James Heddens is a Professor Emeritus from Kent State University.

• Chapter 1 Learning Mathematics 1Societal Needs 2Calls for Change 3National and International Achievement 3The Identification of Standards 5Principles and Standards for School Mathematics 6Implementing the Process Standards 8Describing A Contemporary Mathematics Program 10Levels of Abstraction 11Theoretical Foundations for a ContemporaryMathematics Program 12Research Snapshot 13Theory into Practice 17Closure 18Teaching Competencies and Self-Assessment Tasks 19Chapter 2 Teaching Mathematics 21The NCTM Principles and Standards for School Mathematics 22The NCTM Professional Teaching Standards: Mathematics Teaching Today 23An Example of a Teaching Standard 25Lesson Planning Processes and Purposes 29Issues in Designing Mathematics Lessons 33Mathematics Objectives 35The Role of Motivation and Follow-Up 36The Assessment of Learning and Teaching 37Research Snapshot 38Adjusting Instruction to Meet Individual Needs 38Classroom Instructional Aids 39Closure 40Teaching Competencies and Self-Assessment Tasks 41Chapter 3 Assessing Mathematics 43NCTM Assessment Standards for School Mathematics 44An Example of an Assessment Standard 47A Classroom Assessment Vignette 48Purposes for Assessment 49Assessment Strategies 51Research Snapshot 53Closure 58Teaching Competencies and Self-Assessment Tasks 59Chapter 4 Technology in K–8 Mathematics 61A Brief History of Technology’s Influence on Education 62Calculators and Exploration of Mathematical Concepts 63Research Snapshot 64Calculator Features and Functions 66Examples of Effective Calculator Use 66Tools for Measuring Motion 68Computers and Mathematics Instruction 69Instructional Software in the Mathematics Classroom 71Application Software in the Mathematics Classroom 72Virtual Manipulatives 73Design Issues and Advantages of Virtual Manipulatives 74Classification Attributes for Virtual Manipulatives 75Capsule Descriptions of the National Library of Virtual Manipulatives 76Research Snapshot 80Multimedia in the Mathematics Classroom 81The Internet in the Mathematics Classroom 82Programming Computers and Calculators 83Closure 84Teaching Competencies and Self-Assessment Tasks 85Chapter 5 Problem Solving and Mathematical Language 87The Nature of Problem Solving 88Problem-Solving Strategies 91Research Snapshot 93Approaching Word Problems Effectively 98Logic and Reasoning 101The Language of Logic 101The Language of Mathematics 103Closure 107Practice Exercises for Teachers 108Teaching Competencies and Self-Assessment Tasks 111Sample Instructional and Assessment Activities 112Chapter 6 Number Sense, Numeration, and Place Value 115A Foundation for Mathematics Development 116Patterns and Other Relationships in the Primary Curriculum 117Number Sense 118Number Relations 123Extending Numberness Concepts 125Place-Value Numeration 125Decimal Numeration 126Ancient Numeration Systems 131Research Snapshot 133Number Bases Other Than Ten 134Closure 137Practice Exercises for Teachers 138Teaching Competencies and Self-Assessment Tasks 141Sample Instructional and Assessment Activities 142Chapter 7 Addition and Subtraction of Whole Numbers 145Creating a Sound Base for Addition and Subtraction 146Meaning and Models for Addition 147Developing Basic Addition Facts 148Memorizing Basic Addition Facts Using Structures 149Extending Basic Addition Facts Using Place Value 155Regrouping in the Addition Process 156Meaning and Models for Subtraction 157Research Snapshot 158Developing Basic Subtraction Facts 160Memorizing Basic Subtraction Facts Using Structures 161Extending Basic Subtraction Facts Using Place Value 162Regrouping in the Subtraction Process 163 Closure 164Practice Exercises for Teachers 166Teaching Competencies and Self-Assessment Tasks 167Sample Instructional and Assessment Activities 167Chapter 8 Multiplication and Division of Whole Numbers 171Meaning and Models for Multiplication 172Developing and Memorizing Basic Multiplication Facts 175Expanding Basic Multiplication Facts Through Place Value 178Regrouping in Multiplication 181Meaning and Models for Division 183Research Snapshot 183Developing and Memorizing Basic Division Facts 186Remainders in Division 187Expanding Basic Division Facts Through Place Value 187Closure 190Practice Exercises for Teachers 191Teaching Competencies and Self-Assessment Tasks 192Sample Instructional and Assessment Activities 193Chapter 9 Number Theory and Number Systems 197The Language of Number Theory 199The Sieve of Eratosthenes 200Factor Trees 202Prime Factorization 203Divisibility ‘‘Rules’’ 205Research Snapshot 207Least Common Multiples and Greatest Common Factors 207Number Systems 209Closure 218Practice Exercises for Teachers 218Teaching Competencies and Self-Assessment Tasks 221Sample Instructional and Assessment Activities 222Chapter 10 Algebraic Reasoning: GeneralizingPatterns and Relationships 225The Content of Algebra 226Research Snapshot 227Algebra in Problem Solving 228Promoting Algebraic Thinking in the Lower Elementary Grades 229Enhancing Algebraic Thinking in the Upper Elementary and Middle Grades 232Closure 236Practice Exercises for Teachers 237Teaching Competencies and Self-Assessment Tasks 240Sample Instructional and Assessment Activities 241Chapter 11 Rational Numbers Expressed as Fractions: Concepts 245A Fraction of the History of Fractional Numbers 246Rational Numbers—A Definition and Description 247Research Snapshot 247The Regions Model for Fractional Numbers 248The Groups of Objects Model for Fractional Numbers 250The Number Line Model for Fractional Numbers 251Knowledge and Understandings Prior to Operations on Rational Numbers Expressed as Fractions 253Closure 258Practice Exercises for Teachers 260Teaching Competencies and Self-Assessment Tasks 262Sample Instructional and Assessment Activities 263Chapter 12 Rational Numbers Expressed as Fractions: Operations 267Research Snapshot 268Addition of Fractional Numbers 269Subtraction of Fractional Numbers 274Multiplication of Fractional Numbers 278Division of Fractional Numbers 282Closure 285Practice Exercises for Teachers 286Teaching Competencies and Self-Assessment Tasks 288Sample Instructional and Assessment Activities 288Chapter 13 Rational Numbers Expressed as Decimals 291Models That Give Decimals Meaning 292Place Value—A Foundation for Decimals 293Exponential Notation 295Addition with Decimal Numbers 296Subtraction with Decimal Numbers 297Multiplication with Decimal Numbers 299Division with Decimal Numbers 300Relating Decimals and Fractions 301Scientific Notation 303Ratio, Proportion, and Percent 304Research Snapshot 305Closure 311Practice Exercises for Teachers 312Teaching Competencies and Self-Assessment Tasks 313Sample Instructional and Assessment Activities 314Chapter 14 Data Analysis: Graphs, Statistics, and Probability 317A Child’s View of Statistics 318Measures of Central Tendency in Data 319Research Snapshot 320Organizing and Interpreting Data 321Taking a ‘‘Chance’’ 326Experimental Probability 326Theoretical Probability 327Permutations and Combinations 328Models for Exploring Probability 330Closure 331Practice Exercises for Teachers 332Teaching Competencies and Self-Assessment Tasks 334Sample Instructional and Assessment Activities 335Chapter 15 Measurement 339The Historical Development of Measurement Systems 340Contemporary Measurement Systems 341The Process of Measurement 345Computing with Denominate Numbers 349Money and Time as Measures 350Research Snapshot 352Closure 354Practice Exercises for Teachers 354Teaching Competencies and Self-Assessment Tasks 356Sample Instructional and Assessment Activities 357Chapter 16 Geometry: Basic Concepts and Structures 361Research Snapshot 362Early Experiences in Geometry 363The ‘‘Building Blocks’’ of Geometry 364Extending the Basics—Curves, Regions, and Rays 366Angles and Angle Measure 369Exploring Geometric Constructions 372Symmetry and Transformational Geometry 374Other Devices for Exploring Geometric Concepts 376Closure 377Practice Exercises for Teachers 378Teaching Competencies and Self-Assessment Tasks 381Sample Instructional and Assessment Activities 382Chapter 17 Geometry: Polygons and Polyhedra 385Defining Polygons 386Research Snapshot 386A Closer Look at Triangles 388A Closer Look at Quadrilaterals 394A Closer Look at Circles 395Area, Perimeter, and Circumference Measurement 397Defining Polyhedra 401Closure 409Practice Exercises for Teachers 410Teaching Competencies and Self-Assessment Tasks 414Sample Instructional and Assessment Activities 415Index 417

James Heddens is a Professor Emeritus from Kent State University.

• Chapter 1 Learning Mathematics 1Societal Needs 2Calls for Change 3National and International Achievement 3The Identification of Standards 5Principles and Standards for School Mathematics 6Implementing the Process Standards 8Describing A Contemporary Mathematics Program 10Levels of Abstraction 11Theoretical Foundations for a ContemporaryMathematics Program 12Research Snapshot 13Theory into Practice 17Closure 18Teaching Competencies and Self-Assessment Tasks 19Chapter 2 Teaching Mathematics 21The NCTM Principles and Standards for School Mathematics 22The NCTM Professional Teaching Standards: Mathematics Teaching Today 23An Example of a Teaching Standard 25Lesson Planning Processes and Purposes 29Issues in Designing Mathematics Lessons 33Mathematics Objectives 35The Role of Motivation and Follow-Up 36The Assessment of Learning and Teaching 37Research Snapshot 38Adjusting Instruction to Meet Individual Needs 38Classroom Instructional Aids 39Closure 40Teaching Competencies and Self-Assessment Tasks 41Chapter 3 Assessing Mathematics 43NCTM Assessment Standards for School Mathematics 44An Example of an Assessment Standard 47A Classroom Assessment Vignette 48Purposes for Assessment 49Assessment Strategies 51Research Snapshot 53Closure 58Teaching Competencies and Self-Assessment Tasks 59Chapter 4 Technology in K–8 Mathematics 61A Brief History of Technology’s Influence on Education 62Calculators and Exploration of Mathematical Concepts 63Research Snapshot 64Calculator Features and Functions 66Examples of Effective Calculator Use 66Tools for Measuring Motion 68Computers and Mathematics Instruction 69Instructional Software in the Mathematics Classroom 71Application Software in the Mathematics Classroom 72Virtual Manipulatives 73Design Issues and Advantages of Virtual Manipulatives 74Classification Attributes for Virtual Manipulatives 75Capsule Descriptions of the National Library of Virtual Manipulatives 76Research Snapshot 80Multimedia in the Mathematics Classroom 81The Internet in the Mathematics Classroom 82Programming Computers and Calculators 83Closure 84Teaching Competencies and Self-Assessment Tasks 85Chapter 5 Problem Solving and Mathematical Language 87The Nature of Problem Solving 88Problem-Solving Strategies 91Research Snapshot 93Approaching Word Problems Effectively 98Logic and Reasoning 101The Language of Logic 101The Language of Mathematics 103Closure 107Practice Exercises for Teachers 108Teaching Competencies and Self-Assessment Tasks 111Sample Instructional and Assessment Activities 112Chapter 6 Number Sense, Numeration, and Place Value 115A Foundation for Mathematics Development 116Patterns and Other Relationships in the Primary Curriculum 117Number Sense 118Number Relations 123Extending Numberness Concepts 125Place-Value Numeration 125Decimal Numeration 126Ancient Numeration Systems 131Research Snapshot 133Number Bases Other Than Ten 134Closure 137Practice Exercises for Teachers 138Teaching Competencies and Self-Assessment Tasks 141Sample Instructional and Assessment Activities 142Chapter 7 Addition and Subtraction of Whole Numbers 145Creating a Sound Base for Addition and Subtraction 146Meaning and Models for Addition 147Developing Basic Addition Facts 148Memorizing Basic Addition Facts Using Structures 149Extending Basic Addition Facts Using Place Value 155Regrouping in the Addition Process 156Meaning and Models for Subtraction 157Research Snapshot 158Developing Basic Subtraction Facts 160Memorizing Basic Subtraction Facts Using Structures 161Extending Basic Subtraction Facts Using Place Value 162Regrouping in the Subtraction Process 163 Closure 164Practice Exercises for Teachers 166Teaching Competencies and Self-Assessment Tasks 167Sample Instructional and Assessment Activities 167Chapter 8 Multiplication and Division of Whole Numbers 171Meaning and Models for Multiplication 172Developing and Memorizing Basic Multiplication Facts 175Expanding Basic Multiplication Facts Through Place Value 178Regrouping in Multiplication 181Meaning and Models for Division 183Research Snapshot 183Developing and Memorizing Basic Division Facts 186Remainders in Division 187Expanding Basic Division Facts Through Place Value 187Closure 190Practice Exercises for Teachers 191Teaching Competencies and Self-Assessment Tasks 192Sample Instructional and Assessment Activities 193Chapter 9 Number Theory and Number Systems 197The Language of Number Theory 199The Sieve of Eratosthenes 200Factor Trees 202Prime Factorization 203Divisibility ‘‘Rules’’ 205Research Snapshot 207Least Common Multiples and Greatest Common Factors 207Number Systems 209Closure 218Practice Exercises for Teachers 218Teaching Competencies and Self-Assessment Tasks 221Sample Instructional and Assessment Activities 222Chapter 10 Algebraic Reasoning: GeneralizingPatterns and Relationships 225The Content of Algebra 226Research Snapshot 227Algebra in Problem Solving 228Promoting Algebraic Thinking in the Lower Elementary Grades 229Enhancing Algebraic Thinking in the Upper Elementary and Middle Grades 232Closure 236Practice Exercises for Teachers 237Teaching Competencies and Self-Assessment Tasks 240Sample Instructional and Assessment Activities 241Chapter 11 Rational Numbers Expressed as Fractions: Concepts 245A Fraction of the History of Fractional Numbers 246Rational Numbers—A Definition and Description 247Research Snapshot 247The Regions Model for Fractional Numbers 248The Groups of Objects Model for Fractional Numbers 250The Number Line Model for Fractional Numbers 251Knowledge and Understandings Prior to Operations on Rational Numbers Expressed as Fractions 253Closure 258Practice Exercises for Teachers 260Teaching Competencies and Self-Assessment Tasks 262Sample Instructional and Assessment Activities 263Chapter 12 Rational Numbers Expressed as Fractions: Operations 267Research Snapshot 268Addition of Fractional Numbers 269Subtraction of Fractional Numbers 274Multiplication of Fractional Numbers 278Division of Fractional Numbers 282Closure 285Practice Exercises for Teachers 286Teaching Competencies and Self-Assessment Tasks 288Sample Instructional and Assessment Activities 288Chapter 13 Rational Numbers Expressed as Decimals 291Models That Give Decimals Meaning 292Place Value—A Foundation for Decimals 293Exponential Notation 295Addition with Decimal Numbers 296Subtraction with Decimal Numbers 297Multiplication with Decimal Numbers 299Division with Decimal Numbers 300Relating Decimals and Fractions 301Scientific Notation 303Ratio, Proportion, and Percent 304Research Snapshot 305Closure 311Practice Exercises for Teachers 312Teaching Competencies and Self-Assessment Tasks 313Sample Instructional and Assessment Activities 314Chapter 14 Data Analysis: Graphs, Statistics, and Probability 317A Child’s View of Statistics 318Measures of Central Tendency in Data 319Research Snapshot 320Organizing and Interpreting Data 321Taking a ‘‘Chance’’ 326Experimental Probability 326Theoretical Probability 327Permutations and Combinations 328Models for Exploring Probability 330Closure 331Practice Exercises for Teachers 332Teaching Competencies and Self-Assessment Tasks 334Sample Instructional and Assessment Activities 335Chapter 15 Measurement 339The Historical Development of Measurement Systems 340Contemporary Measurement Systems 341The Process of Measurement 345Computing with Denominate Numbers 349Money and Time as Measures 350Research Snapshot 352Closure 354Practice Exercises for Teachers 354Teaching Competencies and Self-Assessment Tasks 356Sample Instructional and Assessment Activities 357Chapter 16 Geometry: Basic Concepts and Structures 361Research Snapshot 362Early Experiences in Geometry 363The ‘‘Building Blocks’’ of Geometry 364Extending the Basics—Curves, Regions, and Rays 366Angles and Angle Measure 369Exploring Geometric Constructions 372Symmetry and Transformational Geometry 374Other Devices for Exploring Geometric Concepts 376Closure 377Practice Exercises for Teachers 378Teaching Competencies and Self-Assessment Tasks 381Sample Instructional and Assessment Activities 382Chapter 17 Geometry: Polygons and Polyhedra 385Defining Polygons 386Research Snapshot 386A Closer Look at Triangles 388A Closer Look at Quadrilaterals 394A Closer Look at Circles 395Area, Perimeter, and Circumference Measurement 397Defining Polyhedra 401Closure 409Practice Exercises for Teachers 410Teaching Competencies and Self-Assessment Tasks 414Sample Instructional and Assessment Activities 415Index 417