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Math is Figure-Out-Able!
Math teacher educator Pam Harris and her cohost Kim Montague answer the question: If not algorithms, then what? Join them for ~15-30 minutes every Tuesday as they cast their vision for mathematics education and give actionable items to help teachers teach math that is Figure-Out-Able. See www.MathisFigureOutAble.com for more great resources!
Math is Figure-Out-Able!
Ep 241: Algorithm vs Problem Talk vs Problem String
Is mathing just about allowing students to solve problems their way? Aren't Probem Strings and Problem Talks really just the same routine? In this episode Pam and Kim discuss how Problem Strings move beyond Problem Talks to develop student's mathematical reasoning at a more sophisticated level.
Talking Points:
- Example of a Problem Talk
- What do we gain from Problem Talks?
- Example of a Problem String
- Problem Strings are more focused conversations
- Problems Strings allow for constant feedback, a high dose of patterning
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Pam 00:00
Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris, a former mimicker turned mather.
Kim 00:08
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.
Pam 00:15
We know that algorithms are amazing historic 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.
Kim 00:29
In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships.
Pam 00:36
We invite you to join us to make math more figure-out-able.
Kim 00:40
Hey there.
Pam 00:41
Hey, Kim.
Kim 00:42
How's it going?
Pam 00:43
Well, we get to play with some math today.
Kim 00:45
No more poking?
Pam 00:47
No more poking. Last week, we did all the poking today. (unclear).
Kim 00:49
Okay. Alright, let's do some fun.
Pam 00:51
Let's get some mathing down. So, one of the things that I am noticing and have been kind of fussing with for a while. Especially as I travel around. I'll do some mathing with people, and then depending on people's experiences, they will often say, "Oh, yeah, yeah, yeah. We do that." And I'm super curious then because I'm like, "You do what? What do you do?" Oh, we do number talks. And that's not what we do, really. And so, at least, often, it's not what they are referring to. So, then I drill down a little bit, and I'm like, "What do you mean that you do that?" And they're like, "Well, you know, we let kids solve the problem anyway they do. We do these talks once a week. It's really fun." And I'm like, "What do you do the rest of your math time?" "Well, you know, we do traditional." Like, maybe they don't say "traditional". But, you know, "We teach what we're supposed to teach. So, we do this kind of fun thing where we kind of have kids do this, I don't know, creative whatever, and we've taken care of that, and then we get down to the business of actually teaching what we're supposed to teach." I hear that a lot.
Kim 01:49
Mmhm.
Pam 01:50
So, today we thought we would give a really good example of what we mean by... Well, what we mean that mathing and when students are mathing, that it's more than them getting answers. That if we do really good Problem Strings, we get a different result, a different set of mental actions happening in students than if we are either having them solve the problem with an algorithm, some memorized step-by-step procedure.
Kim 02:22
Mmhm.
Pam 02:23
Or even if we're having them do some kind of a number talk, or what I would call a Problem Talk.
Kim 02:28
Mmhm.
Pam 02:28
That there's sort of different things happening. And we thought we'd illustrate that today in maybe a different way than we ever have. For sure that we have never done this on the podcast. So, ya'll, hold on tight because here we go. So, very first, I'm going to suggest the a math problem. So, 18 times 25.
Kim 02:46
Mmhm.
Pam 02:46
And, listeners, if you get a chance, maybe pause the podcast and at least estimate. Like, get some kind of some sense of the relationships. You're welcome to solve the problem any way you want. But we really want you to kind of have some... What? Sense of the magnitude of the relationships that are involved here, the answer, before we kind of superimpose someone else's strategy. So, thinking, thinking. Alright. So, Kim, the first thing I'm going to do is just kind of talk a little bit about what could happen if a student was solving this problem 18 times 25 with the traditional algorithm. By the way, when I was in Croatia, the Croatian teachers taught me the Croatian traditional multiplication algorithm.
Kim 03:27
Oh, fun.
Pam 03:28
Which is so not exactly equivalent to ours. It was really... Yeah, it was quite fun. But similar actions happen in that you're still using the distributive property.
Kim 03:39
Mmhm.
Pam 03:39
You're still treating numbers as if they're digits. And so, let's go ahead and run through what our traditional algorithm looks like. So, I've written 18, "multiplication symbol" 25, drawn the line. If I am an algorithm kid, I could, at this point say the first thing I'm supposed to do is do 5 times 8. And so, I think I need five 8s. So, I could go 8, 16, 24, and I could find those five 8s by adding, skip counting, adding up five 8s. Now, I could think about eight 5s and maybe skip count by 5s. But either way, that's additive reasoning. And I could do all of the single-digit multiplication facts using additive reasoning and get the correct answer. So, I could be looking like I'm a multiplicative reasoner because I'm getting the answer correct. But in reality, all of the single-digit facts could have been happening by me skip counting. So, that's one thing. Another thing that's happening in that algorithm is I'm not thinking about eighteen 25s or twenty-five 18s, I'm thinking about five 8s. And I write down the 0. And then I put up the 4. And then I'm thinking about 5 times 1. And then I'm adding 4. And then I'm writing that down below. And then I'm putting the magic 0, or the turtle lays an egg, or whatever it is that kind of keeps track of that place value place. By the way, in Croatia, they just move the numbers over. They don't even write a 0.
Kim 03:42
Mmhm. Mmhm, I've seen that.
Pam 04:14
They're like, "Of course there's a 0. We just move it over. We don't write a 0." Anyway, then I keep going. So, then, I could do all of the addition by using counting strategies or single-digit additive reasoning. So, I get an answer. I could have been using counting strategies and single-digit multiplicative, additive reasoning by skip counting. At best, in that algorithm, I could be using single-digit multiplicative reasoning. That is the most sophisticated reasoning. And you're like, "Yeah, it's a good thing, Pam, because kids can't do anything these days, and so we've got to give them the most simple, easiest thing to do." But then all we get are answers and kids using less sophisticated reasoning. Which means they haven't built their brains to do something more sophisticated.
Kim 05:44
Yeah. And you started us today by saying, "Check out the problem, and at least like estimate what it might be before we start talking about the answer."
Pam 05:55
Yeah.
Kim 05:55
If kids are drilled into algorithms, I highly doubt that that's their first move is to think about what might be. You know, these are often the kids who round afterwards or fill in the estimation line afterwards.
Pam 06:11
That's
Kim 06:12
Mmhm.
Pam 06:12
what I did. Yeah, when we had the whole reasonable estimation approximation unit, I was like, "This is dumb." Just solve the problem, and then round it off."
Kim 06:19
Yep.
Pam 06:20
Like, if you're going to end up doing the work anyway to do all those steps, why bother estimating to begin with?
Kim 06:26
Yep.
Pam 06:27
Yeah. Anyway, when you're in that fake math land. Oh.
Kim 06:33
Pamela!
Pam 06:33
I know.
Kim 06:34
We got to work on that. Or not. I could totally be the only one who feels the way I feel.
Pam 06:39
Yeah, who knows? We'll keep talking about it. So, one thing that we wanted to suggest today is I could solve that problem, get answers, but not really building my brain. That's one outcome for this problem. Another outcome from this problem is I could say, "Hey, class. Let's do a number talk today." Or I call it a Problem Talk. "The problem today is 18 times 25. What do you got?" Kim, I'm going to call on you 18 times 25. Go.
Kim 07:05
Okay, I'm going to say that... I don't want to talk about 25s. (unclear).
Pam 07:13
You have so many choices. You don't even know what to choose.
Kim 07:15
I'm seriously struggling. I'm going to say I'm going to quadruple the 25.
Pam 07:21
Okay.
Kim 07:22
And I'm going to do a fourth of 18, which is...
Pam 07:27
You can't do a fourth of 18. Just kidding.
Kim 07:31
So, half is 9. Half again is 4 and a 1/2. So, I'm going to quadruple, quarter and get 4.5 times 100, so that is 450.
Pam 07:43
I'm super curious. When you did 4.5 times 100, did you think to yourself, 45, 450?
Kim 07:47
I did. Yeah, I did.
Pam 07:48
Yeah. So, Kim and I often will scale by 10 and then 10 again to get times 100. Yeah, nice. Okay, that's a great strategy. And I might model that on the board by writing the 18 times 25, and then saying we're going to quadruple. So, a scaling arrow times 4, and then a scaling arrow divided by 4, ending up with an equivalent problem. 4.5 times 100 is 450. Cool. I could have also modeled that on the board, if I wanted to, with an open array, an 18 by 25.
Kim 08:15
Mmhm.
Pam 08:16
And I could have cut that 18 in half, and then cut that in half again to make a really skinny rectangle, and moved all those pieces over, and ended up with... That's not tall. 4.5 by 100 (unclear).
Kim 08:29
And that would have been my same strategy represented on two different models.
Pam 08:34
Two different models. Yep. Nice. Cool. Then I might call it another kid, and I might say, "Hey, how are you thinking about 18 times 25?" And at that point... And I'll just throw out a different strategy. I might think about twenty-five 18s. And I might think, well, 25 reminds me of 25%, or 0.25, and that reminds me of one quarter. So, I'm going to solve, not the equivalent problem but a similar problem, a related problem. I'm going to solve one-quarter of 18. It kind of sounds a little bit like what you did. The quarter part. A quarter of 18. Half of 18 is 9. Half of 0 is 4.5. But now, I've got a quarter of 18 is 4.5. That's like 0.25 times 18 is 4.5. And then I might scale that times 100. And now, I have 25 times 18 is 450. So, I'm using similar relationships, but I'm really thinking about it a little bit differently because I was thinking about fractions. Not like... You know, you were quadrupling and quartering, and I was really thinking about a quarter of 18, and then scaling that up.
Kim 08:35
Mmhm.
Pam 08:35
We also might have a kid say... Do you mind just giving us an Over strategy real quick?
Kim 09:01
Yeah. So, I might say I need eighteen 25s, but I know twenty 25s is 500, and I'm going to subtract two 25s, which is 50. So, 500 subtract 50 is 450.
Pam 09:57
Nice. So, I might then put all those on the board, choosing, hopefully, a model that would help us kind of compare those a little bit. And then I might say, "Okay, that's great. Problem Talk done." What do we get out of a Problem Talk, Kim?
Kim 10:14
I get to share my strategy.
Pam 10:16
Okay, yeah. Anything else?
Kim 10:18
I might see my strategy represented on the board because you represented my thinking.
Pam 10:25
Would you agree that I get the I get agency? I get to solve the problem however I want. I have choice.
Kim 10:31
Mmhm, yeah.
Pam 10:31
Rather than thinking that there's only one and only one right way to do it, I get some agency. Like you said, I might get mine on the board.
Kim 10:38
Mmhm.
Pam 10:38
I also, I would suggest, get exposed to the idea that there's more than one way to solve a problem. I don't know that most students seeing Kim's strategy once, or my strategy once, or whoever's strategy once. I don't know that that's enough for most students. I'm going to suggest it's not enough for most students to actually own that strategy.
Kim 10:58
Mmhm.
Pam 10:58
So, I see a lot of people doing number talks where the outcome feels like it should be we just got really flexible.
Kim 11:08
Mmhm.
Pam 11:08
But in reality, most kids didn't learn anything from anybody else. They just kind of once they shared theirs, then they were off on their own, doing whatever because they shared and they were done. And if we really take the time to share everyone's strategy, there's only so many. And now we're going to have duplicates on the board. And we're just gonna kind of call into question the efficacy of that. Like, what do we gain? Do we gain enough? Do we get enough? Yeah, go ahead.
Kim 11:36
And there are some really nice things about a Problem Talk.
Pam 11:40
Keep going.
Kim 11:40
For the purpose of what we're talking about. Kids Learning more strategies is what you're saying, that's not enough for.
Pam 11:50
Yeah, so it depends on your purpose.
Kim 11:51
Yeah, I can see that the way I thought about it was different than you thought about it. Or if the teacher's doing a nice facilitation, I might hear how yours is related to mine, but slightly different.
Pam 12:03
Mmhm.
Kim 12:03
I might see... If you put my strategy on two different models, I might see how it looks on two different models.
Pam 12:12
Mmhm, mmhm.
Kim 12:13
So, there are some things that can be...
Pam 12:15
throw out one more. If we have already developed the strategies that you and I just did. I think we just did three of the major five multiplication strategies. If we've already developed those strategies, now we get a chance to compare. And I left that out of the conversation. But I could have said, "Hey, for these numbers, this problem, which of those strategies do you want your brain to be inclined toward next time?" Let me
Kim 12:36
Mmhm.
Pam 12:36
That sends a real growth mindset message. You have access to this. Send your brain the message, "Ooh, I want my brain to do that next time."
Kim 12:42
Mmhm.
Pam 12:42
But it also gives us the opportunity to talk about the relationships, and for students to go, "Oh, for those numbers, I really like this one." Now, for these numbers, a lot of strategies are actually pretty nice, and so we really kind of care less which strategy wins the comparison. It's more the conversation. We want kids talking about the relationships that are in the different strategies because we gain from that. So, definitely some nice things that we can gain from that. Let's do something. Let's do a Problem String.
Kim 13:11
Okay.
Pam 13:11
And then, listeners, we're going to talk about what do we gain from that experience?
Kim 13:16
Mmhm.
Pam 13:16
Alright, here we go. Kim, first problem. 10 times 25.
Kim 13:21
250.
Pam 13:22
250. And I am going to represent this in a ratio table. So, I'm going to say we're talking about 25. So, I've just written down a ratio table 1 to 25. And then you said 10 times 25, so now I've got 10 to 250.
Kim 13:37
Did you make yours horizontal or vertical?
Pam 13:39
Haha, want to guess?
Kim 13:40
I bet yours is horizontal.
Pam 13:43
Oh, it's vertical.
Kim 13:43
Oh, that's so funny because I'm normally vertical, and I did my horizontally.
Pam 13:46
Oh, that's funny. Yeah, that's funny. Okay, cool. So, I've got 1 to 25 and 10 to 250. Next question. What about 2 times 25?
Kim 13:56
50.
Pam 13:57
That's not hard. So, now I've got 2 to 50. Next question. What about twenty 25s?
Kim 14:03
I'm going to scale from the 2 times 10, and so it's 500.
Pam 14:09
So, 50 times 10, and that gets you the 500. And I'm going to then suggest... Or I'm going to ask. "Did anyone... We have the 10 there. Did anyone think about doubling from the 10 to get to the 20?" I'll trail off. I won't try to say too much in hopes that someone goes, "Oh, yeah, yeah. I did. And then I want to compare those a little bit. I want to say, "Hey, what do you guys think? If you're walking down the street and you had to find 20 times 25, would you prefer to find ten 25s and double or would you prefer to find two 25s and scale times 10?" Have the conversation. In an older grade, we might actually write that down with expressions. We might write down something like 10 times 25 times 2. Or 2 times 25 times 10. And note the commutative property and kind of some associative property stuff going on. And we could kind of talk about properties. Anyway. So, Kim, can you think of an advantage to either one of those? 10 times 25, and then double. Or 2 times 25, and then scale times 10?
Kim 15:14
I think doubling, and then scaling tends to be particularly nice when you have funky numbers because then you're just scaling at the end. You're not like doubling larger numbers. Like, sometimes, if you have like 384 doubled is maybe not as easy for some people if they can double it when it's just a little bit smaller of a number.
Pam 15:38
So, like, if you had to have 384 times 20, you could do 384 times 10, but then you have a big number you have to double. But you could double that smaller number, and then you just scale times 10. Bam. Is pretty readily done?
Kim 15:53
Mmhm.
Pam 15:53
Okay, cool. Next problem. How about if I were to ask you 18 times 25.
Kim 15:59
That is 450.
Pam 16:01
And I know we just did that problem. Maybe you can kind of forget it. What... Is there anything that a kid might use that we already had in this string to help them think about that?
Kim 16:11
Yeah, they probably do the twenty 25s minus the two 25s
Pam 16:16
Bam. So, this string is kind of pointing towards that Over strategy that you used where we did the number talk, the Problem Talk. But we could bring that out. Okay, so we've got the 20. We've got the 2. We could also then say, "Which of those two strategies to find the twenty 25s would help you with this eighteen 25s?" I don't know if that question makes sense. In other words, if I'm looking for eighteen 25s, does it make sense for me to find ten 25s, double to get twenty 25s, then I still have to find two 25s to subtract from the twenty 25s? Or would it make more sense to find the two 25s to get the twenty 25s because then I just already have the 2 and the 20 to subtract to get the eighteen 25s.
Kim 17:01
You mean outside of the Problem String (unclear)?
Pam 17:04
Yes.
Kim 17:04
Okay, so on my own, if I had to solve 18 times 25...
Pam 17:08
Yeah.
Kim 17:09
...it probably wouldn't go 10 of them, 20 of them, back 2 of them because I'd have to find the 2. But it's really nice to have found the 2, found the 20 because then that's all I need.
Pam 17:19
Then you have all you need. And the 2 helped you get the 20. That's kind of nice. Alright, nice.
Pam and Kim 17:20
Yeah.
Kim 17:22
Yeah.
Pam 17:23
Great. Next question, three 25s?
Kim 17:26
75.
Pam 17:27
The next question, thirty 25s?
Kim 17:31
750.
Pam 17:33
Times 10. And the next question, twenty-seven 25s?
Kim 17:39
675.
Pam 17:41
And how did you get that?
Kim 17:42
Thirty 25s minus three 25s.
Pam 17:46
Which was the corresponding 750 minus?
Kim 17:49
75.
Pam 17:50
75. And that got you the 675. Cool. And tell me about that. If you were walking down the street and somebody said 27 times anything?
Kim 18:01
I could think about what it's near, a nice friendly number that it's near. So, 30. And so, I could really find 3 of them, 30 of them, subtract.
Pam 18:10
Bam.
Kim 18:11
That seems like a kind of helpful strategy that we could kind of have kids start to develop. Mmhm. And in this particular string, you've got two instances of that where it could be a conversation with the 2 in the 20 to 18. And then you lob it out again immediately after to see if anyone's picking up what you're having a conversation about just previously.
Pam 18:37
Yeah. And it's a multiple access task because if kids aren't generalizing yet, they're just kind of really focused on the problems, then they're allowed to do that. But we're also challenging all students to really like start to see those patterns and think about the fact that, for example, if I was trying to find 44 times something, I might find 4 times that thing, to then scale it to get 40 times that thing, and then add the 4. And bam, like I've already got kind of what I need.
Kim 19:07
We have talked about this before on some episode. And maybe more than one. That Problem Strings are an art, that there's a nudge, that they're going somewhere, there's a purpose. And I know that some teachers right now are thinking about a Problem String, and they're going, "Wait, but like I let my kids solve it however they want. And even though there might be a goal of the string, they can still solve it however they want." And that's true. They, you know, on their paper, at their desk, whatever, can solve however they want. But the conversation is not about a bunch of different strategies. As you go through the Problem String, there is a focus, and that the direction that the teacher's headed, the conversation supporting that goal. It's like in a Problem Talk, it's there's different conversations about different kinds of strategies. And in a Problem String, the conversation is more narrowed.
Pam 20:06
Especially if you're trying to develop a strategy. If you're trying to develop a model, the conversation can be...
Kim 20:09
Mmhm. Yeah.
Pam 20:11
You can ask for lots of different strategies because you're trying to really get good at the model.
Kim 20:16
Right.
Pam 20:17
Or a big idea. But in this Problem String, we're really trying to focus on a strategy.
Kim 20:21
Yeah.
Pam 20:22
Or building some relationships that kids can use, then we're going to focus the conversation. Not demand what kids do on their paper, but focus the conversation. Yeah.
Kim 20:32
What's really nice about a Problem String too is that you're giving kids the opportunity to do lots of different things. So, in this particular strategy, or in this particular string, that was the work that was on a ratio table had a strategy of Over that you were going for. But there was still some really nice conversation. Or in my head, there was conversation about doubling, about scaling, about times 10, about quarters. Like, all of those things were in my head as I was doing this work. And because there are multiple problems in a Problem String, I'm doing more math than in a Problem Talk where you ask me how I solved that one problem, that one way. I might be a Kim who's sitting on the carpet, and as the kids are talking, I'm staring at the numbers, and I'm thinking about lots of other ways I could solve it. But I might not be. I might be a kid who solved it one time, and then that was the whole thing.
Pam 21:28
Mmhm.
Kim 21:29
So, you get more math, more done in the same amount of time.
Pam 21:33
Yeah, all the
Kim 21:34
Yeah.
Pam 21:34
mathing that you just mentioned that we get out of that Problem String experience, compare that to the mathing or not that we got from a kid solving the problem using the algorithm. Even if we had them solve fifteen or twenty problems using the algorithm, the kinds of mathing that would happen as they're doing that, versus the kind of mathing that's happening as we're taking kids to a Problem String like this. They're getting lots of answers in both cases. But the amount of mathing that's happening is significantly different. Look, what we get for mathing,
Kim 22:08
Yeah. And because kids need more and more and more experiences to really make sense of what's happening around them, the more that we can do with them that's meaningful, the better, for sure.
Pam 22:21
And giving them those meaningful experiences with constant feedback. Like, in a Problem String, they're getting... If I'm doing 1-29, odd, lots of problems, maybe I'm getting correct answer feedback, but that's it. In a Problem String, I'm getting lots of feedback on my thinking over and over and over again. A high dose of patterning. Alright, cool. Ya'll, thanks for joining us. 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. And keep spreading the word that Math is Figure-Out-Able!