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5 Principles of the Modern Mathematics Classroom

5 Principles of the Modern Mathematics Classroom
Creating a Culture of Innovative Thinking

  • Gerald Aungst - School District of Cheltenham Township; Elkins Park, PA

Foreword by Mark Barnes

200 pages | Corwin

Students pursue problems they’re curious about, not problems they’re told to solve.

Creating a math classroom filled with confident problem solvers starts by introducing challenges discovered in the real world, not by presenting a sequence of prescribed problems, says Gerald Aungst. 

In this groundbreaking book, he offers a thoughtful approach for instilling a culture of learning in your classroom through five powerful, yet straightforward principles: Conjecture, Collaboration, Communication, Chaos, and Celebration.

Aungst shows you how to

  • Embrace collaboration and purposeful chaos to help students engage in productive struggle, using non-routine and unsolved problems
  • Put each chapter’s principles into practice through a variety of strategies, activities, and by incorporating technology tools
  • Introduce substantive, lasting cultural changes in your classroom through a manageable, gradual shift in processes and behaviors

Five Principles of the Modern Mathematics Classroom offers new ideas for inspiring math students by building a more engaging and collaborative learning environment.

"Bravo! This book brings a conceptual framework for K-12 mathematics to life. As a parent and as the executive director of Edutopia, I commend Aungst for sharing his 5 principles. This is a perfect blend of inspiring and practical. Highly recommended!”  
Cindy Johanson, Executive Director, Edutopia
George Lucas Educational Foundation

"Aungst ignites the magic of mathematics by reminding us what makes mathematicians so passionate about their subject matter. Grounded in research, his work takes us on a journey into classrooms so that we may take away tips to put into practice today." 
Erin Klein, Teacher, Speaker, and Author of Redesigning Learning Spaces

Take a look at Gerald describe why you need this book!

Foreword by Mark Barnes
About the Author
Chapter 1. Math for the 21st Century and Beyond
Mathematics as Literacy

Mathematics as an Innovation Incubator

Goals of This Book

What Else to Expect

For PLC and Study Groups

Chapter 2. Classroom Environment: A Medium for Change
Why Classroom Culture?

Depth and Rigor

A Focus on the Practices

The 5 Principles of the Modern Mathematics Classroom

For PLC and Study Groups

Chapter 3. Conjecture
The Power of Mysteries

The Curious Brain

Problem Solving Is a Cycle, Not a Recipe

Grades K–3

Grades 2–5

Grades 4–7

Grades 6–9

Grades 8–12

Technology Integration for Conjecture: Case Studies

For PLC and Study Groups

Chapter 4. Communication
The Mysterious Mathemagical Mind Trick

First and Foremost, Communicate

Six Levels of Mastery

Grades K–3

Grades 2–5

Grades 4–7

Grades 6–9

Grades 8–12

Technology Integration for Communication: Case Study

For PLC and Study Groups

Chapter 5. Collaboration
But, Isn’t That Cheating?

Recognizing, Defining, and Representing Problems

Grades K–3

Grades 2–5

Grades 4–7

Grades 8–12

Grades 6–9

Technology Integration for Collaboration: Case Study

For PLC and Study Groups

Chapter 6. Chaos
Features of Effective Math Instruction

Disorganized Chaos

Productive Struggle

Grades K–3

Grades 2–5

Grades 4–7

Grades 6–9

Grades 8–12

Technology Integration for Chaos: Case Studies

For PLC and Study Groups

Chapter 7. Celebration
Today, We Celebrate

Cautions About Celebration

Grades K–3

Grades 2–5

Grades 4–7

Grades 6–9

Grades 8–12

Digital Tools and Resources for Celebration

Technology Integration for Celebration: Case Study

For PLC and Study Groups

Chapter 8. Becoming a Problem-Solving Classroom
Transformation Is Possible


Goal-Setting and Planning

Maintaining Your Momentum

Appendix: Digital Tools

"5 Principles of the Modern Mathematics Classroom is an easy-to-read and use book that provides educators with practical strategies and solutions to immediately incorporate into their mathematics classrooms. The design of the book provides classroom teachers an opportunity for deep reflection of their current mathematics instructional practices to ensure that they are creating students who are mathematical thinkers and problem solvers. 5 Principles of the Modern Mathematics Classroom offers teachers strategies to allow their students to wonder freely, work collaboratively with their peers, communicate efficiently, and participate in experimentation and celebration. Using technology as a way to incorporate the principles of a modern mathematics classroom is briefly explored with short explanations of specific tools for each principle. Overall, Aungst has provided his readers with a five principle framework that when incorporated into a classroom, will transform students into mathematicians."

Stephanie Schwab, STEM Program Administrator
Montgomery County Intermediate Unit

“What does it take to be an excellent teacher? With so many ideas about teaching floating about, it is necessary to capture the essential components and package them for careful consideration. While perspectives for what it actually takes to ensure all students flourish are manifold, it is certain that the aspiring teacher must be constantly reflecting on her practice and expanding her awareness of productive beliefs and actions. To this end, Gerald Aungst has developed a guide ideal for any teacher hoping to find the sweet spot between theory and practice that can easily translate to the classroom. 


Capturing a very complex and hopeful view the nature of a classroom that promotes the success of all students, Gerald Aungst presents illustrations, research, and classroom activities. Simple and engaging, the stories and resources inspire the reader to delve into an entirely unplanned investigation of a new ideas at every turn. The 5 Principles of the Modern Mathematics Classroom are engaging, perplexing, and ripe for exploration and discussion. I'd recommend the read for any teacher looking to better understand what being an excellent teacher means through the lens of the classroom culture and for any PLC looking for a head start in casting a vision for a strong mathematics culture in their school.”

Levi Patrick, Director of Secondary Mathematics Education at the Oklahoma State Department of Education
Co-Founder of

"In 5 Principles of the Modern Mathematics Classroom: Creating a Culture of Innovative Thinking, Gerald Aungst comprehensively describes the characteristics of an environment for mathematical learning which integrates the math practice standards and actively involves students in higher level, inquiry-based thinking.  Aungst’s book holds value for the beginning teacher, who will find a multitude of resources and ideas for establishing an effective mathematics classroom, and for the seasoned teacher, who will be challenged to reflect on her own practice and incorporate the five principles into her math instruction. Coaches and other instructional leaders will find the book well-suited for book study or PLC discussion, with contemplative questions at the conclusion of each chapter. Gerald Aungst’s 5 Principles provides a framework for propelling math learning forward in the twenty-first century."

Louise Kirsh, Math Coach, North Penn School District

"Math is linear... Math is neat and tidy... Math is about getting the right answer... Not in this book! Gerald has done an awesome job of showing us what math learning can look like in a way that embraces the messy inquiry and problem-solving that accompanies real world math. Put this book to work and watch your students be amazing!"

Scott McLeod, J.D., Ph.D., Director of Innovation, Prairie Lakes Area Education Agency, and Founding Director, UCEA Center for the Advanced Study of Technology Leadership in Education
Prairie Lakes Area Education Agency and UCEA Center for the Advanced Study of Technology Leadership in Education

"Mathematical thinking critically impacts a learner's capability to process the world around her. The typical math instruction actually degrades a child's capability to see math in the world and to find delight in the messy construction of analytical, critical and creative processes through mathematical thinking. In 5 Principles of the Modern Math Classroom, Gerald Aungst provides a learning path for teachers that leads them to think together and to practice math instruction that makes math real. And that's what great math teachers do – at any age."

Pam Moran, Superintendent
Albemarle County Public Schools, VA

"Gerald Aungst is as innovative as he is refreshing in his approach to mathematics instruction. The strategies he describes will make an average teacher excellent and an excellent teacher outstanding. The examples in this book can be used immediately in 1-on-1 conversations with teachers, small group professional development, or large group workshops. Each chapter resonates with his approach to the modern mathematics classroom. This book is an excellent resource for every teacher who teaches mathematics."

Sean W. Gardiner, Ed.D, Director of STEM Education
Upper Merion Area School District

"Gerald masterfully reminds us that math education must be more than a “recipe” and challenges us to aid our students in thinking like mathematicians. With practical advice and inspiration, you’ll again find the beauty in math and be empowered to reveal it for your students. Regardless of the age you teach, this book offers a fantastic breakdown of the role of conjecture, collaboration, communication, chaos, and celebration play in transforming your classroom into one where students engage free-range thinking every day."

Kelly Tenkely, Architect of Learning, Founder and Administrator
Anastasis Academy

"Take a good, hard look at your teaching practices! If your math classroom resembles your childhood classroom, then you need to invite Gerald Aungst to the party. His 5 practices will help you transform your teaching so you can develop the critical thinkers of tomorrow. Gone are the days when mathematics instruction was focused solely on procedures. As Gerald describes so astutely in terms of music, we must train our students to be ready to work through unique problems just as we train music students to sight read. Gerald provides sound research and examples from all grade levels, including websites and apps for incorporating technology. He addresses the pressures of state tests and the long-term benefits of an actively-engaged classroom. He supports the premise that ALL students can learn math. Gerald guides you along with reflective questions and suggestions for how to begin. This book is an inspirational must-read, and includes insightful questions for use in a book study. Put it on your reading list and begin the journey."

Mary Palladino, Math Coach

"Bravo! 5 Principles of the Modern Mathematics Classroom brings a conceptual framework for K-12 mathematics to life. As a parent of a struggling math student and as the executive director of Edutopia, a source for what works in education, I commend Gerald Aungst for sharing his five principles of the modern mathematics classroom. This book is a perfect blend of inspiring and practical. Highly recommended!"

Cindy Johanson, Executive Director
Edutopia, George Lucas Educational Foundation

"Gerald Aungst reframes mathematical thinking for the reader, giving space for teachers to go beyond the usual airtight methods for math pedagogy." 

Jose Luis Vilson, Educator and Author of This Is Not A Test

Sample Materials & Chapters

The 5 Principles

Chapter 3 Excerpt

Gerald Aungst

Gerald Aungst has nearly 30 years experience as a professional educator, specializing in design thinking, computer science, mathematics, gifted education, and digital technology. In his various roles as a classroom teacher, gifted support specialist, administrator, curriculum designer, and professional developer, he has worked to create a rich and vibrant learning culture. Gerald is a founder of and 2016, Gerald has served as the gifted support and STEM teacher at Cheltenham Elementary School in the Cheltenham School District, Pennsylvania. He led the development and launch of the school’s... More About Author

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