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Exploring the Power & Purpose of Number Strings with Pam Harris · Rounding Up

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I've struggled when I have a new strategy I want my students to consider,
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and despite my best efforts, it just doesn't surface organically.
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While I didn't want to just tell my students what to do,
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I wasn't sure how to move forward, then I discovered number strings.
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Today we're talking with Pam Harris about the ways number strings enable
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teachers to introduce new strategies while maintaining opportunities for
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students to discover important relationships.
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Welcome to the podcast, Pam. I'm really excited to talk with you today.
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Thanks, Mike. I'm super glad to be on. Thanks for having me.
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Absolutely. So before we jump in, I want to offer a quick note to listeners.
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The routine we're going to talk about today goes by several different names in
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the field. Some folks, including Pam,
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refer to this routine as problem strings and other folks,
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including some folks at the Math learning center,
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refer to them as number strings. For the sake of consistency,
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we'll use the term strings during our conversation today. And Pam,
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with that said, I'm wondering if for listeners,
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without prior knowledge,
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could you briefly describe strings? How are they designed?
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How are they intended to work?
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Yeah,
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if I could tell you just a little of my history when I was a secondary math
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teacher and I dove into research,
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I got really curious how can we do the mental actions that I was seeing my son
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and other people use that weren't the remote memorizing and mimicking I'd gotten
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used to.
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I ran into the work of Kathy Fno and Martin Dlk and young mathematicians at
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work, and they had pulled from the Netherlands strings,
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they called them strings,
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and they were a series of problems that were in a certain order.
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The order mattered, the relationship between the problems mattered,
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and maybe the most important part that I saw was I saw students
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thinking about the problems and using what they learned and saw and heard from
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their classmates in one problem,
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starting to let that impact their work on the next problem.
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And then they would see that thinking made visible and the conversation between
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it and then it would impact how they thought about the next problem. And as I
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saw those students literally learn before my eyes, I was like,
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this is unbelievable. And honestly,
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at the very beginning I didn't really even parse out what was different
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between maybe one of fas no's rich tasks versus herb strings versus just a
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conversation with students.
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I was just so enthralled with the learning because what I was seeing
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were the kind of mental actions that I was intrigued with.
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I was seeing them not only happen live but grow, live, develop,
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like they were getting stronger and more sophisticated because of the series
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of the order the problems were in because that sequence of problems that was
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unbelievable. And I was so excited about that,
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that I began to dive in and get more clear on what is a string of problems.
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The reason I call 'em problem strings is I'm K 12,
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so I will have data strings and geometry strings and pick one trig strings like
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strings with functions in algebra. But for purpose of this podcast,
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there's strings of problems with numbers in them.
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So I have a question, but I think I just want to make an observation first.
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The way you described that moment where students are taking advantage
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of the things that they made sense of in one problem and then the
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next part of the string offers them the opportunity to use that and to see a set
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of relationships.
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I vividly remember the first time I watched someone facilitate a string and
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feeling that same way of this routine really offers kids
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an opportunity to take what they've made sense of and immediately apply it.
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And I think that is something that I cannot say about all the routines that I've
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seen, but it was really so clear.
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I just really resonate with that experience of what will this do for children?
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And if I can offer an additional word in there, it influences their work.
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We're taking the major relationships,
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the major mathematical strategies, and we're high dosing kids with them.
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So we give them a problem,
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maybe a problem or two that has a major relationship involved.
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And then like you said,
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we give them the next one and now they can notice the pattern,
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what they learned in the first one or the first couple,
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and they can let it influence.
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They have the opportunity for it to nudge them to go, well,
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I saw what just happened there. I wonder if it could be useful here.
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I'm going to tinker with that.
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I'm going to play with that relationship a little bit. And then we do it again.
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So in a way, we're taking the relationships that I think for whatever reason,
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some of us can wander through life and we could run into the mathematical
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patterns that are all around us in the low dose that they are all around us,
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but many of us don't pick up on that low dose and connect them and make
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relationships and then let it influence when we do another problem,
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we need a higher dose. I needed a higher dose of those major patterns.
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I think most kids do problem strings or number strings are so brilliant because
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of that sequence and the way that the problems are purposely one after the
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other. Give students the opportunity to, like you said,
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apply what they've been learning instantly. And then not just then,
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but on the next problem and then sometimes in a particular structure we might
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then say, based on what you've been seeing,
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what could you do on this last problem?
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And we might make that last problem even a little bit further away from the
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pattern, a little bit more sophisticated, a little more difficult, a little,
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little less lockstep,
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a little bit more where they have to think outside the box but still could apply
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that important relationship.
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So I have two thoughts, Pam as I listen to you talk. One is that for both of us,
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there's a really clear payoff for children that we've seen in the way that
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strings are designed and the way that teachers can use them to influence
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students' thinking and also help kids build a recognition or high dose a set of
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relationships that are really important.
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The interesting thing is I taught kindergarten through second grade for most of
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my teaching career, and you've run the gamut.
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You've done this in middle school and high school.
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So I think one of the things that might be helpful is to share a few examples of
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what a string could look like at a couple different grade levels.
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Are you okay to share a few?
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You bet. Can I tack on one quick thing before I do?
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Absolutely.
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You mentioned that the payoff is huge for children.
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I'm going to also suggest that one of the things that makes strings really
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unique and powerful in teaching is the payoff for adults.
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Because let's just be clear, most of us now, not all,
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but most of us I think had a similar experience to me that we were in classrooms
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where the teacher said, do this thing.
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That's the definition of math is for you to rote memorize these disconnected
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facts and mimic these procedures. And for whatever reason,
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many of us just believed that and we did it, some people didn't.
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Some of us played with relationships and everything regardless.
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We all kind of had the same learning experience where we may have taken at
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different places, but we still saw the teacher say, do these things,
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rote, memorize, mimic. And so as we now say to ourselves, whoa,
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I've just seen how cool this can be for students and we want to affect our
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practice, we want take what we do,
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do something. We now believe this could be really helpful, like you said,
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for children, but doing that's not trivial,
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but strings make it easier.
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Strings are I think a fantastic differentiated kind of task for
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teachers because a teacher who's very new to thinking and using relationships
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and teaching math a different way than they were taught can dive in and do a
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problem string learn right along with your students. A veteran teacher,
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an expert teacher who's really working on their teacher moves and really owns
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the landscape of learning and all the things still uses problem strings because
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they're so powerful. Like anybody across the gamut can use string.
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I just said problem, strengths, sorry, number strengths, strings,
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all of us no matter where we are in our teaching journey can get a lot out of
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strengths.
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So with all that said, let's jump in.
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Let's talk about some examples across the elementary span.
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Nice. So I'm going to take a young learner, not our youngest,
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but a young learner. I might ask a question like, what is eight plus 10?
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And then if they're super young learners,
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I expect some students might know that 10 plus a single digit is a teen,
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but I might expect many of the students to actually say 8, 9, 10, 11, 12 or 10,
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and they might count by ones given maybe from the larger,
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maybe from the whatever. But anyway, we're going to kind of do that.
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I'm going to get that answer from them. I'm going to write on the board,
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eight plus 10 is 18,
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and then I would have done some number line before this,
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but then I'm going to represent on the board eight plus 10 jump of 10,
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that's 18.
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And then the next problem's going to be something like eight plus nine.
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And I'm going to say, go ahead and solve it any way you want, but I wonder,
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wonder, maybe you could use the first problem,
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maybe not. I'm just going to lightly suggest that you consider what's on the
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board. Let 'em do whatever they do.
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I'm going to expect some students still be counting.
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Some students are going to be like, oh,
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well I can think about nine plus eight counting by ones.
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I think by eight maybe I can think about eight plus eight.
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Maybe I can think about nine plus nine.
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Some students are going to be using relationships, some are counting.
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Kids are over the map. When I get an answer, they're also like 17.
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Then I'm going to say, did anybody use the first problem to help?
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You didn't have to, but did anybody,
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then I'm going to grab that kid and if no one did, I'm going to say,
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could you and pause. Now,
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if no one sparks at that moment,
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then I'm not going to make a big deal of it. I'll just go, okay, alright,
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and I'll do the next problem. And the next problem might be something like,
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what's five plus 10? Again, same thing, we're going to get 15.
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I'm going to st draw it on the board. Oh,
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I should have mentioned when we got to the eight plus nine,
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right underneath that eight jump, 10 land on 18,
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I'm going to draw an eight jump nine shorter jump.
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I'm going to have these lined up land on the 17.
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Then I might just step back and go like 17. That's it's almost where the 18 was.
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Now if kids have noticed, if somebody used that first problem,
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then I'm going to say, well, tell us about that. Well miss,
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we added 10 and that was 18, but now we're adding one less,
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so it's got to be one less. And we go, well,
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is 17 one less than 18? Sure enough. Then I give the next set of problems.
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That might be five plus 10 and then five plus nine,
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and then I might do seven plus 10. Maybe I'll do nine next nine plus 10,
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and then nine plus nine. Then I might end that string. The next problem,
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last problem might be, what is seven plus nine?
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Now notice I didn't give the helper. So this case I might go, Hey,
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I've kind of gave you plus 10. A lot of you use that to do plus nine.
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I gave you plus 10. Some of you use that to do plus nine, I gave you plus 10.
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Some of you used that plus nine for this one. I'm not giving you a helper.
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I wonder if you could come up with your own helper. Now brilliantly,
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what we've done is say to students,
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you've been using what I have up here or not,
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but could you actually think what is the pattern that's happening and create
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your own helper? Now that's meta. Right now we're thinking about our thinking.
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I'm encouraging that pattern recognition in a different way. I'm asking kids,
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what would you create? We're going to share that helper.
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I'm not even having 'em solve the problem.
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They're just creating that helper and then we can move from there.
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So that's an example of a young string that actually can grow up.
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So now I can be in a second grade class and I could ask a similar,
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could you use something that's adding a bit too much to back up?
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But I could do that with bigger numbers.
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So I could start with that eight plus 10, eight plus nine,
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but then the next pair might be 34 plus 10, 34 plus nine,
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but then the next pair might be 48 plus 20 and 48 plus
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19.
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And the last problem of that string might be something like 26 plus
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18.
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So in those cases,
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there's this mental scaffolding that you're creating.
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And I just want to mark this.
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I have a good friend who used to tell me that part of teaching mathematics is
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you can lead the horse to water, you can show them the water,
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they can look at it, but darn it do not push their head in the water.
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And I think what he meant by that is you can't force it,
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but you're not doing that with a string.
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You're creating a set of opportunities for kids to notice you're
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doing all kinds of implicit things to make structure available for kids
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to attend to,
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and yet you're still allowing them the ability to use the strategies that they
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have. We might really want them to notice that,
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and that's beautiful about a string, but you're not forcing.
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And I think it's worth saying that because I could imagine that's a place where
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folks might have questions.
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If the kids don't do the thing that I'm hoping that they would do,
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what should I do?
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Yeah, that's a great question. Let me give you another example.
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And in that example I'll talk about that. So especially as the kids get older,
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I'm going to use the same kind of relationship.
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It's maybe easier for people to hang on to if I stay with the same sort of
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relationship. So I might say, Hey everybody, seven times eight,
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that's a fact I'm noticing most of us just don't have at our fingertips.
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Let's just work on that. What do you know?
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I might get a couple of strategies for kids to think about seven times eight.
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We all agree it's 56. Then I might say, what's 70 times eight?
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And then let kids think about that. Now, this would be the first time I do that,
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but if we've dealt with scaling times 10 at all,
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if I have 10 times the number of whatever the things is,
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then often kids will say, well, I've got 10 times seven to 70,
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so then 10 times 56 is 560. And then the next problem might be,
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I wonder if you could think about 69 times eight.
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If we've got 70 eights,
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can I use that to help me think about 69 eights?
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And I'm saying that in a very specific way to help ping on prior knowledge.
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So then I might do something similar. Well,
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let's pick another often missed facts, I don't know, six times nine.
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And then we could share some strategies on how kids are thinking about that.
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We all agree it's 54. And then I might say, well,
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could you think about six times 90? I'm going to talk about scaling up again.
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So that would be 540. Now I'm going really fast. But then I might say,
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could we use that to help us think about six times 89? I dunno if you noticed,
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but I sort of swapped.
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I'm not thinking about 90 sixes to 89 sixes. Now I'm thinking about six 90
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to help me think about six 80 nines. So that's a little bit of a,
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we have to decide how we're going to deal with that.
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I'll kind of mess around with that.
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And then I might have what we call that clunker problem at the end.
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Notice that I've had a helper seven times eight, 70 times eight.
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A lot of you use that to help you think about 69 times eight that I had a
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helper, six times nine, six times 90.
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A lot of you use that to help you think about six times 89.
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What if I don't give you those helpers? What if I had something like,
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now I'm making this up off the cuff here,
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like nine times 69, 9 times 69.
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Could you use relationships we just did. Now notice,
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Mike,
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I might've had kids solving all those problems using an algorithm. They might've
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been punching their calculator, but now I'm asking the question,
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could you come up with these helper problems?
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Notice how I'm now inviting you into a different space.
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It's not about getting an answer.
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I'm inviting you into what are the patterns that we've been establishing here.
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And so what would be those two problems that would be like the patterns we've
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just been using?
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That's almost like saying when you're out in the world and you hit a problem,
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could you say to yourself, I don't know that one, but what do I know?
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What do I know that could help me get there? And that's math thing.
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I mean, you could have had a kid say, well, I'm not sure about how,
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I don't know the answer to that, but I could do nine times 60, right?
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Or I could do 10 times. I'm thinking 10 times 69. Correct?
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Yes, yes. In fact, when I gave that clunker problem nine times 69,
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I said to myself, oh,
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I shouldn't have said nine because now you could go either direction.
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You could either over either way to find nine, I can do 10 or to find 69,
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I can do 70. And then I thought, ah, we'll go with it because you go either way.
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So I might want to focus it, but I might not.
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And this is a moment where a novice could just throw it out there and then
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almost be surprised. Whoa.
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They could go either direction and an expert could plan.
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And is this the moment where I want lots of different ways to go?
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Or do I want to focus, narrow it a little bit more,
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be a little bit more explicit? It's not that I'm telling kids,
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but I'm having an explicit goal. So I'm maybe narrowing the field a little bit.
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And maybe the problem could have been seven times 69,
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then I wouldn't have gotten that other over, not the 10 to get nine.
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Does that make sense?
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It absolutely does.
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What you really have me thinking about is nct M'S definition of fluency,
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which is accuracy, efficiency and flexibility.
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And the flexibility that I hear coming out of the kinds of things that kids
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might do with a string.
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It's exciting to imagine that that's one of the outcomes you could get from
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engaging with strings.
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Absolutely. Because if you're stuck teaching memorizing algorithms,
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there's no flexibility, like none like zilch.
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But if you're doing strings like this, kids have a brilliant flexibility.
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And one of the conversations I'd want to have here, Mike,
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is if a kid came up with 10 times 69 to help with nine times 69,
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and a different kid came up with nine times 70 to help with nine times 69,
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I would want to just have a brief, which one of those do you like better class?
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And why not that one is better than the other,
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but just to have the comparison conversation. So the kids go, huh,
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I have access to both of those. Well, I wonder when I'm walking down the street,
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I have to answer that one.
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Which one do I want my brain to gravitate towards next time?
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And that's mathematical behavior.
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That's mathematical disposition to do one of the strands of proficiency.
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We want that productive disposition where kids are thinking to themselves,
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I own relationships.
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I just got to pick a good one here to what's the best one I could find here?
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And try that one. Try that one. Ah, I'll go with this one today.
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I love that. As we were talking,
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I wanted to ask you about the design of the string,
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and you started to use some language like helper problems and the clunker.
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And I think that's really the nod to the kinds of features that you would want
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to design into a string.
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Could you talk about either a teacher who's designing their own string,
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what are some of the features or a teacher who's looking at a string that they
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might find in a book that you've written or that they might find in say,
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the Bridges curriculum?
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What are some of the different problems along the way
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that really kind of inform the structure?
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So you might find it interesting that over time we've identified that there's at
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least five major structures, tip strings,
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and the one that I just did with you is kind of the easiest one to facilitate.
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It's the easiest one to understand where it's going,
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and it's the helper clunker structure.
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So the helper clunker structure is all about,
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I'm going to give you a helper problem that we expect all kids can kind of hang
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on. They have some facility with enough that everybody has access to.
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Then we give you a clunker that you could use that helper to inform how you
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could solve that clunker problem. In the first string I did with you,
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I did a helper, clunker helper, clunker helper, clunker, clunker.
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And the second one we did, I did helper, helper, clunker, helper, helper,
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clunker, clunker.
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So you can mix and match kind of helpers and clunkers in that,
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but there are other major structures of strings. If you're new to strings,
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I would dive in and do a lot of helper clunker strings first.
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But I would also suggest I didn't create my own strings for a long
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time.
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I did pre-written Kathy Fno from the Netherlands,
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from the Freal Institute.
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I was doing their strings to get a feel for the mathematical
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relationships for the structure of a string.
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I would watch videos of teachers doing it so I could get an idea of, oh,
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that move right there made all the difference.
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I see how you just invited kids in, not demand what they do.
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The idea of when to have paper and pencil and when not,
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and just lots of different things can come up that if you're having to write the
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string as well, create the string that could feel insurmountable.
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So I would invite anybody out listening that's like, whoa,
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this seems kind of complicated. Feel free to facilitate someone
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strings. Now I like mine. I think mine are pretty good.
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I think Bridges has some pretty good ones,
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but I think you've really gain a lot from facilitating pre-written strings.
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Can I make one quick differentiation that I'm running into more and more?
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So I have had some sharp people say to me, Hey,
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sometimes you have extra problems in your string.
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Why do you have extra problems in your string? And I'll say, well,
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at first I said, what do you mean? I didn't know what they were.
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Are you telling me my string's bad? Why are you dogging my string?
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But what they meant was they thought a string was the
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process a kid or the steps,
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the relationships a kid used to solve last problem. Does that make sense?
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It does.
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And they were like,
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you did a lot of work to just get that one answer down there. And I'm like, no,
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no, no, no. A problem string or a number string.
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A string is an instructional routine. It is a lesson structure.
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It's a way of teaching. It's not a record of the relationships.
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A kid used to solve a problem. In fact, a teacher just asked,
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we run a challenge three times a year. It's just free. I get on and just teach.
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One of the questions was asked was how do we help our kids write their own
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strings? And I was like, oh, no, kids don't write strings.
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Kids solve problems using relationships.
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And so I think what the teachers were saying was, oh,
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I could use that relationship to help me get this one. Oh,
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and then I can use that to solve the problem as if then the lessons
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structure,
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the instructional routine of a string was then what we want kids to do is use
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what they know to logic their way through using mathematical relationships and
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connections to get answers and to solve problems. That record is not a string,
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that record is a record of their work.
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Does that make sense how there's a little difference there?
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It totally does, but I think that's a good distinction. And frankly,
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that's a misunderstanding that I had when I first started working with strings
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as well.
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It took me a while to realize that the point of a string is to unveil a set
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of relationships and then allow kids to take them up and use them.
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And really it's about making these relationships or these problem
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solving strategies sticky. You want them to stick.
22:20
We could go back to what you said.
22:22
We're trying to high dose a set of relationships that are going to help kids
22:26
with strategies not only in this particular string,
22:31
but across the mathematical work they're doing in their school life.
22:35
Yes, very well said. So for example,
22:38
we did an addition over relationship in the addition string that I talked
22:42
through,
22:43
and then we did a multiplication over set of relationships and multiplication.
22:47
We can do the same thing with subtraction.
22:49
We could have a subtraction string where the helper problem is to subtract a bit
22:52
too much. So something like 42 minus 20,
22:56
and then the next problem could be 42 minus 19, and we're using that.
23:00
I'm going to subtract a bit too much. And then how do you adjust?
23:03
And after you've been thinking about addition over subtraction over is quite
23:07
tricky. You're like, wait, why are we adding what we're subtracting?
23:10
And it's not about teaching kids a series of steps.
23:12
It's really helping them reason, well,
23:14
if I give you owe me 19 bucks and I give you a $20 bill,
23:18
what are we going to do? Oh, you got to give me one back.
23:21
Now that's a little harder today because kids don't mess around with money.
23:24
So we might have to do something that feels like they can or help them feel
23:29
money. That's my personal preference.
23:31
Let's do it with money and help them feel money.
23:33
So one of the things I think is unique to my work is as I dove in and started
23:38
facilitating other people's strings and really building my mathematical
23:42
relationships and connections,
23:44
I began to realize that many teachers I worked with,
23:48
myself included thought, whoa, there's just this uncountable,
23:53
innumerable wide universe of all the relationships that are out
23:58
there and there's so many strategies and anything goes and they're all of equal
24:01
value. And I began to realize, no, no, no,
24:05
there's only a small set of major relationships that lead to a small set of
24:09
major strategies. And if we can get those down,
24:13
kids can solve any problem that's reasonable to solve that a calculator,
24:16
but in the process, building their brains to reason mathematically.
24:20
And that's really our goal is to build kids' brains to reason mathematically.
24:24
And in the process we're getting answers. Answers aren't our goal.
24:27
We'll get answers, sure,
24:28
but our goal is to get them to build that small set of relationships because
24:32
that small set of strategies now sets them free to logic their way through
24:37
problems and bam, we've got kids mapping using the mental actions of mapping.
24:42
Absolutely.
24:43
You made me think about the fact that there's a set of relationships that I can
24:48
apply when I'm working with numbers Under 20.
24:50
There's a set of relationships that same set of relationships I can apply
24:56
and make use of when I'm working with multi-digit numbers,
24:59
when I'm working with decimals, when I'm working with fractions,
25:03
it's really the relationships that we want to expose and then
25:08
generalize and recognize this notion of going over or
25:13
getting strategically to a friendly number and then going after that or getting
25:17
to a friendly number and then going back from that.
25:20
That's a really powerful strategy regardless of whether you're talking about
25:25
eight and three or whether you're talking about adding unit fractions together.
25:30
Strings allow us to help kids see how that idea translates
25:35
across different types of numbers.
25:38
And it's not trivial when you change a type of number or the number gets bigger.
25:41
It's not trivial for kids to take this over strategy and to be thinking about
25:45
something like 2,467 plus 1,995,
25:51
and I know I just threw a bunch of numbers out on purpose.
25:53
It's not trivial for them to go, what do I know about those numbers?
25:57
Can I use some of these relationships I've been thinking about? Well,
26:01
2004 67, that's not really close to a friendly number. Well,
26:04
1,995 is bam. Let's just add 2000. Oh, sweet.
26:09
And then you just got to back up five.
26:11
It's not trivial for them to consider what do I know about these two
26:15
numbers and are they close to something that I could use?
26:19
That's the necessary work of building place value and magnitude and
26:24
reasonableness. We've not known how to do that.
26:26
So in some curriculum we create our whole extra unit that's all about place
26:30
value reasonableness. Now we have kids that are learning to rote memorize how to
26:33
estimate by round,
26:34
I mean there's all this crazy stuff that we add on when instead we could
26:38
actually use strings to help kids build that stuff naturally kind of ingrained
26:42
as we are learning something else.
26:45
Can I just say one other thing that we did in my new book?
26:48
Developing Mathematical Reasoning, avoiding the Trap of Algorithms.
26:51
So I actually wrote it with my son who is maybe the biggest impetus to me diving
26:55
into the research and figuring out all of this math and what it means.
27:00
He said as we were writing, he said,
27:02
I think we could make the point that algorithms don't help you learn a new
27:06
algorithm.
27:07
If you learn the addition algorithm and you get good at it and you can do all
27:11
the addition and columns and all the whatever,
27:13
and then when you learn the subtraction algorithm,
27:15
it's a whole new thing. All of a sudden it's a new world and you're doing
27:19
different. It looks the same at the beginning.
27:21
You line those numbers still up and you're still working on that same first
27:25
column, but boy, you're doing all sorts now. You're crossing stuff out.
27:28
You're not just little ones.
27:29
And what algorithms don't necessarily help you learn the next algorithm.
27:33
It's a whole new experience. Strategies are synergistic.
27:37
If you learn a strategy that helps you learn the next set of relationships,
27:42
which then refines to become a new strategy,
27:45
I think that's really helpful to know that we can strategies build on
27:50
each other. There's synergy involved algorithms.
27:53
You got to learn a new one every time.
27:55
And it turns out that memorizing the dictionary of mathematics is fairly
27:59
challenging.
28:01
Indeed, indeed. I tried hard to memorize that. Yeah.
28:05
You said something to me when we were preparing for this podcast that I really
28:09
have not been able to get out of my mind,
28:10
and I'm going to try to approximate what you said.
28:13
You said that during the string,
28:16
as the teacher and the students are engaging with it,
28:19
you want students' mental energy primarily to go into reasoning.
28:23
And I wonder if you could just explicitly say for you at least,
28:27
what does that mean and what might that look like on a practical level?
28:31
So I wonder if you're referring to when teachers will say,
28:34
do we have students write? Do we not have 'em write?
28:37
And I will suggest it depends. It's not if they write, it's what they write.
28:42
That's important. What do I mean by that?
28:44
What I mean is if we give kids paper and pencil,
28:47
there is a chance that they're going to be like, oh, thou shalt get an answer.
28:52
I'm going to write these down and mimic something that I learned last year and
28:55
put their mental energy either into mimicking steps or writing stuff
29:00
down.
29:00
They might even try to copy what you've been representing strategies on the
29:04
board and their mental effort either goes into mimicking or it might go into
29:08
copying.
29:09
What I want to do is free students up that their mental energy is how are you
29:14
reasoning? What relationships are you using? What's occurring to you?
29:18
What's front and center and sort of occurring because we're high dosing you with
29:23
patterns,
29:23
we're expecting those to start happening and I'm going to be saying things
29:27
giving that helper problem, oh, that's occurring to you.
29:30
It's almost like it's your idea, even though I just gave you the helper problem.
29:33
It's letting those ideas bubble up and percolate naturally and then we can use
29:38
those to our advantage. So that's what I mean when I want mental energy into
29:44
what do I know and how can I use what I know to logic my way through this
29:49
problem. And that's math.
29:51
Those are the mental actions of mathematician and that's where I want kids'
29:54
mental energy.
29:56
So I want to pull this string a little bit further. Pun 100% intended there.
30:01
Apologies to listeners.
30:03
What I find myself thinking about is there've got to be some
30:08
do's and don'ts for how to facilitate a string that support the kind of
30:13
reasoning and experience that you've been talking about.
30:17
I wonder if you could talk about what you've learned about what you want to do
30:22
as a facilitator when you're working with a string and maybe what you don't want
30:25
to do.
30:25
Yeah, absolutely.
30:26
So a good thing to keep in mind is you want to keep a string snappy.
30:30
You don't want a lot of dead space. You don't want to put,
30:33
one of the things that we see novice, well,
30:35
even sometimes not novice teachers do, it's not very helpful,
30:40
is they will put the same weight on all the problems.
30:44
So I'll just use the example eight plus 10, eight plus nine, they'll, well,
30:48
let me do a higher one. Seven times eight 70 times eight, they'll say, okay,
30:52
you guys, seven times eight, let's really work on that. That's super hard.
30:57
And kids are like, it's 56.
30:59
Maybe they have to do a little bit of reasoning to get it because it is an often
31:02
miss fact, but I don't want to land on it,
31:05
especially what was the one we did before? 34 plus 10? I don't want to be like,
31:08
okay guys, if the last problem on my string is 26 plus 18,
31:13
I don't want to spend a ton of time. All right,
31:15
everybody really put all your mental energy in 36 plus 10 or whatever I said,
31:19
or let's do the seven times eight one again. So okay, everybody,
31:21
seven times eight, how are you guys thinking about that? We're missing it.
31:24
I might put some time into sharing some strategies that kids use to come up with
31:28
seven times eight because we know it's often missed.
31:30
But then when I do 70 times eight, if I'm doing this string,
31:34
kids should have some facility with times 10. I'm not going to be like, okay,
31:39
alright you guys, let's see what your strategies are. Right? Everybody ready?
31:41
You better write something down on your paper. Take your time,
31:44
tell your neighbor how it's times 10.
31:48
So you don't want to put the same weight as in emphasis
31:53
and time wait time,
31:55
either one on the problems that are kind of the gimmies,
31:59
we're pretty sure everybody's got this one. Let's move on and apply it now in
32:03
the next one. So there's one thing, keep it snappy.
32:06
No one has a sense of what the patterns are.
32:09
It's probably not the right problem string. Just bail on it, bail on it.
32:12
You're like, let me rethink that. Lemme kind of see what's going on.
32:16
If on the other hand everybody's just like, well duh, it's this and duh,
32:19
it's that. Then it's also probably not the right string.
32:22
You probably want to up the ante somehow.
32:25
So one of the things that we did in our problem string books is we would give
32:28
you a lesson and give you what we call the main string and we would write up
32:32
that and some sample dialogues and what the board could look like when you're
32:35
done and lots of help. But then we would give you two echo strings here,
32:39
two strings that get at the same relationships with about the same kind of
32:43
numbers,
32:44
but they're different and it will give you two extra experiences to kind of hang
32:47
there if you're like, I think my kids need some more with exactly this,
32:52
but we also then gave you two next step strings that sort of up the ante.
32:57
These are just little steps that are just a little bit more to crunch on
33:02
before you go to the next lesson.
33:04
That's a bit of a step up that's now going to help everybody increase.
33:08
Maybe the numbers got a little bit harder. Maybe we're shifting strategy,
33:11
maybe we're going to use a different model.
33:13
I might do the first set of strings on an area model.
33:15
If I'm doing multiplication,
33:16
I might do the next set of strings in a ratio table and I want kids to get used
33:21
to both of those.
33:23
When we switch up from the ape string to the next string,
33:27
kind of think about only switching one thing, don't up the numbers,
33:31
change the model and change the strategy at the same time.
33:34
Keep two of those constant, stay with the same model. Maybe up the numbers,
33:38
stay with the same strategy. Maybe if you're going to change strategies,
33:42
you might back up the numbers a little bit.
33:44
Stick with the model for a minute before you switch the model before you go up
33:48
the numbers. So those are three things to consider.
33:50
Kind of only change up one of them at a time or kids are going to be like, wait,
33:55
what kids will get higher dosed with the pattern?
33:58
You want them to see better if you only switch one thing at a time.
34:02
Part of what you had me thinking was it's helpful whether you're constructing
34:06
your own string or whether you're looking at a string that's in a textbook or a
34:10
set of materials,
34:12
it's still helpful to think about what are the variables at play here?
34:16
I really appreciated the notion that they're not all created equal.
34:20
There are times where you want to pause and linger a little bit that you don't
34:24
need to spend that exact same amount of time on every clunker and every helper.
34:29
There's a critical problem that you really want to invest some time in at one
34:33
point in the string. And I appreciated the way you described.
34:37
You're playing with the size of the number or the complexity of the number,
34:41
the shift in the model,
34:43
and then being able to look at those kinds of things and say,
34:45
what all is changing? Because like you said,
34:49
we're trying to kind of walk this line of creating a space of
34:53
discovery where we haven't suddenly turned the volume up to
34:58
11 and made it really go from like, oh, we discovered this thing,
35:02
now we're at full complexity and yet we don't want to have it turned down to,
35:07
it's not even discovery because it's so obvious that I knew it immediately.
35:11
There's not really anything even to talk about.
35:13
Nice. Yeah, and I would say we want to be right on the edge of kid zone proximal
35:17
development, right on the edge.
35:19
Right on the edge where they have to grapple with what's happening.
35:24
And I love the word grapple.
35:25
I've been in martial arts for quite a while and grappling makes you stronger.
35:30
I think sometimes people hear the word struggle and they're like,
35:33
why would you ever want kids to struggle?
35:35
I don't know that I've met anybody that ever heards the word grapple as a
35:37
negative thing. When you grapple, you get stronger, you learn.
35:40
So I want kids right on that edge where they are grappling and succeeding.
35:44
They're getting stronger. They're not just like,
35:47
let me just have you guess what's in my head. You're off in the field and sure,
35:50
hope you figure out math guys today.
35:52
It's not that kind of discovery that people think it is. It really is.
35:56
Let me put you in a place where you can use what you know to notice maybe a new
36:00
pattern and use it maybe in a new way.
36:02
And now you own those relationships and let's build on that.
36:06
And it continues to go from there. When you just said the equal weight thing,
36:11
let me just, if I can, there's another,
36:13
so I mentioned that there's at least five structures of problem strings.
36:17
Lemme just mention one other one that we like to give you an example of how the
36:22
weight could change in a string.
36:25
So if I have an equivalent structure,
36:27
an equivalent structure looks like I give a problem,
36:30
and an example of that might be 15 times 18. Now I'm not going to give a helper,
36:35
I'm just going to give 15 times 18 if I'm going to do this string.
36:38
We would have developed a few strategies before. Now kids would have some
36:41
partial products going on. I would probably hope they would have an over,
36:45
I would've done partial products over and probably what I call five is half a
36:49
10. So for 15 times 18, they could use any one of those.
36:53
They could break those up.
36:54
They could think about 20 fifteens to get rid of the extra two to have 18, 15.
36:58
So in that case, I'm going to go find a partial product,
37:01
an over and a five is half a 10,
37:03
and I'm going to model those and I'm going to go, alright, everybody clear?
37:05
Everybody clear on this answer? Then the next problem I give.
37:08
So notice that we just spent some time on that,
37:10
unlike those helper clunker strings where the first problem was like a give me,
37:15
we didn't spend time on that,
37:16
that was going to help us with the next one. In this case,
37:18
this one's a bit of a clunker. We're starting with one,
37:21
kids are having to dive in, chew on. Then I give the next problem 30 times nine.
37:26
So I had 15 times 18 now 30 times nine. Now kids get a chance to go, oh,
37:30
that's not too bad. That's just three times nine times 10. So that's 270. Wait,
37:35
that was the answer to the first problem. That was probably just coincidence.
37:38
Or was it?
37:40
And now especially if I have represented that 15 times 18,
37:43
one of those strategies with an area model with an open array.
37:47
Now when I draw the 30 by nine, I will purposely say, okay,
37:52
we have the 15 by 18 up here. That's what that looked like.
37:55
I'll just use that to kind of make sure the 30 by nine looks like it should.
37:58
How could I use the 15 by 18? Oh, I could double the 15. Okay,
38:02
well here's the 15. I'm going to double that. Alright, there's the 30. Well,
38:05
how about the nine? Oh, I get half. You think I should half? Okay.
38:08
Well I guess half of 18. That's nine. So I've just helped them.
38:12
I've brought out, because I'm inviting them to help me draw it on the board,
38:15
they're thinking about, oh, I just half that side, doubled that side.
38:20
Did we lose in the area? Oh, maybe that's why the products are the same.
38:24
The areas of those two rectangles are the same.
38:27
And then I give the next problem.
38:28
Now I give another kind of clunker problem and then I give its equivalent.
38:32
And again, we just sort of notice did it happen again?
38:36
And then I might give another one and then I might end the string with something
38:39
like 3.5 times. I'm thinking off the cuff here,
38:44
16. So 3.5 times 16 kids might say, well,
38:49
I could double 3.5 to get seven and I could half the 16 to get eight,
38:53
and now I'm landing on seven times eight.
38:55
And that's another way to think about 3.5 times 16. Anyway.
38:59
So equivalent structure is also a brilliant structure that we
39:03
use primarily when we're trying to teach kids what I call the most
39:07
sophisticated of all of the strategies. So like in addition,
39:11
give and take I think is the most sophisticated addition in subtraction,
39:14
constant difference in multiplication. There's a few of 'em.
39:18
There's doubling and having,
39:19
I call it flexible factoring to develop those strategies.
39:23
We often use the equivalent structure like what's happening here?
39:27
So there's just a little bit more about structure.
39:30
There's a bit of a persona that I've noticed that you take on
39:35
when you're facilitating a string.
39:38
I'm wondering if you can talk about that or if you could maybe explain a little
39:43
bit,
39:43
because I've heard it a couple different times and it makes me want to
39:48
lean in as a person who's listening to you.
39:51
And I suspect that's part of its intent when it comes to facilitating a string.
39:56
Can you talk about this?
39:58
So I wonder if what you're referring to, sometimes people will say,
40:01
you're just pretending you don't know what we're talking about. And I will say,
40:05
no, no, I'm actually intensely interested in what you're thinking.
40:10
I know the answer, but I'm intensely interested in what you're thinking.
40:14
So I'm trying to say things like I wonder,
40:17
I wonder if there's something up here you could use to help. I don't know.
40:20
Maybe not.
40:23
What kind of clunker could or helper could you write for this clunker?
40:27
So I don't know if that's what you're referring to,
40:28
but I'm trying to exude curiosity and
40:33
belief that what you are thinking about is worth hearing about.
40:38
And I'm intensely interested in how you're thinking about the problem and
40:43
there's something worth talking about here.
40:46
Is that kind of what you're referring to?
40:48
Absolutely. Okay.
40:50
We're at the point in the podcast that always happens,
40:54
which is I would love to continue talking with you and I suspect there are
40:59
people who are listening who would love for us to keep talking or at the end of
41:03
our time,
41:04
what resources would you recommend people think about if they really
41:09
want to take a deeper dive into understanding strings,
41:12
how they're constructed, what it looks like to facilitate them.
41:16
Perhaps they're a coach and they're thinking about how might I apply this set of
41:19
ideas to educators who are working with kindergartens and first graders,
41:24
and yet I also coach teachers who are working in middle school and high school.
41:28
What kind of resources or guidance would you offer to folks?
41:32
So the easiest way dive in immediately would be my brand new book from
41:36
Corwin. It's called Developing Mathematical Reasoning,
41:39
avoiding the Trap of Algorithms.
41:41
There's a section in there all about strings.
41:44
We also do a walkthrough where you get to feel a problem string in a K two
41:49
class and a three five. And well, what we really did was counting strategies,
41:53
additive reasoning, multiplicative reasoning, proportional reasoning,
41:56
and functional reasoning.
41:57
So there's a chapter in there where you go through a functional reasoning
42:00
problem string.
42:01
So you get to feel what is it like to have a string with real kids?
42:05
What's on the board? What are kids saying? And then we link to videos of those.
42:09
So from the book, you can go and see those live with real kids, expert teachers,
42:13
like facilitating good strings if anybody's middle school,
42:17
middle school coaches.
42:18
I've got building powerful numeracy and lessons and activities for building
42:21
powerful numeracy. Half of the books are all problem strings,
42:24
so lots of good resources if you'd like to see them live.
42:28
You could go to math is figureoutable.com/ps
42:33
and we have videos there that you can watch of problem strings happening.
42:37
If I could mention just one more,
42:39
when we did the K 12 developing mathematical reasoning,
42:42
avoiding the trap of algorithms that we will now have grade band
42:47
companion books coming out in the fall of 25.
42:51
The K two book will come out in the spring of 26.
42:54
The three five book will come out in the fall of 26.
42:56
The six eight book will come out and then six months after that,
42:59
the nine 12 companion book will come out.
43:02
And those are what to do to build reasoning,
43:05
lots of problem strings and other tasks,
43:08
rich tasks and other instructional routines to really dive in and help your
43:11
students. Reason like math people reason, because we are all math people.
43:17
I think that's a great place to stop. Pam, thank you so much for joining us.
43:21
It's been a pleasure talking with you.
43:23
Mike. It was a pleasure to be on. Thanks so much.
43:28
This podcast is brought to you by the Math Learning Center and the Meyer Math
43:31
Foundation dedicated to inspiring and enabling all individuals to discover
43:36
and develop their mathematical confidence and ability.