Pam  00:01

Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris, a former mimicker turned mather. Because we know that algorithms are amazing human achievements, and we also know they are not good teaching tools because mimicking step-by-step procedures can actually trap students into using less sophisticated reasoning than the problems are intended to develop, and trap students into taking a digit focus, piecing everything into its pieces and parts, and never reasoning about the problem as a whole, and it can trap them into an identity of not being a math person. 

Kim  00:09

And I'm Kim Montague, a reasoner who now knows how to share her thinking with others. At Math is Figure-Out-Able, we are on a mission to improve math teaching. Yeah. In this podcast, we hope you teach mathing, building relationships with your students, and grappling with mathematical relationships.

Pam  00:57

We invite you to join us to make math more figure-out-able. Hey, Kim. I got to raise my desk. Just a second. 

Kim  01:03

Okay. 

Pam  01:05

I know. I just stood up, and it's when I'm... You don't care. But I needed to be higher. It's higher now. Okay, there we go. Let's... 

Kim  01:12

You got it. 

Pam and Kim  01:13

Hey. Hi.

Pam  01:14

is it going?  How

Kim  01:15

It's good. 

Pam  01:16

Oh, hey. 

Kim  01:17

We're wrapping up a great series today. We've been talking about the traps and algorithms. You've shared about the two traps, so far, that kids have to deal with with algorithms, and today we're going to tackle the third. So, if you have not heard the previous episodes, I invite you to go back and take a listen, even now before you hit the third. Or you can go back afterwards and hear about the three major traps of algorithms. 

Pam  01:44

Yeah. So, Kim, this has been work for a very long time that I've been pouring into my brand new book, Developing Mathematical Reasoning - Avoiding the Trap of Algorithms. In these episodes, we're scratching the surface, giving everybody a feel for the three major traps that I think afflict all algorithms. In the book, I go into all the traps of a lot of the algorithms that we use and what to do instead. Really, here, we're just kind of skimming the surface a little bit. But let's skim the surface on a super important trap which has everything to do with our identities as mathers and how we think and reason mathematically. Yeah. Alright, so let's dive in.

Kim  01:47

Mmhm. 

Pam  01:47

Let's do a quick string, Problem String. 

Kim  01:54

Mmhm.

Pam  01:56

We're going to get some reasoning flowing, and then we're going to talk about how if we were to have solved some of these problems using an algorithm, how we could actually have been trapped in a new way. Alright, so here we go. Kim, if you were to go to the bulk food aisle at the grocery store.

Kim  02:51

Mmhm. 

Pam  02:51

Could you imagine something that would cost $8.00 for 5 pounds? 

Kim  02:58

Okay.

Pam  02:59

Maybe 5 ounces? 5 liters? I don't know. Pick your (uncleaer). Okay. It's got to be $8.00 for 5 something. Can you? Is there something that you can think about that's...

Kim  03:08

I was thinking 5 pounds, but that's... I don't know. It's like special flour. I don't know. 

Pam  03:12

Oh, how about salt for a water softener. Isn't that about $8.00?

Kim  03:18

Oh, I have... That is not my job in this house I have absolutely no idea.

Pam  03:21

Oh, you know what? 

Kim  03:21

Yeah.

Pam  03:22

No, I think I'm crazy because I think it's $8.00 for like 25 pounds. 

Kim  03:26

Yeah.

Pam  03:27

Those bags are heavier than 5 pounds. I was just thinking of something I knew was $8.00

Kim  03:31

I'm honestly am just picturing like a random bag. I don't know what's in it, but I'm picturing a bag. (unclear).

Pam  03:35

Okay, so you don't care what's in it.

Kim  03:37

(unclear). $8.00 tag on it. Yeah, I don't know. I'm going to have to check it out when I go to the bulk food aisle next. 

Pam  03:43

So, if you went to the bulk food aisle, and you put the thing in the bag, and you walked over and you stuck it on this scale, and then it came with the little stickers in it, then you could see that it cost you $8.00 for 5 pounds. And you don't even know what's in the bag, but you can kind of picture?

Kim  03:57

Yeah.

Pam  03:57

Yeah. And so, 5 pounds like that's... I mean, you're strong, you're strong.

Kim  04:03

Yes.

Pam  04:03

I remember when you did all that weight lifting. You got really upper body. Your body, yeah. Yeah, are you still? (unclear).

Kim  04:08

I'm lazy. (unclear).

Pam  04:09

No, you're just running now? 

Kim  04:12

I work a lot. 

Pam  04:13

I'm starting to do so more.

Kim  04:14

We're busy. 

Pam  04:15

I'm starting to do a little bit more weight lifting. 

Kim  04:17

That's great. 

Pam  04:18

Yeah, give me a few months before you check on that. 

Kim  04:21

I'll follow up.

Pam  04:22

Anyway. Alright, so if whatever it is that we just bought was 8 pounds for... Or, sorry, $8.00 for 5 pounds.

Kim  04:29

Okay.

Pam  04:29

What if I had $16.00? How many pounds could I get? 

Kim  04:33

Oh, okay. So, it's twice as much money, so twice as many pounds, so 10.

Pam  04:39

Okay, so $16.00 for 10 pounds. 

Kim  04:41

Mmhm.

Pam  04:41

What if I only wanted to buy 1 pound. So, it's $8.00 for 5 pounds, but I only want to buy 1 pound. 

Kim  04:49

Mmhm.

Pam  04:50

Okay.

Kim  04:51

You just asked me for $16.00, and I found that it was 10 pounds. 

Pam  04:58

Your welcome.

Kim  04:58

So, I'm going to use that 10 pounds. Yeah. I was just thinking, I'm glad you asked me that one. So, I'm going to use the 10 pounds, and I'm going to divide by 10.

Pam  05:07

Okay.

Kim  05:07

So, 16.00 divided by 10 is $1.60,

Pam  05:13

Cool, so $1.60 would get your 1 pound of whatever that stuff is that we're buying. 

Kim  05:17

Yeah.

Pam  05:17

Okay, cool.

Kim  05:18

Mmhm.

Pam  05:19

Nice. What if I wanted 32... No, I had $32.00 what if I had $32.00?

Kim  05:24

Mmhm. 

Pam  05:25

How many pounds could I get? 

Kim  05:26

Okay, you just asked me about $16.00 and that was 10 pounds, so I'm going to double it. $32.00 would be 20 pounds. 

Pam  05:35

Okay, cool. And what if I really like walked in to the store and I had $33.60, $33.60. How many pounds? Maybe I had a coupon. Maybe that's all that's left on my credit card. I don't know.

Kim  05:53

Maybe that's your receipt, and you wondered how much you bought? 

Pam  05:55

Oh, very nice. That's excellent. Thank you for helping my context there. 

Kim  05:58

Mmhm.

Pam  05:59

So, if the receipt said that I spent $33.60 because I just was scooping stuff in the bag. 

Kim  06:04

Have you every sent your kids to the bulk food aisle and they roll up with, "Hey, you're going to get this bag. Happened to me.

Pam  06:11

There you go. That is good context. I like it. Alright, so they roll up, and you look at it, and it's $33.60 that's how much you're about to pay. 

Kim  06:18

Yeah.

Pam  06:19

That they stuck in the bag. So, how many pounds are you hefting, right now?

Kim  06:23

That's a lot. I'm going to say that $32.00 I spent was 20 pounds.

Pam  06:28

Mmhm.

Kim  06:28

And the previous $1.60 was 1 pound. So, add those together is $33.60 and 21 pounds.

Pam  06:37

Alright, so $33.60 Yeah, you'd be heft. It's 21 pounds. Can you picture that bag? 

Kim  06:43

It's like, a big bag of... Yeah, I don't know.

Pam  06:47

Heavy, yeah. I'm not sure anybody would get 21 pounds in the bulk. 

Kim  06:51

Maybe, like masa or rice or something?

Pam  06:54

Masa? What is that?

Kim  06:55

Oh, that's like for like tamales. It's like the corn meal, I think is what.

Pam  07:00

Ah, (unclear).

Kim  07:01

I mean I don't make, so I don't, but.

Pam  07:04

You just eat them. 

Kim  07:05

I actually don't. Oh. Along with your chips and salsa?

Pam  07:08

Oh, yeah. When people are eating tamales, you're eating the chips and salsa next to them. Mmhm, for sure. Okay, yeah, that makes sense. Cool. Do you make spicy salsa?

Kim  07:16

Oh, the hotter the better. Give me all the jalapenos! Give me all! Yes! 

Pam  07:20

Really?

Kim  07:20

I want my mouth to burn!

Pam  07:24

You take the heat. You take the heat, and I'll take the non.

Kim  07:27

I do. And I have one sweet little friend in my house who cannot handle any hot. 

Pam  07:32

Oh, he's my friend. 

Kim  07:34

None, none. Makes me so sad.

Pam  07:36

Yeah, we're not big heat people over here.

Kim  07:38

Oh, love it.

Pam  07:39

Though. I will tell you we've been eating at our neighborhood Tex Mex place, and I can now take some heat. Which doesn't that mean that I've killed some taste buds or something?

Kim  07:48

I don't know. No. (unclear).

Pam  07:49

Yeah, whatever. Alright, cool. So, Kim, would you consider that when you found when your kids rolled up with that bag and on the tag it said $33.60 got to be 21 pounds. Would you consider that we just solved the proportion 8 to $33.60. So, I've just written 8 "fraction bar" 33.6 equals 5 to x. 

Kim  08:16

Mmhm.

Pam  08:18

So, we just solved that proportion. Now, I helped you, right? Like, I gave you some helper problems. I kind of walked you there. But we were developing mathematical reasoning, so that we're developing proportional reasoning, so that you could like reason about these, about this ratio, and keep the ratios equivalent, and get to an equivalent ratio to where you found that x was equal to 21. Yeah?

Kim  08:41

Mmhm.

Pam  08:42

Alright. So, listeners, I would just invite you to consider that proportion 8 "fraction bar" 33.6 equals 5 "fraction bar" x. If I'm in a traditional classroom...

Kim  08:56

You're crying. You're crying (unclear).

Pam  08:58

Well, you've just been handed a problem where if I'm going to cross, multiply, and divide, I now have a fraction. Excuse me, a decimal multiplication problem. 

Kim  09:06

Mmhm.

Pam  09:06

And so, now I get to ask myself, "Is this where I butt cheek the decimal? Or is this where I line them up?"

Kim  09:11

Yeah.

Pam  09:11

And maybe I'll do both. Like, if we've heard some teachers just say, always line them up, and then butt cheek if you're multiplying. So, you've got that decimal multiplication problem, then you're going to have to do long division. If you do that decimal multiplication problem, in that multiplication problem, you could use a bunch of digit. Sorry, a bunch of additive reasoning. You could take those that digit focus, and you could just, in that multiplication algorithm, do a bunch of digit focused, single-digit multiplication where you are either using additive reasoning or at best, single digit multiplicative reasoning, trapped in less sophisticated reasoning. So, notice, if I'm cross multiplying and dividing, the best reasoning that I'm using is multiplicative. I'm not reasoning proportionally at all, even though this is a proportion, and I'm solving a proportion. I should be developing proportional reasoning. Let's just review. It's counting strategies to additive reasoning to multiplicative reasoning to proportional reasoning. So, if I'm solving a proportion, I should be developing and using proportional reasoning. But if I'm cross, multiplying, and dividing, I'm using less sophisticated reasoning in the multiplication algorithm. I just made my hands go shorter, smaller, so I'm now in multiplicative reasoning. And if when I do that multiplication algorithm and then division algorithm I'm using additive skip counting, then I'm using additive thinking. That less sophisticated. And in the addition and subtraction I do in those algorithms, I could be using counting strategies.

Kim  09:13

Right.

Pam  09:19

So, I could be getting a correct answer to solving this proportion and using less sophisticated reasoning than the problem is calling for. So, that's our first trap that we talked about a couple episodes ago. Secondly, every one of those algorithms turns the problem into digits. That's the trap we talked about last week. Notice, that I'm not thinking about the ratio of 8 to 5 or the ratio of 8 to $33.60. I'm not thinking about that as a ratio. I'm literally thinking about cross, multiply, and divide. I'm drawing a bat and a ball or whatever other dumb thing that we've heard that people like. Ya'll, remember, if you're creating a story, chances are super high you've turned math into fake math. So, I've kind of got... I've definitely have that digit trap where I'm turning everything into just pieces and parts, and I'm not really reasoning about the whole thing. Here's the last trap that I'd like you to consider. I invite you to consider that if we turn mathematics into rote memorizing and mimicking steps of algorithms, then we've changed the definition of math, mathing, into memorizing and mimicking.

Kim  11:45

Right.

Pam  11:46

And if we make that the definition of mathematics... So, trap number three is the Identity Definition of Math Trap. If we make, if we have this distorted view of what math is, which it isn't, but we've turned it into rote memorizing and mimicking, then how does our identity deal with that? Well, one way it could deal with that is that we could say, "Okay, that's what math is. It's rote memorizing and mimicking. And bam, I'm good at that. Give me some more. Bring some more steps on." Well, frankly, that was me. In fourth grade, I was the first in my class to do long division. I didn't know what I was doing, but, man, I can do those steps and smile and circle my answer at the end, and I had classmates next to me going, "What's the next? Bring down? That's a mathematical operation?" Like they were trying to memorize the steps. I have a younger sister. Now, I have four sisters, so you'll never know which one this is. But I have a younger sister who cried every day before she went to fourth grade, and my mom would... I was about to say, "force her in the car, kind of". Like, take her to school crying because she could not get long division. Like, it was super, super traumatic. And we sat and worked with her, and it was not a happy time. And her identity took the hit of saying, "If that's math, then I don't do that. Like, I'm not a math person. We don't do math in our family." The third trap is all about the definition of math, and if we create the definition of math to be what it isn't, which is rote memorizing and mimicking algorithms, that we get these distorted views where I had the distortion of, "Sweet! I'm good at this stuff!" What was I good at? Fake math? Ah! Imagine the hit it took when I realized I was actually good at fake math and not good at real mathing at all. Or we get this identity of, "Well, if that's what math is, then who wants to be good at that? I'm not a math person. We don't do math in our family." We have a sort of national identity that there's a very few people that have the math gene, and everybody else doesn't. No, no, no. Everybody can math! Yeah, so this is the third kind of major trap, and I call it the Definition of Math Identity Trap, where we've got the definition wrong, and so now we assume these crazy identities. Well, let me mention one other identity that we could assume. Those who actually see the shark. That low doses of the patterns out there, they're able to just naturally kind of do. They have the natural inclination or natural talent for math. All of a sudden, they become this weird elite in our...

Kim  14:20

Right. 

Pam  14:21

In our society. People would hear that I was a high school math teacher, and they would almost like take a step back. Like, "Oh, you're one of those." Like, we have this sort of strange... Yeah, just weird elite. That's how we get this idea that there is a math gene and only some people, some special called people could do math. No, no, no. Everybody can math. We start to actually know it's a thing like and have the actual definition of math, mathing, the real mental actions that mathematicians do. Or we get these kind of really weird identity traps, and we get trapped into identities that aren't... I don't know. That we can stay out of.

Kim  15:01

Yeah. And some people... You know, for some young learners, that happens early. Like, really early. First grade. You know, second grade. Some it's not until later. Fifth grade, seventh grade. And for you, it was as an adult, right?

Pam  15:16

Sure enough, yes.

Kim  15:17

So, it's not like you are making it out of school, and then, like you're off scot free. Like, at some point you were like, "Wait a second," and then you had to take a look at your learning and really work hard and like set kind of a little bit of what you thought you knew aside to like take on the learning. And I think what you're suggesting is that we can acknowledge that that's a thing and help people earlier.

Pam  15:43

Absolutely. You know, reminds me of my neighbor, Russ. Who's no longer my neighbor. 

Kim  15:47

Oh, we love Russ. 

Pam  15:49

Russ, come back! Come back! Super, super guy. When he came over and he says, "Hey, I'm about to do this..." I think he said, "more cool occupations than anybody ever". And at this point, he was doing some studying to take a test to be a... Sorry, a bank something. I don't even know it was finance. I don't think it was like bank. Some finance something. And he goes, "I just don't understand eighths." And I was like, "What do you mean?" And he goes, "0.375 and point... Like, is there? There's got to be some rhyme or reason." He said something like, "You know, I just was never really good at math or not a math person." Something, something. And I said, "Russ, like you're a carpenter." Like, he's super handy. And I said, "You know like, if you're measuring one-eighth." And he goes, "Wait, wait, wait. Like, one-eighths and eighths? They're the same?" Because he was totally... Someone had said "eighths", and then connected only to the decimal representations, because he was dealing with points and percentage points and stuff. And he's like, "Wait, wait, wait. Those are the same?" And I was like, "Yeah." He goes, "Oh my gosh!" And he just like left. Like.

Kim  16:46

Yeah, yeah. 

Pam  16:47

He totally, "Oh, okay. Then..." Because he had so much experience measuring with eighths and fourths and all the things with with our customary system because he'd had that experience. I knew how good he was. I just had to connect it a little bit. But how tragic it is that this guy is actually super good at all that stuff would have this identity of like, "Oh. Well, you know like, not that real math stuff. I mean, I can do Carpenter. I can measure. But not that other stuff.

Kim  17:13

It's so pervasive that people say that, and other people go, "Oh." And like, "Like, yeah. Okay." I mean.

Pam  17:23

We just take it. Yeah. 

Kim  17:24

Yeah, it's oh so sad. Yeah. 

Pam  17:27

So, everybody, there are numerous traps of the algorithms, and in my new book Developing Mathematical Reasoning - Avoiding the Trap of Algorithms, I'm going to dive deeper into the three traps that I've just talked about into the major algorithms that we teach and what to do instead. Ya'll, we can actually help people math the way mathy people math. We can all be mathers doing the mental actions of mathematicians because we're not trapped by rote memorizing and mimicking algorithms.

Kim  18:03

Yeah, so get a copy of this fantastic book. Grab it for a book study with your friends, your teams, your leaders. Yeah.

Pam  18:12

Suggest it your leaders. I think this... We've got a book study companion guide that's going to be super helpful. In the book are QR codes where you can watch videos of kids who are trapped in less sophisticated reasoning, kids who are using the appropriate level of reasoning. We've got videos of Problem Strings where people are developing the reasoning before your eyes, using the sophisticated level of reasoning that we're actually intending to develop. It is a phenomenal resource. I'm super, super grateful to everybody who's helped write. My son, Cameron, who you've heard about on the podcast, was a huge help in writing the book. Kim, he did a great job helping review stuff and helping me come up with all the things. So grateful to Corwin. They've been fantastic. Looking forward to the four book series that will come after this book that will focus on the grade levels. There'll be a K-2, 3-5, 6-8, 9-12 that will really focus on the major models and strategies in those grade bands and how to high dose kids with the major patterns to develop the kind of reasoning in those grade bands. Super excited about it. Thanks to everybody who's been major influence. And yeah, go check it out, everybody.

Kim  19:25

Yeah. 

Pam  19:26

Alright, 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. Let's keep spreading the word that Math is Figure-Out-Able.