We Reason & We Prove for ALL Mathematics

Dr. Fran Arbaugh is an associate professor of mathematics education at Penn State University, having begun her career as a university mathematics teacher educator at the University of Missouri. She is a former high school mathematics teacher, received a M.Ed. in Secondary Mathematics Education from Virginia Commonwealth University and a PhD in Curriculum & Instruction (Mathematics Education) from Indiana University - Bloomington. Fran's scholarship is in the area of professional learning opportunities for mathematics teachers and mathematics teacher educators, and her work is widely published for both research and practitioner audiences. She is a Past-President of the Association of Mathematics Teacher Educators (ATME) and served as a Co-Editor of the Journal of Teacher Education. Margaret (Peg) Smith is a Professor Emerita at University of Pittsburgh. Over the past three decades she has been developing research-based materials for use in the professional development of mathematics teachers. She has coauthored several books including Five Practices for Orchestrating Productive Discussions (with Mary Kay Stein), the middle and high school versions of the Taking Action series (with Melissa Boston, Fredrick Dillon, Stephen Miller, and Lynn Raith), and The 5 Practices in Practice: Successfully Orchestrating Mathematics Discussion in Your Classroom series (with Victoria Bill, Miriam Gameron Sherin, and Michael Steele). In 2006 she received the Chancellor's Distinguished Teaching Award given annually to honor outstanding faculty at the University of Pittsburgh. In 2009 she received the award for Excellence in Teaching in Mathematics Teacher Education from AMTE. In April 2019 she received the Lifetime Achievement Award from NCTM. Justin Boyle is an assistant professor at the University of Alabama. He is interested in learning how best to develop secondary mathematics teachers, so that they are prepared to engage their future students in becoming intellectually curious about mathematics. In particular, he uses reasoning-and-proving as a way to investigate and discuss the truth of mathematical statements, concepts and objects. Gabriel J. Stylianides is Professor of Mathematics Education at the University of Oxford (UK) and Fellow of Oxford's Worcester College. A Fulbright scholar, he received MSc degrees in mathematics and mathematics education, and then his PhD in mathematics education, at the University of Michigan. He has conducted extensive research in the area of reasoning-and-proving at all levels of education, including teacher education and professional development. He was an Editor of Research in Mathematics Education and is currently an Editorial Board member of the Elementary School Journal and the International Journal of Educational Research. He received an American Educational Research Association Publication Award for his 2009 article "Reasoning-and-proving in Mathematics Textbooks." Michael D. Steele is a Professor and Chairperson of the Department of Educational Studies in Teachers College at Ball State University. He is a Past President of the Association of Mathematics Teacher Educators, current director-at-large of the National Council of Teachers of Mathematics, and editor of the journal Mathematics Teacher Educator. A former middle and high school mathematics and science teacher, Dr. Steele has worked with preservice secondary mathematics teachers, practicing teachers, administrators, and doctoral students across the country. He has published several books and research articles focused on supporting mathematics teachers in enacting research-based effective mathematics teaching practices. Dr. Steele's work focuses on supporting secondary math teachers in developing mathematical knowledge for teaching, integrating content and pedagogy, through teacher preparation and professional development. He is the co-author of NCTM's Taking Action: Implementing Effective Mathematics Teaching Practice in Grades 6-8. He is a co-author of several research-based professional development volumes, including The 5 Practices in Practice: Successfully Orchestrating Mathematics Discussions in Your High School Classroom, Mathematics Discourse in Secondary Classrooms, and We Reason and Prove for All Mathematics. He directed the NSF-funded Milwaukee Mathematics Teacher Partnership, an initiative focused on microcredential-based teacher professional development and leadership. His research focuses on teacher learning through case-based professional development, and he has been an investigator on several National Science Foundation-funded projects focused on teacher learning and development. He also studies the influence of curriculum and policy in high school mathematics, with a focus on Algebra I policy and practice, and is the author of A Quiet Revolution: One District's Story of Radical Curricular Change in Mathematics, a resource focused on reforming high school mathematics teaching and learning. He works regularly with districts across the country to design and deploy teacher professional development to strengthen effective secondary teaching practice. Dr. Steele was awarded the inaugural Best Reviewer award for Mathematics Teacher Educator and was author of the 2016 Best Article in Journal of Research in Leadership Education. He is an active member of and regular presenter for the National Council of Teachers of Mathematics, the National Council of Supervisors of Mathematics, and the Association of Mathematics Teacher Educators. He reviews regularly for major mathematics education and teacher education journals.

Preface Acknowledgements About the Authors Chapter 1 Setting the Stage Are Reasoning and Proving Really What You Think? Supporting Background and Contents of This Book What is Reasoning and Proving in Middle and High School Mathematics? Realizing the Vision of Reasoning-and-Proving in Middle and High School Mathematics Discussion Questions Chapter 2 Convincing Students Why Proof Matters Why Do We Need to Learn How To Prove? The Three Task Sequence Engaging in the Three Task Sequence, Part 1: The Squares Problem Engaging in the Three Task Sequence, Part 2: Circle and Spots Problem Engaging in the Three Task Sequence, Part 3: The Monstrous Counterexample Analyzing Teaching Episodes of the Three Task Sequence: The Cases of Charlie Sanders and Gina Burrows Connecting to Your Classroom Discussion Questions Chapter 3 Exploring the Nature of Reasoning-and-Proving When is an Argument a Proof? The Reasoning-and-Proving Analytic Framework Developing Arguments Developing a Proof Reflecting on What You've Learned about Reasoning and Proving Revisiting the Squares Problem from Chapter 2 Connecting to Your Classroom Discussion Questions Chapter 4 Helping Students Develop the Capacity to Reason-and-Prove How Do You Help Students Reason and Prove? A Framework for Examining Mathematics Classrooms Determining How Student Learning is Supported: The Case of Vicky Mansfield Determining How Student Learning is Supported: The Case of Nancy Edwards Looking Across the Cases of Vicky Mansfield and Nancy Edwards Connecting to Your Classroom Discussion Questions Chapter 5 Modifying Tasks to Increase the Reasoning-and-Proving Potential How Do You Make Tasks Reasoning-and-Proving Worthy? Returning to the Effective Mathematics Teaching Practices Examining Textbooks or Curriculum Materials for Reasoning-and-Proving Opportunities Revisiting the Case of Nancy Edwards Continuing to Examine Tasks and Their Modifications Re-Examining Modifications Made to Tasks Through a Different Lens Comparing More Tasks with their Modifications Strategies for Modifying a Task to Enhance Students' Opportunities to Reason-and-Prove Connecting to Your Classroom Discussion Questions Chapter 6 Using Context to Engage in Reasoning-and-Proving How Does Context Affect Reasoning-and-Proving? Considering Opportunities for Reasoning-and-Proving Solving the Sticky Gum Problem Analyzing Student Work from the Sticky Gum Problem Analyzing Two Different Classroom Enactments of the Sticky Gum Problem Connecting to Your Classroom Discussion Questions Chapter 7 Putting it All Together Key Ideas at the Heart of this Book Tools to Support the Teaching of Reasoning-and-Proving Putting the Tools to Work Moving Forward in Your PLC Discussion Questions Appendix A Developing a Need for Proof: The Case of Charlie Sanders Appendix B Motivating the Need for Proof: The Case of Gina Burrows Appendix C Writing and Critiquing Proofs: The Case of Vicky Mansfield Appendix D Pressing Students to Prove It: The Case of Nancy Edwards Appendix E Making Sure that All Students Understand: The Case of Calvin Jenson Appendix G Helping Students Connect Pictorial and Symbolic Representations: The Case of Natalie Boyer References

"Simply stated, this book is a must-have for preservice and inservice mathematics teachers and teacher leaders who are looking to enhance their understanding of how reasoning-and-proving are critical processes for increasing proficiency across all mathematics content domains. This expert author team has illustrated a clear vision and plan, supported by key strategies and exceptional tools, for guiding teacher teams as they help their students learn how to make conjectures and develop and judge the effectiveness of their arguments and proofs. This book is an exceptionally useful and timely resource for schools and districts that are looking to connect and deepen their professional focus with the Effective Mathematics Teaching Practices (NCTM, 2014) and other evidence-based practices." -- Jonathan (Jon) Wray, Coordinator of Secondary Mathematics, Howard County Public Schools (MD) "Reasoning-and-proving are central to investigating ideas, solving problems, and establishing mathematics knowledge at all levels. Built around rich classroom cases, this book provides research-supported frameworks and practical resources for teachers to deepen their understanding and develop practices to aid students in reasoning-and-proving as powerful mathematical thinkers." -- Daniel Heck, Vice President "We Reason & We Prove for ALL Mathematics provides an enlightening and engaging examination of reasoning-and-proving in secondary mathematics classrooms. Filled with carefully designed tasks and task sequences, along with illustrative classroom cases, it clearly articulates the nature of reasoning-and-proving, what students need to know and understand about it, and how teachers can support this learning. The thought-provoking discussion questions and recommended classroom activities support readers' implementation of reasoning-and-proving activities into their own classrooms. We Reason & We Prove for ALL Mathematics is an outstanding resource for practice-based learning on this essential component of mathematics learning. I recommend it most highly." -- Diane J. Briars, PhD, Mathematics Education Consultant "The authors of We Reason & We Prove for ALL Mathematics have taken aim at a long-standing challenge in mathematics education: helping students become proficient with mathematical reasoning and proof. In so doing they have produced a book that will be useful to teachers and scholars alike in addressing a topic that is both difficult to teach and difficult to learn. This volume blends knowledge obtained through rigorous research with practical wisdom derived from extensive experience. Building upon a solid foundation of prior research on students' mathematical reasoning, the authors offer a collection of narrative cases and mathematics activities designed to deepen the understanding of teachers in ways that will enhance the teaching and learning of proof and reasoning." -- Edward A. Silver, Senior Associate Dean for Research & Graduate Studies, William A. Brownell Collegiate Professor of Education, & Professor of Mathematics "Grounded in the research on effective mathematics teaching practices and connected to the mathematical content taught in middle and high school, We Reason & We Prove for ALL Mathematics offers exceptional guidance, superb exemplars, and important classroom discussion questions to support student reasoning-and-proving. The ideas in this book are what we need to move away from repeat-after-me mathematics toward a convince-me mathematics-totally transforming mathematics classrooms and increasing students' opportunities to engage in doing authentic mathematics." -- Jennifer Bay-Williams, PhD, Mathematics Educator & Professor, Co-Author of Teaching Student-Centered Mathematics: Developing Appropriate Instruction Series