Foundations for Teaching Elementary and Middle School Mathematics

Chapter 1 A Fresh Perspective of Mathematical Literacy 
Introduction
Teaching and Learning through Mathematical Activity
Intellectual Development in Mathematics
Genesis 
Mathematical Modeling
Discourse
Specialized Knowledge for Teaching Elementary Mathematics
Counting 
Patterns
Measurement and Estimation
Change and Invariance
Learning Progressions 
What Is Algebra, Really? 
Conclusion
Chapter 2 Constructing a Mathematics Learning Environment 13 Introduction 
Mathematics as Problem-Solving
Mathematics as Communication
Building Meta-representational Competency
English Language Learners
Learners with Special Needs
Mathematics as Collaboration
Learning to Learn
Student Disengagement in the Mathematics Classroom
Teaching Mathematics by Design
Lesson Design
A Few Words about Assessment
Conclusion
Chapter 3 Mathematical Foundations for Early Childhood and Kindergarten 
Introduction
Observations  
Checklists 
Children’s Work 
Children with Special Needs 
English Language Learners 
Standards and Learning Outcomes for Early Childhood and Kindergarten 
Number and Operations
Counting 
Matching and Correspondence  
Subitizing
Cardinality 
Data 
Sorting and Classification 
Comparison (Relative Magnitude)  
Measurement
Area
Capacity and Volume
Weight
Geometry
Attributes
Patterns
Relations of Objects in Space
Exploring Part–Part–Whole Relationships
A Culmination of Ideas: Modeling Addition and Subtraction
Exercises for the Teacher
Chapter 4 Mathematical Foundations for First Grade Introduction 
Assessment in First Grade
Number Talks (Math Talks)
Exit Tickets
Standards and Learning Outcomes for First Grade
Number and Operations
Initial Base Ten Concepts
Place Value Notation
The Number Line
Early Algebra 
Number Sentences and Equivalence
Addition sentences
Subtraction sentences
Geometry and Measurement
Length 
Area 
Volume
Time 
Introducing Fraction as Measure 
Data 
Sorting and Classification 
Tally Graphs 
A Culmination of Ideas: A First Look at Modeling Arithmetic with Area 
Exercises for the Teacher
Chapter 5 Mathematical Foundations for Second Grade 93 Introduction 
Assessment in Second Grade
Word Walls
Word Cards
Standards and Learning Outcomes for Second Grade
Number and Operations
Solving Equations
Odd and Even Numbers
Standard Algorithms for Addition and Subtraction
Modeling Addition with Base Ten
Modeling Subtraction with Base Ten 
Patterns 
Unit Fractions 
Early Algebra  
Addition and Subtraction Operations as Functions 
Geometry and Measurement  
Length
Area 
Data 
A Culmination of Ideas: The Double Number Line
Exercises for the Teacher  
Chapter 6 Mathematical Foundations for Third Grade 
Introduction 
Assessment in Third Grade
Authentic Assessment
Journals as Formative Assessment
Standards and Learning Outcomes for Third Grade 
Number and Operations
Base Ten and Place Value
Developing Multiplicative Reasoning Using Array Models
Developing Division Reasoning Using Array Models
Multiplication and Division Contexts
Developing Understanding of the Rational Numbers
Fractions on the Number Line
Geometry
Relative Magnitude
Estimation
Computational Estimation
Spatial Reasoning
Area
Volume  
Early Algebra  
Functional Thinking and the Number Line 
A Culmination of Ideas: Vector Diagrams
Addition 
Subtraction 
Exercises for the Teacher 
Chapter 7 Mathematical Foundations for Fourth Grade 
Introduction  
Assessment in Fourth Grade   
Rubrics 
Elements of a rubric 
Constructing a rubric 
Standards and Learning Outcomes for Fourth Grade 
Number and Operations 
Factors and Multiples 
Prime Numbers 
Making Arrays 
The Greatest Common Factor and the Least Common Multiple
Factor Trees 
The Ladder Method
Fractions as Measure   
Equivalence
Relative Magnitude 
Area model for Adding Fractions 
Area model for Subtracting Fractions
Area Model for Multiplying Fractions
Fractions, Place Value, and Decimals  
Percentages 
Area Model for Multiplication of Whole Numbers
The Lattice Algorithm 
Napier’s Bones
Models for Division of Whole Numbers 
Array Model 
Base Ten Model 
Geometry 
The Circle
Angles  
Symmetry
Rotation
Reflection 
Translation   
Dilation 
Early Algebra
Rules of the form y 5 mx + b in Tables 
Flow Diagrams  
Data  
Stem-and-Leaf Plots 
A Culmination of Ideas: From Dilations to Algebra
Exercises for the Teacher
Chapter 8 Mathematical Foundations for Fifth Grade 
Introduction
Assessment in Fifth Grade
Formative Assessment
Standards and Learning Outcomes for Fifth Grade 
Number and Operations
Fraction as Measure Revisited
Mixed Numbers
Fraction Division
The Common Denominator Algorithm
Adding and Subtracting Mixed Numbers
Multiplying Mixed Numbers
Dividing Mixed Numbers 
Decimals
Adding and Subtracting Decimals 
Multiplying Decimals 
Dividing Decimals
Geometry and Early Algebra
Ratio and Proportion
Percentages as Proportions 
Ratio Boxes
The Coordinate (Cartesian) Plane
Scatterplots
A Culmination of Ideas: The Mathematics of Change
Exercises for the Teacher
Chapter 9 Mathematical Foundations for Sixth Grade
Introduction
Assessment in Sixth Grade
Standards and Learning Outcomes for Sixth Grade
Number and Operations
Equivalent Expressions
Square Roots
Integers (Signed Numbers)
Addition Using Vector Diagrams 
Subtraction Using Vector Diagrams 
Multiplication Using Vector Diagrams
Division Using Vector Diagrams  
Geometry and Early Algebra 
The Pythagorean Theorem 
Taxicab versus Euclidean Distance 
Proportionality  
Inequalities
Constructing Area Formulas
The parallelogram 
The triangle
The trapezoid 
The circle 
Data 
Distribution
Mean and Median
Box Plots
Probability 
The Probability Continuum 
Theoretical versus Experimental Probability
Sample Spaces
A Culmination of Ideas: The Geometric Mean 
Exercises for the Teacher
Chapter 10 Mathematical Foundations for Middle School 
Introduction
Lesson Design in Middle School 
Standards and Learning Outcomes for Middle School
Number and Operations and Algebra 
Infinite Series
Performing long division to calculate infinite series 
Irrational Numbers
Square Roots 
Nonlinear Relationships
Quadratic relations
Exponential relations
Geometry and Proportionality
Area
Volume 
Formal Symmetry 
Rigid Transformations and Congruence
Right Triangle Trigonometry
Data
Lines of Best Fit
Mean Absolute Deviation
Probability
Probability Rules
Independent Events
Dependent Events 
Mutually Exclusive Events
Exercises for the Teacher
Curriculum 
Resources 
Bibliography