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Mathematical Argumentation in Middle School-The What, Why, and How

Mathematical Argumentation in Middle School-The What, Why, and How
A Step-by-Step Guide With Activities, Games, and Lesson Planning Tools

September 2017 | 192 pages | Corwin

Get them talking: Your formula for bringing math concepts to life!

Want your middle schoolers to intelligently engage with mathematical ideas? Ready to help them construct and critique viable arguments that meet tough Standards for Mathematical Practice 3 standards? Look no further. This research-based gem will help you foster the critical reasoning and argumentation skills every student needs for intelligent discourse within our modern society. Learn how to bring mathematical argumentation alive in your classroom—all within a thoroughly explained four-part model that covers generating cases, conjecturing, justifying, and concluding. 

Filled with content-focused and classroom-ready games, activities, vignettes, sample tasks, and links to online tools and a rich companion website, this innovative guide will help you 

  • Immediately engage students in fun, classroom-ready argumentation activities
  • Plan lessons that foster lively, content-driven, viable argumentation
  • Help students explore mathematical ideas and take ownership of their learning
  • Facilitate deep mathematical understanding
  • Promote students’ precise use of mathematical language to construct, justify, and critique mathematical ideas and mathematical statements or the arguments of others.
  • Encourage logical, clear connections between abstract ideas for enhanced 21st century skills 

This guide delivers all the tools you need to get serious about mathematical argumentation and bring well-planned, well-constructed mathematical discourse to life in your classroom today!

About the Authors
Chapter 1. Mathematical Argumentation: Why and What
Argumentation Is Important!

What Argumentation Is—and Is Not

A Four-Part Model of Argumentation

About Truth

Teaching as Disciplined Improvisation

Improvisation for Argumentation and Norm Setting

Sharing Mathematical Authority

Getting Started With Argumentation

Argumentation Lessons Versus Argumentation in Lessons

Working Together

Chapter 2. Generating Cases
What Does It Mean to Generate Cases?

An Activity Rich in Argumentation and Content

Vignette: Small Groups Generate Cases

Teaching Moves

Establishing Norms



Working Together

Chapter 3. Conjecturing
What Does It Mean to Conjecture?

Vignette: Conjecturing Together

Teaching Moves

Establishing Norms



Working Together

Chapter 4. Justifying
What Does It Mean to Justify?

Vignette: Justifying Multiple Conjectures

Teaching Moves for Eliciting Justifications

Vignette: Critiquing and Connecting Arguments

Teaching Moves for Critiquing and Connecting Arguments

Establishing Norms



Working Together

Chapter 5. Representations in Justifications
What Are Representations?

Vignette: Visual Representations Foster Participation

Vignette: Gestures Enable a Unique Contribution

Teaching Moves

Using Dynamic Digital Tools

Establishing Norms



Working Together

Chapter 6. Levels of Justification
Four Levels of Justification

Level 0: No Justification

Level 1: Case-Based Justifications

Level 2: Partially Generalized Justifications Based on Cases

Level 3: Fully Generalized Justifications

A Rubric for Levels

Teaching Moves for Transitions Between Levels

Working Together

Chapter 7. Concluding
What Does It Mean to Conclude?

Vignettes: Concluding

Teaching Moves

Establishing Norms



Working Together

Chapter 8. Planning
How Can You Plan for Students’ Argumentation?

Written Lesson Plans

Visualizing a Lesson

Vignette: Visualizing Justification

Digital Tools

Updating and Sharing Lesson Plans

Advice on Planning

Working Together


If you share my belief that “construct viable arguments and critique the reasoning of others” are perhaps the nine most important words in the Common Core era, then Mathematical Argumentation in Middle School is just what you need. This powerful and practical book takes us through an accessible process of generating cases, making conjectures, and justifying that fully supports bringing SMP #3 to life in our classrooms.

Steve Leinwand
American Institutes for Research

This great resource gives teachers tools to implement the four cycles of mathematical argumentation and help students develop a “variety of expertise,” as described in the Standards of Mathematical Practice. As students cycle through the phases, they are guided in building “mathematical authority” as independent thinkers and creators of mathematical ideas. I recommend this book to any teacher who wants to amp up the math discussion in their classroom.

Annette Hilts
Vallejo City Unified School District

Now more than ever, we need to provide all children with opportunities to learn to think critically and participate in thoughtful, productive debate in today’s society. This book translates the mathematical practice of argumentation into a four-stage process that can be applied across a wide range of mathematical content. This process utilizes an innovative, research-based approach based on improv games that opens access for all students to participate in the process of mathematical argumentation. Finally, there is a practical guide for making argumentation an everyday practice in mathematics classrooms!

Kristen Bieda
Michigan State University

Sample Materials & Chapters

Table of Contents


Chapter 1

Jennifer Knudsen

Jennifer Knudsen has been working in mathematics education since her days as a Peace Corps volunteer in Kenya and as a teacher in in New York City Public Schools. She has focused on students’ engagement in mathematics as an equity issue throughout her career, including work on numerous curriculum and professional development projects. She directs the Bridging Professional Development project as part of her role as a senior mathematics educator at SRI International. She holds a B.A. from The Evergreen State College, where she learned to love mathematical argumentation. She lives in Austin, Texas with her husband and daughter. More About Author

Harriette Stevens

Harriette S. Stevens attended the University of Kansas where she received her Bachelor of Arts in Applied Mathematics and Master of Arts in Education, with a concentration in Mathematics. She received her Doctorate in Education, with a focus on curriculum and instructional design, from the University of San Francisco. She was the director of a mathematics professional development program for K-12 teachers at the University of California, Berkeley’s Lawrence Hall of Science.  In this capacity, she worked in partnership with several urban-school districts, and designed PD and instructional materials to help improve teachers’... More About Author

Teresa Lara-Meloy

Teresa Lara-Meloy is passionate about finding better ways of teaching middle school math and improving ways to support teachers. As Math Ed Researcher at SRI International, she designs technology-integrated curricular and professional development materials. She received her M.Ed. from Harvard's Graduate School of Education. She is a member of the NCSM and TODOS. She has co-authored articles on technology in education and the role of technology in supporting the participation of English Language Learners in math class. More About Author

Hee-Joon Kim

Hee-Joon Kim, Ph.D. is a mathematics education researcher at SRI International located in Menlo Park, CA. Her research focuses on understanding classroom discourse that supports mathematical argumentation in middle school. She has expertise in designing curriculum materials with dynamic tools for students in middle grades. She has been involved in research-based professional development projects that focus on improving classroom practices that support conceptual understanding and promote equity. She received a B.S. in Mathematics at Ewha Womans University in South Korea and a Ph.D. in Mathematics Education at the University of Texas at... More About Author

Nicole Shechtman

Nicole Shechtman, Ph.D., is a senior educational researcher at SRI International located in Menlo Park, CA. Her research and evaluation work explores critical issues in mathematics teaching and learning, innovative uses of educational technology, and the development of social and emotional competencies, such as effective communication, teamwork, and everyday problem-solving. She holds a PhD in psychology from Stanford University. More About Author

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