Pam Harris, Exploring the Power & Purpose of Number Strings ROUNDING UP: SEASON 4 | EPISODE 4
I've struggled when I have a new strategy I want my students to consider and despite my best efforts, it just doesn't surface organically. While I didn't want to just tell my students what to do, I wasn't sure how to move forward. Then I discovered number strings.
Today, we're talking with Pam Harris about the ways number strings enable teachers to introduce new strategies while maintaining opportunities for students to discover important relationships.
BIOGRAPHY
Pam Harris, founder and CEO of Math is Figure-out-able™, is a mom, a former high school math teacher, a university lecturer, an author, and a mathematics teacher educator. Pam believes real math is thinking mathematically, not just mimicking what a teacher does. Pam helps leaders and teachers to make the shift that supports students to learn real math.
RESOURCES
Young Mathematicians at Work by Catherine Fosnot and Maarten Dolk
Procedural fluency in mathematics: Reasoning and decision-making, not rote application of procedures position by the National Council of Teachers of Mathematics
Bridges number string example from Grade 5, Unit 3, Module 1, Session 1 (BES login required)
Developing Mathematical Reasoning: Avoiding the Trap of Algorithms by Pamela Weber Harris and Cameron Harris
Math is Figure-out-able!™ Problem Strings
TRANSCRIPT
Mike Wallus: Welcome to the podcast, Pam. I'm really excited to talk with you today.
Pam Harris: Thanks, Mike. I'm super glad to be on. Thanks for having me.
Mike: Absolutely.
So before we jump in, I want to offer a quick note to listeners. The routine we're going to talk about today goes by several different names in the field. Some folks, including Pam, refer to this routine as “problem strings,” and other folks, including some folks at The Math Learning Center, refer to them as “number strings.” For the sake of consistency, we'll use the term “strings” during our conversation today.
And Pam, with that said, I'm wondering if for listeners, without prior knowledge, could you briefly describe strings? How are they designed? How are they intended to work?
Pam: Yeah, if I could tell you just a little of my history. When I was a secondary math teacher and I dove into research, I got really curious: How can we do the mental actions that I was seeing my son and other people use that weren't the remote memorizing and mimicking I'd gotten used to?
I ran into the work of Cathy Fosnot and Maarten Dolk, and [their book] Young Mathematicians at Work, and they had pulled from the Netherlands strings. They called them “strings.” And they were a series of problems that were in a certain order. The order mattered, the relationship between the problems mattered, and maybe the most important part that I saw was I saw students thinking about the problems and using what they learned and saw and heard from their classmates in one problem, starting to let that impact their work on the next problem. And then they would see that thinking made visible and the conversation between it and then it would impact how they thought about the next problem.
And as I saw those students literally learn before my eyes, I was like, “This is unbelievable!” And honestly, at the very beginning, I didn't really even parse out what was different between maybe one of Fosnot's rich tasks versus her strings versus just a conversation with students. I was just so enthralled with the learning because what I was seeing were the kind of mental actions that I was intrigued with. I was seeing them not only happen live but grow live, develop, like they were getting stronger and more sophisticated because of the series of the order the problems were in, because of that sequence of problems. That was unbelievable. And I was so excited about that that I began to dive in and get more clear on: What is a string of problems?
The reason I call them “problem strings” is I'm K–12. So I will have data strings and geometry strings and—pick one—trig strings, like strings with functions in algebra. But for the purposes of this podcast, there's strings of problems with numbers in them.
Mike: So I have a question, but I think I just want to make an observation first.
The way you described that moment where students are taking advantage of the things that they made sense of in one problem and then the next part of the string offers them the opportunity to use that and to see a set of relationships. I vividly remember the first time I watched someone facilitate a string and feeling that same way, of this routine really offers kids an opportunity to take what they've made sense of and immediately apply it. And I think that is something that I cannot say about all the routines that I've seen, but it was really so clear. I just really resonate with that experience of, what will this do for children?
Pam: Yeah, and if I can offer an additional word in there, it influences their work. We're taking the major relationships, the major mathematical strategies, and we're high-dosing kids with them. So we give them a problem, maybe a problem or two, that has a major relationship involved. And then, like you said, we give them the next one, and now they can notice the pattern, what they learned in the first one or the first couple, and they can let it influence. They have the opportunity for it to nudge them to go, “Hmm. Well, I saw what just happened there. I wonder if it could be useful here. I'm going to tinker with that. I'm going to play with that relationship a little bit.” And then we do it again. So in a way, we're taking the relationships that I think, for whatever reason, some of us can wander through life and we could run into the mathematical patterns that are all around us in the low dose that they are all around us, but many of us don't pick up on that low dose and connect them and make relationships and then let it influence when we do another problem.
We need a higher dose. I needed a higher dose of those major patterns. I think most kids do. Problem strings or number strings are so brilliant because of that sequence and the way that the problems are purposely one after the other. Give students the opportunity to, like you said, apply what they've been learning instantly [snaps]. And then not just then, but on the next problem and then sometimes in a particular structure we might then say, “Mm, based on what you've been seeing, what could you do on this last problem?” And we might make that last problem even a little bit further away from the pattern, a little bit more sophisticated, a little more difficult, a little less lockstep, a little bit more where they have to think outside the box but still could apply that important relationship.
Mike: So I have two thoughts, Pam, as I listen to you talk.
One is that for both of us, there's a really clear payoff for children that we've seen in the way that strings are designed and the way that teachers can use them to influence students' thinking and also help kids build a recognition or high-dose a set of relationships that are really important.
The interesting thing is, I taught kindergarten through second grade for most of my teaching career, and you've run the gamut. You've done this in middle school and high school. So I think one of the things that might be helpful is to share a few examples of what a string could look like at a couple different grade levels. Are you OK to share a few?
Pam: You bet. Can I tack on one quick thing before I do?
Mike: Absolutely.
Pam: You mentioned that the payoff is huge for children. I'm going to also suggest that one of the things that makes strings really unique and powerful in teaching is the payoff for adults. Because let's just be clear, most of us—now, not all, but most of us, I think—had a similar experience to me that we were in classrooms where the teacher said, “Do this thing.” That's the definition of math is for you to rote memorize these disconnected facts and mimic these procedures. And for whatever reason, many of us just believed that and we did it. Some people didn't. Some of us played with relationships and everything. Regardless, we all kind of had the same learning experience where we may have taken at different places, but we still saw the teacher say, “Do these things. Rote memorize. Mimic.”
And so as we now say to ourselves, “Whoa, I've just seen how cool this can be for students, and we want to affect our practice.” We want to take what we do, do something—we now believe this could be really helpful, like you said, for children, but doing that's not trivial. But strings make it easier. Strings are, I think, a fantastic differentiated kind of task for teachers because a teacher who's very new to thinking and using relationships and teaching math a different way than they were taught can dive in and do a problem string. Learn right along with your students. A veteran teacher, an expert teacher who's really working on their teacher moves and really owns the landscape of learning and all the things still uses problem strings because they're so powerful. Like, anybody across the gamut can use strings—I just said problem strings, sorry—number strengths—[laughs] strings, all of us no matter where we are in our teaching journey can get a lot out of strings.
Mike: So with all that said, let's jump in. Let's talk about some examples across the elementary span.
Pam: Nice. So I'm going to take a young learner, not our youngest, but a young learner. I might ask a question like, “What is 8 plus 10?” And then if they're super young learners, I expect some students might know that 10 plus a single digit is a teen, but I might expect many of the students to actually say “8, 9, 10, 11, 12,” or “10, 11,” and they might count by ones given—maybe from the larger, maybe from the whatever. But anyway, we're going to kind of do that. I'm going to get that answer from them. I'm going to write on the board, “8 plus 10 is 18,” and then I would have done some number line work before this, but then I'm going to represent on the board: 8 plus 10, jump of 10, that's 18. And then the next problem's going to be something like 8 plus 9. And I'm going to say, “Go ahead and solve it any way you want, but I wonder—maybe you could use the first problem, maybe not.” I'm just going to lightly suggest that you consider what's on the board. Let them do whatever they do. I'm going to expect some students to still be counting. Some students are going to be like, “Oh, well I can think about 9 plus 8 counting by ones.” I think by 8—”maybe I can think about 8 plus 8. Maybe I can think about 9 plus 9.” Some students are going to be using relationships, some are counting. Kids are over the map.
When I get an answer, they're all saying, like, 17. Then I'm going to say, “Did anybody use the first problem to help? You didn't have to, but did anybody?” Then I'm going to grab that kid. And if no one did, I'm going to say, “Could you?” and pause.
Now, if no one sparks at that moment, then I'm not going to make a big deal of it. I'll just go, “Hmm, OK, alright,” and I'll do the next problem. And the next problem might be something like, “What's 5 plus 10?” Again, same thing, we're going to get 15. I'm going to draw it on the board.
Oh, I should have mentioned: When we got to the 8 plus 9, right underneath that 8, jump, 10 land on 18, I'm going to draw an 8 jump 9, shorter jump. I'm going to have these lined up, land on the 17. Then I might just step back and go, “Hmm. Like 17, that's almost where the 18 was.” Now if kids have noticed, if somebody used that first problem, then I'm going to say, “Well, tell us about that.” “Well, miss, we added 10 and that was 18, but now we're adding 1 less, so it's got to be 1 less.” And we go, “Well, is 17 one less than 18? Huh, sure enough.”
Then I give the next set of problems. That might be 5 plus 10 and then 5 plus 9, and then I might do 7 plus 10. Maybe I'll do 9 next. 9 plus 10 and then 9 plus 9. Then I might end that string. The next problem, the last problem might be, “What is 7 plus 9?” Now notice I didn't give the helper. So in this case I might go, “Hey, I've kind of gave you plus 10. A lot of you use that to do plus 9. I gave you plus 10. Some of you use that to do plus 9, I gave you plus 10. Some of you used that plus 9. For this one, I'm not giving you a helper. I wonder if you could come up with your own helper.”
Now brilliantly, what we've done is say to students, “You've been using what I have up here, or not, but could you actually think, ‘What is the pattern that's happening?’ and create your own helper?” Now that's meta. Right? Now we're thinking about our thinking. I'm encouraging that pattern recognition in a different way. I'm asking kids, “What would you create?” We're going to share that helper. I'm not even having them solve the problem. They're just creating that helper and then we can move from there.
So that's an example of a young string that actually can grow up. So now I can be in a second grade class and I could ask a similar [question]: “Could you use something that's adding a bit too much to back up?” But I could do that with bigger numbers. So I could start with that 8 plus 10, 8 plus 9, but then the next pair might be 34 plus 10, 34 plus 9. But then the next pair might be 48 plus 20 and 48 plus 19. And the last problem of that string might be something like 26 plus 18.
Mike: So in those cases, there's this mental scaffolding that you're creating. And I just want to mark this. I have a good friend who used to tell me that part of teaching mathematics is you can lead the horse to water, you can show them the water, they can look at it, but darn it, do not push their head in the water. And I think what he meant by that is “You can't force it,” right?
But you're not doing that with a string. You're creating a set of opportunities for kids to notice. You're doing all kinds of implicit things to make structure available for kids to attend to—and yet you're still allowing them the ability to use the strategies that they have. We might really want them to notice that, and that's beautiful about a string, but you're not forcing. And I think it's worth saying that because I could imagine that's a place where folks might have questions, like, “If the kids don't do the thing that I'm hoping that they would do, what should I do?”
Pam: Yeah, that's a great question. Let me give you another example. And in that example I'll talk about that.
So especially as the kids get older, I'm going to use the same kind of relationship. It's maybe easier for people to hang on to if I stay with the same sort of relationship. So I might say, “Hey everybody. 7 times 8. That's a fact I'm noticing most of us just don't have [snaps] at our fingertips. Let's just work on that. What do you know?” I might get a couple of strategies for kids to think about 7 times 8. We all agree it's 56.
Then I might say, “What's 70 times 8?” And then let kids think about that. Now, this would be the first time I do that, but if we've dealt with scaling times 10 at all, if I have 10 times the number of whatever the things is, then often kids will say, “Well, I've got 10 times 7 is 70, so then 10 times 56 is 560.” And then the next problem might be, “I wonder if you could think about 69 times 8. If we've got 70 eights, can I use that to help me think about 69 eights?” And I'm saying that in a very specific way to help ping on prior knowledge. So then I might do something similar. Well, let's pick another often missed facts, I don't know, 6 times 9. And then we could share some strategies on how kids are thinking about that. We all agree it's 54. And then I might say, “Well, could you think about 6 times 90?” I'm going to talk about scaling up again. So that would be 540. Now I'm going really fast. But then I might say, “Could we use that to help us think about 6 times 89?” I don’t know if you noticed, but I sort of swapped. I'm not thinking about 90 sixes to 89 sixes. Now I'm thinking about 6 nineties to help me think about 6 eighty-nines. So that's a little bit of a—we have to decide how we're going to deal with that. I'll kind of mess around with that. And then I might have what we call that clunker problem at the end. “Notice that I've had a helper: 7 times 8, 70 times 8. A lot of you use that to help you think about 69 times 8. Then I had a helper: 6 times 9, 6 times 90. A lot of you use that to help you think about 6 times 89. What if I don't give you those helpers? What if I had something like”—now I'm making this up off the cuff here, like—“9 times 69. 9 times 69. Could you use relationships we just did?”
Now notice, Mike, I might've had kids solving all those problems using an algorithm. They might've been punching their calculator, but now I'm asking the question, “Could you come up with these helper problems?” Notice how I'm now inviting you into a different space. It's not about getting an answer. I'm inviting you into, “What are the patterns that we've been establishing here?” And so what would be those two problems that would be like the patterns we've just been using? That's almost like saying when you're out in the world and you hit a problem, could you say to yourself, “Hmm, I don't know that one, but what do I know? What do I know that could help me get there?” And that's math-ing.
Mike: So, you could have had a kid say, “Well, I'm not sure about how—I don't know the answer to that, but I could do 9 times 60, right?” Or “I could do 10 times”—I'm thinking—“10 times 69.” Correct?
Pam: Yes, yes. In fact, when I gave that clunker problem, 9 times 69, I said to myself, “Oh, I shouldn't have said 9 because now you could go either direction.” You could either “over” either way. To find 9 I can do 10, or to find 69 I can do 70. And then I thought, “Ah, we'll go with it because you can go either way.” So I might want to focus it, but I might not. And this is a moment where a novice could just throw it out there and then almost be surprised. “Whoa, they could go either direction.” And an expert could plan, and be like, “Is this the moment where I want lots of different ways to go? Or do I want to focus, narrow it a little bit more, be a little bit more explicit?” It's not that I'm telling kids, but I'm having an explicit goal. So I'm maybe narrowing the field a little bit. And maybe the problem could have been 7 times 69, then I wouldn't have gotten that other “over,” not the 10 to get 9. Does that make sense?
Mike: It absolutely does. What you really have me thinking about is NCTM’s [National Council of Teachers of Mathematics’] definition of “fluency,” which is “accuracy, efficiency, and flexibility.” And the flexibility that I hear coming out of the kinds of things that kids might do with a string, it's exciting to imagine that that's one of the outcomes you could get from engaging with strings.
Pam: Absolutely. Because if you're stuck teaching memorizing algorithms, there's no flexibility, like none, like zilch. But if you're doing strings like this, kids have a brilliant flexibility. And one of the conversations I'd want to have here, Mike, is if a kid came up with 10 times 69 to help with 9 times 69, and a different kid came up with 9 times 70 to help with 9 times 69, I would want to just have a brief conversation: “Which one of those do you like better, class, and why?” Not that one is better than the other, but just to have the comparison conversation. So the kids go, “Huh, I have access to both of those. Well, I wonder when I'm walking down the street, I have to answer that one: Which one do I want my brain to gravitate towards next time?” And that's mathematical behavior. That's mathematical disposition to do one of the strands of proficiency. We want that productive disposition where kids are thinking to themselves, “I own relationships. I just got to pick a good one here to—what's the best one I could find here?” And try that one, then try that one. “Ah, I'll go with this one today.”
Mike: I love that.
As we were talking, I wanted to ask you about the design of the string, and you started to use some language like “helper problems” and “the clunker.” And I think that's really the nod to the kinds of features that you would want to design into a string. Could you talk about either a teacher who's designing their own string—what are some of the features?—or a teacher who's looking at a string that they might find in a book that you've written or that they might find in, say, the Bridges curriculum? What are some of the different problems along the way that really kind of inform the structure?
Pam: So you might find it interesting that over time, we've identified that there's at least five major structures to strings, and the one that I just did with you is kind of the easiest one to facilitate. It's the easiest one to understand where it's going, and it's the helper-clunker structure. So the helper-clunker structure is all about, “I'm going to give you a helper problem that we expect all kids can kind of hang on.” They have some facility with, enough that everybody has access to. Then we give you a clunker that you could use that helper to inform how you could solve that clunker problem. In the first string I did with you, I did a helper, clunker, helper, clunker, helper, clunker, clunker. And the second one we did, I did helper, helper, clunker, helper, helper, clunker, clunker. So you can mix and match kind of helpers and clunkers in that, but there are other major structures of strings.
If you're new to strings, I would dive in and do a lot of helper-clunker strings first. But I would also suggest—I didn't create my own strings for a long time. I did prewritten [ones by] Cathy Fosnot from the Netherlands, from the Freudenthal Institute. I was doing their strings to get a feel for the mathematical relationships for the structure of a string. I would watch videos of teachers doing it so I could get an idea of, “Oh, that move right there made all the difference. I see how you just invited kids in, not demand what they do.” The idea of when to have paper and pencil and when not, and just lots of different things can come up that if you're having to write the string as well, create the string, that could feel insurmountable.
So I would invite anybody out listening that's like, “Whoa, this seems kind of complicated,” feel free to facilitate someone else’s prewritten strings. Now I like mine. I think mine are pretty good. I think Bridges has some pretty good ones. But I think you'd really gain a lot from facilitating prewritten strings.
Can I make one quick differentiation that I'm running into more and more? So I have had some sharp people say to me, “Hey, sometimes you have extra problems in your string. Why do you have extra problems in your string?” And I'll say—well, at first I said, “What do you mean?” Because I didn't know what they were talking about. Are you telling me my string's bad? Why are you dogging my string? But what they meant was, they thought a string was the process a kid—or the steps, the relationships a kid used to solve the last problem. Does that make sense?
Mike: It does.
Pam: And they were like, “You did a lot of work to just get that one answer down there.” And I'm like, “No, no, no, no, no, no. A problem string or a number string, a string is an instructional routine. It is a lesson structure. It's a way of teaching. It's not a record of the relationships a kid used to solve a problem.” In fact, a teacher just asked—we run a challenge three times a year. It's free. I get on and just teach. One of the questions that was asked was, “How do we help our kids write their own strings?” And I was like, “Oh, no, kids don't write strings. Kids solve problems using relationships.” And so I think what the teachers were saying was, “Oh, I could use that relationship to help me get this one. Oh, and then I can use that to solve the problem.” As if, then, the lesson’s structure, the instructional routine of a string was then what we want kids to do is use what they know to logic their way through using mathematical relationships and connections to get answers and to solve problems. That record is not a string, that record is a record of their work. Does that make sense, how there's a little difference there?
Mike: It totally does, but I think that's a good distinction. And frankly, that's a misunderstanding that I had when I first started working with strings as well. It took me a while to realize that the point of a string is to unveil a set of relationships and then allow kids to take them up and use them. And really it's about making these relationships or these problem solving strategies sticky, right? You want them to stick. We could go back to what you said. We're trying to high-dose a set of relationships that are going to help kids with strategies, not only in this particular string, but across the mathematical work they're doing in their school life.
Pam: Yes, very well said. So for example, we did an addition “over” relationship in the addition string that I talked through, and then we did a multiplication “over” set of relationships and multiplication. We can do the same thing with subtraction. We could have a subtraction string where the helper problem is to subtract a bit too much. So something like 42 minus 20, and then the next problem could be 42 minus 19. And we're using that: I'm going to subtract a bit too much and then how do you adjust? And hoo, after you've been thinking about addition “over,” subtraction “over” is quite tricky. You're like, “Wait, why are we adding what we're subtracting?” And it's not about teaching kids a series of steps. It's really helping them reason. “Well, if I give you—if you owe me 19 bucks and I give you a $20 bill, what are we going to do?” “Oh, you’ve got to give me 1 back.” Now that's a little harder today because kids don't mess around with money. So we might have to do something that feels like they can—or help them feel money. That's my personal preference. Let's do it with money and help them feel money.
So one of the things I think is unique to my work is as I dove in and started facilitating other people's strings and really building my mathematical relationships and connections, I began to realize that many teachers I worked with, myself included, thought, “Whoa, there's just this uncountable, innumerable wide universe of all the relationships that are out there, and there's so many strategies, and anything goes, and they're all of equal value.” And I began to realize, “No, no, no, there's only a small set of major relationships that lead to a small set of major strategies.” And if we can get those down, kids can solve any problem that's reasonable to solve without a calculator, but in the process, building their brains to reason mathematically. And that's really our goal, is to build kids' brains to reason mathematically. And in the process we're getting answers. Answers aren't our goal. We'll get answers, sure. But our goal is to get them to build that small set of relationships because that small set of strategies now sets them free to logic their way through problems. And bam, we've got kids math-ing using the mental actions of math-ing.
Mike: Absolutely. You made me think about the fact that there's a set of relationships that I can apply when I'm working with numbers Under 20. There's a set of relationships, that same set of relationships, I can apply and make use of when I'm working with multidigit numbers, when I'm working with decimals, when I'm working with fractions. It's really the relationships that we want to expose and then generalize and recognize this notion of going over or getting strategically to a friendly number and then going after that or getting to a friendly number and then going back from that. That's a really powerful strategy, regardless of whether you're talking about 8 and 3 or whether you're talking about adding unit fractions together. Strings allow us to help kids see how that idea translates across different types of numbers.
Pam: And it's not trivial when you change a type of number or the number gets bigger. It's not trivial for kids to take this “over” strategy and to be thinking about something like 2,467 plus 1,995—and I know I just threw a bunch of numbers out, on purpose. It's not trivial for them to go, “What do I know about those numbers? Can I use some of these relationships I've been thinking about?” Well, 2,467, that's not really close to a friendly number. Well, 1,995 is. Bam. Let's just add 2,000. Oh, sweet. And then you just got to back up 5. It's not trivial for them to consider, “What do I know about these two numbers, and are they close to something that I could use?” That's the necessary work of building place value and magnitude and reasonableness. We've not known how to do that, so in some curriculum we create our whole extra unit that's all about place value reasonableness. Now we have kids that are learning to rote memorize, how to estimate by round. I mean there's all this crazy stuff that we add on when instead we could actually use strings to help kids build that stuff naturally kind of ingrained as we are learning something else.
Can I just say one other thing that we did in my new book? Developing Mathematical Reasoning: Avoiding the Trap of Algorithms. So I actually wrote it with my son, who is maybe the biggest impetus to me diving into the research and figuring out all of this math-ing and what it means. He said, as we were writing, he said, “I think we could make the point that algorithms don't help you learn a new algorithm.” If you learn the addition algorithm and you get good at it and you can do all the addition and columns and all the whatever, and then when you learn the subtraction algorithm, it's a whole new thing. All of a sudden it's a new world, and you're doing different—it looks the same at the beginning. You line those numbers still up and you're still working on that same first column, but boy, you're doing all sorts—now you're crossing stuff out. You're not just little ones, and what? Algorithms don't necessarily help you learn the next algorithm. It's a whole new experience. Strategies are synergistic. If you learn a strategy, that helps you learn the next set of relationships, which then refines to become a new strategy. I think that's really helpful to know, that we can—strategies build on each other. There's synergy involved. Algorithms, you got to learn a new one every time.
Mike: And it turns out that memorizing the dictionary of mathematics is fairly challenging.
Pam: Indeed [laughs], indeed. I tried hard to memorize that. Yeah.
Mike: You said something to me when we were preparing for this podcast that I really have not been able to get out of my mind, and I'm going to try to approximate what you said. You said that during the string, as the teacher and the students are engaging with it, you want students' mental energy primarily to go into reasoning. And I wonder if you could just explicitly say, for you at least, what does that mean and what might that look like on a practical level?
Pam: So I wonder if you're referring to when teachers will say, “Do we have students write? Do we not have them write?” And I will suggest: “It depends. It's not if they write; it's what they write that's important.”
What do I mean by that? What I mean is if we give kids paper and pencil, there is a chance that they're going to be like, “Oh, thou shalt get an answer. I'm going to write these down and mimic something that I learned last year.” And put their mental energy either into mimicking steps or writing stuff down. They might even try to copy what you've been representing strategies on the board. And their mental effort either goes into mimicking, or it might go into copying.
What I want to do is free students up [so] that their mental energy is, how are you reasoning? What relationships are you using? What's occurring to you? What's front and center and sort of occurring? Because we're high-dosing you with patterns, we're expecting those to start happening, and I'm going to be saying things, giving that helper problem. “Oh, that's occurring to you? It's almost like it's your idea—even though I just gave you the helper problem!” It's letting those ideas bubble up and percolate naturally and then we can use those to our advantage. So that's what I mean when [I say] I want mental energy into “Hmm, what do I know, and how can I use what I know to logic my way through this problem?” And that's math-ing. Those are the mental actions of mathematicians, and that's where I want kids' mental energy.
Mike: So I want to pull this string a little bit further. Pun 100% intended there. Apologies to listeners.
What I find myself thinking about is there've got to be some do's and don'ts for how to facilitate a string that support the kind of reasoning and experience that you've been talking about. I wonder if you could talk about what you've learned about what you want to do as a facilitator when you're working with a string and maybe what you don't want to do.
Pam: Yeah, absolutely. So a good thing to keep in mind is you want to keep a string snappy. You don't want a lot of dead space. You don't want to put—one of the things that we see novice, well, even sometimes not-novice, teachers do, that’s not very helpful, is they will put the same weight on all the problems.
So I'll just use the example 8 plus 10, 8 plus 9, they'll—well, let me do a higher one. 7 times 8, 70 times 8. They'll say, “OK, you guys, 7 times 8. Let's really work on that. That's super hard.” And kids are like, “It's 56.” Maybe they have to do a little bit of reasoning to get it, because it is an often missed fact, but I don't want to land on it, especially—what was the one we did before? 34 plus 10. I don't want to be like, “OK, guys, phew.” If the last problem on my string is 26 plus 18, I don't want to spend a ton of time. “All right, everybody really put all your mental energy in 36 plus 10” or whatever I said. Or, let's do the 7 times 8 one again. So, “OK, everybody, 7 times 8, how are you guys thinking about that?” Often we're missing it. I might put some time into sharing some strategies that kids use to come up with 7 times 8 because we know it's often missed. But then when I do 70 times 8, if I'm doing this string, kids should have some facility with times 10. I'm not going to be like, “OK. Alright, you guys, let's see what your strategies are. Right? Everybody ready? You better write something down on your paper. Take your time, tell your neighbor how….” Like, it's times 10. So you don't want to put the same weight—as in emphasis and time, wait time—either one on the problems that are kind of the gimmes, we're pretty sure everybody's got this one. Let's move on and apply it now in the next one. So there's one thing. Keep it snappy. If no one has a sense of what the patterns are, it's probably not the right problem string. Just bail on it, bail on it. You're like, “Let me rethink that. Let me kind of see what's going on.” If, on the other hand, everybody's just like, “Well, duh, it's this” and “duh, it's that,” then it's also probably not the right string. You probably want to up the ante somehow.
So one of the things that we did in our problem string books is we would give you a lesson and give you what we call the main string, and we would write up that and some sample dialogs and what the board could look like when you're done and lots of help. But then we would give you two echo strings. Here are two strings that get at the same relationships with about the same kind of numbers, but they're different and it will give you two extra experiences to kind of hang there if you're like, “Mm, I think my kids need some more with exactly this.” But we also then gave you two next-step strings that sort of up the ante. These are just little steps that are just a little bit more to crunch on before you go to the next lesson that's a bit of a step up, that's now going to help everybody increase. Maybe the numbers got a little bit harder. Maybe we're shifting strategy. Maybe we're going to use a different model. I might do the first set of strings on an area model if I'm doing multiplication. I might do the next set of strings in a ratio table. And I want kids to get used to both of those.
When we switch up from the 8 string to the next string, kind of think about only switching one thing. Don't up the numbers, change the model, and change the strategy at the same time. Keep two of those constant. Stay with the same model, maybe up the numbers, stay with the same strategy. Maybe if you're going to change strategies, you might back up the numbers a little bit, stick with the model for a minute before you switch the model before you go up the numbers. So those are three things to consider. Kind of—only change up one of them at a time or kids are going to be like, “Wait, what?” Kids will get higher dosed with the pattern you want them to see better if you only switch one thing at a time.
Mike: Part of what you had me thinking was it's helpful, whether you're constructing your own string or whether you're looking at a string that's in a textbook or a set of materials, it's still helpful to think about, “What are the variables at play here?” I really appreciated the notion that they're not all created equal. There are times where you want to pause and linger a little bit that you don't need to spend that exact same amount of time on every clunker and every helper. There's a critical problem that you really want to invest some time in at one point in the string. And I appreciated the way you described, you're playing with the size of the number or the complexity of the number, the shift in the model, and then being able to look at those kinds of things and say, “What all is changing?” Because like you said, we're trying to kind of walk this line of creating a space of discovery where we haven't suddenly turned the volume up to 11 and made it really go from like, “Oh, we discovered this thing, now we're at full complexity,” and yet we don't want to have it turned down to, “It's not even discovery because it's so obvious that I knew it immediately. There's not really anything even to talk about.”
Pam: Nice. Yeah, and I would say we want to be right on the edge of kids’ own proximal development, right on the edge. Right on the edge where they have to grapple with what's happening. And I love the word “grapple.” I've been in martial arts for quite a while, and grappling makes you stronger. I think sometimes people hear the word “struggle” and they're like, “Why would you ever want kids to struggle?” I don't know that I've met anybody that ever hears the word “grapple” as a negative thing. When you “grapple,” you get stronger. You learn. So I want kids right on that edge where they are grappling and succeeding. They're getting stronger. They're not just like, “Let me just have you guess what's in my head.” You're off in the field and, “Sure hope you figure out math, guys, today.” It's not that kind of discovery that people think it is. It really is: “Let me put you in a place where you can use what you know to notice maybe a new pattern and use it maybe in a new way. And poof! Now you own those relationships, and let's build on that.” And it continues to go from there.
When you just said—the equal weight thing, let me just, if I can—there's another, so I mentioned that there's at least five structures of problem strings. Let me just mention one other one that we like, to give you an example of how the weight could change in a string. So if I have an equivalent structure, an equivalent structure looks like: I give a problem, and an example of that might be 15 times 18. Now I'm not going to give a helper; I'm just going to give 15 times 18. If I'm going to do this string, we would have developed a few strategies before now. Kids would have some partial products going on. I would probably hope they would have an “over,” I would've done partial products over and probably, what I call “5 is half a 10.”
So for 15 times 18, they could use any one of those. They could break those up. They could think about twenty 15s to get rid of the extra two to have 18, 15. So in that case, I'm going to go find a partial product, an “over” and a “5 is half a 10,” and I'm going to model those. And I'm going to go, “Alright, everybody clear? Everybody clear on this answer?” Then the next problem I give—so notice that we just spent some time on that, unlike those helper clunker strings where the first problem was like a gimme, nobody needed to spend time on that. That was going to help us with the next one.
In this case, this one's a bit of a clunker. We're starting with one that kids are having to dive in, chew on. Then I give the next problem: 30 times 9. So I had 15 times 18 now 30 times 9. Now kids get a chance to go, “Oh, that's not too bad. That's just 3 times 9 times 10. So that's 270. Wait, that was the answer to the first problem. That was probably just coincidence. Or was it?” And now especially if I have represented that 15 times 18, one of those strategies with an area model with an open array, now when I draw the 30 by 9, I will purposely say, “OK, we have the 15 by 18 up here. That's what that looked like. Mm, I'll just use that to kind of make sure the 30 by 9 looks like it should. How could I use the 15 by 18? Oh, I could double the 15? OK, well here's the 15. I'm going to double that. Alright, there's the 30. Well, how about the 9? Oh, I could half? You think I should half? OK. Well I guess half of 18. That's 9.”
So I've just helped them. I've brought out, because I'm inviting them to help me draw it on the board. They're thinking about, “Oh, I just half that side, double that side. Did we lose any area? Oh, maybe that's why the products are the same. The areas of those two rectangles are the same. Ha!” And then I give the next problem. Now I give another kind of clunker problem and then I give its equivalent. And again, we just sort of notice: “Did it happen again?” And then I might give another one and then I might end the string with something like 3.5 times—I'm thinking off the cuff here, 16. So 3.5 times 16. Kids might say, “Well, I could double 3.5 to get 7 and I could half the 16 to get 8, and now I'm landing on 7 times 8.” And that's another way to think about 3.5 times 16. Anyway, so, equivalent structure is also a brilliant structure that we use primarily when we're trying to teach kids what I call the most sophisticated of all of the strategies. So like in addition, give and take, I think, is the most sophisticated addition. In subtraction, constant difference. In multiplication, there's a few of them. There's doubling and having, I call it flexible factoring to develop those strategies. We often use the equivalent structure, like what's happening here? So there's just a little bit more about structure.
Mike: There's a bit of a persona that I've noticed that you take on when you're facilitating a string. I'm wondering if you can talk about that or if you could maybe explain a little bit because I've heard it a couple different times, and it makes me want to lean in as a person who's listening to you. And I suspect that's part of its intent when it comes to facilitating a string. Can you talk about this?
Pam: So I wonder if what you're referring to, sometimes people will say, “You're just pretending you don't know what we're talking about.” And I will say, “No, no, I'm actually intensely interested in what you're thinking. I know the answer, but I'm intensely interested in what you're thinking.” So I'm trying to say things like, “I wonder.” “I wonder if there's something up here you could use to help. I don't know. Maybe not. Mm. What kind of clunker could—or helper could you write for this clunker?”
So I don't know if that's what you're referring to, but I'm trying to exude curiosity and belief that what you are thinking about is worth hearing about. And I'm intensely interested in how you're thinking about the problem and there's something worth talking about here. Is that kind of what you're referring to?
Mike: Absolutely.
OK. We're at the point in the podcast that always happens, which is: I would love to continue talking with you, and I suspect there are people who are listening who would love for us to keep talking. We’re at the end of our time. What resources would you recommend people think about if they really want to take a deeper dive into understanding strings, how they're constructed, what it looks like to facilitate them. Perhaps they're a coach and they're thinking about, “How might I apply this set of ideas to educators who are working with kindergartners and first graders, and yet I also coach teachers who are working in middle school and high school.” What kind of resources or guidance would you offer to folks?
Pam: So the easiest way to dive in immediately would be my brand-new book from Corwin. It's called Developing Mathematical Reasoning: Avoiding the Trap of Algorithms. There's a section in there all about strings. We also do a walk-through where you get to feel a problem string in a K–2 class and a 3–5 [class]. And well, what we really did was counting strategies, additive reasoning, multiplicative reasoning, proportional reasoning, and functional reasoning. So there's a chapter in there where you go through a functional reasoning problem string. So you get to feel: What is it like to have a string with real kids? What's on the board? What are kids saying? And then we link to videos of those. So from the book, you can go and see those, live, with real kids, expert teachers, like facilitating good strings. If anybody's middle school, middle school coaches: I've got building powerful numeracy and lessons and activities for building powerful numeracy. Half of the books are all problem strings, so lots of good resources.
If you'd like to see them live, you could go to mathisfigureoutable.com/ps, and we have videos there that you can watch of problem strings happening.
If I could mention just one more, when we did the K–12, Developing Mathematical Reasoning, Avoiding the Trap of Algorithms, that we will now have grade band companion books coming out in the fall of ’25. The K–2 book will come out in the spring of ’26. The [grades] 3–5 book will come out in the fall of ’26. The 6–8 book will come out and then six months after that, the 9–12 companion book will come out. And those are what to do to build reasoning, lots of problem strings and other tasks, rich tasks and other instructional routines to really dive in and help your students reason like math-y people reason because we are all math-y people.
Mike: I think that's a great place to stop. Pam, thank you so much for joining us. It's been a pleasure talking with you.
Pam: Mike, it was a pleasure to be on. Thanks so much.
Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability.
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