Computer-Enabled Mathematics: Integrating Experiment and Theory in Teacher Education (Education in a Competitive and Globalizing World)

COMPUTER-ENABLED MATHEMATICS: INTEGRATING EXPERIMENT  AND THEORY IN TEACHER EDUCATION......Page 3
COMPUTER-ENABLED MATHEMATICS: INTEGRATING EXPERIMENT  AND THEORY IN TEACHER EDUCATION......Page 5
CONTENTS......Page 7
PREFACE......Page 9
1. INTRODUCTION......Page 13
2. BASIC SUMMATION FORMULAS......Page 14
3. ON THE GEOMETRIC MEANING AND INDUCTIVE  CONJECTURING OF FORMULA (4)......Page 16
4. ON THE DEFICIENCY OF INDUCTIVE REASONING......Page 19
5. CHECKERBOARD PROBLEM AND ITS DIFFERENT EXTENSIONS......Page 21
5.1. Finding the Number of Squares on the Checkerboard......Page 24
5.2. Finding the Number of Prisms within a Cube......Page 25
5.3. Counting Rectangles with Special Properties on the Checkerboard......Page 27
6.1. Comparing Different Sets of Rectangles on the Checkerboard......Page 34
6.2. Comparing the Number of Squares to the Number of Rectangles on the Checkerboard......Page 36
6.3. Harmonic Series and the Method of d’Oresme......Page 38
6.4. Comparing the Number of Cubes to the Number of Prisms in the Rubik’s Cube......Page 40
6.5. Converging Series......Page 41
6.6. Comparing the Rate of Growth of Squares and Rectangles with Special Properties......Page 44
7. ACTIVITY SET......Page 48
1. INTRODUCTION......Page 49
2. A LOCUS APPROACH TO QUADRATIC EQUATIONS WITH PARAMETERS......Page 50
3. QUADRATIC EQUATIONS  WITH A HYPERBOLA-LIKE LOCUS......Page 54
4. QUADRATIC EQUATIONS WITH TWO PARAMETERS......Page 59
5. CUBIC EQUATIONS WITH TWO PARAMETERS......Page 67
6. SYSTEMS OF SIMULTANEOUS EQUATIONS WITH PARAMETERS......Page 73
7. ACTIVITY SET......Page 79
1. INTRODUCTION......Page 81
2. SPREADSHEET MODELING OF LINEAR EQUATIONS......Page 83
3. INEQUALITIES AS TOOLS IN MODELING NON-LINEAR PROBLEMS......Page 86
4. GEOMETRIC CONTEXT AS A SPRINGBOARD  INTO NEW USES OF INEQUALITIES......Page 90
4.1. Finding Rectangle with the Largest Perimeter......Page 91
4.2. Finding Rectangle with the Smallest Perimeter......Page 94
4.4. Alternative Approaches to Problem 3......Page 95
5. TRANSITION TO THREE-DIMENSIONAL MODELING......Page 96
6. DISCUSSION OF MODELING DATA  AND ITS ALTERNATIVE INTERPRETATION......Page 100
7. FORMAL AND INFORMAL APPROACHES TO SOLVING THREE-DIMENSIONAL PROBLEMS......Page 102
7.2. Finding the Prism with the Largest Surface Area Using a Combination of Formal Argument and Numeric Computation......Page 104
7.3. A Three-Dimensional Geometric Problem in the Plane......Page 105
8. ACTIVITY SET......Page 106
1. INTRODUCTION......Page 107
2. FORMAL DEFINITION OF GEOMETRIC PROBABILITY......Page 108
3. SPREADSHEET AS A GEOMETRIC MEDIUM......Page 109
4. COMPARING PROBABILITIES OF TWO EVENTS  THROUGH GEOMETRIZATION......Page 113
5. A SYSTEMATIC APPROACH TO GENERATING  “JUMPING” FRACTIONS......Page 116
6. EXPLAINING THE BEHAVIOR OF FRACTIONS  THROUGH THE BEHAVIOR OF FUNCTIONS......Page 118
7. FUNCTIONS WITH PARAMETERS AND GEOMETRIC PROBABILITY......Page 120
8. CALCULATING GEOMETRIC PROBABILITY THROUGH  A SPREADSHEET-BASED SIMULATION......Page 124
9. LOCI OF TWO-VARIABLE INEQUALITIES AS IMAGES OF POINTS AND PICK’S FORMULA......Page 125
10. ADVANCED EXPLORATIONS WITH GEOMETRIC PROBABILITIES......Page 126
11. ACTIVITY SET......Page 139
1. INTRODUCTION......Page 141
2. TWO WAYS OF DEFINING PERMUTATIONS......Page 142
3. COUNTING PERMUTATIONS OF REPEATED OBJECTS......Page 144
4. DEFINING COMBINATIONS THROUGH A PARTIAL  DIFFERENCE EQUATION......Page 145
5. NUMERICAL MODELING AS A WAY OF MAKING  MATHEMATICAL CONNECTIONS......Page 146
6. COMBINATIONS WITH REPETITIONS......Page 151
7.1. Identities Involving C(n, r)......Page 154
7. 2. Identities Involving......Page 159
8. CONNECTING NUMBERS WITH DIFFERENT  COMBINATORIAL MEANING......Page 162
9. PARTITIONING PROBLEMS AND RECURSIVE REASONING......Page 163
10. MODELING THE SUMS OF PERFECT POWERS......Page 168
11. CONNECTING THE SUMS OF PERFECT POWERS  TO COMBINATIONS......Page 170
12. EULERIAN NUMBERS......Page 174
13. CLOSED FORMULAS FOR THE SUMS OF PERFECT POWERS......Page 179
14. ACTIVITY SET......Page 181
1. INTRODUCTION......Page 183
2. THE SPIRAL OF THEODORUS MOTIVATES CONCEPT LEARNING WITH TECHNOLOGY......Page 185
3. THE CONSTRUCTION OF THE SPIRAL OF THEODORUS  BY USING A SPREADSHEET......Page 186
4. PARAMETERIZATION OF RECURRENCE RELATION (2)......Page 189
5. GENERATING POLYGONAL NUMBERS THROUGH  A SIEVE-LIKE PROCESS......Page 194
6. DEVELOPING CLOSED AND RECURSIVE FORMULAS  FOR POLYGONAL NUMBERS......Page 196
7. INTERPRETING SUMMATION FORMULAS FOR POLYGONAL NUMBERS IN THE LANGUAGE OF SULVASUTRAS......Page 198
8. THE SPIRAL MOTIVATES TRANSITION FROM SUMMATION  TO ESTIMATION......Page 202
9.1. Revisiting Geometry through Verifying Base Clause......Page 204
9.2. Posing Problems in the Context of Inductive Transfer......Page 206
10. THE HARMONIC SERIES REVISITED......Page 209
11. ACTIVITY SET......Page 210
1. INTRODUCTION......Page 211
2. ONE EQUATION—FOUR SOLUTIONS......Page 215
3. A COMPUTER-SUPPORTED GRAPHICAL DEMONSTRATION......Page 219
4. A FORMAL GEOMETRIC DEMONSTRATION......Page 220
5. INTRODUCING A PARAMETER IN EQUATION (8)......Page 222
5.2. Exploring the Case a = 0......Page 223
6. DOUBLE PARAMETERIZATION OF EQUATION (8)......Page 224
7. TRIPLE PARAMETERIZATION OF EQUATION (8)......Page 226
8. EXPLORING THE EQUATION  cossin......Page 228
8.2. Second Solution to Equation (33)......Page 230
8.3. Demonstration of the Case a = 2, b = -2......Page 231
8.4. Third Solution to Equation (33).......Page 233
9. ACTIVITY SET......Page 235
1. INTRODUCTION......Page 237
2. SETTING A CONTEXT AND INTRODUCING MODELING TOOLS......Page 239
3. MODELING VS. COUNTING......Page 240
4. POLYGONAL NUMBERS REVISITED......Page 242
5. CHANGE OF MODEL AFFECTS CONTEXT......Page 243
6. CHANGE OF CONTEXT REQUIRES NEW MODEL......Page 245
7. INTRODUCING NEW MODELING TOOLS......Page 250
8. QUADRATIC FUNCTIONS AS TOOLS OF SUMMATION......Page 251
9. FINDING THE SUM OF ROOM NUMBERS THAT BELONG  TO THE SAME FLOOR......Page 253
10. REFINING OLD MODELS TO MATCH NEW CONTEXT......Page 255
11. INTERPRETING THE RESULTS  OF COMPUTATIONAL EXPERIMENTS......Page 259
12. ACTIVITY SET......Page 262
1. INTRODUCTION......Page 263
2.3. Programming Details for Figure 1.9......Page 264
2.5. Programming Details for Figures 1.10 and 1.13......Page 265
3.4. Programming Details for Figure 3.6......Page 266
4.3. Programming Details for Figure 8.10......Page 267
4.8. Spreadsheet Programming for Figure 8.22......Page 268
REFERENCES......Page 269
INDEX......Page 271