unnecessary work. Let's not..." I think you just kind of said. If reading and the language is a difficult thing for you. I mean, I have a daughter with dyslexia, so that's a thing. Then we don't  want the burden of language to impact the mathing that you can do.

Kim  9:26  
Right. 

Pam  9:26  
So, we want to keep all that alive. But here's maybe the biggest point. The bigger point is, in reality, we need kids diving in to making sense of phenomena, and situations, and realizing is this context here additive? Is it multiplicative? Is it proportional? Like, what's actually happening here, even young, super young kids that are still in counting strategies. Like, are the puppies coming in the room? Are they leaving the room? Am I popping the balloons? Are the balloons flying away? Am I blowing up balloons? Like, what's actually happening? And feeling the phenomena, the actually the occurrence that we're representing with numerals, and equations, and functions, and graphs. Like, when they're diving in and making sense of all that, then I would suggest we get more of what you were doing where they're the same. And maybe they're making up their own scenarios, their own situations to those naked contextual problems. The best way to teach word problems is to teach reasoning.

Kim  10:31  
Yeah. So, I'm going to take you back to maybe my childhood. So...

Pam  10:38  
Just a few years ago.

Kim  10:39  
Yeah, just a few. Maybe four or five. And because I played with numbers, when I would get a piece of paper that had 1-29 Odd, I didn't own all the most sophisticated strategies at that point, but I played with number. And so, when I looked at that paper, I would go, "Okay, which ones do I want to solve first? Or in what order?" And, you know, there was a little bit of like playfulness to the paper. I didn't always go 1, 2, 3, 4. And so I would look for like categories. 

Pam  11:04  
You're such a rule breaker.

Kim  11:07  
I didn't. People do this right? Like, I didn't go in order. I looked for like what types of problems had some similarities to them. And I would do maybe those kinds of problems. And then, you know, so there was examination. There was playfulness. There was categorizing. There was when I got into a type of problem, then I would go like, "How do I want to mess with the numbers? Do I want to..." You know, I did maybe in my early, early years, a lot of adding left to right. And so, I was kind of tinkering, and holding number, and jotting some things down. And so, the time that I was spending, on average, in a 1-29 problem was maybe a little bit more than somebody else who was stacking and following steps because, you know, people could do that pretty quickly. And then when I got to the problems at the bottom, for me to read them and like consider them and think about how I want to solve them, was probably a bit longer than 1-29 Odd, but I don't know that it was like that much longer. So, in the work that I was doing in the first bit of the page, like I already had a little bit of stamina for working on math. Like, I dug in and I did some things. And so, I want to juxtapose that with my very first year of teaching. You know, we've talked a little bit about distortions, and one of them for me was that I understood math, and I did math, and I played with math, but I didn't really. I knew that lots of people did or didn't. Like, it wasn't something that we talked about. And so, I was like, "Hey, when I teach, I must teach the way that I was taught. And some people will do the things that I did and some won't." And so when I taught, it was very traditional. And so, you know, in the 1-29 portion of the paper, I would teach kids that first year or two to do some traditional steps. And when my kids could do those quickly, they were just banging those problems out. And so, when they got to the last few problems and they had to like actually engage with them for a longer amount of time, I would see kids like quit on it. Like, if it didn't have a subtraction symbol, or an addition symbol, or they had to like really like put some mental energy into it, I would see kids just kind of like, give up. And that didn't make sense to me in my first year because I was like, "Let's read the thing." And these are kids that like had some reading skills, and that wasn't the challenge was like reading. It was that they were like, "I don't know what I'm supposed to do." And, "This feels like it's taking too long." And I was just struck by their lack of stamina, their lack of willingness to dive in and to tackle, and so when you say kids, the best way to teach word problems is to teach reasoning. I am 100% on board for a couple of reasons. The most of which is that we give kids the opportunity to just hang on, to like give them something to chew on, so that they understand that Math is about like working on and working with.

Pam  14:18  
Putting forth mental effort. 

Kim  14:20  
Yeah.

Pam  14:20  
Not just putting forth effort to mimic.

Kim  14:23  
Yeah, yeah. 

Pam  14:24  
Those are different mental actions. Mimicking steps, different mental actions than... 

Kim  14:29  
For sure.

Pam  14:30  
...reasoning through using relationships, logicing your way through, 

Pam and Kim  14:33  
Yeah. 

Kim  14:34  
For sure. 

Pam  14:34  
Yeah. So, I'm aware that we both just said something like, "The best way to teach word problems is to teach reasoning," and that might sound like the best way to teach word problems is to give them some problem solving strategies, like pull you sort of thing. You know like, what's reasonable. And that's not what we mean. So, let's dive in and do an example of when I say the best way to teach word problems is to teach reasoning. Well, in fact, let me say. Let's illustrate that. But let me say one other quick thing. Sometimes people will say to me, "Pam, you sure do an awful lot of naked number stuff. Where are your word problems? And I'll say, "Oh, that's..." Because, to me, they're like the same.

Kim  15:14  
Mmhm.

Pam  15:14  
Like in the I'm going to help you reason additively, multiplicatively, proportionally, so that when you are in additive situations, your brain is reasoning additively. So, let's do an example of that. So, let's say that we had... We're suggesting that you teach reasoning for a problem like 36 plus 19. And so, if we are working with kids and we're helping them learn reasoning, then a kid is going to attack that problem not by saying, "Ooh, I better line those up. What were the... What did my teacher tell me on Tuesday? You know, I start with these small..." Not those steps. But rather, if we've been teaching reasoning, then a young student goes, "Mmm, let's..." Well, young. I said young because in my head, I'm like, they haven't... Their first gut instinct isn't the algorithm, so we've sort of caught them. We're teaching reasoning. That first young student is going to go, "36 plus 19. Alright, what do I know about 36? I have some sense of it's more than 30, and it's less than 40." And it's kind of in this area. And I kind of like 36. Like, I have this sense of the magnitude, and where it lies, and kind of neighborhood and nearness to things enough to go, "But now I'm adding. Adding, that's gaining. I'm going to the right on a number line. It's getting bigger." I have this feel for this additive thing that I'm getting bigger. "19. What do I know about 19?" And one kid might go, "Let's see if I'm supposed to get bigger than 36. Ooh, I can get bigger by 10." And a kid might add 10, and then know that they're going to have to add the rest of the 9. Or another kid might go. "19. Starting at I can picture 36, but I'm going to add 20. I'm not going to add 19. I'm add too much." And that kid might think I'm going to go a bit too... Add 20, then I'd have to adjust from there. But another kid might go, "36 plus 19. 36. I know where that is. But 19, I know where that is. Man, if I had like these piles of marbles, 36 and 19, I can just move a marble, and that would... I'd end up with 35 in this pile and 20 in that pile. 35 and 20. I have some feel for 30 and 20, and I got this 5 left over." Like, all of those feelings, that gut intuition that I'm pulling things together, I'm getting bigger than 36. I wish you could see me. I'm doing all these actions in the air

Pam  17:33  
When a kid then hits a problem, and they're like, "I'm on page 36, and I need to read 19 more pages tonight." Or, "I've read for 36 minutes, and I need to read 19 more minutes." Or give me another one, Kim. Oh, for you. I'm thinking about Kim. "I ran 36 miles. I need to run 19 more miles." Maybe it was kilometers? 

Kim  17:53  
Minutes. Maybe minutes. 

Pam  17:54  
Okay, minutes, okay. Like, pick your thing. When a kid hits that word problem, now they're like, "Okay. I've been in this cognitive space before. I have thought about starting at 36.

Pam  18:05  
Whatever's happening. I'm gaining I'm feeling like the this is adding on. I'm going to the right on a number line. 19. What can I do with that 19?" If we've taught, if we've literally helped them build those mathematical relationships, when they hit a word problem and the word problem says

Pam  18:26  
you're on page 36 and... No, that's bad. Off the cuff. Off the cuff. You have $36.00, and you give Kim $19.00. I don't know why Kim needed 19 of my dollars. 

Kim  18:36  
I'll take it! 

Pam  18:37  
Then, all of a sudden, that felt different. 

Kim  18:40  
Yeah. 

Pam  18:40  
And I didn't have to memorize. "Give" is one of those key words. "Loan" is one of those key words. "Slipped under the car seat." Like, you can't memorize all the words that could ever be... slipped under car seat was the weirdest (unclear).

Kim  18:55  
It's like losing change in the 80s. (unclear).

Pam  18:57  
There you go.

Pam  18:58  
Yeah, okay. Yeah, I'm old. It's true. When I'm in a word problem, in a story problem, in a context, and I feel the actions that I have felt before, now I just continue to reason about it.

Kim  19:11  
Yeah. What you were describing right now is like it's taking me to a place in my head where I'm like, "Yes! Yes! This is like how it feels." Like, this is exactly like when I look at a problem, I just like zone in and just get so entrenched in what is happening. And it is like it's so being cognitively involved, both in understanding the magnitude of the numbers like you described and how you're feeling about the operations. So, when somebody's reading it, the feeling comes out. And I am so happy that we are helping develop that in students because it's how I felt, and when I talk to my kids, it's how they feel. And I just think many, many, many more people deserve to like feel that ownership of the things that they're looking at.

Pam  20:07  
Feel the phenomena. Feel the operation. Feel the sense of. For example, if I'm in... Well, I was about to tell you where in the curriculum, but I'll just say. If I were to say, hey, it's 4 scoops to 5 gallons for this recipe. And it's 3 scoops for... I should have made this up earlier. 4 gallons for this other recipe. Which one's more orangey? Then I want a kid to go, "Ooh, it's like 4 to 5 and 3 to 4. This feels like when I've thought about equivalent orangeyness before. This feels like where I've thought about equivalent situations where if I scale the amount of scoops, I've got to scale the amount of water in order for it to be the equivalent orangeyness. This feels like other situations where I've felt this idea of scaling in tandem. Ooh, it feels like I could do that here. I'm feeling the phenomena of equivalent ratios. And ooh, where have I done that? What have I done? And because it's not, "Ooh, what word have I memorized, and so what formula am I going to plug it into?" But I'm actually feeling the sense of, "Ooh, it's like this ratio idea. And do I want equivalency? Do I want more than? Do I want more orangey?" I can feel that in order to do that, then I have a sense of what that would look like. 

Kim  21:24  
Yeah. 

Pam  21:25  
It's almost like if I have really built in kids proportional relations. And a kid can say, "Well, yeah, I have a feel for what y equals 27x feels like. Like, it's 1 to 27, and 2 to 54, and 10 to 270. And I can see that line. If I graph it, it's steep because it's 1 to 27. Like, that's a steep rate. When they feel that, then there might be a day where they look at data and they see 1 to 28 and 2 to 55, and they go, "Whoa! That almost feels like 1 to 27 and 2 to 54. It's so close. 1 to 28 and 2 to 55. How close is it? Well, both points just shifted up 1." And if you could see my body, I just took my arm to be a line in the sky. In the sky. In the air. Where am I? And so, I've got this line, and both points just, and I just shifted my arm up one because both points just shifted up 1. Ooh, well, I've done that before. I felt where I've had data shift. And I've had linear equations be the model for that data. And when we shifted up, oh, I know what that  looks like. That looks like in a function where I've just taken that f(x), and I've just added one to all those values. So, I could say that if f(x) was my original function, then f(x) all plus 1 is that transformed function.

Kim  22:44  
Mmhm. 

Pam  22:45  
So, there are times where... Let me give you one more brief example. And I think I've told this story a little bit, but I'm going to tell in a kind of a different purpose today. One day, my third who probably had the best teaching and everything because I'd worked with his teachers the longest, and he had you. Yay. Came home as a senior. He was taking calculus and physics. And he said, "Hey, Mom, I need your help." And I was like, "Okay." You know like, whatever. And he goes, "Yeah, angular velocity and momentum." And I thought to myself, "I haven't done that since high school. I never taught it. When I took it in high school, it was a small part of a test. I never really understood what was going on. I memorized my way through it. I may have even missed those questions on the test. Still got the A whatever. So, in this moment... Now, this is still early. This is where I've done work with younger grade teachers, and I'm starting to do work with older grades. I'm really turning my attention to secondary math. But I kind of haven't done a lot of work there yet to really teach the way we are now advocating. And he says, "I need your help with this." And I was like, "Oh, ask me after dinner." And I think to myself, "I just kind of hope he forgets, and he'll just ask his teacher tomorrow, and whatever," because I've never  done it. Well, after dinner, he doesn't forget. And so he says, "Hey Mom. You know, I need your help with this." In my head, I was like, "Okay, then I'm going to need the formula. And my goal is to look at a couple of examples and see where I'm going to put the numbers in the formula. And then I've got enough algebra that we can then solve for the missing value of whatever. Not because I meant that to be in my head.

Kim  24:09  
Right. 

Pam  24:10  
But because in the moment to moment decision making, I leaned on my prior experience. And my prior experience with physics was, you find the formula and you plug stuff in. And he goes. I said, "So, you need help. Okay, go get your textbook." And he goes, "Textbook. I think I have a physics." And I was like, "Seriously? Go get your textbook." And he's like, "Mom, I we don't need the textbook here." And I was like, "Dude, I haven't..." You know, all the things. "I haven't taught this. I've never taught it, and I haven't done it since high school. I didn't understand it when I did it. I'm going to need your textbook." And he goes, "Just help me understand the question. So ya'll, think of a word problem. It was totally a word problem. It wasn't word problem. It wasn't a naked formula or whatever. It was a word problem. And he goes, Just help me see if we can understand what's going on. And so, "Fine." We pulled up the problem, and he goes, "Okay, this? And I was like, "Yeah, so then that." And he goes, "Okay, then I think this, right?" And I was like, "Well, but and this. Wait that.? And he's like, "Ooh, but then it would be this. And then that's the answer." And I was like, "Oh my gosh. Physics is Figure-out-Able. 

Kim  25:04  
Yeah. 

Pam  25:04  
Like, it was in that moment where I was like, "Gah!" Like, just because I hadn't thought about it before didn't mean that it wasn't figure-out-able. But maybe more important for today's episode, it was a word problem where I thought, "Here's how you..." That my experience was here's how you do these physics word problems. You pluck the numbers. You put them in the formula where they go. You solve for it. And you're done. Or, actually, it's figure-out-able. I just needed to feel the phenomena of angular velocity and momentum. And bam, we could go from there.

Kim  25:36  
And I'm going to wonder... Maybe this is not true, but I'm going to wonder if you thought you were going to plug in some stuff and do it. I'm quoting "do it". 

Pam  25:46  
Yeah.

Kim  25:47  
But really because you two owned equivalence, when you said, "This, so this," I'm wondering if that was where you were thinking about equivalence.

Pam  25:58  
Mmm, could be. 

Kim  25:58  
And so, you know, we talk about this a little bit with young students that when we're talking about additive, and multiplicative, proportional strategies, the... Let's just stick with the four operations. The equivalence strategies are the most sophisticated. And I think it's because to own deeply equivalence takes some... It's weightier, right? It's you have to understand the situation.

Pam  26:28  
More cognitively complex.

Kim  26:30  
Yeah. And in order to really understand the situation and to be able to manipulate it, that takes some. Yeah, so it's more complex. And so, like if you have a problem with like 12 times 18, you might say, "Okay, it's got an equal sign, and so maybe I'm supposed to find the answer." But in order to think, "Well, that's equivalent to something like 6 times 36," you have to be involved in understanding what 12 times 18 is.

Pam  26:59  
You said that really fast. Sorry, keep going, keep going.

Kim  27:02  
Okay. No, no. Okay,

Kim  27:03  
so 12 times 18, you have to recognize that's twelve 18s. And I could find an answer to twelve 18s. But also, I could know an equivalent problem is 6 times 36. Six 36s is equivalent to twelve 18s. And so, you might manipulate that number, manipulate that problem and with equivalence to say, "I'm going to find a different problem that I want to solve." And those strategies are complex, and they take kids who are thinking and reasoning, and it's about the 12s and the 18s and the relationships between the problems. And so, when kids have been asked to reason, they're able to do that kind of work, and it starts by saying equal doesn't mean just "do it". It's about building equivalence in your students.

Pam  27:53  
Yeah, and you had me thinking about if I can think about twelve 18s as half as many groups that are twice as big. 6 groups of 36. 

Kim  28:03  
Mmhm.

Pam  28:04  
That's kind of a way that you can. And to do that, we're sort of, like you said, kind of outside the problem almost. It's not like "do it". The equal sign means "do it". The equal sign means find something equivalent. What do I know? How can I reason? We want to put kids in reasoning land, so everybody has the opportunity to think like Kim just did. And if we're in reasoning land, word problems kind of become...

Kim  28:27  
Yeah.

Pam  28:28  
...a non issue. They become just the same as everything else. A given. We'll have to take the reading. You know, we have to help kids read a little bit. 

Kim  28:35  
Sure.

Pam  28:36  
But if we take any difficulties with reading out, it's just we're still reasoning. We're reasoning everywhere. Because Math is Figure-Out-Able. Alright, ya'll, thanks for tuning in and teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutable.com. Thanks for spreading the word that Math is Figure-Out-Able!