The Stenhouse Blog Tour: Math Exchanges

"Learning occurs when children are actively engaged in their environment and work to create a system of meaning and deep understanding."  Kassia O. Wedekind 

Sometimes the right book comes along at the right time.  That’s what happened when I read Kassia Omohundro Wedekind’s new book, Math Exchanges.  Right now in our school, we are in the process of transitioning from using a math program to really focusing on Ohio's New Model Curriculum (developed from the common core), data from assessments (such as the Developmental Math Assessment), and student performance to plan instruction.  In the transition I'm spending much time reading and rethinking math instruction in our classroom.

In her book, Kassia focuses on the problem solving conversations teachers have with small groups during math workshop.  What are children saying?  How are they figuring things out?  How is the teacher supporting math learning? While Kassia's book, Math Exchanges, focuses on the significance of small group conversations in developing mathematicians it also helped me to rethink the way we talk about mathematics in our classroom community.  Kassia reminds us, "It is the role of the students to raise their own questions, generate their own hypotheses and model possibilities, test them out for viability, and defend and discuss them in the community. (p. 9)"

When I was asked to be a part of the Stenhouse Blog Tour for Kassia’s book I was honored.  I’ve been having conversations with Kassia (@kassiaowedekind) on Twitter, and following her blog, Math Exchanges, for quite some time.  I've already learned so much from Kassia and enjoyed so many conversations, but to have an opportunity to talk with her about her new book was a bonus.  (Speaking of a bonus...remember to leave a comment to possibly win a copy of Kassia's new book.)     

Our Interview

 Me:  In your book you talk about the difference between “mathematics” (ideas and procedures created in the past by others) and “mathematizing” (constructing one’s own ideas and understandings of math).  What do students need in math workshop for this type of learning to occur? 

Kassia:  Lucy Calkins said, “teach the writer, not the writing.” When I began to think in terms of “teach the mathematician, not the math,” my workshop changed. Put this quote on your wall, in your planning notebook, someplace you can look at it when you’re feeling, “I just need to show them how to…” or “I’ll just teach them this easier way to....” Those thoughts creep into my mind from time to time. When I find myself getting into this kind of thinking, I tell myself: Honor the child’s thinking. Meet her where she is. Build from there. I know I can’t rush a child into building understanding overnight. That’s not how true learning works.

When your focus is on “teaching the mathematician,” your workshop will change. Here are some changes that occurred in my workshop when I made this shift:

1. Focus on fewer problems. Giving just a couple of problems and spending significant time asking “How did you solve the problem?” “How did you figure it out?” “Did you solve it like Clara or did you use a different strategy?” “Why do you think Abdel decided to start counting at 52 instead of 6?” Teach deeply with fewer problems.
2.  Simplify. I used to spend a lot of time planning complicated centers for my students to do while I met with small groups. I hardly had anytime left for looking at student work and my notes in order to plan strategically for the small groups with which I’d work. I learned to simplify my centers or tasks. Giving students collections to count, starting a counting journal (where students count by different numbers and find patterns), writing their own story problems, playing simple math games. Focus on planning simple, truly independent centers or tasks that provide meaningful practice and exploration. This will free your planning time for focusing on thinking about where students are in their thinking and how to move them forward.
3. Choice. Elements of choice are important for students to feel ownership and self-efficacy in the workshop. Choice of tasks, choice of numbers for a problem (“Choose the numbers that you feel ready to take on. Not too easy, not too hard.”), choice of partners. You don’t need to offer all of these choices all the time, but it is important to always have some element of choice in the workshop.
4. Joy! How do you feel during your math workshop? How do your students feel? When things are going well, I feel a kind of joyful buzz in the air. Kids are talking to each other. They are trying new ideas. They feel important. Children are excited to meet together in small group math exchanges. If I’m feeling like this most of the time, then I know I’m on the right path. If not, I look for ways to change what is going on.

Me:  Tools are obviously flexibly used in your math workshop to solve problems and think about math.  How do you create environment in which students know they can flexibly utilize multiple tools/strategies in math exchanges?  

Kassia:  First you want to create an organized space where kids can easily access tools. Children should know what tools are available and the ways in which they can be used. Give time in the beginning of the year for free exploration of the tools. Have kids make labels for the various tools so they feel ownership over the tools and learn how to care for and organize them.

Help students identify what tools and strategies that help them most effectively and efficiently solve problems. “Jaime, why did you choose to use the sticks of ten Unifix cubes to solve that problem?” Or “I noticed that you used to use … and now you use … Why did you decide to change tools/strategies?” You want to highlight strategic tool use that is appropriate and efficient. You want kids to use the tools when they need them, and then feel comfortable dropping the tools for more efficient strategies at some point. Help them understand and talk about when and why they changed tools and strategies. You’re helping children to self-monitor for changes in their thinking and strategies.

Me:  You stated, “the role of teachers is to provide problem-solving experiences that invite questions, wonderings, and space for grappling with new thinking.”  One of the pieces I appreciated was reading many examples of language you use with young learners in math exchanges.  Can you share a few types of feedback you used to give and what you say now that encourages students to questions, wonder, and grapple with math learning?

Kassia:  What an important question! I learned a lot about academic language from Peter Johnston’s book, Choice Words. I re-read this book a couple of times a year (it helps that it’s a short book!) to remind myself of language I want to use. And then I write it down. It takes a while for new language to stick in my mind, so I may need to write down a couple of questions or phrases that I want to use until it becomes natural for me. So, here are some examples I’ve been thinking about lately. There are many more in the book!

Me:  I realized when I was reading your book I needed to learn to look at what students do and say in math much more closely to determine what they know and what is next.  What resources/practices have helped you to know and notice important shifts in thinking?

Kassia:  Yes, I’ll tell you about two books that changed my thinking and practice as a math teacher.

1. Young Mathematicians at Work: Constructing Number Sense, Addition and Subtraction by Catherine Fosnot and Maarten Dolk.  This book examines the “landscape of learning,” the mathematical strategies, big ideas and models that young children construct as they learn mathematics. It taught me what to look for as I analyze children’s work and words, and how to help children make steps forward in constructing meaning. Young Mathematicians at Work is one of those books that perfectly balances research with practice, and offers a lot of classroom vignettes that you can implement in your classroom.
2. Children’s Mathematics: Cognitively Guided Instruction by Thomas Carpenter, Elizabeth Fennema, Megan Franke, Linda Levi and Susan Empson.  This is the foundational work of the idea that children construct understanding through experience with problem-solving. The authors explain the problem types you can work with and the development of strategies towards efficiency and understanding.  

Me:  You talk a lot about “the stories of math” in life.  How do you help students to find the math stories in their lives and share them in the classroom community?

Kassia:  I think we all live rich mathematical lives, whether we are aware of them of not. I certainly didn’t think of my life in these terms before I started teaching! It happened slowly. First I’d be in a grocery store and see the perfect array of plums that I needed to take a picture of to share with my third graders who were just beginning to explore strategies for multiplication. Then I’d be at the metro stop and see these hexagon tiles on the floor and start to wonder, “why hexagons?” So, I think sharing this passion with my students caused them to start thinking in these terms as well. I invite students to bring in stories and items from their mathematical lives. I give them time to share them and a space to display pictures and items where other kids can interact with them, ask questions and offer comments on the items and pictures shared. (There’s a post about a Multiplication Museum my kids made on my blog that shows a picture of this kind of thing).

I’ve just returned to the classroom this year as a kindergarten teacher. It’s made me start thinking about how to help our youngest students discover and share their mathematical lives as well. The best parallel I have is to how I teach writing. I learned from Katie Wood Ray and Matt Glover (in Already Ready and their many other books) to say to my students “You could make a book about…” or “You could make a book like that” in reference to kinds of picture books we study or experiences my students have. I’ve started looking for opportunities to say things like that about the math my students are discovering in their play. After they became interested in make long rows of animal toys on parade I said, “Wow! How many animals are in the parade? Would you like to take a picture of this so you could make a sign telling people about how many and what kinds of animals are in the parade?”

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I want to thank Kassia for taking the time to answer these questions.  Reading her book has helped me to shift my thinking as I move from teaching "mathematics" to supporting "mathematizing".   Kassia explains, "The difference between "mathematics" ideas and procedures created in the past by others, and "mathematizing," constructing one's own ideas and understanding of math (p. 40)."  I'm looking forward to the journey.

Remember to leave a comment or question for your chance to win a free copy of Kassia's book from Stenhouse.  Also, if you haven't had a chance to stop by the other blogs on the tour you will want to take time to visit each blog.  I have enjoyed each day's conversation.

Monday, October 3rd:  Catching Readers Before They Fall hosted by Stenhouse authors Pat Johnson and Katie Keier.  This post discusses some ways to encourage students to talk together about math among other conversation points. 

Tuesday, October 4th:  Our Camp Read-A-Lot hosted by first grade teacher Laura Komos.  This post discusses ways to assess the small group conversations and student thinking.  The rubric shared one way to think about the work students do in small group math exchanges.  

Wednesday, October 5th: Here  

Thursday, October 6th:  Elementary My Dear, Or Far From It, hosted by first grade teacher Jenny Orr