Children’s Reasoning While Building Fraction Ideas

TABLE OF CONTENTS......Page 6
FOREWORD......Page 8
ACKNOWLEDGEMENTS......Page 12
INTRODUCTION......Page 14
 REFERENCES......Page 17
 1.1 OVERVIEW......Page 18
 1.2 PERSPECTIVE......Page 19
 1.4 FRACTIONS IN THE CURRICULUM......Page 20
 1.6 TASKS......Page 21
 1.9 ORGANIZATION OF CHAPTERS......Page 22
 1.10 CUISENAIRE® RODS AS TOOLS......Page 23
  1.12.1 Reasoning by Cases......Page 24
  1.12.4 Reasoning using the Generic Example......Page 25
 REFERENCES......Page 26
 2.1 INTRODUCTION......Page 29
  Task 1: I Claim That the Light Green Rod Is Half as long as the Dark Green Rod. What Do You Think? What Would You Do to Convince Me?......Page 30
  Task 2: Someone Told Me That the Red Rod Is Half as long as the Yellow Rod. What Do You Think?......Page 31
  Task 4: Someone Told Me That the Red Rod Is One Third as long as the Dark Green Rod. What Do You Think?......Page 32
  Task 5: What Number Name Would I Have to Give to Dark GreenIf I Wanted Red to be One?......Page 33
 2.4 STUDENT-CREATED TASKS......Page 34
 REFERENCES......Page 36
  Task: If I Call the Blue Rod One, What Rod Will I Call One Half?......Page 37
 3.3 STUDENTS DEVELOP AND DEFEND THEIR SOLUTION......Page 38
 3.4 REVIEWING THE REASONING......Page 42
 REFERENCES......Page 43
 4.2 A DESIGN TASK......Page 44
 4.3 GENERALIZING THE SOLUTION......Page 46
 REFERENCES......Page 47
 5.2 ANOTHER TASK WITH NO SOLUTION......Page 49
 5.3 A WHOLE CLASS DISCUSSION......Page 50
 REFERENCES......Page 54
  Task 1: What If I Change the Name of the Orange Rod to Six? What NumberName Would I Call the Yellow Rod?......Page 56
  Task 3: I’d Like You to Make the Yellow and Light Green One and Then Tell Me What the Number Name Would be for Red Also.......Page 57
  Task 4: I’m Going to Call the Orange Ten… and I’m Wondering If You CouldTell Me the Number Name for White.......Page 58
 QUESTIONS FOR DISCUSSION OF CHAPTER 6......Page 59
 REFERENCES......Page 60
 7.1 INTRODUCTION......Page 61
 7.3 EXPLORING THE UNIT: THE CANDY BAR METAPHOR......Page 62
 7.4 WHAT ROD OR TRAIN OF RODS CAN BE USED FOR THE UNIT?......Page 65
 7.5 LAURA AND JESSICA’S MODEL TO COMPARE 1/2 AND 1/3......Page 71
 7.6 CONCLUSION......Page 73
 NOTES......Page 75
 REFERENCES......Page 76
 8.1 INTRODUCTION......Page 77
 8.2 EXPLORING THE QUESTION: “HOW MUCH LARGER IS 1/2 THAN 1/3?”......Page 79
 8.3 STUDENTS CONSIDER THE CLAIM THAT 1/2 IS BIGGER THAN 1/3 BY 1/3......Page 81
 8.4. STUDENTS PRESENT A SECOND MODEL – GIVING THE DARKGREEN ROD THE NUMBER NAME 1......Page 84
 8.5 STUDENTS RETURN TO THE MODEL WITH ORANGE AND RED AS THE UNIT......Page 86
 8.6 AGREEING THAT 1/2 IS 1/6 BIGGER THAN 1/3 FOR THE ORANGEAND RED ROD TRAIN REPRESENTING THE UNIT 1......Page 90
 QUESTIONS FOR DISCUSSION OF CHAPTER 8......Page 91
 REFERENCES......Page 92
 9.1 INTRODUCTION......Page 94
 9.2 STUDENTS’ ARGUMENTS USING DIFFERENT MODELS FOR THE UNIT......Page 95
 9.3 STUDENTS AGREE AND DISAGREE WITH THE TWO MODELS......Page 99
 9.4 CONCLUSION......Page 103
 REFERENCES......Page 104
  Task 1: If I Give This Orange Rod the Number Name One, What NumberName Would I Give to White?......Page 106
  Task 3: I’m Calling the Orange One, What Number Name WouldI Give to Two Whites?......Page 107
  Task 4: Is 1/5=2/10?......Page 108
  Task 5: Which Is Larger, One Half or Two Thirds, and by How Much?......Page 109
 10.4 REVIEWING THE REASONING......Page 110
 REFERENCES......Page 111
 11.2 DOUBLING CONJECTURE......Page 112
 11.3 BUILDING THE PROPOSED MODEL......Page 116
 11.4 EXTENDING THE DOUBLING CONJECTURE TO OTHER MODELS......Page 118
 11.5 CONJECTURING ABOUT THE DIFFERENCE BETWEENONE FOURTH AND ONE NINTH......Page 119
 11.6 A MODEL TO DETERMINE THE DIFFERENCE BETWEENONE FOURTH AND ONE NINTH......Page 121
 11.7 SUMMARY......Page 122
 QUESTIONS FOR DISCUSSION OF CHAPTER 11......Page 123
 REFERENCE......Page 124
 12.1 INTRODUCTION......Page 125
  Task 2: Which Is Larger, Two Thirds or Three Fourths, and by How Much?......Page 126
 12.4 DAVID AND MEREDITH ADOPT THE UPPER BOUND ARGUMENT......Page 128
 12.6 ERIK USES A LOWER BOUND ARGUMENT......Page 129
 REFERENCE......Page 131
  Task 1: Which Is Larger, One Half or One Third, and by How Much?......Page 132
  Task 3: Which Is Larger, One Half or Two Thirds, and by How Much?......Page 134
  Task 4: Which Is Larger, One Half or Three Fourths, and by How Much?......Page 137
 13.4 REVIEWING THE REASONING......Page 141
 REFERENCE......Page 142
  Task 1: Which Is Bigger, One Half or One Quarter, and by How Much?......Page 143
  Task 2: Which Is Larger, Two Thirds or Three Fourths, and by How Much?......Page 145
  Task 3: Compare One Half or Two Fifths. Which Is Larger, and by How Much?......Page 146
  Task 3: Which Is Larger, Two Thirds or Three Fourths, and by How Much?......Page 149
 14.4 REVIEWING THE REASONING......Page 153
 REFERENCES......Page 154
 15.1 INTRODUCTION......Page 156
  Task 1: Which of the Following Fractions Are Bigger or Smaller: One Half,One Third, One Fourth, and One Fifth?......Page 157
 15.3 DAVID REPRESENTS UNIT FRACTIONS WITH A DRAWING......Page 158
 15.4 ERIC PLACES THE NUMBER 1/3 ON THE NUMBER LINE SEGMENT......Page 159
 15.5 STUDENTS PLACE THE FRACTIONS 1/4. 1/5 AND 1/10 ONTHE NUMBER LINE SEGMENT......Page 161
 15.6 STUDENTS CREATE THEIR PERSONAL NUMBER LINE SEGMENTS BETWEEN ZERO AND ONE......Page 164
 15.7 REVIEWING THE STUDENTS’ REASONING......Page 166
 REFERENCES......Page 167
 16.2 ARGUMENTATION AND REASONING......Page 169
 16.3 THE ARGUMENTATION SESSION......Page 170
 16.4 CONSIDERING THE DENSITY OF NUMBERS BETWEEN 0 AND 1......Page 171
 16.5 ERIK CHALLENGES THE CLAIM ABOUT DENSITY......Page 172
 16.6 ALAN MAKES A CLAIM......Page 173
 16.7 THE IDEAS OF ERIK, ALAN, AND MICHAEL......Page 174
 16.8 ALAN’S METAPHOR TO BACK HIS CLAIM......Page 175
 16.9 BRIAN’S CONTRIBUTION TO ALAN’S CLAIM......Page 176
 16.10 ERIK’S COUNTERCLAIM......Page 177
 16.11 ALAN’S AND ERIK’S ARGUMENTS......Page 178
 16.13 SUPPORT FROM AUDRA, JESSICA, BETH, AND MARK......Page 179
 16.14 CLARIFICATION FROM DAVID AND MICHAEL......Page 180
 16.15 ALAN’S DRAWING AND JUSTIFICATION......Page 182
 16.16 BRIAN’S SUPPORT FOR ALAN......Page 184
 16.18 SUMMARY AND CONSENSUS......Page 185
 REFERENCES......Page 186
 17.1 INTRODUCTION......Page 188
  Task 1: Meredith’s Set of Partitioned Line Segments......Page 189
  Task 3: Erik and Alan discuss Meredith’s Representation......Page 190
  Task 4: Alan’s Partitioned Line Segments......Page 192
 17.4 STUDENTS CHALLENGE EACH OTHER......Page 194
  Task 6: Sarah, Beth and Audra Show Where They Would Place 3/3......Page 195
 17.5 REVIEWING THE REASONING......Page 196
 REFERENCE......Page 197
 18.1 INTRODUCTION......Page 198
 18.3 PLACING ONE HALF ON THE NUMBER LINE......Page 199
 18.4 QUESTIONING THE PLACEMENT OF 1/2......Page 201
 18.7 POSITIONING MIXED NUMBERS ON THE NUMBER LINE......Page 202
 SUMMARY......Page 203
 REFERENCES......Page 204
 19.1 SUMMARY......Page 206
 19.2 ANALYTICS......Page 207
  19.3.2 Varied Forms of Reasoning......Page 208
  19.3.3 Reasoning-Eliciting Tasks......Page 209
 REFERENCES......Page 210
 CHAPTER TWO: ESTABLISHING A MATHEMATICAL COMMUNITY......Page 212
 CHAPTER SEVEN: ESTABLISHING THE IMPORTANCE OF THE UNIT......Page 213
 CHAPTER TWELVE: THE DEVELOPMENT OF UPPER AND LOWER BOUNDS ARGUMENTS WHILE COMPARING FRACTIONS......Page 214
 CHAPTER SEVENTEEN: COMPARING AND ORDERING FRACTIONS......Page 215
 CHAPTER EIGHTEEN: EXTENDING FRACTION PLACEMENT FROM A LINE SEGMENT TO A NUMBER LINE......Page 216
 NOTE......Page 217
APPENDIX B: FRACTION DISSERTATIONS FROM THE COLTS NECK INTERVENTION......Page 218
APPENDIX C: VIDEO NARRATIVES BY CHAPTER AND AUTHOR(S)......Page 219
INDEX......Page 220