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Acknowledgments

You can’t write a book like this in a vacuum. We have met and worked with countless educators over the years and have discussed the ideas in this book with them. You know who you are and we hope you hear your voice in these pages. Thank you for your contribution to these thoughts and ideas. We credit you and appreciate you deeply.

We would also like to thank Erin Null, who started with one vision of this book, received a draft of another interpretation, and worked with us to land on the third—and we think best—vision. Your patience, support, and diligence pushed us continually forward. We also extend a profound debt of gratitude to Paula Stacey, who asked us the questions we needed to answer in a way that made the manuscript better. As we finish the series of books, we recognize Amy Marks, Tori Mirsadjadi, Jessica Vidal, and the entire Corwin team for helping to keep our work consistent and presenting it so carefully.

Thank you also to Debbie Thomas, Barbara Dougherty, Kimberly Rimbey, Julie McNamara, Linda Levi, Terrie Galanti, Johnny Lott, and Jeff Shih, whose thoughtful comments challenged us to know better and do better. We are indebted to you for your positive feedback, but even more so for your constructive criticism. Any flaws that remain are ours alone.

I would first like to thank Linda and Sara for trusting me enough to invite me to join this project. I appreciate your confidence, but more than anything, I appreciate your friendship. I also want to thank Dr. Megan Murray of the University of Hull, who introduced me to the idea that addition and subtraction problems weren’t all the same. To the staff of GBW, Julie, Pat, Kim and her students, and Michelle and her friend the second-grade teacher: Thank you for sharing your time and brilliant students who have helped us collect and interpret their thinking from many different angles. I want to thank my daughter Cassandra and her friends for fielding random questions about math, chemistry, and physics at all hours of the day and night. The book is better because of your guidance. Finally, thank you to my husband, Greg, who watched me take the big leap of writing three books and never questioned my sanity.

—Kim

Thanks to Linda for starting us down this road with rich conversations. Thanks to Kim for coming on this journey with us. I appreciate your knowledge, your experience, your care, and your friendship. Thanks to Margie Mason, who first brought me into the world of mathematics education, and to all my friends and colleagues in this community, including those at ETA hand2mind and ORIGO Education, who have encouraged and supported me along the way. Thank you to the teachers who came before me, particularly my mother and grandmother, for showing me that learning is important and good teaching is invaluable. And thanks to Bill, for loving and supporting me always.

—Sara

Writing a book is always a challenge! While it seems that writing on a topic you feel passionate about should be easier, it is actually a bigger challenge because you want to get it right. I thank Kim and Sara for their vision and our many long conversations. I learned so much from both of you. I want to thank the elementary teachers and coaches with whom I work who challenge my thinking and force me to make ideas clearer. I thank my colleagues Ruth Harbin Miles, Annemarie Newhouse, and Jerry Moreno, whose friendship I value and who make this career a joy.

—Linda

Publisher’s Acknowledgments

Corwin gratefully acknowledges the contributions of the following reviewers:

Kevin Dykema

Middle School Math Teacher

Mattawan Middle School

Mattawan, MI

Julie McNamara

Assistant Professor of Mathematics Education

California State University, East Bay

Hayward, CA

Kimberly Rimbey

Executive Director of Curriculum, Instruction, and Assessment

Buckeye Elementary School District

Buckeye, AZ