[00:00:01.220] - Pam
Hey, fellow math-ers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris, a former mimicker turned math-er.

[00:00:10.280] - Kim
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.

[00:00:17.750] - Pam
We know that algorithms are really cool historic achievements, but may we suggest that they are terrible teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop.

[00:00:32.500] - Kim
In this podcast, we help you teach math-ing, building relationships with your students, and grappling with mathematical relationships.

[00:00:39.140] - Pam
Y'all, thanks for joining us to make math more figure-out-able. Hey, Kim.

[00:00:43.570] - Kim
Hello.

[00:00:45.010] - Pam
How's it going?

[00:00:46.130] - Kim
It's good.

[00:00:47.090] - Pam
Yeah? You done any running lately?

[00:00:49.160] - Kim
Yes. And I have to tell you that sitting for a long time after you run in the morning. Whoo!

[00:00:54.230] - Pam
Ooh, yeah, yeah, yeah. Yeah.

[00:00:55.750] - Kim
I love to run. I'm a terrible foam roll stretcher. It's awful.

[00:01:01.270] - Pam
Foam rolling stretching is what... How do I say it? It's that two-edged sword.

[00:01:07.040] - Kim
You're supposed to do it. (unclear).

[00:01:08.080] - Pam
Like, well, it feels, it feels so good, but like getting on the floor to do it is a whole. At least for me. I don't know. It's like a whole thing.

[00:01:15.460] - Kim
I'm like, "Do as I say, not as I do," to my son.

[00:01:18.100] - Pam
Hey, so I've been lifting weights for a couple of years and I kind of let the cardio go, which is interesting.

[00:01:24.190] - Kim
Yeah.

[00:01:24.670] - Pam
But my cardiologist the other day was like, "You can't do that." I was like, "Really?" He's like, "No." I'm like, but when I lift weights, I do it right in a row. I get my heart rate up. He goes, "No." So I'm back to ellipticaling, I'll have you know.

[00:01:37.970] - Kim
Very good. Very good.

[00:01:38.690] - Pam
You know, it kind of stinks when you sort of start over. It's not ground zero because I was getting some decent cardio because I was not sucking air that bad. But yeah there's something about cardio that's just...

[00:01:50.350] - Kim
I love it. Love it. Love it. Love it.

[00:01:52.270] - Pam
Yeah, it's funny that like the third day I did it, I was like, "Why did I stop?"

[00:01:56.770] - Kim
Yeah.

[00:01:56.770] - Pam
So, now I'm doing both. Now, I'm doing both.

[00:01:58.700] - Kim
Excellent.

[00:01:59.660] - Pam
Alright.

[00:02:00.220] - Kim
Okay, let's get going. I have a Problem String for you today. And I want to tell the listeners you have no idea. Sometimes, you know, we'll take notes, and like whatever, but.

[00:02:10.500] - Pam
I'll suggest this is the string I'm going to do today, and you glance at it. But not today. No, nope, I have not seen it.

[00:02:15.890] - Kim
It's on a Post-it note in front of me. Nowhere in our notes.So.

[00:02:19.720] - Pam
I can tell my heart to calm down because Math is Figure-Out-Able.

[00:02:22.790] - Kim
It is. It is.

[00:02:23.560] - Pam
And no one's going to judge me for like taking my time and thinking, right?

[00:02:27.640] - Kim
Yeah.

[00:02:27.950] - Pam
Yeah. It's not about speed.

[00:02:29.140] - Kim
Should we talk about the string ahead of time? Should we just dive in and talk about it?

[00:02:33.320] - Pam
It's up to you, I think.

[00:02:34.240] - Kim
I think let's just get started, and then we can kind of unpack and talk about what you're thinking afterwards.

[00:02:38.840] - Pam
I will say I know the topic of what we're going to do.

[00:02:42.040] - Kim
Yeah.

[00:02:42.560] - Pam
You might want to... This might be the podcast that you actually write down.

[00:02:48.000] - Kim
Yeah.

[00:02:48.540] - Pam
Yeah. So, if you're listening.

[00:02:50.520] - Kim
You're driving right now, like push pause. Come back and do it later.

[00:02:53.130] - Pam
You could listen to it now if you're never going to listen to it again. But if you get a chance to do it with paper and pencil. Oh, I just said pencil.

[00:03:00.830] - Kim
Yeah, you did. True belief.

[00:03:03.410] - Pam
Okay. Do you like Ticonderoga recyclable ones? Am I even saying that right?

[00:03:08.560] - Kim
I don't know that I have one. Did you know that there is a Ticonderoga Facebook page? I got a text. I got a text from Martha on our team the other night and she was like, "Check this out."

[00:03:21.670] - Pam
Do you know I was an hour and a half? No, 2. 2 hours away from the Ticonderoga factory when I was in Vermont, and we actually considered going there just to take a picture to send you. And I just couldn't be in the car any longer. We've done way too much driving (unclear).

[00:03:36.170] - Kim
You know, this is not a me thing, right? It's an every teacher thing.

[00:03:39.050] - Pam
I'm learning that. I'm learning that.

[00:03:40.800] - Kim
(unclear).

[00:03:40.820] - Pam
I kind of would say it to teachers like, "Psh, whatever, Kim." And they would be like, "No, that's actually a thing."

[00:03:45.930] - Kim
It's what we do. Okay. Listen, we have several problems in our string, so we got to get going.

[00:03:50.630] - Pam
Dang it. Alright, get going.

[00:03:51.410] - Kim
Alright, here you go. Your first problem.

[00:03:52.810] - Pam
Come on, Kim. Come on, Kim. Move on.

[00:03:55.220] - Kim
Control the situation. Alright, first problem. 245 plus 6 equals 245 plus x.

[00:04:05.240] - Pam
I think...

[00:04:05.740] - Kim
You're solve solving for x.

[00:04:06.440] - Pam
Yeah, I think x has to be 6.

[00:04:08.720] - Kim
Why is that?

[00:04:09.740] - Pam
Because I can kind of ignore the 245s. They're both the same. It's almost like I could think of a mobile, and if they're balanced, then I can kind of be like, "I don't need to think about those 245, but I got a 6 and I got an X. Ah, so 6 has to be x or x has to be 6. Okay.

[00:04:24.210] - Kim
That makes sense. Alright, here you go. Next one. 192 plus 31 equals x plus 192.

[00:04:35.120] - Pam
Okay, so I'm feeling some commutative property here. 192 plus something is 31. So, then something plus 192, that something also has to be 31.

[00:04:45.510] - Kim
And did you do the same thing where you just kind of ignored or did you...

[00:04:48.250] - Pam
Totally, yeah. Well, really, I noticed the 192s were the same.

[00:04:53.010] - Kim
Mmhm.

[00:04:53.080] - Pam
And so, I was like, "I can ignore those. Ah." And then I thought about the communitive thing. But yeah, definitely just kind of ignored what was the same.

[00:04:58.400] - Kim
Something about ignoring.

[00:04:59.370] - Pam
Both sides of the mobile. Yeah.

[00:05:01.010] - Kim
Okay. Ready for the next one?

[00:05:03.540] - Pam
Yep.

[00:05:04.030] - Kim
26 plus 14 plus X.

[00:05:07.830] - Pam
Oh.

[00:05:09.710] - Kim
Mmhm. Is equal to 26 plus 39.

[00:05:13.810] - Pam
Ooh, I kind of like this one. Okay. So, I got 26 something on the left-hand side, plus 2 things on the left-hand side are equal to 26 plus 39 on the right-hand side. So, I'm going to do the ignore thing again and sort of think about 14 plus x has to be 39.

[00:05:31.710] - Kim
Okay, so I just wrote down...

[00:05:33.150] - Pam
So, the 2 things have to be equal to the 1 thing because I'm ignoring the 26s.

[00:05:37.120] - Kim
Mmhm.

[00:05:37.280] - Pam
So, if I've got 14 plus something is 39, then I've also kind of got 14s on both sides that I can kind of ignore. Because I can think about 14 plus x. There's a 14. But I can also think about 14 plus... Is that... Help me. I can do it. 15 plus 23. Wow, that's harder than it should have been. Uh, no. (unclear)

[00:06:03.230] - Kim
Wait, wait. You equal to 39?

[00:06:05.790] - Pam
Yeah, 39 is equivalent to 14 plus 20. Oh my gravy. Plus 20. I can do it. I can do it. 5.

[00:06:15.670] - Kim
Yeah. So, did you... I'm wondering if you then on your paper wrote off to the side from the original problem 14 plus x equals 39. And then did you kind of split the 39 into a 14 and a 25?

[00:06:27.450] - Pam
Yeah.

[00:06:28.000] - Kim
Yeah, so then you can see that 14 that you want to ignore.

[00:06:31.180] - Pam
And I'm going to ignore that 14. And then I have just x equals 25.

[00:06:34.400] - Kim
Yeah, nice.

[00:06:35.280] - Pam
Yeah.

[00:06:35.920] - Kim
Okay, this ignoring thing seems to be nice.

[00:06:38.840] - Pam
Okay.

[00:06:39.140] - Kim
So, you've got the next problem, ready?

[00:06:40.900] - Pam
Okay.

[00:06:42.100] - Kim
175 plus 32 equals 179 plus x. Say that again. 175 plus 32 is equal to 179 plus x.

[00:06:55.180] - Pam
So, I think I'm going to ignore some 175s.

[00:06:58.880] - Kim
Okay.

[00:06:59.500] - Pam
So, I have 32 equals,um, that 170. The difference between 175 and 179 is 4. So, I've got 32 is equivalent to 4 plus x. So, what plus 4 is 32? So, x is 28. And, you know... Oh, sorry.

[00:07:17.640] - Kim
No, go ahead.

[00:07:18.520] - Pam
Well, I lost my train of thought.

[00:07:19.920] - Kim
Can I ask you a question?

[00:07:21.360] - Pam
Yeah.

[00:07:21.560] - Kim
Well, I was going to ask you because you said that you ignored the 175, and I'm imagining that maybe you wrote what I wrote, which is I split the 179 into a 175 and a 4.

[00:07:31.060] - Pam
I kind of didn't that time.

[00:07:32.900] - Kim
Oh, interesting. Okay.

[00:07:34.820] - Pam
That time, I don't know, it was close enough that I just really thought about the difference between 175 and 179.

[00:07:41.140] - Kim
Okay. So, then you thought about 32 equals 4 plus x. And at that point, I didn't hear you say you ignored the 4.

[00:07:48.730] - Pam
Mm, I didn't.

[00:07:49.790] - Kim
You just kind of like you just could tell that it was going to be 28?

[00:07:53.460] - Pam
Well, it's like 4 plus something is 32.

[00:07:56.290] - Kim
Okay.

[00:07:57.010] - Pam
Yeah, that just fell out. But I'm wondering now if that idea of saying what's the difference between 175 and 179, was that 4? If I could have done that when I ended up with 14 plus x equals 39? I could have said what's the difference between 14 and 39? And that's 25.

[00:08:14.730] - Kim
Wait, wait. 14? Where's 14? Oh, you mean in the previous problem?

[00:08:17.950] - Pam
Previous problem, sorry.

[00:08:18.990] - Kim
Okay, yeah, yeah.

[00:08:19.330] - Pam
Yeah, when I had 14 plus x is 39, I could have said to myself, "What's the difference between 14 and 39?"

[00:08:25.090] - Kim
So, sometimes thinking about the difference in the span between the numbers, and sometimes you're just ignoring what they have in common.

[00:08:33.130] - Pam
Yeah, yeah.

[00:08:33.810] - Kim
Nice.

[00:08:34.070] - Pam
I'm not sure when I'm doing which, but yep.

[00:08:36.030] - Kim
Okay, we ready?

[00:08:37.200] - Pam
Yep.

[00:08:37.650] - Kim
Okay, so we've got another one. x plus 16.

[00:08:41.520] - Pam
Okay.

[00:08:41.830] - Kim
Equals x plus x.

[00:08:44.700] - Pam
I like this one. I like to think about this one as there's sort of on the right-hand side, there's a number plus a number. And on the left-hand side, there's a number plus 16. So, if there's a number plus a number, and that's equivalent to a number plus 16, the number's got to be 16. I don't know another way to say that, but...

[00:09:08.070] - Kim
I mean, could you also say that if you just ignore that they have x's on both sides?

[00:09:13.320] - Pam
I could. That's not what I did. But yeah, I could. I could ignore the x's on both sides. Yeah. Yep.

[00:09:18.370] - Kim
It's kind of like sometimes you're thinking about what they have in common.

[00:09:22.140] - Pam
And sometimes I'm thinking about the difference.

[00:09:23.750] - Kim
Uh-huh, yeah.

[00:09:24.370] - Pam
Yeah, that's nicely said.

[00:09:25.830] - Kim
Alright, here we go. We've got 2x equals x plus 6.

[00:09:32.160] - Pam
X plus 6. So, I'm kind of thinking about the difference between 2x and x.

[00:09:39.570] - Kim
Mmhm.

[00:09:39.590] - Pam
And, so I'm just left with x equals 6.

[00:09:42.710] - Kim
Okay.

[00:09:44.570] - Pam
Yeah. Yep.

[00:09:46.870] - Kim
Next problem.

[00:09:47.590] - Pam
Okay.

[00:09:48.070] - Kim
X plus 24 equals x plus x plus x.

[00:09:53.810] - Pam
Let me make sure I got it right. x plus 24 equals x plus x plus x.

[00:09:58.010] - Kim
Uh-huh.

[00:09:59.450] - Pam
Okay, so I think in this case I'm going to ignore x on the left and x on the right because I can. And now, I end up with... And I literally just kind of squiggled through. I squiggled through the x on the left and squiggled through the x on the right. Now, I've got 24 is equivalent to x plus x. So, something plus itself is 24. I'm thinking divide 24 in half, that's 12. So, x is equal to 12. Or I actually wrote down 12 is equal to x.

[00:10:26.490] - Kim
Why did you ignore the x's on both sides?

[00:10:28.640] - Pam
Well, actually, if I really said what I thought. The very first thing I thought is, so I've got 3 x's on one side and x on the other side. And then I went, "Ooh, but I could just ignore those x's and then I don't have to deal with 3 of them. That's kind of what I did.

[00:10:45.670] - Kim
Yeah, I think it's a pretty interesting idea to focus on... Instead of focus on everything, it's like if something fell away, like what do I really need to focus on?

[00:10:57.700] - Pam
Mm, yeah, yeah.

[00:10:59.280] - Kim
All right, 2 more. Ready?

[00:11:01.900] - Pam
Yep.

[00:11:02.140] - Kim
Okay, 2x plus 20.

[00:11:04.140] - Pam
2x plus 20.

[00:11:06.040] - Kim
Is equal to 3x.

[00:11:08.680] - Pam
Okay, 2x plus 20 is equal to 3x.

[00:11:11.440] - Kim
What do you think about that?

[00:11:13.020] - Pam
Well, I'm seeing 2x plus 20 is equal to 2x plus x. Or kind of like I'm thinking about the difference between 2x and 3x. And 3x's is 1 more x than 2x. So, I'm going to have 20 equals that 1x that was the difference between 3x's and 2x's.

[00:11:31.980] - Kim
So, again, kind of...

[00:11:33.540] - Pam
Comparing.

[00:11:34.140] - Kim
...Focusing on what's left. Yeah.

[00:11:35.640] - Pam
Yeah.

[00:11:35.840] - Kim
Comparing thing.

[00:11:37.280] - Pam
Yeah.

[00:11:37.400] - Kim
Alright.

[00:11:38.480] - Pam
The difference between them, uh-huh.

[00:11:39.980] - Kim
Mmhm.

[00:11:40.300] - Pam
Okay.

[00:11:40.920] - Kim
Last problem. You ready?

[00:11:41.980] - Pam
Whoo! Yep.

[00:11:43.120] - Kim
You're doing great. 5x plus 19.

[00:11:45.600] - Pam
Okay.

[00:11:46.840] - Kim
Equals 2x plus 32. I wonder if you focus on...

[00:11:55.610] - Pam
I'm feeling like focusing on the fact that I've got 5x's on the left and 2x's on the right.

[00:12:03.650] - Kim
Okay.

[00:12:03.850] - Pam
So, I can kind of ignore 2 of the x's on both sides.

[00:12:07.790] - Kim
Okay.

[00:12:08.150] - Pam
That would... The difference between 5x and 2x is 3x, so I've got 3x on the left plus 19.

[00:12:13.070] - Kim
Okay.

[00:12:13.870] - Pam
Equal to those 2x's I'm ignoring, so it's just 32. So, 3x plus 19 is 32.

[00:12:19.510] - Kim
Okay.

[00:12:20.160] - Pam
And then I'm asking myself, what's the difference between 19 and 32?

[00:12:26.460] - Kim
For what purpose?

[00:12:28.060] - Pam
Because I feel like I've got 19 on one side and 32 on the other, so I can sort of ignore 19s.

[00:12:35.840] - Kim
Mmhm.

[00:12:36.480] - Pam
So, I need to know if I'm going to ignore 19 on the 32 side, I need to know what's left over.

[00:12:41.940] - Kim
Mmhm.

[00:12:42.020] - Pam
So, the difference between 19 and 32.

[00:12:44.250] - Kim
Mmhm.

[00:12:45.070] - Pam
And I'm literally thinking kind of as a number line in my head. There's like 1, and 10, and 2, that's 13. I don't know. Is that very additive thinking of me? The difference between 19 and 32 is 13. So, now I end up with 3x is equivalent to 13. That doesn't feel very even, Kim. Did I do it right?

[00:13:03.150] - Kim
Mmhm.

[00:13:05.200] - Pam
Yeah, okay. So, now I've got 3 times something.

[00:13:07.940] - Kim
Oh, wait, wait. What did you... Oh, yeah, 13.

[00:13:09.790] - Pam
3x equals 13. So, I'm just going to say 3 times something is 13. 3 times thirteen-thirds is 13. So, I'm going to say x is equivalent to thirteen-thirds.

[00:13:22.060] - Kim
Nice. Alright. I probably could have given you a better number at the end there.

[00:13:26.710] - Pam
Ah, it's... I mean, and if we want, we could think about what 13... Like, I'm thinking about thirteen-thirds. That's equivalent to 9 divided by 3 plus... Oh, wait. Sorry. 12 divided by 3 plus another 1 divided by 3. So, that's like 4 and a third? That would be, you know. So, thirteen-thirds, a 4 and a third. Either way. 4.3 repeating. Nice Problem String, Kim.

[00:13:54.610] - Kim
Thanks. So, often when we're talking about equation solving, what grade are we talking here? Just like freshmen? So, 8th grade?

[00:14:03.520] - Pam
This could be 7th grade. There's a world where this is 7th grade. For sure 8th grade.

[00:14:08.840] - Kim
For sure 8th grade.

[00:14:09.220] - Pam
For sure 9th grade. Yeah.

[00:14:10.720] - Kim
Yeah. Often we say to kids, "Here's what you do."

[00:14:15.510] - Pam
For sure. Like apply the equivalence properties and do the thing. I used to sell in my early days. I used to tell students your job is to get x alone.

[00:14:26.750] - Kim
Yeah.

[00:14:28.360] - Pam
And then do a bunch of stuff. And, in fact, I'm imagining there's a lot of people that heard this particular string, and they're like, "Wait, what are you doing?" Because, you know, for instance, when we got to the problem 2x plus 20 equals 3x, you could have just said, "Well, yeah. Subtract 2x from both sides. And I could have done that. But it's not what I was thinking about. I have a goal now to think relationally, not think procedurally, not just do stuff, not just do stuff because I was told to do it, but to actually think about the relationships. That might not even be the best one. Like, the 5x plus 19 equals 2x plus 32. I mean, I could subtract 2x from both sides, but I like thinking about kind of ignoring the 2x's. You could be like, "Well, Pam. That's like subtracting the 2x's." But it's not what I'm thinking about. I'm not just doing. I'm thinking about, "Well, if these are equivalent, and I've got 2x's on both sides, I can just ignore those. What am I left with? What's the difference between 5x and 2x? Ah, 3x. I think we invite everyone to consider that the way that I just kind of thought through those--thank you Kim and Brandon. We'll give credit--is the way that a lot of people actually think when they look like they're doing the steps.

[00:15:43.400] - Kim
Yeah.

[00:15:43.820] - Pam
But they're actually thinking the way I just tried to think out loud. And I'll be all really vulnerable here and say I did not used to think this way until Brandon Pelter and Kim heard me doing some equation solving using a double open number line. And they said, "Sure, you can do that. And maybe it's better in some cases. But in some cases, don't you want to just kind of ignore what's the same on both sides?" And I was like, "What?" And man, you guys had to work with me a little bit because the way of thinking was so foreign, it just had never occurred to me. And boy, my brain traveled the mental path of solving equations using those equivalence properties and just bam, bam, bam, bam, that I had a hard time stopping my brain from traveling that path. At first. Until I did a little bit of work and started bringing the Relational Thinking that I've had from earlier topics and thinking, "Huh. Like, could I think that way? And is it advantageous to think that way?" So, we'll throw that out. We're super curious what you guys think. Give us a response. Dive into the Math is Figure-Out-Able Teacher Facebook group. Or if you're lucky enough, blessed enough. What's a good word? Fortunate. That's the word I was looking for. If you're fortunate to be in one of our Building Powerful Mathematics workshops or our teacher or leader coaching group, we'd love to hear from you. What do you think about thinking about relationships and using those to solve equations? There you go. Yeah.

[00:17:08.250] - Kim
So, I'm going to say one more thing.

[00:17:10.480] - Pam
Sure.

[00:17:10.760] - Kim
That when we talk about strategies, we know that the equivalence strategies, we believe, are the most sophisticated strategies.

[00:17:20.090] - Pam
Mmhm.

[00:17:20.160] - Kim
And so, you know, if a teacher says, "Well, I just tell them to subtract 2x from both sides, and you have to write it this way, and then you draw the line, and then you bring it down," I wonder what that's doing to students who are already thinking relationships and are thinking about equivalence. What that forced method of recording tells them about what math is. When they already are thinking about relationships and what we can do to develop relationships with students. And if we are building equivalency in our students earlier on, how many of our students can think about that even into solving equations? Yeah.

[00:17:58.760] - Pam
Yeah. Nice.

[00:17:59.500] - Kim
Yeah.

[00:17:59.980] - Pam
I would invite you, Kim, to think... I appreciate you saying kids who are thinking relationally and you're like sort of forcing the other... Yeah, just like what you just said. That ow does that impact how they think about math? And what about kids who aren't thinking relationally? Like, it just solidifies that math are these steps to memorize. I'm not a math-er. Everybody understands what they're doing. This doesn't mean anything to me. It's not related to my life. But man, when you start thinking, when you start asking, "How are you thinking about this? And kids are like, "Well, I'm seeing it this way." That is a human endeavor. That is where you get to say, "I'm thinking about it this way. This is what my brain is doing." And that is true engagement. That is identity as a thinker, reasoner, math-er, somebody who is capable of doing this kind of thinking. That really develops math-ing.

[00:18:56.810] - Kim
Yeah.

[00:18:57.120] - Pam
Yeah. Around the world.

[00:18:59.050] - Kim
Okay, well, maybe we will do some more equivalence.

[00:19:02.590] - Pam
More advanced ones. We should.

[00:19:04.140] - Kim
Yeah.

[00:19:04.460] - Pam
Yeah, let's do that. Alright, y'all. 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!