## 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:

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

## Cars, coffee, and climbing stairs: Inviting students into the story

Here’s another important contribution from our friend and colleague Marcelle Good — a 6th grade teacher at School of the Future (Brooklyn) and a Math for America Master Teacher.

In this post she illustrates the role of number strings in helping students to reason quantitatively. This idea  — codified as one of eight Standards for Mathematical Practice — means that students of all ages can “make sense of quantities and their relationships
in problem situations…the ability to contextualize.”

It also suggests that students have developed the “habit of creating a coherent representation of the problem at hand….considering the units involved, attending to the meaning of quantities, not just how to compute them.” In other words, giving students the chance to situate numbers and other values in a story and using those stories to make sense of the mathematics.

To be honest, I was slow to come to the idea of loving context in number strings — the numbers were so beautiful on their own!  An even bigger issue was that the context never seemed to be taken up by students during independent work or when the problems got more complex.

Recently with my students, though, I came to really appreciate the power of story as a referent for kids. When my entire class tried to convince me that since 100 cars would have 400 tires, 99 cars would have 399, I knew I had a problem.

With just a bare number ratio table, my kids would not have found their way out of this misconception. But, with the context in mind, one of my 7th graders explained to the class, “This is how you think about it: You have 100 cars for some reason. Some jerk comes and steals one of your cars. He doesn’t drive off with one tire — he drives away with four tires, so you have 396 left.”

Before another example, some background about me and my students. At my school, School of the Future (located in East New York) I am really struggling to help students access grade-level content. When they enter my school in 6th grade, most of my students are 3 to 4 years below grade level. To address the challenge, our approach has been to take the long view: we are not too concerned with getting them to do 6th grade math in 6th grade. Instead, our goal is to get them to do 8th grade math by 8th grade.

That wasn’t our initial approach. Originally, we tried to teach them grade level content, and scaffold the work by re-teaching or reviewing topics like double-digit multiplication or generating equivalent fractions. What we found, as a school community,  was that this approach was not working, and just not enough. Students were entering 7th grade with a partial understanding of 6th grade, and still lacking a weak foundation.

As a result, I spend a lot of time figuring out exactly what my students know.

Do they have a concept of how the number ten works in addition and why it’s so powerful?

What ideas so they have about multiplication?

Do they have a concept of area?

Do they count only by ones, or do they have some strategies?

Once I know where they are, I work to address their needs in class by meeting them where they’re at. My students who are the least ready for 6th grade material are also assigned to a numeracy class, in addition to their regular 6th grade math class. And in this class, I give myself permission to teach what would be considered K-5 mathematics.

Now, as I prepare to lead number strings where I know the math is more challenging for students, my first question is always, “What story can you tell about what happened?”

Today, the story was that I had to buy coffee for a meeting (this part was true), and that I had bought 4 cups of fancy coffee for \$7. The next part of the string was: Then, something happened and I had bought 8 cups. Turn and talk to your partner to tell them a story about what might have happened.

Students said things like, “Then you got back to school and 4 new teachers showed up to the meeting so you had to go back.” We are on the 5th floor of my building, so the students were feeling pretty bad for me.

They groaned when I put a 10 on the chart under the cups of coffee, possibly a sign that they were invested in the context with me. “What story can you tell now?” I asked them. KellyAnn said, “The principal called you and said we had visitors and that we needed two more cups.” Andrew said, “Now we have to figure out what two cups cost, I’m not sure about that.”

A few takeaways from this lesson:

• Students who struggled with the math were able to get started on the problem. I’d much rather have students dealing with, “How much for two cups?” then, “What should I do?”
• This string had a tone of, “What’s going to happen next?” When they found out that I needed 10 cups of coffee now, we started thinking about all the stairs I had to climb (and this became fodder for a new string).
• The decision to draw pictures of the situation felt authentic and not a tool to use with students who needed “remedial” math education — because we were all imagining it together. There was no point where I felt like I needed to offer a picture, we were just right there in it together.
• The pictures were a key to transfer. Students who struggled to reason on their own during independent and partner work could be prompted to draw a picture and suddenly they were able to reason through problems.

One student’s notebook who struggled a lot with proportional reasoning, but then was able to draw pictures to work through the story.

When we think about context in number strings, this question, “What happened next?” gives students an entry point to get started, and often, this translates so quickly to a picture. My students now know they have been invited into a world where maybe we can have a garage with 100 cars, or I can spend an entire day just on coffee runs, and we can wonder about how many stairs we have climbed. After many such invitations, they’re willing to go there with me to think about the math. The numbers are beautiful on their own, but that’s because they tell us a story.

## Feet, inches and yards: Conversions on a ratio table

This post is from our friend and colleague Kathy Minas, a 4th grade teacher at PS 158, and an avid strings enthusiast.

Recently, our fourth grade team met with Math in the City co-director and staff developer Kara Imm, who has been working with our school for several years. We wanted to explore ways to introduce Common Core 4th grade standards of measurement and conversion by using strings. What follows is the string we designed together and my notes about how to lead it with kids.

Why this string? Why a ratio table?

The purpose of this string is to introduce students to conversions, using the most common units of measurement, inches and feet. We wanted to ensure that students had a familiar model that visually captured the (twelve-to-one) relationship between these units, which led us to using the ratio table. Our students have worked with the ratio table before in both our multiplication and division units of study. The model allowed them to represent and maintain the relationship between known units in order to multiply, divide, break apart, or even add groups with ease.

Performing measurement conversions on a ratio table also supported students to monitor the reasonableness of their answers. In addition, it encouraged them to keep the relationship between units in mind.  Instead of memorizing whether they needed to multiply or divide feet to get inches, they simply trusted the existing relationships on the table that we build together.  In fact, student were much more flexible about how to convert and did not rely on a memorized rule or catchy mnemonic to solve these problems.

We build ratio table together with the class, instead of revealing the entire completed ratio table all at once. We add values to the ratio table — one at a time, increasingly more complex — and ask students to determine the corresponding number of inches or feet. Some values are added to the ratio table as students explain their process.

The string:

Begin by naming a true statement to ground the conversation:

Mathematicians, we know that there are 12 inches in one foot.

Draw a ratio table and label the columns, number of inches and number of feet. If this piece of social knowledge is not known by your students, having a 12-inch ruler to see and touch is also useful at this moment.

Since we know that there are 12 inches in one foot, how many inches are there in 3 feet? How do you know?

Student responses may include:

• I know that there are 12 inches in one foot, so there are 36 inches in 3 feet because you multiply the number of feet by three, which means you have to multiply the number of inches by three.

Note: In order to move away from additive reasoning on the ratio table  also known as repeated addition or “chunking” I purposely do not record the unit rate of  12 inches in 1 foot on the table until the students bring it up as part of their reasoning. I’m nudging students from additive to multiplicative reasoning on the ratio table. You may wish to ask students to visualize three feet, “What does this look like? What does this make you think of?”

A fourth grader (and sports enthusiast) in my class introduced the unit of a yard immediately, so I added it as a third column to the ratio table and made a post it note about his unique contribution. I left the yards column blank for most of the string, but then we returned to it later to reason about this third unit.

Mathematicians, how many feet are there in 60 inches? How do you know?

Student responses might sound like:

• I know that there are 12 inches in 1 foot, 60 inches is 5 times greater than 12 inches, so 60 inches are equivalent to 5 feet.
• I know that 36 inches is equivalent to 3 feet, and 60 inches is 24 more inches than 36. 24 inches is equivalent to 2 feet, so I need to add 2 feet to 3 feet and that’s 5 feet.

So, how many feet are there in 72 inches? How do you know?

Students will likely reason:

• I know that 72 inches is 12 more inches than 60 inches, which means I need to add one more foot to 5 feet, which is 6 feet.
• I know that 72 inches is equal to 36 inches times 2, if I double the 36 inches, I have to double the 3 feet.

Follow up question, “How many yards are there in 72 inches or in 6 feet? How do you know?”

Okay, how many inches are there in 9 feet? Can you picture this?  What are you thinking?

I heard students say:

• I know that 9 feet is three times 3 feet, so there are 108 inches in 9 feet because I have to triple 36 inches.
• I know that 6 feet is 3 feet away from 9 feet, so I need to add 36 more inches to 72 inches, which is 108.

Follow up question, “How many yards are there in 72 inches or in 6 feet? How do you know?”

Once we were done with the inches and feet component of the string, we tackled the yards. We returned to the relationship between feet and yards. I asked students to consider,

If we know that there are 3 feet in 1 yard and we only have 1 foot, what part of a whole yard do we have?

If we have 2 feet, do we have a whole yard yet? What part of a whole yard do we have?

Later I said,

Using all you know about the relationship between feet and yards and all you know about fractions, if we have 5 feet, how many yards do we have?

My students tackled this problem with ease:

• I know that there is 1/3 yard in 1 foot, so there are 5/3 yards in 5 feet.
• So, there is 1 yard in 3 feet and 2/3 yard in 2 feet, which means that in 5 feet there is 1 2/3 yards. I just put them together to make 5.

As big ideas or important strategies come up, my colleagues and I have begun annotating the strings poster so that kids can both see and hear these ideas.  I listen carefully for students to make these contributions we write and display them so that the ideas are shared and accessible to all kids, even if they are still emergent.  Sometimes, when students are ready, I nudge kids towards a generalization, which helps us move beyond the specific string and into other related quantities and relationships. Examples of this practice of annotation are below:

## A “juicy” dilemma

This number string is inspired by an investigation (Comparing and Scaling, Investigation 1.2) from the Connected Math Project curriculum for 7th graders.  It can be used as a way to extend and solidify ideas that will develop as kids investigate, or it could be used to launch the task. In the investigation, students are asked to compare the following orange juice recipes, according to their “oranginess.” Continue reading “A “juicy” dilemma”

## A Number String for Angle Measures — Before We Kick the Bucket

This post was co-created by Jesse Burkett, Ranona Bowers, and Adrian Sperduto — teachers in the Hazelwood School District and their colleague Cheryl Montgomery of the Parkway School District (both in St. Louis, MO). They were recently selected as participants in the Mathematicians in Residence program — a three year, three district project involving almost 100 teachers and eventually a summer math academy for approximately 200 students. Jesse, Ranona, Cheryl and Adrian just completed a two-week professional development academy, focused on numeracy routines in grades K – 5.  Jesse Burkett, a 4th grade teacher at Brown Elementary School, led the writing for his colleagues. Continue reading “A Number String for Angle Measures — Before We Kick the Bucket”

## What’s in a name?

This post is by reader and strings enthusiast Laura Bofferding, Assistant Professor, Purdue University

I was first introduced to number strings by Jennifer DiBrienza (one of the teachers highlighted in Young Mathematicians at Work: Constructing Number Sense, Addition and Subtraction by Fosnot and Dolk) when we both worked as teaching assistants for an elementary mathematics methods course. Captivated by their complexity and ability to hook students of all ages, I began to use number strings myself with teacher candidates. Now, as an assistant professor at Purdue University, I routinely explore them with my undergraduate mathematics methods students and require them to try a number string in their practicum classrooms. As I have looked for resources and talked with colleagues about this practice over the past few years, I’ve noticed an increased focus on mathematics instructional routines that people refer to as math talks, number talks, number strings, math strings, cluster problems, and problem strings. Some of these things are not like the others…

## (Re)Thinking Context

A recent conversation between two New York City math educators, Carol Mosesson-Teig and Kara Imm

Kara,

I have been thinking a lot about the strings that you modeled last week for the city-wide math content seminar and thought I’d share some of my thoughts.

Those of you who are fans of the middle school curriculum Connected Math Project (CMP) will especially appreciate this string. I was preparing for a visit to MS 22, a middle school in the South Bronx —  my collaborator Erica Berger, a thoughtful and dedicated teacher, asked me to design a string to introduce linear relationships and to prepare students for a potentially messy investigation in CMP.

Some would say number strings are “curriculum neutral” or “curriculum impartial.”  That is, they are not tied to or loyal to any one curriculum — the routine of number strings can be a helpful and grounding experience for all kids. The challenge, for all of us, is locating and/or designing strings that will support the lesson or investigation that follows.

## New percentage number strings

We recently saw the problem above on a New York CCSS-M 6th grade benchmark test. A few teachers I was working with argued about what the question required.  Some teachers thought this called for converting a percent into a decimal and multiplying.  Another teacher reasoned persuasively, “I don’t want kids to do any calculations here.  None. Without a calculator they would be tedious and time consuming. And I also don’t want them setting up some equation like .12x = 3.84.  Unless they are really fluent in algebra and division of decimals, and there is a lot that can go wrong there.”

## The “King of Strings” teaches us that strings are maatwerk

I recently reached out to Willem Uittenbogaard. Willem was one of the original collaborators between Math in the City (founded by Cathy Fosnot) and the Freudenthal Institute in the Netherlands.  He spent two years in New York City — working with teachers to develop the idea that realistic contexts in mathematics problems help children to build on their understanding of the world. He also taught many New York City teachers how to lead number strings. I was one of those teachers. I was lucky enough to be spend two weeks of the summer of 1999 with Willem, as he challenged me to solve mathematics mentally through number strings. Willem went on to co-author all of the Minilessons Resource books for the Contexts for Learning Mathematics series.

## Challenge problems, helper problems

When designing strings, there are typically one or more helper problems before a challenge problem.  For example, you might start an addition string on compensation (a big idea) with:

50 + 20 = (helper problem)

48 + 22 = (challenge problem)

## 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”