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

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!

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.