Over the past few decades there has been increased interest in how students and teachers think and learn about negative numbers from a variety of perspectives. In particular, there has been debate about when integers should be taught and how to teach them to best support students’ learning. This book brings together recent work from researchers to illuminate the state of our understanding about issues related to integer addition and subtraction with a goal of highlighting how the variety of perspectives support each other or contribute to the field in unique ways. In particular, this book focuses on three main areas of integer work: students’ thinking, models and metaphors, and teachers’ thinking. Each chapter highlights a theoretically guided study centered on integer addition and subtraction. Internationally known scholars help connect the perspectives and offer additional insights through section commentaries. This book is an invaluable resource to those who are interested in mathematics education and numerical thinking.

The book synergizes research on number across two disciplines—mathematics education and psychology. The underlying problem the book addresses is how the brain constructs number. The opening chapter frames the problem in terms of children’s activity, including mental and physical actions. Subsequent chapters are organized into sections that address specific domains of number: natural numbers, fractions, and integers. Chapters within each section address ways that children build upon biological primitives (e.g., subitizing) and prior constructs (e.g., counting sequences) to construct number. The book relies on co-authored chapters and commentaries at the end of each section to create dialogue between junior faculty and senior researchers, as well as between psychologists and mathematics educators. The final chapter brings this work together around the framework of children’s activity and additional themes that arise in the collective work. The book is aimed to appeal to mathematics educators, mathematics teacher educators, mathematics education researchers, educational psychologists, cognitive psychologists, and developmental psychologists.

The subject of algebra has always been important in American secondary mathematics education. However, algebra at the elementary level has been garnering increasing attention and importance over the past 15 years. There is consequently a dire need for ideas, suggestions and models for how best to achieve pre-algebraic instruction in the elementary grades. Planting the Seeds of Algebra will empower teachers with theoretical and practical knowledge about both the content and pedagogy of such instruction, and show them the different faces of algebra as it appears in the early grades. The book will walk teachers of young children through many examples of K-6 math lessons and unpack, step by step, the hidden connections to higher algebra. After reading this book, teachers will be better equipped ...

Explains how children between the ages of four and eight construct a deep understanding of numbers and the operations of addition and subtraction.

Strengthen your mathematics lessons through collaborative planning Teaching by Design in Elementary Mathematics is a series of comprehensive professional development guides that help teachers investigate how students learn. Grounded in the latest research, this book is one of three volumes focused on grade-appropriate number and operations topics aligned with the Common Core State Standards. The capstone activity of each book guides the group through the co-creation and implementation of a prototype lesson. The teacher teams then evaluate the impact of the lesson on student learning and work together to revise it for maximum effectiveness. Through the process, teachers develop: Deeper content knowledge of important mathematical concepts Improved understanding of how students learn these mathematical ideas A stronger foundation for developing effective lessons and improving instruction Enhanced collaboration skills Each volume includes a large assortment of reproducible handouts as well as built-in facilitation notes. Teachers will also find helpful resources that address the issue of finding time for school-based professional development and teacher collaboration.

Explains how children between the ages of four and eight construct a deep understanding of numbers and the operations of addition and subtraction.

Strengthen your mathematics lessons through collaborative planning Teaching by Design in Elementary Mathematics is a series of comprehensive professional development guides that help teachers investigate how students learn. Grounded in the latest research, this book is one of three volumes focused on grade-appropriate number and operations topics aligned with the Common Core State Standards. The capstone activity of each book guides the group through the co-creation and implementation of a prototype lesson. The teacher teams then evaluate the impact of the lesson on student learning and work together to revise it for maximum effectiveness. Through the process, teachers develop: Deeper content knowledge of important mathematical concepts Improved understanding of how students learn these mathematical ideas A stronger foundation for developing effective lessons and improving instruction Enhanced collaboration skills Each volume includes a large assortment of reproducible handouts as well as built-in facilitation notes. Teachers will also find helpful resources that address the issue of finding time for school-based professional development and teacher collaboration.

High above the noise and traffic of metropolitan Phoenix, Native American rock art offers mute testimony that another civilization once thrived in the Arizona desert. In the city's South Mountains, prehispanic peoples pecked thousands of images into the mountains' boulders and outcroppings—images that today's hikers can encounter with every bend in the trail. Todd Bostwick, an archaeologist who has studied the Hohokam for more than twenty years, and Peter Krocek, a professional photographer with a passion for archaeology, have combed the South Mountains to locate nearly all of the ancient petroglyphs found in the canyons and ridges. Their years of learning the landscape and investigating the ancient designs have resulted in a book that explores this wealth of prehistoric rock art within its natural and cultural contexts, revealing what these carvings might mean, how they got there, and when they were made. Landscape of the Spirits is the first book to cover these ancient images and is one of the most comprehensive treatments of a rock art location ever published. It conveys the range of different rock art elements and compositions found in the South Mountains—animals, humans, and geometric shapes, as well as celestial and calendrical markings at key sites—through accurate descriptions, drawings, and photographs. Interpretations of the petroglyphs are based on Native American ethnographic accounts and consider the most recent theories concerning shamanism and archaeoastronomy. Written in a simple and accessible style, Landscape of the Spirits is an indispensable volume for anyone exploring the South Mountains, and for rock art enthusiasts everywhere who wish to broaden their understanding of the prehistoric world. It is both an authoritative overview of these ancient wonders and an unprecedented benchmark in southwestern rock art research at a single geographic location.

Strengthen mathematics lessons through collaborative learning with this research-based professional development program. Included are grade-appropriate number and operations topics aligned with the Common Core State Standards.