This book's 28 chapters are adapted and updated from articles published in NCTM's "Journal for Research in Mathematics Education" between 2000 and 2010. The authors have rewritten and revised their work to make it clear, understandable, and--most of all--useful for mathematics teachers today. To help teachers even more, these articles have been chosen for their relevance to the eight Standards for Mathematical Practice in the Common Core State Standards. The book's three sections present chapters that relate to reasoning and proving; communicating, sense making, and using tools strategically; and modeling and problem solving. Teachers who are seeking to help students develop these mathematical practices will find insights and support in this survey of research. The chapters cover a wide range of topics, approaches, and settings, including--(1) a case study of a third-grade teacher who sought to create a math-talk learning community in an urban classroom; (2) an examination of middle school students' problem-solving behaviors from a reading comprehension perspective; (3) a meta-analysis of the effects of calculator use in K--12 classrooms; an exploration of the strategies that high school geometry students employ when using a dynamic software program; and (4) an analysis of a professional development initiative designed to help teachers select and implement cognitively challenging tasks. Mathematics teachers at all levels will find examples of research that is relevant to the challenges they face. This book enables researchers and teachers to meet on common ground to improve mathematics education for all students. Chapters include: (1) Creating a Need for Proof (Gabriel J. Stylianides and Andreas J. Stylianides); (2) Checking Students' Proofs for Correctness (Annie Selden and John Selden); (3) Proof Validation through the Lens of a Mathematician: What Practitioners Can Learn (Keith Weber); (4) Teachers and Proof in School Mathematics (Eric J. Knuth); (5) Proof Exercises and How They Challenge the Work of Students and Teachers in High School Geometry (Patricio G. Herbst); (6) Proof in School Mathematics as Early as the Elementary Grades (Andreas J. Stylianides); (7) Recognizing and Fostering Mathematical Reasoning in Elementary School Classrooms (David A. Reid); (8) How Generalizing Can Foster Proving and Vice Versa: A Case with Linear Functions (Amy B. Ellis); (9) The Linear Imperative: Why Students Overuse Linearity (Wim Van Dooren, Dirk De Bock, and Lieven Verschaffel); (10) Mapping Mathematics Classroom Discourse and Its Implications for Models of Teaching (Mary P. Truxaw and Thomas C. DeFranco); (11) Describing Levels and Components of a Math-Talk Learning Community (Kimberly Hufferd-Ackles, Karen C. Fuson, and Miriam Gamoran Sherin); (12) Lifting the Labels: A Cautionary Story about Stories We Tell about Mathematics Students (Miriam Ben-Yehuda, Ilana Lavy, Liora Linchevski, and Anna Sfard); (13) Making Sense of Graphs (Susan N. Friel, Frances R. Curcio, and George W. Bright); (14) What We Can Learn from How Professionals Read Graphs (Wolff-Michael Roth); (15) Measurement Estimation as a Vehicle for Making Sense of Measurement (Elana Joram and Anthony J. Gabriele); (16) Using School-Based Mathematical Concepts in the Workplace: An Example from Nursing (Richard Noss, Celia Hoyles, and Stefano Pozzi); (17) The Effects of Calculators on Students' Achievement and Attitude Levels in K-12 Mathematics Classes (Aimee J. Ellington); (18) Low-Achieving Students Using Graphing Software to Solve Problems (Michal Yerushalmy); (19) Using a Dynamic Software Program for High School Geometry: A Look at Strategies Students Employ (Karen F. Hollebrands); (20) The Model Method: Singapore Children's Visual Representation Tool for Solving Algebraic Word Problems (Swee Fong Ng); (21) Conceptual Model-Based Problem Solving: Emphasizing Prealgebraic Conceptualization of Mathematical Relations (Yan Ping Xin); (22) Problem-Solving Behaviors of Middle School Students: Implications for the Classroom (Stephen J. Pape); (23) The Practice of Representing and Modeling with Diagrams, Symbols, and Words (Curtis Pyke); (24) The Role of Students' Criteria for Good Algebraic Representations in the Construction of Modeling Knowledge (Andrew Izsák); (25) Knowledge Connectedness and the Quality of Student and Teacher Mathematical Knowledge (Michael J. Lawson and Mohan Chinnappan); (26) The Role of Covariational Reasoning in Understanding and Using the Function Concept (Marilyn P. Carlson and Kevin C. Moore); (27) Learning Mathematics in a Classroom Community of Inquiry: Examples from a Secondary Mathematics Classroom (Merrilyn Goos); and (28) Selecting and Implementing Cognitively Challenging Instructional Tasks: A Pathway for Engaging Students in the Standards for Mathematical Practice (Melissa D. Boston).