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This book is for preservice and in-service teachers’ teaching elementary and middle- school mathematics. The focus of our book is to develop mathematical ideas by sharing ways students respond to modeling mathematical ideas and allowing teachers to discuss student strategies through case scenarios from lesson studies taken from the elementary and middle-grade classrooms.

With the recent release of the Common Core State Standards (CCSS) (NGACBP & CCSSI, 2010), researchers and mathematics educators are looking at the standards to help teachers map out the learning progression that will guide the sequence of mathematical concepts crucial in building mathematical understanding. Learning progression extends “previous learning while avoiding repetition and large gaps” (Hunt Institute, 2012, p. 8) and guides “a path through a conceptual corridor in which there are predictable obstacles and landmarks” (Confrey, 2012, p. 4).

Understanding the learning progression is important for teachers for it serves as the guiding post for analyzing student learning and tailoring their teaching sequence. This essential instructional practice leads to important professional development focused on vertical articulation around important math concepts and worthwhile tasks with cognitive demand at multiple levels. Teachers need to be engaged in understanding the learning progression across standards. Our proposed book will provide the support with the focus on the learning progression for grades 3-8 by presenting rich problemsolving tasks that model mathematics across the vertical progression. We hope this book will provide a guide for navigating through the mathematics learning progression through meaningful conceptual tasks and formative and summative assessments.

All of the Lesson Study Vignettes are from our past lesson studies with classroom teachers in the past eight years. These lesson studies were supported in part by the Virginia Department of Education, State Council for Higher Education in Virginia, and the National Science Foundation through a variety of Mathematics Science Partnership grants. We have conducted multiple professional development courses focused on these important math practices and strands and essential themes including computational fluency, building number sense, rational numbers and proportional reasoning, STEM in high schools and have used this curriculum.

We want to thank all the teachers who participated in our professional development institutes and lesson study over the eight years. We have learned a great deal about students’ mathematical thinking, and how teachers can skillfully incorporate rich tasks in their classrooms to develop mathematical hearts and minds. We also would like to express gratitude to all our school partnership mathematics professional development leaders (Patti Freeman, Spencer Jamieson, Kim Leong, Katherine Meints, Theresa Wills, Linda Gillen, Mimi Corcoran, and Courtney Baker) whom we have had the fortunate opportunities to work with. Finally, we would not be able to do this kind of work without the support of our graduate students and our doctoral students (Lesley King, Sara Birkhead, Kathleen Matson, Monique Williams, Terrie Galanti, Dasha Gerasimova, and Kim Fair) who have dedicated countless hours on observing and supporting lesson studies.

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