Elementary and Middle School Mathematics: Teaching Developmentally

Cover
Title Page
Copyright Page
About the Authors
Contents
Preface
SECTION 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
			Principles and Standards for School Mathematics
			Common Core State Standards
			Principles to Actions
		An Invitation to Learn and Grow
			Becoming a Teacher of Mathematics
		Reflections on Chapter 1
			Writing to Learn
			For Discussion and Exploration
		Resources for Chapter 1
			Recommended Readings
	CHAPTER 2 Exploring What it Means to Know and Do Mathematics
		What Does It Mean to Do Mathematics?
			Verbs of Doing Mathematics
		An Invitation to Do Mathematics
			Searching for Patterns
			Analyzing a Situation
			Generalizing Relationships
			Experimenting and Explaining
			Where Are the Answers?
		What Does It Mean to Be Mathematically Proficient?
			Relational Understanding
			Mathematical Proficiency
		How Do Students Learn Mathematics?
			Constructivism
			Sociocultural Theory
			Implications for Teaching Mathematics
		Connecting the Dots
		Reflections on Chapter 2
			Writing to Learn
			For Discussion and Exploration
		Resources for Chapter 2
			Recommended Readings
	CHAPTER 3 Teaching through Problem Solving
		Problem Solving
			Teaching for Problem Solving
			Teaching about Problem Solving
			Teaching through Problem Solving
		Features of Worthwhile Tasks
			High Levels of Cognitive Demand
			Multiple Entry and Exit Points
			Relevant Contexts
			Evaluating and Adapting Tasks
		Developing Concepts and Procedures through Tasks
			Concepts
			Procedures
			What about Drill and Practice?
		Orchestrating Classroom Discourse
			Classroom Discussions
			Questioning Considerations
			How Much to Tell and Not to Tell
			Writing to Learn
		Problem Solving for All
		Reflections on Chapter 3
			Writing to Learn
			For Discussion and Exploration
		Resources for Chapter 3
			Recommended Readings
	CHAPTER 4 Planning in the Problem-Based Classroom
		A Three-Phase Lesson Format
			The Before Phase of a Lesson
			The During Phase of a Lesson
			The After Phase of a Lesson
		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 of the Lesson
			Step 6: Plan the During Phase of the Lesson
			Step 7: Plan the After Phase of the Lesson
			Step 8: Reflect and Refine
		More Options for the Three-Phase Lesson
			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
		Reflections on Chapter 4
			Writing to Learn
			For Discussion and Exploration
		Resources for Chapter 4
			Recommended Readings
	CHAPTER 5 Creating Assessments for Learning
		Integrating Assessment into Instruction
			What Is Assessment?
			What Should Be Assessed?
		Assessment Methods
			Observations
			Interviews
			Tasks
		Rubrics and Their Uses
			Generic Rubrics
			Task-Specific Rubrics
		Writing as an Assessment Tool
		Student Self-Assessment
		Tests
			Improving Performance on High-Stakes Tests
		Communicating Grades and Shaping Instruction
		Reflections on Chapter 5
			Writing to Learn
			For Discussion and Exploration
		Resources for Chapter 5
			Recommended Readings
	CHAPTER 6 Teaching Mathematics Equitably to All Children
		Mathematics for ALL Students
		Providing for Students Who Struggle and Those with Special Needs
			Prevention Models
			Implementing Interventions
			Teaching and Assessing Students with Learning Disabilities
			Teaching Students with Moderate/Severe Disabilities
		Culturally and Linguistically Diverse Students
			Culturally Responsive Instruction
			Focus on Academic Vocabulary
			Facilitating Engagement during Instruction
			Implementing Strategies for English Language Learners
		Providing for Students Who Are Mathematically Gifted
		Creating Gender-Friendly Mathematics Classrooms
			Gender Differences
			What Can You Try?
		Reducing Resistance and Building Resilience
		Reflections on Chapter 6
			Writing to Learn
			For Discussion and Exploration
		Resources for Chapter 6
			Recommended Readings
	CHAPTER 7 Using Technological Tools to Teach Mathematics
		Tools and Technology
		Technology-Supported Learning Activities
		Calculators in Mathematics Instruction
			When to Use a Calculator
			Benefits of Calculator Use
			Graphing Calculators
			Portable Data-Collection Devices
		Appropriate and Strategic Use of Digital Tools
			Concept Instruction
			Problem Solving
			Drill and Reinforcement
		Guidelines for Selecting and Using Digital Resources for Mathematics
			Guidelines for Using Digital Content
			How to Select Appropriate Digital Content
		Mathematics Resources on the Internet
			How to Select Online Resources
			Emerging Technologies
		Reflections on Chapter 7
			Writing to Learn
			For Discussion and Exploration
		Resources for Chapter 7
			Recommended Readings
SECTION II: Development of Mathematical Concepts and Procedures
	CHAPTER 8 Developing Early Number Concepts and Number Sense
		Promoting Good Beginnings
		The Number Core: Quantity, Counting, and Knowing How Many
			Quantity and the Ability to Subitize
			Early Counting
			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
			Data Collection and Analysis
		Reflections on Chapter 8
			Writing to Learn
			For Discussion and Exploration
		Resources for Chapter 8
			Literature Connections
			Recommended Readings
	CHAPTER 9 Developing Meanings for the Operations
		Teaching Operations through Contextual Problems
		Addition and Subtraction Problem Structures
			Change Problems
			Part-Part-Whole Problems
			Compare Problems
			Problem Difficulty
		Teaching Addition and Subtraction
			Contextual Problems
			Model-Based Problems
			Properties of Addition and Subtraction
		Multiplication and Division Problem Structure
			Equal-Group Problems
			Comparison Problems
			Area and Array Problems
			Combination Problems
		Teaching Multiplication and Division
			Contextual Problems
			Remainders
			Model-Based Problems
			Properties of Multiplication and Division
		Strategies for Solving Contextual Problems
			Analyzing Context Problems
			Multistep Problems
		Reflections on Chapter 9
			Writing to Learn
			For Discussion and Exploration
		Resources for Chapter 9
			Literature Connections
			Recommended Readings
	CHAPTER 10 Developing Basic Fact Fluency
		Developmental Phases for Learning the Basic Facts
		Teaching and Assessing the Basic Facts
			Different Approaches to Teaching the Basic Facts
			Teaching Basic Facts Effectively
			Assessing Basic Facts Effectively
		Reasoning Strategies for Addition Facts
			One More Than and Two More Than
			Adding Zero
			Doubles
			Combinations of 10
			Making 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, 0, 1
			Nifty Nines
			Derived Multiplication Fact Strategies
			Division Facts
		Reinforcing Basic Fact Mastery
			Games to Support Basic Fact Fluency
			About Drill
			Fact Remediation
		Reflections on Chapter 10
			Writing to Learn
			For Discussion and Exploration
		Resources for Chapter 10
			Literature Connections
			Recommended Readings
	CHAPTER 11 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
		Developing Base-Ten Concepts
			Grouping Activities
			Grouping Tens to Make 100
			Equivalent Representations
		Oral and Written Names for Numbers
			Two-Digit Number Names
			Three-Digit Number Names
			Written Symbols
		Patterns and Relationships with Multidigit Numbers
			The Hundreds Chart
			Relationships with Benchmark Numbers
			Connections to Real-World Ideas
		Numbers Beyond 1000
			Extending the Place-Value System
			Conceptualizing Large Numbers
		Reflections on Chapter 11
			Writing to Learn
			For Discussion and Exploration
		Resources for Chapter 11
			Literature Connections
			Recommended Readings
	CHAPTER 12 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
			Creating a Supportive Environment
			Models to Support Invented Strategies
		Development of Invented Strategies for Addition and Subtraction
			Single-Digit Numbers
			Adding Two-Digit 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
		Reflections on Chapter 12
			Writing to Learn
			For Discussion and Exploration
		Resources for Chapter 12
			Literature Connections
			Recommended Readings
	CHAPTER 13 Developing Strategies for Multiplication and Division Computation
		Student-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
		Student-Invented Strategies for Division
		Standard Algorithm for Division
			Begin with Models
			Develop the Written Record
			Two-Digit Divisors
		Computational Estimation in Multiplication and Division
			Suggestions for Teaching Computational Estimation
			Computational Estimation Strategies
		Reflections on Chapter 13
			Writing to Learn
			For Discussion and Exploration
		Resources for Chapter 13
			Literature Connections
			Recommended Readings
	CHAPTER 14 Algebraic Thinking, Equations, and Functions
		Strands of Algebraic Thinking
		Structure in the Number System: Connecting Number and Algebra
			Number Combinations
			Place-Value Relationships
			Algorithms
		Structure in the Number System: Properties
			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
			Describing 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
		Reflections on Chapter 14
			Writing to Learn
			For Discussion and Exploration
		Resources for Chapter 14
			Literature Connections
			Recommended Readings
	CHAPTER 15 Developing Fraction Concepts
		Meanings of Fractions
			Fraction Constructs
			Why Fractions Are Difficult
		Models for Fractions
			Area Models
			Length Models
			Set Models
		Fractional Parts
			Fraction Size Is Relative
			Partitioning
			Sharing Tasks
			Iterating
			Fraction Notation
		Equivalent Fractions
			Conceptual Focus on Equivalence
			Equivalent Fraction Models
			Developing an Equivalent-Fraction Algorithm
		Comparing Fractions
			Comparing Fractions Using Number Sense
			Using Equivalent Fractions to Compare
			Estimating with Fractions
		Teaching Considerations for Fraction Concepts
		Reflections on Chapter 15
			Writing to Learn
			For Discussion and Exploration
		Resources for Chapter 15
			Literature Connections
			Recommended Readings
	CHAPTER 16 Developing Fraction Operations
		Understanding Fraction Operations
			A Problem-Based Number-Sense Approach
		Addition and Subtraction
			Contextual Examples and Invented Strategies
			Models
			Estimation and Informal Methods
			Developing the Algorithms
			Fractions Greater Than One
			Addressing Misconceptions
		Multiplication
			Contextual Examples and Models
			Estimation and Invented Strategies
			Developing the Algorithms
			Factors Greater Than One
			Addressing Misconceptions
		Division
			Contextual Examples and Models
			Answers That Are Not Whole Numbers
			Estimation and Invented Strategies
			Developing the Algorithms
			Addressing Misconceptions
		Reflections on Chapter 16
			Writing to Learn
			For Discussion and Exploration
		Resources for Chapter 16
			Literature Connections
			Recommended Readings
	CHAPTER 17 Developing Concepts of Decimals and Percents
		Extending the Place-Value System
			The 10-to-1 Relationship—Now in Two Directions!
			The Role of the Decimal Point
		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
			Density of Decimals
		Computation with Decimals
			Addition and Subtraction
			Multiplication
			Division
		Introducing Percents
			Physical Models and Terminology
			Percent Problems in Context
			Estimation
		Reflections on Chapter 17
			Writing to Learn
			For Discussion and Exploration
		Resources for Chapter 17
			Literature Connections
			Recommended Readings
	CHAPTER 18 Ratios, Proportions, and Proportional Reasoning
		Ratios
			Types of Ratios
			Ratios Compared to Fractions
			Two Ways to Think about Ratio
		Proportional Reasoning
			Proportional and Nonproportional Situations
			Additive and Multiplicative Comparisons in Story Problems
			Covariation
		Strategies for Solving Proportional Situations
			Rates and Scaling Strategies
			Ratio Tables
			Tape or Strip Diagram
			Double Number Line Diagrams
			Percents
			Equations
		Teaching Proportional Reasoning
		Reflections on Chapter 18
			Writing to Learn
			For Discussion and Exploration
		Resources for Chapter 18
			Literature Connections
			Recommended Readings
	CHAPTER 19 Developing Measurement Concepts
		The Meaning and Process of Measuring
			Concepts and Skills
			Introducing Nonstandard Units
			Introducing Standard Units
			The Role of Estimation and Approximation
		Length
			Comparison Activities
			Using Physical Models of Length Units
			Conversion
			Making and Using Rulers
		Area
			Comparison Activities
			Using Physical Models of Area Units
			The Relationship between Area and Perimeter
			Developing Formulas for Area
			Areas of Rectangles, Parallelograms, Triangles, and Trapezoids
			Circumference and Area of Circles
		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
			Solving Problems with Time
		Money
			Recognizing Coins and Identifying Their Values
		Reflections on Chapter 19
			Writing to Learn
			For Discussion and Exploration
		Resources for Chapter 19
			Literature Connections
			Recommended Readings
	CHAPTER 20 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
			Investigations, Conjectures, and the Development of Proof
		Transformations
			Line Symmetry
			Rigid Motions
			Congruence
			Similarity
			Using Transformations and Symmetries
		Location
			Measuring Distance on the Coordinate Plane
		Visualization
			Two-Dimensional Imagery
			Three-Dimensional Imagery
			The Platonic Solids
		Reflections on Chapter 20
			Writing to Learn
			For Discussion and Exploration
		Resources for Chapter 20
			Literature Connections
			Recommended Readings
	CHAPTER 21 Developing Concepts of Data Analysis
		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
			Beyond One Classroom
		Data Collection
			Collecting Data
			Using Existing Data Sources
		Data Analysis: Classification
			Attribute Materials
		Data Analysis: Graphical Representations
			Creating Graphs
			Analyzing Graphs
			Bar Graphs
			Pie Charts/Circle Graphs
			Continuous Data Graphs
			Bivariate Graphs
		Data Analysis: Measures of Center and Variability
			Measures of Center
			Understanding the Mean: Two Interpretations
			Choosing a Measure of Center
			Variability
		Interpreting Results
		Reflections on Chapter 21
			Writing to Learn
			For Discussion and Exploration
		Resources for Chapter 21
			Literature Connections
			Recommended Readings
	CHAPTER 22 Exploring Concepts of Probability
		Introducing Probability
			Likely or Not Likely
			The Probability Continuum
		Theoretical Probability and Experiments
			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
		Common Misconceptions about Probability
		Reflections on Chapter 22
			Writing to Learn
			For Discussion and Exploration
		Resources for Chapter 22
			Literature Connections
			Recommended Readings
	CHAPTER 23 Developing Concepts of Exponents, Integers, and Real Numbers
		Exponents
			Exponents in Expressions and Equations
			Order of Operations
			Integer Exponents
			Scientific Notation
		Positive and Negative Numbers
			Contexts for Exploring Positive and Negative Numbers
			Meaning of Negative Numbers
			Models for Teaching Positive and Negative Numbers
		Operations with Positive and Negative Numbers
			Addition and Subtraction
			Multiplication and Division
		Real Numbers
			Rational Numbers
			Square Roots and Cube Roots
		Reflections on Chapter 23
			Writing to Learn
			For Discussion and Exploration
		Resources for Chapter 23
			Literature Connections
			Recommended Readings
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
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	B
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Credits