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We Reason & We Prove for ALL Mathematics
Develop concrete instructional strategies that support your students’ capacity to reason-and-prove across all mathematical content areas in 6-12 classrooms, while becoming adept at reasoning-and-proving.
- Grade Level: PreK-12
- ISBN: 9781506378190
- Published By: Corwin
- Series: Corwin Mathematics Series
- Year: 2018
- Page Count: 272
- Publication date: August 31, 2018
Price: $38.95
Description
Sharpen concrete teaching strategies that empower students to reason-and-prove
How do teachers and students benefit from engaging in reasoning-and-proving? What strategies can teachers use to support students’ capacity to reason-and-prove? What does reasoning-and-proving instruction look like?
We Reason & We Prove for ALL Mathematics helps mathematics teachers in grades 6-12 engage in the critical practice of reasoning-and-proving and support the development of reasoning-and-proving in their students. The phrase “reasoning-and-proving” describes the processes of identifying patterns, making conjectures, and providing arguments that may or may not qualify as proofs – processes that reflect the work of mathematicians. Going beyond the idea of “formal proof” traditionally relegated only to geometry, this book transcends all mathematical content areas with a variety of activities for teachers to learn more about reasoning-and-proving and about how to support students’ capacities to engage in this mathematical thinking through:
- Solving and discussing high-level mathematical tasks
- Analyzing narrative cases that make the relationship between teaching and learning salient
- Examining and interpreting student work that features a range of solution strategies, representations, and misconceptions
- Modifying tasks from curriculum materials so that they better support students to reason-and-prove
- Evaluating learning environments and making connections between key ideas about reasoning-and-proving and teaching strategies
We Reason & We Prove for ALL Mathematics is designed as a learning tool for practicing and pre-service mathematics teachers and can be used individually or in a group. No other book tackles reasoning-and-proving with such breadth, depth, and practical applicability. Classroom examples, case studies, and sample problems help to sharpen concrete teaching strategies that empower students to reason-and-prove!
Key features
Includes:
- Vignettes, teaching takeaways, and key vocabulary terms
- Pause and consider sections and discussion questions
- In-depth analysis of student work
- Companion website contains downloadable tasks and case studies for analysis
Author(s)
Fran Arbaugh
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) S. Smith
Margaret (Peg) Smith is a Professor Emerita at University of Pittsburgh. Over the past two decades she has been developing research-based materials for use in the professional development of mathematics teachers. She has authored or coauthored over 90 books, edited books or monographs, book chapters, and peer-reviewed articles including the best seller Five Practices for Orchestrating Productive Discussions (co-authored with Mary Kay Stein). She was a member of the writing team for Principles to Actions: Ensuring Mathematical Success for All and she is a co-author of two new books (Taking Action: Implementation Effective Mathematics Teaching Practices Grades 6-8 & 9-12) that provide further explication of the teaching practices first describe in Principles to Actions. She was a member of the Board of Directors of the Association of Mathematics Teacher Educators (2001-2003; 2003 – 2005), of the National Council of Teachers of Mathematics (2006-2009), and of Teachers Development Group (2009 – 2017).
Justin Boyle
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
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
Table of Contents
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
Reviews
"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)
Board of Directors, National Council of Teachers of Mathematics
"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"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."
Horizon Research, Inc.
"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
Past President, National Council of Teachers of Mathematics (NCTM)
"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
University of Michigan
"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
University of Louisville
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