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• Title Page
• Copyright Page
• About the Authors
• About the Contributors
• Brief Contents
• Contents
• Preface
• PART I Teaching Mathematics: Foundations and Perspectives
• CHAPTER 1 Teaching Mathematics in the 21st Century
• Becoming an Effective Teacher of Mathematics
• A Changing World
• Factors to Consider
• The Movement toward Shared Standards
• Mathematics Content Standards
• The Process Standards and Standards for Mathematical Practice
• How to Effectively Teach the Standards
• An Invitation to Learn and Grow
• Becoming a Teacher of Mathematics
• Resources for Chapter 1
• Self Check
• CHAPTER 2 Exploring What It Means to Know and Do Mathematics
• What Does It Mean to Do Mathematics?
• Goals for Students
• An Invitation to Do Mathematics
• Where Are the Answers?
• What Does It Mean to Know Mathematics?
• Relational Understanding
• Mathematical Proficiency
• How Do Students Learn Mathematics?
• Constructivism
• Sociocultural Theory
• Implications for Teaching Mathematics
• Connecting the Dots
• Resources for Chapter 2
• Self Check
• CHAPTER 3 Teaching through Problem Solving
• Problem Solving
• Teaching for Problem Solving
• Teaching about Problem Solving
• Teaching through Problem Solving
• Teaching Practices for Teaching through Problem Solving
• Ensuring Success for Every Student
• Tasks That Promote Problem Solving
• High-Level Cognitive Demand
• Multiple Entry and Exit Points
• Relevant Contexts
• Evaluating and Adapting Tasks
• Developing Procedural Fluency
• Example Tasks
• What about Drill and Practice?
• Orchestrating Classroom Discourse
• Classroom Discussions
• Questioning Considerations
• How Much to Tell and Not to Tell
• Writing
• Resources for Chapter 3
• Self Check
• CHAPTER 4 Planning in the Problem-Based Classroom
• A Three-Phase Lesson Format
• The Before Lesson Phase
• The During Lesson Phase
• The After Lesson Phase
• Process for Preparing a Lesson
• Step 1: Determine the Learning Goals
• Step 2: Consider Your Students’ Needs
• Step 3: Select, Design, or Adapt a Worthwhile Task
• Step 4: Design Lesson Assessments
• Step 5: Plan the Before Phase
• Step 6: Plan the During Phase
• Step 7: Plan the After Phase
• Step 8: Reflect and Refine
• High-Leverage Routines
• 3-Act Math Tasks
• Number Talks
• Worked Examples
• Warm-ups and Short Tasks
• Learning Centers
• Differentiating Instruction
• Open Questions
• Tiered Lessons
• Parallel Tasks
• Flexible Grouping
• Planning for Family Engagement
• Communicating Mathematics Goals
• Family Math Nights
• Homework Practices
• Resources for Families
• Involving All Families
• Resources for Chapter 4
• Self Check
• CHAPTER 5 Creating Assessments for Learning
• Integrating Assessment into Instruction
• What Are the Main Assessment Types?
• What Should Be Assessed?
• Assessment Methods
• Observations
• Questions
• Interviews
• Tasks
• Rubrics and Their Uses
• Generic Rubrics
• Task-Specific Rubrics
• Student Self-Assessment
• Tests
• Expanding the Usefulness of Tests
• Improving Performance on High‐Stakes Tests
• Communicating Grades and Shaping Instruction
• Grading
• Shaping Instruction
• Resources for Chapter 5
• Self Check
• CHAPTER 6 Teaching Mathematics Equitably to All Students
• Mathematics for Each and Every Student
• Providing for Students Who Struggle and Those with Special Needs
• Multitiered System of Support: Response to Intervention
• Implementing Interventions
• Teaching and Assessing Students with Learning Disabilities
• Adapting for Students with Moderate/Severe Disabilities
• Culturally and Linguistically Diverse Students
• Funds of Knowledge
• Mathematics as a Language
• Culturally Responsive Mathematics Instruction
• Teaching Strategies That Support Culturally and Linguistically Diverse Students
• Focus on Academic Vocabulary
• Foster Student Participation during Instruction
• Implementing Strategies for English Learners
• Providing for Students Who Are Mathematically Gifted
• Acceleration and Pacing
• Depth
• Complexity
• Creativity
• Strategies to Avoid
• Reducing Resistance and Building Resilience
• Give Students Choices That Capitalize on Their Unique Strengths
• Nurture Traits of Resilience
• Make Mathematics Irresistible
• Give Students Leadership in Their Own Learning
• Resources for Chapter 6
• Self Check
• PART II Teaching Student-Centered Mathematics
• CHAPTER 7 Developing Early Number Concepts and Number Sense
• Promoting Good Beginnings
• The Number Core: Quantity, Counting, and Cardinality
• Quantity and the Ability to Subitize
• Counting
• Cardinality
• Thinking about Zero
• Numeral Writing and Recognition
• Counting On and Counting Back
• The Relations Core: More Than, Less Than, and Equal To
• Developing Number Sense by Building Number Relationships
• Relationships between Numbers 1 through 10
• Relationships for Numbers 10 through 20 and Beyond
• Number Sense in Their World
• Calendar Activities
• Estimation and Measurement
• Represent and Interpret Data
• Resources for Chapter 7
• Self Check
• CHAPTER 8 Developing Meanings for the Operations
• Developing Addition and Subtraction Operation Sense
• Addition and Subtraction Problem Structures
• Teaching Addition and Subtraction
• Properties of Addition and Subtraction
• Developing Multiplication and Division Operation Sense
• Multiplication and Division Problem Structures
• Teaching Multiplication and Division
• Properties of Multiplication and Division
• Strategies for Teaching Operations through Contextual Problems
• Resources for Chapter 8
• Self Check
• CHAPTER 9 Developing Basic Fact Fluency
• Teaching and Assessing the Basic Facts
• Developmental Phases for Learning Basic Facts
• Approaches to Teaching Basic Facts
• Teaching Basic Facts Effectively
• Assessing Basic Facts Effectively
• Reasoning Strategies for Addition Facts
• One More Than and Two More Than (Count On)
• Adding Zero
• Doubles
• Combinations of 10
• 10 +
• Making 10
• Use 10
• Using 5 as an Anchor
• Near-Doubles
• Reasoning Strategies for Subtraction Facts
• Think-Addition
• Down under 10
• Take from 10
• Reasoning Strategies for Multiplication and Division Facts
• Foundational Facts: 2, 5, 10, 0, and 1
• Nines
• Derived Multiplication Fact Strategies
• Division Facts
• Reinforcing Basic Fact Mastery
• Games to Support Basic Fact Fluency
• About Drill
• Fact Remediation
• Resources for Chapter 9
• Self Check
• CHAPTER 10 Developing Whole-Number Place-Value Concepts
• Pre-Place-Value Understandings
• Developing Whole-Number Place-Value Concepts
• Integrating Base-Ten Groupings with Counting by Ones
• Integrating Base-Ten Groupings with Words
• Integrating Base-Ten Groupings with Place-Value Notation
• Base-Ten Models for Place Value
• Groupable Models
• Pregrouped Models
• Nonproportional Models
• Activities to Develop Base-Ten Concepts
• Grouping Activities
• Grouping Tens to Make 100
• Equivalent Representations
• Reading and Writing Numbers
• Two-Digit Number Names
• Three-Digit Number Names
• Written Symbols
• Place Value Patterns and Relationships—A Foundation for Computation
• The Hundreds Chart
• Relative Magnitude Using Benchmark Numbers
• Approximate Numbers and Rounding
• Connections to Real‐World Ideas
• Numbers Beyond 1000
• Extending the Place-Value System
• Conceptualizing Large Numbers
• Resources for Chapter 10
• Self Check
• CHAPTER 11 Developing Strategies for Addition and Subtraction Computation
• Toward Computational Fluency
• Connecting Addition and Subtraction to Place Value
• Three Types of Computational Strategies
• Direct Modeling
• Invented Strategies
• Standard Algorithms
• Development of Invented Strategies in Addition and Subtraction
• Creating a Supportive Environment for Invented Strategies
• Models to Support Invented Strategies
• Adding and Subtracting Single-Digit Numbers
• Adding Multidigit Numbers
• Subtraction as “Think-Addition”
• Take-Away Subtraction
• Extensions and Challenges
• Standard Algorithms for Addition and Subtraction
• Standard Algorithm for Addition
• Standard Algorithm for Subtraction
• Introducing Computational Estimation
• Understanding Computational Estimation
• Suggestions for Teaching Computational Estimation
• Computational Estimation Strategies
• Front-End Methods
• Rounding Methods
• Compatible Numbers
• Resources for Chapter 11
• Self Check
• CHAPTER 12 Developing Strategies for Multiplication and Division Computation
• Invented Strategies for Multiplication
• Useful Representations
• Multiplication by a Single-Digit Multiplier
• Multiplication of Multidigit Numbers
• Standard Algorithms for Multiplication
• Begin with Models
• Develop the Written Record
• Invented Strategies for Division
• Standard Algorithm for Division
• Begin with Models
• Develop the Written Record
• Two-Digit Divisors
• A Low-Stress Approach
• Computational Estimation
• Teaching Computational Estimation
• Computational Estimation Strategies
• Resources for Chapter 12
• Self Check
• CHAPTER 13 Algebraic Thinking, Equations, and Functions
• Strands of Algebraic Thinking
• Connecting Number and Algebra
• Number Combinations
• Place-Value Relationships
• Algorithms
• Properties of the Operations
• Making Sense of Properties
• Applying the Properties of Addition and Multiplication
• Study of Patterns and Functions
• Repeating Patterns
• Growing Patterns
• Relationships in Functions
• Graphs of Functions
• Linear Functions
• Meaningful Use of Symbols
• Equal and Inequality Signs
• The Meaning of Variables
• Mathematical Modeling
• Algebraic Thinking across the Curriculum
• Geometry, Measurement and Algebra
• Data and Algebra
• Algebraic Thinking
• Resources for Chapter 13
• Self Check
• CHAPTER 14 Developing Fraction Concepts
• Meanings of Fractions
• Fraction Constructs
• Fraction Language and Notation
• Fraction Size Is Relative
• Models for Fractions
• Area Models
• Length Models
• Set Models
• Fractions as Numbers
• Partitioning
• Iterating
• Magnitude of Fractions
• Equivalent Fractions
• Conceptual Focus on Equivalence
• Equivalent Fraction Models
• Fractions Greater than 1
• Developing an Equivalent-Fraction Algorithm
• Comparing Fractions
• Comparing Fractions Using Number Sense
• Using Equivalent Fractions to Compare
• Teaching Considerations for Fraction Concepts
• Fraction Challenges and Misconceptions
• Resources for Chapter 14
• Self Check
• CHAPTER 15 Developing Fraction Operations
• Understanding Fraction Operations
• Effective Teaching Process
• Addition and Subtraction
• Contextual Examples
• Models
• Estimation
• Developing the Algorithms
• Fractions Greater Than One
• Challenges and Misconceptions
• Multiplication
• Contextual Examples and Models
• Estimation
• Developing the Algorithms
• Factors Greater Than One
• Challenges and Misconceptions
• Division
• Contextual Examples and Models
• Answers That Are Not Whole Numbers
• Estimation
• Developing the Algorithms
• Challenges and Misconceptions
• Resources for Chapter 15
• Self Check
• CHAPTER 16 Developing Decimal and Percent Concepts and Decimal Computation
• Extending the Place-Value System
• The 10-to-1 Relationship—Now in Two Directions!
• The Role of the Decimal Point
• Measurement and Monetary Units
• Precision and Equivalence
• Connecting Fractions and Decimals
• Say Decimal Fractions Correctly
• Use Visual Models for Decimal Fractions
• Multiple Names and Formats
• Developing Decimal Number Sense
• Familiar Fractions Connected to Decimals
• Comparing and Ordering Decimal Fractions
• Computation with Decimals
• Addition and Subtraction
• Multiplication
• Division
• Introducing Percents
• Physical Models and Terminology
• Percent Problems in Context
• Estimation
• Resources for Chapter 16
• Self Check
• CHAPTER 17 Ratios, Proportions, and Proportional Reasoning
• Ratios
• Types of Ratios
• Ratios Compared to Fractions
• Two Ways to Think about Ratio
• Proportional Reasoning
• Types of Comparing Situations
• Covariation
• Strategies for Solving Proportional Situations
• Rates and Scaling Strategies
• Ratio Tables
• Tape or Strip Diagram
• Double Number Line Diagrams
• Equations (Cross Products)
• Percent Problems
• Teaching Proportional Reasoning
• Resources for Chapter 17
• Self Check
• CHAPTER 18 Developing Measurement Concepts
• The Meaning and Process of Measuring
• Concepts and Skills
• Introducing Nonstandard Units
• Introducing Standard Units
• Developing Unit Familiarity
• Measurement Systems and Units
• The Role of Estimation and Approximation
• Strategies for Estimating Measurements
• Measurement Estimation Activities
• Length
• Comparison Activities
• Using Physical Models of Length Units
• Making and Using Rulers
• Conversion
• Area
• Comparison Activities
• Using Physical Models of Area Units
• The Relationship between Area and Perimeter
• Developing Formulas for Perimeter and Area
• Volume and Capacity
• Comparison Activities
• Using Physical Models of Volume and Capacity Units
• Developing Formulas for Volumes of Common Solid Shapes
• Weight and Mass
• Comparison Activities
• Using Physical Models of Weight or Mass Units
• Angles
• Comparison Activities
• Using Physical Models of Angular Measure Units
• Using Protractors
• Time
• Comparison Activities
• Reading Clocks
• Elapsed Time
• Money
• Recognizing Coins and Identifying Their Values
• Resources for Chapter 18
• Self Check
• CHAPTER 19 Developing Geometric Thinking and Geometric Concepts
• Geometry Goals for Students
• Developing Geometric Thinking
• The van Hiele Levels of Geometric Thought
• Implications for Instruction
• Shapes and Properties
• Sorting and Classifying
• Composing and Decomposing Shapes
• Categories of Two- and Three-Dimensional Shapes
• Construction Activities
• Applying Definitions and Categories
• Exploring Properties of Triangles
• Midsegments of a Triangle
• Exploring Properties of Quadrilaterals
• Exploring Polygons
• Circles
• Investigations, Conjectures, and the Development of Proof
• Transformations
• Symmetries
• Composition of Transformations
• Congruence
• Similarity
• Dilation
• Location
• Coordinate Plane
• Measuring Distance on the Coordinate Plane
• Visualization
• Two-Dimensional Imagery
• Three-Dimensional Imagery
• Resources for Chapter 19
• Self Check
• CHAPTER 20 Developing Concepts of Data and Statistics
• What Does It Mean to Do Statistics?
• Is It Statistics or Is It Mathematics?
• The Shape of Data
• The Process of Doing Statistics
• Formulating Questions
• Classroom Questions
• Questions beyond Self and Classmates
• Data Collection
• Sampling
• Using Existing Data Sources
• Data Analysis: Classification
• Attribute Materials
• Data Analysis: Graphical Representations
• Creating Graphs
• Bar Graphs
• Pie Charts and Circle Graphs
• Continuous Data Graphs
• Bivariate Data
• Data Analysis: Measures of Center and Variability
• Measures of Center
• Understanding the Mean
• Choosing a Measure of Center
• Variability
• Analyzing Data
• Interpreting Results
• Resources for Chapter 20
• Self Check
• CHAPTER 21 Exploring Concepts of Probability
• Introducing Probability
• Likely or Not Likely
• The Probability Continuum
• Theoretical Probability and Experiments
• Process for Teaching Probability
• Theoretical Probability
• Experiments
• Why Use Experiments?
• Use of Technology in Experiments
• Sample Spaces and the Probability of Compound Events
• Independent Events
• Area Representation
• Dependent Events
• Simulations
• Student Assumptions Related to Probability
• Resources for Chapter 21
• Self Check
• CHAPTER 22 Developing Concepts of Exponents, Integers, and Real Numbers
• Exponents
• Exponents in Expressions and Equations
• Order of Operations
• Exploring Exponents on the Calculator
• Integer Exponents
• Scientific Notation
• Positive and Negative Numbers
• Contexts for Exploring Positive and Negative Numbers
• Meaning of Negative Numbers
• Tools for Illustrating Positive and Negative Numbers
• Operations with Positive and Negative Numbers
• Addition and Subtraction
• Multiplication
• Division
• Real Numbers
• Rational Numbers
• Irrational Numbers
• Supporting Student Reasoning about Number
• Resources for Chapter 22
• Self Check
• APPENDIX A Standards for Mathematical Practice
• APPENDIX B NCTM Mathematics Teaching Practices from Principles to Actions
• APPENDIX C Guide to Blackline Masters
• APPENDIX D Activities at a Glance
• References
• Index
• A
• B
• C
• D
• E
• F
• G
• H
• I
• J
• K
• L
• M
• N
• O
• P
• Q
• R
• S
• T
• U
• V
• W
• Y
• Z
• Credits