I’ve known Rachel Carr for many, many years. She shepherded me through that exhilarating and exhausting first year of teaching and has remained a mentor and role model ever since. This summer Rachel attended a summer institute at Math in the City, and we got to work together again — making sense of the landscape of learning for rational number. I am continually struck by how a teacher with so much experience and insight still considers herself a learner.

This week, I offered to lead a number string in Rachel’s 5th grade class. Her students were delightful, eager and enthusiastic. I learned that most of them were “quick” numerically and relied primarily on algorithms to get the job done. So my goal was to disrupt some of this **—** to decouple their belief that being fast at math meant being good at it; to slow down the conversation so that more kids could be apart of it; and to insist that the solving of problems was simply the beginning of our work together (see Boaler,

Click to access Mathematics-of-Hope-Paper.pdf

)

The string we did is below, a modification of one from *Minilessons for Operations with Fractions, Decimals and Percents *(Imm, Fosnot & Uttenbogaard, 2008).

1/2 + 1/5 =

2/5 + 1/2 =

2/5 + 3/10 =

2 1/2 + 1 1/5 + 3/20 =

A few days later Rachel presented me with a beautiful stack of letters. “That’s what we always do when a visitor comes,” she explained, matter-of-factly. I read each letter carefully, looking for evidence of how Rachel’s students were thinking about, and relating to, mathematics. Here are a few poignant examples….

This note from **Henry** typifies several of the students **—** kids who love math, are really good at it, and find it easy. So this letter, in my mind, serves as *an invitation to make math challenging* and make sure that *effortless* does not equate with *uninteresting*:

**Tu** is another such student who offers an invitation to teachers **—** to keep kids “feeling good” and “always enjoying” mathematics. But to also ensure that “*multiple choices*” are a very tiny part of their overall mathematical experience.

**Melody** strikes me as a student who is asking big questions and grappling with big ideas. Little of this kind of thinking gets taken up by standardized test measures, but is so very important to doing mathematics. A question like hers **—** why do people share? **—** fits beautifully at the center of a class’s study of rational number: *Why do people share? What does it mean to share fairly? What does this look like mathematically?* Melody also reminds me of how important it is to invite kids’ questions and then make use of them as a teacher. Imagine a math class where our questions drive our learning.

Then, there were a series of letters about what it feels like to do mathematics. **Cassie’s** assessments that “we have to use a lot of thinking from our brain” sounds about right to me. So, in this way, she provides a direct challenge **—** keep me thinking, but please keep it interesting, fun and not “kind of boring.”

Cassie’s belief that using different strategies to solve a problem is a “terrific way” is especially important. This ability to think flexibly and see relationships between strategies should be an explicit goal for math instruction. While** Eli** doesn’t exactly call math boring, he does echo Cassie’s sentiments about math:

He represents several challenges that we face **—** How do we invite a not-a-big-fan-of-thinking to think? How do we help him to love creating new ideas and taking up others’ ideas? How do we create a classroom community of kids who love to think?

Lastly, there is my friend **Savannah**, who reminds us of the diversity of learners in any classroom:

Savannah’s letter underscores the idea that slowing down and letting kids think is essential. Maybe the most important thing we can do, aside from building a culture of respect. When I posed the first problem, there were so many kids who were ready to share. I could have called on any of them. But I waited. And then waited some more. And I then explicitly stated that we were waiting so that everyone had a chance to think. That Savannah could “* think for a long time and get the answer*” feels especially important to me now.

Together, these letters and my morning with class 505 reinforce our beliefs about why number strings are so important:

- We are trying to build a classroom community where kids’ ideas (not ours) are at the center. Our discourse reflects this intention (“Show us….Convince us….We need to understand your idea….What do we think?”)
- There is no evidence that being quick at math means being good at math
**—**Let’s slow it down so that we can all be apart of it. - Humans think differently and that’s kind of wonderful
**—**sameness is not all that interesting. - Our best conversations begin, not end, once we agree on an answer. Now we can study and make sense of our strategies and explore some big ideas (
*Will this always work? Why is this happening? We all agree it is 270, so let’s hear some strategies.*) - Number strings create a ritual of
*making thinking visible*(see also http://www.amazon.com/Making-Thinking-Visible-Understanding-Independence/dp/047091551X). Once ideas are seen and heard, they can be shared, understood, and taken up by others.

Thanks for sharing, Kara.

My pleasure. I thought we could all learn something from these letters.