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!
Cuprins
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
Despre autor
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.
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