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Fluency in mathematics is more than adeptly using basic facts or implementing algorithms. It is not about speed or recall. Real fluency is about choosing strategies that are efficient, flexible, lead to accurate solutions, and are appropriate for the given situation. Developing fluency is also a matter of equity and access for all learners. The landmark book "Figuring Out Fluency in Mathematics Teaching and Learning" offered educators the inspiration to develop a deeper understanding of procedural fluency, along with a plethora of pragmatic tools for shifting classrooms toward a fluency approach. Now, teachers have the chance to apply that inspiration through explicit instruction and practice every day with the classroom companion "Figuring Out Fluency: Addition and Subtraction with Whole Numbers." With this book, teachers can: (1) dive deeper into the Significant Strategies for fluency explained in the anchor book; (2) learn how these strategies grow from and relate to the basic fact strategies children learn; (3) access over 100 strategy-aligned and classroom-ready activities for fluency instruction and practice in adding and subtracting multi-digit whole numbers, including worked examples, routines, games, and centers; (4) find activities for assessing all components of addition and subtraction fluency plus support for engaging families; and (5) download all of the needed support tools, game boards, and other resources from the companion website for immediate implementation. Give each and every student the knowledge and power to become skilled and confident mathematical thinkers and doers. [For "Figuring Out Fluency in Mathematics Teaching and Learning, Grades K-8: Moving beyond Basic Facts and Memorization. Corwin Mathematics Series," see ED615102.]

Word Problems those are also named as "Story Problems" which are well known to the students around the world. In this book there are 26 chapters which encompass diversified problems of four basic mathematical rules--addition, subtraction, multiplication and division. These problems may help students to practice more and more as well as may help the teachers to create worksheet and questions.

The "Reasoning Algebraically about Operations Casebook" was developed as the key resource for participants' Developing Mathematical Ideas seminar experience. The thirty-four cases, written by teachers describing real situations and actual student thinking in their classrooms, provide the basis of each session's investigation into the generalizations underlying the study of the operations in the elementary and middle grades and teaching strategies that support students' efforts to make sense of the concepts. Reading and discussing the cases under the guidance of the facilitator actively engages participants in their own learning enterprise as they--(1) learn to recognize the key mathematical ideas with which students are grappling; (2) consider the types of classroom settings and teaching strategies that support the development of student understanding; (3) become aware of how core mathematical ideas develop across the grades; (4) work on mathematical concepts and gain better understanding of mathematical content; and (5) discover how to continue learning about children and mathematics. The casebook is composed of eight chapters: the first seven consist of classroom cases from kindergarten through grade 7; chapter 8 is an essay providing an overview of the research related to the situations described in the first seven chapters. The chapters are as follows: (1) Chapter 1: Discovering rules for odds and evens; (2) Chapter 2: Finding relationships in addition and subtraction; (3) Chapter 3: Reordering terms and factors; (4) Chapter 4: Expanding the number system; (5) Chapter 5: Doing and undoing, staying the same; (6) Chapter 6: Multiplying in clumps; (7) Chapter 7: Exploring rules for factors; and (8) Chapter 8: The World of Arithmetic from Different Points of View (Stephen Monk). [For part 1 "Number and Operations, Part 1: Building a System of Tens Casebook," see ED566650. For part 2: Making Meaning for Operations. Casebook," see ED594600.]

Educational strategies have evolved over the years due to research breakthroughs and the application of technology. By using the latest learning innovations, curriculum and instructional design can be enhanced and strengthened. Also, as learners move away from traditional scholarly media and toward technology-based education, students gain an advantage in learning about their world and how to interact with modern society. "Learning Strategies and Constructionism in Modern Education Settings" is a critical scholarly resource that enhances the competencies of educational professionals by providing practical advice on providing an innovative educational process to promote the cognitive growth of individuals, regardless of special needs or obstacles. The book features coverage on a variety of topics including integration approaches of digital media in the teaching/learning process, the role of parents for developing digital literacy in their young children, and the effectiveness of using technology tools to teach mathematics. As a publication focused on education advancements through technology, the book serves as a useful resource for academicians, educators, school administrators, and individuals seeking current research on education technologies. This book contains the following chapters: (1) SMART Pedagogy: (Re)Defining Pedagogy (Linda Daniela and Miltiadis D. Lytras); (2) Integration Approaches of Digital Media in the Teaching/Learning Process (Zanda Rubene); (3) A Social Constructionist Model for Human-Machine Ecosystems (Neus Lorenzo Galés and Ray Gallon); (4) Positive Interdependence in Blended Learning Environments: Is It Worth Collaborating? (Calixto Gutiérrez-Braojos, Jesús Montejo-Gámez, Ana E. Marín Jiménez, and Asunción Martínez Martínez); (5) A Continuous Assessment Strategy Using Fuzzy Logic (Abraham Varghese, Jincy S. George, and Joseph George); (6) Pedagogy of Inclusion: A Quest for Inclusive Teaching and Learning Through the Theory of Constructionism (Maluleka Khazamula Jan); (7) The Role of Parents for Developing Digital Literacy of 0-5 Year Olds (Linda Daniela and Arta Rudolfa); (8) Digital Technologies in Kindergarten: Paths of Kindergarten Teachers and Potentialities for Children (Rita Brito and Patrícia Dias); (9) Comparing the Effectiveness of Using Tablet Computers for Teaching Addition and Subtraction (Nicholas Zaranis); (10) Gamification in Service Learning: An Innovative Experience (Laura Varela-Candamio, Joaquín Enríquez-Díaz, and Marcos Rouco-Couzo); (11) Toward New Method for Adaptive Learning (Souhaib Aammou, Youssef Jdidou, and Kaoutar El Bakkari); (12) Learning Models and Strategies and the Constructionism in Modern Education Settings: With Applications in Modern Learning of Biology (Mariana Iancu); (13) The Assessment of Students' Competencies and Constructionism with Examples in Biological and Natural Sciences (Mariana Iancu); and (14) Learning by Working: Examining Examples of Good Practice in Organizing Work Placements in Vocational Education (Anita Lice). A section about the contributors and an index are included.

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. [Individual chapters are indexed in ERIC.]

Learn the most effective ways to teach elementary math, no matter how much experience you have with the subject. In this book, Fuchang Liu takes you through many common mistakes in math instruction and explains the misunderstandings behind them. He points out practices that should be avoided, helping you to adjust your lessons so that all students can achieve success. You'll discover how to: (1) Increase your confidence with core math principles and reasoning; (2) Set your students on the path toward eventually developing more complex math skills; (3) Improve student achievement by approaching problems in logical yet creative ways; (4) Overcome common challenges faced by students and teachers; and (5) Teach problem solving for different learning styles. Every chapter reconsiders well-established ways of teaching all areas of elementary math, from addition and subtraction to statistics and graphs. Helpful examples and tips are scattered throughout the book, offering revisions to the way these topics are often presented in the classroom. Also included are group study ideas for principals and instructional coaches so your school or district can work on the book together. With this practical guide, you'll be ready to help students truly develop their math understanding.

The "Making Meaning for Operations Casebook" was developed as the key resource for participants' Developing Mathematical Ideas seminar experience. The twenty-nine cases, written by teachers describing real situations and actual student thinking in their classrooms, provide the basis of each session's investigation of specific mathematical concepts and teaching strategies. By reading and discussing the cases under the guidance of the facilitator, participants are actively engaged in their own learning enterprise and will: (1) learn to recognize the key mathematical ideas with which your students are grappling; (2) consider the types of classroom settings and teaching strategies that support the development of student understanding; (3) become aware of how core mathematical ideas develop across the grades; and (4) work on mathematical concepts and gain better understanding of mathematical content; and discover how to continue learning about children and mathematics. The casebook is composed of eight chapters: the first seven consist of classroom cases from kindergarten through grade 7; chapter 8 is an essay providing an overview of the research related to the situations described in the first seven chapters. The chapters are as follows: (1) Chapter 1 Making meaning for whole number addition and subtraction; (2) Chapter 2 Making meaning for multiplication and division; (3) Chapter 3 When dividing doesn't come out evenly; (4) Chapter 4 Greater than, less than, equal to; (5) Chapter 5 Combining shares, or adding fractions; (6) Chapter 6 Taking portions of portions, or multiplying fractions; (7) Chapter 7 Expanding ideas about division in the context of fractions; and (8) Chapter 8 Highlights of related research. [For "Number and Operations, Part 1: Building a System of Tens Casebook," see ED566650. For "Number and Operations, Part 3: Reasoning Algebraically about Operations Casebook," see ED592077.]

Do your students have the incorrect idea that addition "makes numbers bigger" and subtraction "makes numbers smaller"? Do they believe that subtraction is always "taking away"? What tasks can you offer--what questions can you ask--to determine what your students know or don't know--and move them forward in their thinking? This book focuses on the specialized pedagogical content knowledge that you need to teach addition and subtraction effectively in prekindergarten-grade 2. The authors demonstrate how to use this multifaceted knowledge to address the big ideas and essential understandings that students must develop for success with these computations--not only in their current work, but also in higher-level mathematics and a myriad of real-world contexts. Explore rich, research-based strategies and tasks that show how students are reasoning about and making sense of addition and subtraction. Use the opportunities that these and similar tasks provide to build on their understanding while identifying and correcting misunderstandings that may be keeping them from taking the next steps in learning. You have essential understanding. It's time to put it into practice in your teaching. The Putting Essential Understanding into Practice Series moves NCTM's Essential Understanding Series into the classroom. The new series details and explores best practices for teaching the essential ideas that students must grasp about fundamental topics in mathematics--topics that are challenging to learn and teach but are critical to the development of mathematical understanding. Classroom vignettes and samples of student work bring each topic to life, and questions for reader reaction open it up for hands-on exploration. Each volume underscores connections with the Common Core State Standards for Mathematics while highlighting the knowledge of learners, curriculum, instructional strategies, and assessment that pedagogical content knowledge entails. Resources and tasks are available at nctm.org/more4U. Maximize the potential of student-centered learning and teaching by putting essential understanding into practice.

What is the relationship between addition and subtraction? How do individuals know whether an algorithm will always work? Can they explain why order matters in subtraction but not in addition, or why it is false to assert that the sum of any two whole numbers is greater than either number? It is organized around two big ideas and supported by smaller, more specific, interconnected ideas (essential understandings). Gaining an understanding about addition and subtraction is essential as they are the foundation for students' later learning of multiplication and division. This book contains three chapters: (1) Addition and Subtraction: The Big Ideas and Essential Understandings; (2) Connections: Looking Back and Ahead in Learning; and (3) Challenges: Learning, Teaching, and Assessing.

Useful for small groups or one-on-one instruction, this book offers successful math interventions and response to intervention (RTI) connections. Teachers will learn to target math instruction to struggling students by: (1) Diagnosing weaknesses; (2) Providing specific, differentiated instruction; (3) Using formative assessments; (4) Offering corrective feedback; and (5) Motivating students by using games. Taylor-Cox emphasizes four main goals for math instructors. They must help students achieve: accuracy, efficiency, flexibility, and fluency in solving math problems. Integral to each of these goals is ensuring that students understand math "concepts." "Math Interventions: Building Number Power, Grades PreK-2" provides math teachers with specific strategies for imparting those concepts. Contents include: (1) Early Number Concepts; (2) Numbers and Number Relationships Concepts; and (3) Addition and Subtraction Concepts.

Children's mathematical misconceptions very often arise as a result of poor communication. This practical and innovative book presents a range of creative strategies to help teachers communicate effectively in the mathematics classroom, offering some new ways of presenting the fundamental concepts and principles of mathematics, and clearly demonstrating that the most effective form of communication is not always verbal. Each chapter focuses on a theme or concept central to the numeracy strategy, such as subtraction, shape and space, constructing and connecting 2D and 3D shapes, data-logging and graph interpretation, and mathematical reasoning. Each chapter supports teachers' subject knowledge as well as suggesting a range of communication strategies (e.g. speaking, listening, drawing, role play, and visual methods such as posters and ICT work) to help teachers and children to share and build on their understanding. All of the activities have been tried and tested in classrooms across Europe. Written by a team of highly experienced teachers, the book will be valuable reading for pre- and in-service teachers, teacher educators, and anyone who has an interest in extending the ways children actively engage with mathematics. Teachers with EAL pupils in their class may find it of particular value in communicating mathematical concepts to those who find English difficult. The book contains five chapters: (1) Understanding Subtraction (Anne Cockburn); (2) Communicating Ideas About Space and Shape (Brigitte Spindeler and Bernd Wollring); (3) Constructing and Connecting 2- and 3-D Shapes (Diana Hunscheidt and Andrea Peter-Koop); (4) Exploring Movement through ICT (Fiona Thangata and Alan Pagden); and (5) Investigating Decision Making and Mathematical Reasoning (Milan Hejn). [This book contains a DVD ROM.]

This book is about playing with mathematics, teaching: (1) How to multiply and divide large numbers mentally; (2) Making addition and subtraction easy; (3) Tricks for understanding fractions and decimals; and (4) How to quickly check answers for each calculation. Described as a foolproof way to do multiplication, division, factoring estimating, and more, the book is designed to assist even those who have struggled with math in the past. Following a preface and introduction, the book includes: (1) Multiplication: Getting Started; (2) Using a Reference Number; (3) Numbers Above the Reference Number; (4) Multiplying Above & Below the Reference Number; (5) Checking Your Answers; (6) Multiplication Using Any Reference Number; (7) Multiplying Lower Numbers; (8) Multiplication by 11; (9) Multiplying Decimals; (10) Multiplication Using Two Reference Numbers; (11) Addition; (12) Subtraction; (13) Simple Division; (14) Long Division by Factors; (15) Standard Long Division Made Easy; (16) Direct Long Division; (17) Checking Answers (Division); (18) Fractions Made Easy; (19) Direct Multiplication: (20) Putting It All into Practice; and Afterword. Eleven appendices are included: (1) Using the Methods in the Classroom; (2) Working Through a Problem; (3) Learn the 13, 14 and 15 Times Tables; (4) Tests for Divisibility; (5) Keeping Count; (6) Plus and Minus Numbers; (7) Percentages; (8) Hints for Learning; (9) Estimating; (10) Squaring Numbers Ending in 5; and (11) Practice Sheets. An index is included. [This book is also available as an e-book.]

This book explores how primary school children with dyslexia or dyspraxia and difficulty in math can learn math and provides practical support and detailed teaching suggestions. It considers cognitive features that underlie difficulty with mathematics generally or with specific aspects of mathematics. It outlines the ways in which children usually learn basic mathematics and the special cognitive needs of this population. Sections address the following broad areas: definitions and premises; basic counting and the early stages of addition and subtraction; addition and subtraction with larger numbers; and multiplication and division. Individual chapters examine the following topics: (1) background information; (2) teaching premises; (3) counting; (4) counting in basic calculation; (5) defining the difficulties; (6) an understanding-based approach to teaching the number structures; (7) basic calculation to twenty; (8) two-digit addition and subtraction; (9) more on two-digit addition and subtraction; (10) theoretical debates about multiplication and division; (11) an understanding based approach to multiplication and division for dyslexic and dyspraxic children; and (12) multiplication and division with larger numbers. (Contains references.) (DB)

This book contains a collection of addition and subtraction lessons that support curriculum and instruction. The activities are rich with child-friendly contexts that are fun and engaging and give students opportunities to gain computational skills while developing their own confidence in and enjoyment of mathematics. Three elements are regarded as essential for the understanding of addition and subtraction: (1) counting; (2) the decomposition of numbers; and (3) the place value system of tens. The lessons in this book provide a wide variety of addition and subtraction experiences in contextual situations. Activities include playing a game; reading a piece of literature; working with personal information such as names, body measurement, pockets, and telephone numbers; interpreting newspaper data; working with money for shopping and making a penny collection; and writing word problems about real and imagined experiences. In order to help teachers with planning and teaching the lessons, each lesson is organized into sections that include an overview, materials, time, teaching directions, teaching notes, a vignette describing the actual lesson, follow-up suggestions, and questions and discussion. (KHR)

This textbook is a course in elementary arithmetic. It stresses mental as well as written exercises. Addition and subtraction are first presented together, and later multiplication and division.

In the foreword to this book, Inhelder states that its essential contribution is that "it transposed to actual reality Piaget's central epistemological thesis according to which mathematical thought is, from its most elementary manifestations, the product of the subject's activity that he characterized by the term 'reflective abstraction.'" The book advocates eliminating all traditional instruction in first grade arithmetic and replacing it with two kinds of activities: situations in daily living (such as voting) and group games. The sections in the book focus on: (1) theoretical foundation, with chapters on Piaget's theory of number and on the importance of social interaction; (2) objectives, including chapters on autonomy, numerals and place value, addition, and subtraction; (3) activities and principles of teaching; (4) how things appeared to the teacher, including a month-to-month account of using Piagetian-based games; and (5) evaluation and prospects for the future. The book, designed for preservice as well as in-service teachers, is based on research, as well as on personal experiences at the University of Geneva. (MNS)

RECENT DEMOCRATIZATION Our understanding of recent democratization, both in theoretical and empirical terms, has been strongly influenced by the experiences of Latin America and southern Europe. This is largely because the global wave of democratization began in these two regions. At the same time, scholars with expertise in these areas had an analytical advantage: they had long wrestled with issues of regime change, in whatever direction. Indeed, when a group of specialists in these two regions first convened at the Social Science Research Council to think together about democratization, the countries they studied were for the most part not yet moving from authoritarian to democratic rule. The contributions of the transitions school, as members of this group came to be known, have been considerable. However, a question remains. Do the approaches, the concepts, and the arguments that have developed in the course of analyzing democratization in the “south” still stand, once we broaden our geographical and, therefore, historical and variable horizons? Does the addition of new cases, in short, lead us to rethink the origins, the practices, and the sustainability of new democracies? The purpose of this chapter is to provide an answer to this question, by adding the postcommunist experience to the equation. In particular, I will conduct a conversation between two bodies of research: analyses of Latin America and southern Europe, which have collectively constituted the reigning wisdom in the field of democratization studies, and analyses of postcommunist politics in general and Russian politics in particular. [ABSTRACT FROM AUTHOR]