Number string structure and design

How are number strings designed? Typically, people tend to describe number strings as having the following structure

Entry problem

Helper problems

Challenge problem (or clunker)

 

This post from Math Coach on Demand (which also has a bunch of addition and subtraction number strings) describes the structure like this:

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Again, the concept of helper problems. But is there just one “formula” for a number string? Continue reading “Number string structure and design”

30 ways to make number strings more inclusive

As a teacher in inclusive settings, number strings were a critical part of my daily mathematical work.  Routines are highly effective in inclusive classrooms, particularly routines like number strings which externalize complex cognitive processes.  By that delicious turn of phrase, I mean that a number string is not the kind of routine that teaches low-level thinking like memorization.  Instead, it allows kids to participate in the strategic thinking of other kids, giving them access to complex processes that too often only go on in individual minds. Continue reading “30 ways to make number strings more inclusive”

Encouraging Strings Discussions

Defining the Problem:

In a recent workshop, a third grade teacher shared the following: “I think I’m doing it right, I ask a lot of questions like, ‘How did you solve?’ and ‘Can you explain your strategy?’ or ‘What do you think about _______’s approach?’ But the kids often just stare at me blankly. Only a handful will jump into the conversation. What am I doing wrong?”

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The Power of Strings

I’m embarrassed to say this, I have a vivid memory of around my fourth year of teaching 5th grade and my second year of using number strings during a staff development meeting with a very patient and kind staff developer, “I’m not sure I understand why I am teaching strings. I do the string, the kids do it, we discuss it, there’s a chart up and then poof it disappears and I see no transfer.” It took me a while to understand the purpose and powerfulness of strings. Here are some of my initial questions about strings and my responses to those thoughts after much practice with strings. Continue reading “The Power of Strings”

Closed to Open Array

After looking over beginning-of-the-year assessments, a new 4th grade teacher was concerned that half of her students were still unsure of the open array model. Some were simply still not convinced of the empty boxes! As her coach, we planned a number string together that would engage the students through questioning, get to know the students even more through open discussions (since it’s still September), and to help each student “hook in” and trust the open array. I modeled this string, hoping to model the questioning but to also model for the teacher who is coming across the open array for the very first time.

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Photo number strings for multiplication

Here are two photos I snapped as I walked by a 99 cent store in LA. Beautiful arrays, no?  Image

I am thinking about how to use these kinds of images as the anchors for number strings, particularly for intervention work with older students.  Sometimes older kids need work thinking about multiplication, but in an age-appropriate way.  What kind of questions do you think of with this image?  One could most simply begin by asking what kids noticed about the image.  That would bring most of the interesting mathematics forward, I think. Beginning perhaps with how many boxes of hot chocolate do you see (nice numbers)?  And then, considering this is a 99 cent store, how much would it cost to buy all of this chocolate.  It reminds me of some work that Pamela Harris suggests in her book on Powerful Numeracy, in which she asks kids what is 99 plus any number?  A 99 cent store is a great way to think about what is 99 times any number?

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Go-to questions for teaching number strings

My four-year-old son likes to walk over to the magnetic rekenrek (math rack) I keep on the fridge, move some of the red beads to the right, and then ask me, “How many beads you see Momma?” and “How did you get that?”  My son has my routine down pat.  When I ask him or his brother questions on the math rack, I keep my questions very consistent.  I put up a particular number and ask, “How many beads” and then when they answer, “How did you get that? or “How did you know?”  or “How did you see that?”

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Teaching mathematics is a very complex act- when teaching a number string, you are listening carefully to students’ responses, carefully representing their thinking, and always at the same time looking for ways to connect to the larger mathematical goals of the string.  As the mathematics educator Deborah Ball so beautifully wrote, we teach mathematics with  “ears to the ground, listening to students, eyes are focused on the mathematical horizon” (Ball, 1993, p. 376).  I found that as my mind does so many things simultaneously, it helps to keep my own words pretty simple.  When I am teaching strings without the rekenrek, thus most number strings, I tend to use these phrases a lot:

“Give me a thumb when you are ready”

“What did you get?”

“How did you get that?”

“Talk to your partner about this one”

“Can anyone restate her/his strategy?”

“Does this match what you were thinking?”(about the representation I made of their work)

Much of my talk when representing strategies is my repeating the words of a student who shared a strategy, simply because it helps me remember the strategy if I restate it while I am drawing a representation of it on the board.

My aim is to get strategies up on the board, and then to make sure that at some point in the string, I ask a question that moves the discussion to the level of generalization. The best questions are closely connected to the mathematics of the string, and the strategies of the students, but I seem to often say,

“Will this strategy always work?”

“Can we name and define this strategy?”

“Are there helpful patterns in this number string?”

Often, a student will begin this generalization process for you, and you just need to follow their thinking.  They may say, I think that this strategy works because . . . , and at that point I write their words down verbatim on the side of the board, leaving space for the group to refine the generalization.

Just like my son learned his questions from me, I learned them from watching other people teach strings.  Are there other go-to questions for teaching number strings?