Title Page 1 Dedication 2 About the Authors 3 Table of Contents 4 Introduction 10 Purposes of Mathematics 10 Perspective 11 The What and When of Mathematics Programming 12 A Primary Purpose 13 A Basic Understanding for Teachers 15 Section 1: Problem Solving Precedes Computation 19 Framework for Alternative Representations 20 Utilizing Alternative Representations in Problem Solving 26 What is a Problem? 27 Information Processing in Word Problem Activities 30 Word Problems and Conjunctive/Disjunctive Relationship 31 Selecting the Best Question for a Word Problem 34 Word Problems Using Cloze Procedure 35 Views of Mathematics 38 Problem Solving Precedes Computation 39 Semantics 41 Active and Passive Problem Solving 45 Problem to Match the Question 46 Information Sets to Complete a Problem Activity 46 Display Activities for Problem Creation 46 Developing Vocabulary 47 Teaching Vocabulary 49 Instruction in Vocabulary 50 Syntax 53 The Elements of a Word Problem 54 Classifying Word Problems by Mathematical Constructs 56 Section 2: Developing Word Problems for Diagnostic Feedback 59 Problem Solving, Cognition and Language Complexity 59 Word Problem Solving Activities 60 Problem Characteristics 62 Direct Word Problems 63 Indirect Word Problems 65 Two Step Problems 67 Word Problems Made From Nonsense Words 68 Problem Formats 69 Script/display/picture format 69 Write format 72 Story format 73 Sentence format 74 Nominal Numbers 76 Organizing Quantitative Information 77 Section 3: Connections to Other Subjects 79 Arithmetic Activities and Word Problems Related to Community Concerns 79 Addition 79 Subtraction 80 Multiplication 81 Division 83 Quantitative and Qualitative Distractors 84 Contiguity and Non-Contiguity 85 Definite and Indefinite Quantifiers 86 Word Problems to Address Emotions 86 Formula Types of Word Problems 88 Machines and formulas 88 Work and simple machines 89 Effort and resistance 93 Pre-Algebra Thinking 94 Extended Problem Activities 94 Related Problems 95 Solving a Problem 95 Being a Problem Solver 95 Long-Term Problem Solving 96 Executive Processes 99 Summary 101 Section 4: Arithmetic Computation 102 Preparing to Compute 105 Patterns 105 Pattern Traits - Identify a Pattern and Original Learning 106 Pattern Traits - Identify a Pattern and Intradimensional Shift 107 Pattern Traits - Identify a Pattern and Extradimensional Shift 107 Pattern Traits - Copy a Pattern and Extradimensional Shift 108 Pattern Traits - Extend a Pattern and Extradimensional Shift 108 Sequences 110 Counting 111 Counting Forward and Backward 111 Cardinal Property 113 Skip Counting 113 Naming the Numbers 114 Section 5: Knowing About and Being Able To Do 117 Curricula Choices 119 Alternative Representations 121 Background for the Operations 122 Relations 123 Counting 123 A Pendulum 125 A Balance Scale 125 Counting the 10's and Accounting for the 10's 126 Unusual Combinations of 10's 129 Place Value 130 Estimation 131 Expanded Notation 133 Representations of Quantity 134 Section 6: Communicating Mathematics 137 Knowing About and Doing Addition 140 Addition - Things to Know About Addition 141 Addition - Things to Know When Doing Addition 146 A + H: Memorization of Basic Facts 148 Subtraction - Things to Know About Subtraction 150 Subtraction - Things to Know When Doing Subtraction 152 YAP and YAN 155 Multiplication - Things to Know About Multiplication 156 Multiplication - Things to Know When Doing Multiplication 158 Division - Things to Know About Division 162 Division - Things to Know When Doing Division 163 Section 7: Teaching the Operations Using Whole Numbers 168 Probability Control 171 Controlled Repetition 172 Active versus Passive Activities 172 Error Detection Activities 173 Teaching Addition 173 Teaching Subtraction 180 Alternative Algorithms 184 Left-to-right 184 Without renaming 184 Teaching Multiplication 185 Alternative Representations 189 Array multiplication 194 Two or more digit combinations 192 Estimation 195 Algorithmic Variations 196 Teaching Division 196 Remainders 202 Moving Over 203 Regrouping Partial Dividends 204 Alternative Algorithms 207 Section 8: Hand-Held Calculators 209 Activity-Based Computer Participation 209 Evaluation 216 Section 9: Concluding Comments 218 Appendix 219 References 233.