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We Reason & We Prove for ALL Mathematics: Building Students' Critical Thinking, Grades 6-12

by Fran Arbaugh, Margaret S Smith, Justin D Boyle, Gabriel J Stylianides and Michael Steele Corwin Press
Pub Date:
Pbk 272 pages
AU$75.00 NZ$77.39
Product Status: Not Our Publication - we no longer distribute
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Sharpen concrete teaching strategies that empower students to reason-and-prove

What does reasoning-and-proving instruction look like and how can teachers support students’ capacity to reason-and-prove? Designed as a learning tool for mathematics teachers in grades 6-12"," this book transcends all mathematical content areas with a variety of activities for teachers that include

  •  Solving and discussing high-level mathematical tasks

  • Analyzing narrative cases that make the relationship between teaching and learning salient

  • Examining and interpreting student work

  • Modifying curriculum materials and evaluating learning environments to better support students to reason-and-prove

No other book tackles reasoning-and-proving with such breath", depth," and practical applicability.


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

"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."

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 GÇô 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 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 GÇô 2005), of the National Council of Teachers of Mathematics (2006-2009), and of Teachers Development Group (2009 GÇô 2017).

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 of Mathematics Education and Chair of the Department of Curriculum and Instruction in the School of Education at the University of Wisconsin-Milwaukee. He is currently the President-Elect of the Association of Mathematics Teacher Educators. 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 for the past two decades. He has published several books and journal articles focused on supporting mathematics teachers in enacting research-based effective mathematics teaching practices. He is the co-author of NCTM's Taking Action: Implementing Effective Mathematics Teaching Practices in Grades 6-8 and Mathematics Discourse in Secondary Classrooms, two research-based professional development resources for secondary mathematics teachers. He is also 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.