Pam  00:01

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

Kim  00:10

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  00:17

We know that algorithms are amazing human achievements, but 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 it can trap students into a digit-oriented perspective.

Kim  00:37

Mmm! In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships. 

Pam  00:45

Ya'll, we invite you to join us to make math more figure-out-able. Dun dun dum

Kim  00:51

Hi there. So, last week we... Well, two weeks ago, we started talking about your brand new book, Developing Mathematical Reasoning - Avoiding the Trap of Algorithms. 

Pam  01:00

Oh, I'm so excited (unclear).

Kim  01:00

I know. It's so great. And so, last week, we started talking about the first trap. A major trap that affects all algorithms, and that is the less sophisticated reasoning trap. 

Pam  01:09

Yeah, (unclear).

Kim  01:16

It's rough. It's rough, right? 

Pam  01:17

So important. And, you know, you really have to understand the development of mathematical reasoning. You have to have a feel for counting to additive to multiplicative to proportional to functional, to kind of then go, "Oh, wow. It can really trap kids into using less sophisticated reasoning than the problems are intended to develop." So, that one's a little bit kind of have to have some background. Ya'll, today, let's dive into the second trap. I don't know if it's... They're not in a specific order. But trap number two, another trap, that I think fits all algorithms. Like, pick an algorithm, and it can trap kids into this second trap. And I think this one you actually can maybe... It doesn't take as much background to understand this trip. Trip? Trippy trap. We can go down this trip and get trapped in... Okay, I'll stop. So, today I'm going to do this a little bit differently than we did in the last episode. I want to do a Problem String with you first and develop some reasoning, and then kind of compare that to an algorithm. And notice, how the algorithm can trap kids into... How it can trap kids differently than what we just did in the Problem String. So, are you game for that? 

Kim  02:35

Sure.

Pam  02:36

When we do that, then we'll talk about the trap afterwards. 

Kim  02:38

Sounds good. 

Pam  02:39

Okay, cool. So, first problem. Kim, what is 56 minus 20?

Kim  02:46

That would be 36.

Pam  02:48

Alright, so I've just drawn on my paper, 56 minus 20. I wrote that problem down. And then I drew a number line that started on 56, and I jumped backwards 20. And you said that that would land on 36, so I finished the equation equals 36.

Kim  03:00

Yep. 

Pam  03:01

Next problem. How about 56 minus 19?

Kim  03:03

Mmhm. So, 56 minus 20 was 36, so I'm going to add 1 back to get to 37. 

Pam  03:14

Yeah, but we're subtracting. Why are you adding?

Kim  03:17

Because I built off of the minus 20, and 20 subtract 20 is subtracting 1 more than 19. 

Pam  03:25

Okay.

Kim  03:26

So, I'm adjusting by saying I'm going to add one back. (unclear). 

Pam  03:30

Yeah, you subtracted too much. Yeah. Subtracted too much.

Kim  03:33

Mmhm. 

Pam  03:34

On my paper, I went ahead and drew 56 minus 20 as that big jump of 20 back.

Kim  03:38

Mmhm. 

Pam  03:38

If I were to have instead not jumped that big jump of 20 back, and only jumped a jump of 19, it would have been a shorter jump.

Kim  03:46

Mmhm.

Pam  03:46

Which means that the ending place would have been further to the right.

Kim  03:50

Mmhm. 

Pam  03:50

And further to the right 1 from 36 is, like you said, 37. Cool.

Kim  03:55

Mmhm.

Pam  03:55

Next problem 274 minus 100

Kim  04:00

That would be 174.

Pam  04:03

And I've just drawn... You didn't need to do this on your paper. Nobody did. 274. Big jump of 100 back and landing on 174. Cool, next problem. 274 subtract 98. 

Kim  04:15

Mmhm. So, here's why it might have been useful to write it on your paper, because you could subtract 100 to get to 174, but then add 2 back to get to 176 because 100 was too much, so you can add those 2 back.

Pam  04:32

Nice. Cool. How about 563 subtract 300?

Kim  04:40

That would be 263.

Pam  04:42

So, I've just drawn an even bigger jump back. 563 subtract 300. And it's not all that bigger because I don't have that big of a paper, but it's a little bigger. It's as bigger as it can get. And what did you say you landed on 200? 

Kim  04:57

263.

Pam  04:58

Okay, cool. Hey, Kim, do you want to guess what the next problem is?

Kim  05:03

We've done back 19 back... I'm going to go with you wanted me to do 273. I mean 297.

Pam  05:13

297. That was an excellent guess. Excellent guess. So, we had minus 300. 

Kim  05:16

Yeah.

Pam  05:16

I'm actually going to ask you minus 295. 

Kim  05:16

(unclear). Ah, okay. Alright. So, then...

Pam  05:25

But that was a good guess. That was good guess.

Kim  05:26

Back 300 is too much by 5, so I'm going to add 5 back. And I got 268.

Pam  05:34

Because 263 and 5 is 268. Cool. Alright, so if you jump too far back, then you adjust up because you subtracted a bit too much. That makes lots of sense. How about 7.2, 7 and 2/10, subtract 4.

Kim  05:52

Mmhm. So, I'm going to call that 3. I actually thought about like $3.20, so 3 and 2/10.

Pam  05:59

Okay, and I've drawn 7.2 with a little jump back of 4 and landed on 3.2 or $3.20. How about 7.2, or 7 and 2/10 subtract $3.97. Hey, there's your 7.

Kim  06:15

Thanks. Alright, so if I subtracted $4.00 to get $3.20, then I'm going to... $3.97 is just $0.03 less, so I'm going to add $0.03 back and get $3.23.

Pam  06:31

Because $3.20 plus $0.03 is $3.23. And to be clear, we often have kids kind of mess up the place value when they do that a little bit.

Kim  06:41

Yeah.

Pam  06:42

Which, let's just point out, is a brilliant opportunity for us to have a great conversation about place value.

Kim  06:47

Yeah.

Pam  06:48

Which I want to come up in just a second. Alright, last problem. What is 8.25, 8 and a 1/4, minus 5?

Kim  06:57

Okay. So, that would be 3.25.

Pam  07:00

Okay, if 8.25 minus 5 is 3.25, what's 8.25 minus 4.75?

Kim  07:09

Okay. So, I subtracted 5. I'm adding another quarter back to get 3.5.

Pam  07:17

Because 3.25 and a 0.25 is 3.5.

Kim  07:19

Mmhm.

Pam  07:20

Okay. Alright, Kim.

Kim  07:23

Can I tell you that I was recently in a second grade classroom and started developing this strategy with them, and they were brilliant. 

Pam  07:32

Oh, nice. 

Kim  07:33

They could absolutely talk about why you would add a little bit back and kind of where the amounts were on a number line. I was so impressed with them. Just so proud, yeah. They did great, yeah.

Pam  07:47

So, maybe keep talking. Like, could you look at the problems that we just did and kind of just talk about some....

Kim  07:53

Yeah, so... 

Pam  07:55

...patterns you're seeing.

Kim  07:55

Yeah, you gave me kind of paired problems.

Pam  07:58

Mmhm.

Kim  07:58

And in the helper, the first one, subtracting a really nice friendly number. So, like 20 and 100, 300, even 4 was kind of a friendly number.

Pam  08:08

Mmhm.

Kim  08:09

Then those helper problems that I subtracted a little bit too much helped with the... We call it a "clunker problem", the one behind it that is kind of a cranky number. So, 56 minus 20 helped subtract 56 minus 19 because you could back up a little too much, and then adjust.

Pam  08:29

Mmhm.

Kim  08:29

And then back 100.

Pam  08:31

Before you move on? 

Kim  08:32

Yeah? 

Pam  08:33

Can you keep that train of thought because I just want to interrupt you just a little bit 

Kim  08:35

Sure. 

Pam  08:36

So, to do that, you had to think about 56 minus 20 first.

Kim  08:40

Mmhm. 

Pam  08:40

That's noteworthy.

Kim  08:42

Yep.

Pam  08:43

That you weren't just dealing with digits, but you were actually thinking about 56 and 20. Okay, cool.

Kim  08:48

Yeah, the full magnitude of the number, the value of 20. 

Pam  08:52

Yeah.

Kim  08:52

Not just 2-0. 

Pam  08:53

Yeah, yeah. 

Kim  08:54

So, then when you gave me 274 minus 100, I could use that to help with 274 minus 98. So, I was thinking about how far apart 98 and 100 were. So, I subtracted too much, and then added some back. So, I was thinking about 98 and how far away it was from 100 instead of 9-8 and 1-0-0.

Pam  09:18

Yeah, so it's not a bunch of digits. You had to think about hundreds minus hundreds. And you had to think, then, about the connection or relationship between 100 and 98, how far, and then which direction to go. 

Kim  09:28

Mmhm.

Pam  09:28

Cool. And then, similarly, did you have to do something... I'm going to skip down to the 7.2 or 7 and 2/10 minus 4. 

Kim  09:36

Yeah. 

Pam  09:36

Is that where you lined them up, or is that where you butt cheek the decimal? 

Kim  09:40

Oh, geez. No, I...

Pam  09:43

Which rule? Which rule? 

Kim  09:44

Right? 

Pam  09:45

Flip a coin. Flip a coin 

Kim  09:46

Yeah.

Pam  09:46

Or were you... In fact, I think... Did you bring up money first? 

Kim  09:50

I did. Yeah, I said I think about it like $7.20 minus $4.00. Which made it nice because then when you gave me 3.97 or 3 and 97/00, then I could think about that like, $3.97. And it's really nice to think about how far away $3.97 is from $4.00 (unclear).

Pam  10:10

Yeah, it's just those $0.03, yeah. And then you could just tack that $0.03 back on to $3.20 that you had before, and not... Like, again. Like 3.2 and 0.03. Am I lining? Am I butt cheeking? Like, how are these all related? Before we do the last problem because I do want to do that. I just want to point out that in this problem, you just dealt with place value, and reasonableness, and rounding, and connections between numbers, and a sense of ish. Like, so Jo Boaler, in her Math-Ish book talks about get your ish number. And in a huge way, you were kind of... I was sort of providing for you an ish number.

Kim  10:52

Yeah.

Pam  10:52

And then you used that, but you had to think about the relationships between them. 

Kim  10:56

Well, and I'm going to point out one more.

Pam  10:58

Okay. 

Kim  10:58

That when I was thinking about 295 and 300.

Pam  11:01

Mmhm.

Kim  11:01

I was thinking about the distance meaning for subtraction. Like, I was finding how far apart 295 and 300 were instead of, oh my heavens, 3-0-0 minus 2-9-5,

Pam  11:14

borrow from a bunch of zeros. Oh, nicely said.

Kim  11:17

Ick!

Pam  11:18

Oh, that was good guess. I hadn't thought of that. Oh, that's good, Kim, I'll bring that up next time I use that example.

Kim  11:24

Happy to serve.

Pam  11:26

Yeah, because wait till you see where I'm going to go with that in just a second. 

Kim  11:29

Okay.

Pam  11:29

That makes my next one even better. So, then I kind of wish for the last problem that I hadn't given you the helper. I wish... Very last one, 8.25 minus 5. I wish I had just given you 8.25 minus 4.75. 

Kim  11:43

Okay. 

Pam  11:43

And asked you to come up with the helper. 

Kim  11:45

Yeah.

Pam  11:46

That would have been maybe a better teacher move on my part, so do you mind if I just give you one extra problem? 

Kim  11:51

Sure enough.

Pam  11:52

that.  (unclear) we're kind of doing this. So, let's see. What if I were to ask you something like 6 and 1/5 minus 3... No. Minus... Yeah, yeah. Sorry, sorry. 3 and 4/5. So, 6 and a 1/5 minus 3 and 4/5. I wonder if you could help me come up with a helper problem for

Kim  12:00

Yeah, I'm going to subtract 4. Wait, did you say... I didn't even write down what you told me? 6 and 1/5 minus? 

Pam  12:15

3 and 4/5. 

Kim  12:16

3 and 4/5. 

Pam  12:17

Yeah. 

Kim  12:17

So, I'm going to subtract 4. 

Pam  12:19

Okay.

Kim  12:19

So, I'm going to get 2 and a 1/5. And then the next thing I'm thinking about is how far away are that 3 and 4/5. And that 4 that I already subtracted.

Pam  12:28

Mmhm. 

Kim  12:29

And I subtracted 1/5 too much. 

Pam  12:31

Okay. 

Kim  12:32

So, I'm going to add back 1/5. And so, I have 2 and 2/5.

Pam  12:36

Because 2 and a 1/5 and one more 1/5 is 2 and 2/5. Yeah. 

Kim  12:40

Mmhm.

Pam  12:40

Okay, cool. When you just did that, when I gave you the problem 6 and a 1/5 minus 3 and 4/5 you had to come up with your own helper. How did you come up with the helper of 4?

Kim  12:54

Because I thought if I could subtract a little bit extra, a little bit too much. I didn't really want to think about... I mean, I guess I could have subtracted 3, and then four-fifths, but I... You know, I really do tend to like Over.

Pam  13:07

Well especially because...

Kim  13:08

(unclear) adding back... We just did some work with that, but adding back a fifth is is pretty nice.

Pam  13:14

And, like you said, we had just done work where I kind of had gotten your brain, and hopefully everybody in the... Well I high-dosed everybody in the podcast with this idea of subtracting a bit too much, and then you could just kind of adjust from there. 

Kim  13:26

Yeah. 

Pam  13:26

So, what you just did, and if anybody was actually doing the mental actions of solving these problems as we were going, is you were developing senses of place value, and nearness, and neighborhood, and how things are related, and how you can adjust. All very mathy kinds of things that we need students to do as they're developing additive reasoning and more.

Kim  13:50

Pam, I'm sitting here thinking about what somebody would be taught to do in class for 6 and 1/5.

Pam  13:57

That's where we're going right now.

Kim  13:58

Okay, alright. Okay, sorry.

Pam  14:00

Well, I want to do the whole numbers first, and then we'll go to fractions. So, Kim, everybody on the podcast, one of the problems that I asked you to solve was 563 minus 295.

Kim  14:10

Mmhm. 

Pam  14:11

You chose to solve it by thinking about 563 minus 300.

Kim  14:14

Mmhm.

Pam  14:15

And then adjusting 5 up. But what if was trapped in the algorithm?

Kim  14:20

Mmhm.

Pam  14:21

So, if I write down 5-6-3.

Kim  14:23

Mmhm. 

Pam  14:24

Line, little subtraction symbol. 2-9-5, big line. Then what am I supposed to do? I'm supposed to start from the least consequential numbers in the problem, right? Not the 500 and the 200 or 500 and almost 300. Not that. No, no, no. But I'm going to start with the least consequential numbers, which are the end 3 and 5. 563, that 3. 295, that 5. And I can't now... Then I say to myself, 3 minus 5. Whoops, can't do that. And now I've got to like turn that 3 into 13 because I grabbed a 10 from the 60 or maybe just a 1 from the 6. That leaves me a 5 or a 50 depending on if you're thinking about place value. So, complicated right now. Then I can think about 13 minus 5. So, I've grabbed, I've borrowed, or regrouped. I now have a 13 minus 5. Could you agree with me that if I'm a student that's mimicking these steps of the algorithm, at this point, I could say to myself, "13 minus 5. Ready, begin. 13, 12, 11, 10..." And I could literally count back. You can't see me, but my fingers are popping up, and I'm counting back 5. There's a couple different ways to count back. But either way, I'm counting by ones, and I end with 8, and I write down an 8.

Kim  15:38

Mmhm. 

Pam  15:38

And then I go over where I borrow the sugar from next door. Or however that dumb thing goes. And now, I've got 5 minus 9. Oh, crumb. So, now I've got to make that into 15, borrow from the 5 next to it or regroup, however you're... Ya'll, the words don't matter so much because I'm just performing a bunch of... I'm mimicking a bunch of steps. So, now I've got 15 minus 9. Could you agree with me that I could go 15, 14, 13, 12... I could count by ones, and I could get that correct, and I can then continue counting by ones, get the entire subtraction problem correct. My teacher could say, "Yay! You've got subtraction down. You are thinking additively about subtraction. You're an additive reasoner. You're reasoning about subtraction. Sweet!" When in reality, I was using less sophisticated reason counting by ones. But maybe today's focus. So, that was a last week's focus. I'm using less sophisticated reason the problems are intended to develop. But notice, that the only problems I ever solved were things like 13 minus 5, 15 minus 9, and then I would end with 4 minus 2. They were always either single-digit minus single-digit or teens minus single-digit. 

Kim  16:29

Mmhm. Right.

Pam  16:53

If I'm stuck in the algorithm, I will never ever do anything more than digit stuff. Maybe the max will be teens minus a digit.

Kim  17:02

Right. 

Pam  17:02

And I can count by ones. 

Kim  17:03

Mmhm.

Pam  17:04

The algorithms trap students into a digit oriented focus.

Kim  17:10

Mmhm.

Pam  17:10

Where the numbers are pulled apart, and I'm only dealing with just a bunch of digits, and I don't ever think about those bigger relationships that you were talking about earlier like 274 minus 100 or 563 minus 300 or...

Kim  17:24

Right.

Pam  17:24

We can get to the fractions now. 8 and a 1/4 minus 5. Or the last one we did 6 and a 1/5 minus 4. Like, we don't ever have kids reasoning about those sort of bigger kind of ish numbers. We don't have kid... Thank you, Jo Boaler. We don't have kids. We haven't high dosed them enough to to invite them to grapple with those numbers in a meaningful ish, round kind of way...

Kim  17:54

Right. 

Pam  17:54

...to reason about them. Yeah. You were going to say something earlier about the fractions. Do you want to? Do you remember what it was? 

Kim  17:59

Yeah, I was... When I wrote down the problem, I wrote... You know, when I wrote 6 and a 1/5 minus 3 and 4/5, I started kind of reminiscing about what many kids would have been asked to do. And, you know, I'm assuming that that's change it to an improper fraction to then subtract.

Pam  18:21

That could definitely be one way of doing that, mmhm. Yeah. Another way. So, if we change both of those into improper fractions, then we would end up with a numerator and denominator where we would just... It would have the common denominators. We would just subtract the numerators. And when we subtract... I haven't done it, so I'd have to (unclear).

Kim  18:39

Well, let me tell you what the numbers would be. 

Pam  18:41

Is it 31 and...

Kim  18:42

31/5 and 19/5. So, then you're subtracting 31 minus 19.

Pam  18:47

And probably from a digit oriented perspective, right?

Kim  18:50

Right, right.

Pam  18:50

Yeah, yep. Or I might say to myself, instead of 6 and a 1/5, I'm going to regroup and turn that into 5 and 6/5.

Kim  18:59

Right. 

Pam  18:59

Minus 3 and 4/5. And then I might just do 6 minus 4 and 5 minus 3. And never... And always only be thinking about digits.

Kim  19:08

Correct.

Pam  19:08

Trapped in this digit perspective. 

Kim  19:11

And...

Pam  19:11

Teachers... Oh, you do your "and" first. (unclear).

Kim  19:14

I was going to say and when you're only thinking about digits, you have no idea when you look up if it's even close.

Pam  19:21

Yeah, you're kind of reading off the answer. You're not... Yeah. Yeah, like you said before, when you did... I don't remember which problem it was. Like, 563 minus 300, you're already clear, kind of. Well, I think that was less. Anyway. If you're doing 563 minus 295, and you think about it as 563 minus 300, you're already in the ballpark.

Kim  19:43

Right.

Pam  19:44

You already have an ish. That's a way of thinking about an ish number. And we're high dosing kids with this Over pattern, so they have a strategy for thinking about that. They're already partially reasonable. Teachers, you know how many times you've said to your students, "Oh, my gosh! Your answer isn't even reasonable."

Kim  20:02

Yes.

Pam  20:03

I invite you to consider it's because we took them out of reasoning land.

Kim  20:07

Right.

Pam  20:07

And we put them into a digit focused set of steps to mimic. And they did this stuff with the digits. And it's not that they're unwilling to wonder if their answer is reasonable. We haven't helped their brain grow, so that their brain can think of whether their answer is reasonable. And how do we help their brain grow? By doing problem strings like we just did where kids are actually invited to grapple with the relationships in a way that gives them the strategies with which to do it. We don't just leave them. We don't just go, "Hey, everybody, go find your ish number. Good luck. Come on. Do it. Like, go." Too many kids, probably including me, wouldn't have a strategy until we high dose them with the major patterns and relationships, so that they go, "Oh, I could just go a little over, and then now I have a good ish number." Yeah?

Kim  21:01

Yeah. 

Pam  21:01

Alright, so one of the traps of the algorithms is digit oriented focus where students aren't actually reasoning about what the problems are asking to reason for. Bam. Like I did in the last episode, I invite you to consider, do you have an algorithm that you're like, This one. This one doesn't." Secondary teachers, it's not always digits, but it's parsing out the problem into too many pieces, so that I never consider the whole. So, if you're thinking about an algorithm that involves polynomials, consider, does it break the polynomial into the terms, and I never think about the entire polynomial. I only think about terms. And maybe even the coefficient and the variable in the term. So, it's not just digits, but it's kind of breaking the problem into pieces and parts, and doing stuff with the pieces and parts, and then reading off the answer at the end. That's the same. When I refer to a "digit trap", I'm also referring to a trap that includes parts like that. Breaking the thing into pieces. 

Kim  21:31

Mmhm.  Yeah. 

Pam  22:08

Alright, ya'll, thank you 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!