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5 Principles of the Modern Mathematics Classroom

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Table of Contents

Foreword by Mark Barnes Preface Acknowledgements 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 Self-Assessment Goal-Setting and Planning Maintaining Your Momentum Appendix: Digital Tools References Index

About the Author

Gerald Aungst is the supervisor of gifted education and elementary mathematics for the School District of Cheltenham Township, in Pennsylvania. Prior to his service as an administrator, Gerald taught mathematics at the elementary level for eighteen years in both the regular classroom and as a gifted support specialist. He also ran a summer remedial program for elementary students for four years and served as the lead teacher for science during a major curriculum transition. Most recently Gerald has been part of his district administrative team leading the transition to the Pennsylvania Core Standards and training teachers in high quality math instructional practices. A longtime advocate of the power of technology to empower students and enhance learning, Gerald was the first in his school to embrace tools like blogging and wikis to allow students to collaborate across schools and to give parents a window into the classroom. Gerald is a cofounder of the websites All About Explorers ( and Connected Teachers ( He has been a member of the Alliance for Excellent Education's Project 24 Team of Experts and the Digital Learning Day planning team. He also served on the national Administrator Task Force for the National Association for Gifted Children (NAGC) and has been a member of the board of directors for the Pennsylvania Association for Gifted Children (PAGE). ? Gerald is an experienced speaker, and was a member of the keynote panel and a featured presenter at Digital Learning Day 2013. He has also presented at national and state conferences, including the International Society for Technology in Education (ISTE), FETC, EduCon, the Pennsylvania Educational Technology Expo and Conference (PETE&C), ITEC, NAGC, and the Carnegie-Mellon Institute for Talented Elementary and Secondary Students. Gerald is also on the organizing team for Edcamp Philly. ? Gerald holds a bachelor's degree from Penn State University and a master of education degree from Widener University. Gerald blogs at his website ( and is a regular contributor to Brilliant or Insane ( and his writing has appeared in ASCD's Educational Leadership magazine and a chapter in the book, Teaching Literacy in the Digital Age: Inspiration for All Levels and Literacies. You can find Gerald on Twitter at @geraldaungst.


"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 "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 "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 "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 "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 "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 "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 "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 "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 "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 "This is the book that teachers have been waiting for - evidence-based practices that are easy to implement. Aungst walks us through simple ways to change our mindset about math, encourage students to embrace mathematical challenges and integrate technology in our daily practices. So many ideas for every grade level!" -- Annie Beninghof, Teacher, Speaker, and Author "Gerald has written a book about math instruction and learning, but the strategies can easily trickle to other disciplines in school. He not only shares great ideas about how we can create meaningful learning opportunities and environments for students, he clearly articulates why this is important. As someone who is not a math teacher, I found what Gerald wrote to be applicable to much of the work that I am doing in all areas, and really felt that the environment of math learning that he shared is much better than the one I experienced as a student." -- George Couros, Innovative Teaching, Learning, and Leadership Consultant "Aungst ignites the magic of mathematics for his readers through five beautifully articulated guiding principles. Reminding us of what makes mathematicians so passionate about their subject matter, Gerald not only grounds his work in research but also takes us on a journey into several classrooms in each vignette so that we may take away actual, practical tips to put into practice today." -- Erin Klein, Teacher, Speaker, and Author of Redesigning Learning Spaces "A dynamic read for math educators K-12, Aungst challenges the status quo and many traditional methods of classroom instruction that are still prevalent in many of our nation's classrooms. In doing so, he makes a case for the vitality of a student-centered, rich classroom culture, filled with collaboration, problem-solving, rigorous curriculum, and high-quality communication, while celebrating student success at every turn. Moving from theory to practice, and providing a myriad of concrete methods and examples, 5 Principals of the Modern Mathematics Classroom is a must-have tool for every math teacher's professional toolbox!" -- Thomas C. Murray, State and District Digital Learning Director "In this book, Gerald Aungst expertly weaves math & literacy to promote a culture of thoughtful inquiry, problem-solving, and innovative thinking. He consistently demonstrates deep respect for the topic, the teachers, and the students, carefully interspersing stories, strategies, questions, tips, and tools in just right places throughout each chapter to maximize understanding using a gradual release of responsibility model. My eyes are forever opened to an instructional world where math and literacy not only coexist - but work in concert to elevate our instructional power potential!" -- Dr. Mary Howard, Author "Math is an area in which critical thinking skills are vital. This book will assist educators in encouraging students to think critically when dealing with math problems. Learning to incorporate the 5C's into the math classroom is sure to enhance both teaching and learning! . . . I would purchase this book. We always welcome material that will enhance our students' ability to learn and develop critical thinking skills! (And even have some "fun" while learning!!!)" -- Susan E. Schipper, First Grade Teacher "I absolutely recommend publication! Elementary teachers are overwhelmed with how to teach multi-step problems to young children. Knowing they (the teachers) don't have to be the Know-All of the class, but to allow the students the opportunity to think, possibly fail, repeat, provides the teacher time to watch the students and help direct their learning." -- Lyneille Meza, Coordinator of Data & Assessment "This book is practical and forward thinking. It should be required reading for undergraduate college math methods classes." -- Christine Ruder, 3rd Grade Teacher "This book is great for our educational world right now. There have been many who are talking about the changes in our math classrooms to help promote more problem solving and not just answering questions. This is a great start for teachers to begin to make changes in their classrooms for students to become great problem solvers . . . I would purchase this book for the math team to have them begin the discussions of helping students become problem solvers." -- Anthony Purcell, 6th Grade Math Teacher "One of the major strengths is that the book is written in teacher friendly language and that it gives practical examples. This content is much needed for today's teachers who are either stuck in a traditional way of teaching OR wanting to change and are looking for ideas . . . I would definitely purchase this book and would strongly consider it as a supplementary, recommended text for classes I teach as an adjunct faculty member." -- Barbara Fox, Adjunct Faculty

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