COMPUTER-ENABLED MATHEMATICS: INTEGRATING EXPERIMENT AND THEORY IN TEACHER EDUCATION -- COMPUTER-ENABLED MATHEMATICS: INTEGRATING EXPERIMENT AND THEORY IN TEACHER EDUCATION -- CONTENTS -- PREFACE -- Chapter 1 THE MULTIPLICATION TABLE FROM AN ADVANCED STANDPOINT -- 1. INTRODUCTION -- 2. BASIC SUMMATION FORMULAS -- 3. ON THE GEOMETRIC MEANING AND INDUCTIVE CONJECTURING OF FORMULA (4) -- 4. ON THE DEFICIENCY OF INDUCTIVE REASONING -- 5. CHECKERBOARD PROBLEM AND ITS DIFFERENT EXTENSIONS -- 5.1. Finding the Number of Squares on the Checkerboard -- 5.2. Finding the Number of Prisms within a Cube -- 5.3. Counting Rectangles with Special Properties on the Checkerboard -- 6. COMPARING THE RATES OF GROWTH OF DIFFERENT FAMILIES OF GEOMETRIC FIGURES -- 6.1. Comparing Different Sets of Rectangles on the Checkerboard -- 6.2. Comparing the Number of Squares to the Number of Rectangles on the Checkerboard -- 6.3. Harmonic Series and the Method of d'Oresme -- 6.4. Comparing the Number of Cubes to the Number of Prisms in the Rubik's Cube -- 6.5. Converging Series -- 6.6. Comparing the Rate of Growth of Squares and Rectangles with Special Properties -- 7. ACTIVITY SET -- Chapter 2 ALGEBRAIC EQUATIONS WITH PARAMETERS -- 1. INTRODUCTION -- 2. A LOCUS APPROACH TO QUADRATIC EQUATIONS WITH PARAMETERS -- 3. QUADRATIC EQUATIONS WITH A HYPERBOLA-LIKE LOCUS -- 4. QUADRATIC EQUATIONS WITH TWO PARAMETERS -- 5. CUBIC EQUATIONS WITH TWO PARAMETERS -- 6. SYSTEMS OF SIMULTANEOUS EQUATIONS WITH PARAMETERS -- 7. ACTIVITY SET -- Chapter 3 INEQUALITIES AND SPREADSHEET MODELING -- 1. INTRODUCTION -- 2. SPREADSHEET MODELING OF LINEAR EQUATIONS -- 3. INEQUALITIES AS TOOLS IN MODELING NON-LINEAR PROBLEMS -- 4. GEOMETRIC CONTEXT AS A SPRINGBOARD INTO NEW USES OF INEQUALITIES -- 4.1. Finding Rectangle with the Largest Perimeter.

4.2. Finding Rectangle with the Smallest Perimeter -- 4.3. Geometric Proof of the Arithmetic Mean-Geometric Mean Inequality -- 4.4. Alternative Approaches to Problem 3 -- 5. TRANSITION TO THREE-DIMENSIONAL MODELING -- 6. DISCUSSION OF MODELING DATA AND ITS ALTERNATIVE INTERPRETATION -- 7. FORMAL AND INFORMAL APPROACHES TO SOLVING THREE-DIMENSIONAL PROBLEMS -- 7.1. Finding the Prism with the Smallest Surface Area -- 7.2. Finding the Prism with the Largest Surface Area Using a Combination of Formal Argument and Numeric Computation -- 7.3. A Three-Dimensional Geometric Problem in the Plane -- 8. ACTIVITY SET -- Chapter 4 GEOMETRIC PROBABILITY -- 1. INTRODUCTION -- 2. FORMAL DEFINITION OF GEOMETRIC PROBABILITY -- 3. SPREADSHEET AS A GEOMETRIC MEDIUM -- 4. COMPARING PROBABILITIES OF TWO EVENTS THROUGH GEOMETRIZATION -- 5. A SYSTEMATIC APPROACH TO GENERATING "JUMPING" FRACTIONS -- 6. EXPLAINING THE BEHAVIOR OF FRACTIONS THROUGH THE BEHAVIOR OF FUNCTIONS -- 7. FUNCTIONS WITH PARAMETERS AND GEOMETRIC PROBABILITY -- 8. CALCULATING GEOMETRIC PROBABILITY THROUGH A SPREADSHEET-BASED SIMULATION -- 9. LOCI OF TWO-VARIABLE INEQUALITIES AS IMAGES OF POINTS AND PICK'S FORMULA -- 10. ADVANCED EXPLORATIONS WITH GEOMETRIC PROBABILITIES -- 11. ACTIVITY SET -- Chapter 5 COMBINATORIAL EXPLORATIONS -- 1. INTRODUCTION -- 2. TWO WAYS OF DEFINING PERMUTATIONS -- 3. COUNTING PERMUTATIONS OF REPEATED OBJECTS -- 4. DEFINING COMBINATIONS THROUGH A PARTIAL DIFFERENCE EQUATION -- 5. NUMERICAL MODELING AS A WAY OF MAKING MATHEMATICAL CONNECTIONS -- 6. COMBINATIONS WITH REPETITIONS -- 7. COMBINATORIAL IDENTITIES AND MATHEMATICAL INDUCTION PROOF -- 7.1. Identities Involving C(n, r) -- 7. 2. Identities Involving -- 8. CONNECTING NUMBERS WITH DIFFERENT COMBINATORIAL MEANING -- 9. PARTITIONING PROBLEMS AND RECURSIVE REASONING -- 10. MODELING THE SUMS OF PERFECT POWERS.

11. CONNECTING THE SUMS OF PERFECT POWERS TO COMBINATIONS -- 12. EULERIAN NUMBERS -- 13. CLOSED FORMULAS FOR THE SUMS OF PERFECT POWERS -- 14. ACTIVITY SET -- Chapter 6 HISTORICAL PERSPECTIVES -- 1. INTRODUCTION -- 2. THE SPIRAL OF THEODORUS MOTIVATES CONCEPT LEARNING WITH TECHNOLOGY -- 3. THE CONSTRUCTION OF THE SPIRAL OF THEODORUS BY USING A SPREADSHEET -- 4. PARAMETERIZATION OF RECURRENCE RELATION (2) -- 5. GENERATING POLYGONAL NUMBERS THROUGH A SIEVE-LIKE PROCESS -- 6. DEVELOPING CLOSED AND RECURSIVE FORMULAS FOR POLYGONAL NUMBERS -- 7. INTERPRETING SUMMATION FORMULAS FOR POLYGONAL NUMBERS IN THE LANGUAGE OF SULVASUTRAS -- 8. THE SPIRAL MOTIVATES TRANSITION FROM SUMMATION TO ESTIMATION -- 9. PROOF OF PROPOSITION 6 AS AN AGENCY FOR PROBLEM POSING -- 9.1. Revisiting Geometry through Verifying Base Clause -- 9.2. Posing Problems in the Context of Inductive Transfer -- 10. THE HARMONIC SERIES REVISITED -- 11. ACTIVITY SET -- Chapter 7 COMPUTATIONAL EXPERIMENTS AND FORMAL DEMONSTRATION IN TRIGONOMETRY -- 1. INTRODUCTION -- 2. ONE EQUATION-FOUR SOLUTIONS -- 3. A COMPUTER-SUPPORTED GRAPHICAL DEMONSTRATION -- 4. A FORMAL GEOMETRIC DEMONSTRATION -- 5. INTRODUCING A PARAMETER IN EQUATION (8) -- 5.1. Exploring the Case a = 3 -- 5.2. Exploring the Case a = 0 -- 5.3. Exploring the Case a = 1 -- 6. DOUBLE PARAMETERIZATION OF EQUATION (8) -- 7. TRIPLE PARAMETERIZATION OF EQUATION (8) -- 8. EXPLORING THE EQUATION cossin -- 8.1. First Solution to Equation (33) -- 8.2. Second Solution to Equation (33) -- 8.3. Demonstration of the Case a = 2, b = -2 -- 8.4. Third Solution to Equation (33). -- 9. ACTIVITY SET -- Chapter 8 DEVELOPING MODELS FOR COMPUTATIONAL PROBLEM SOLVING -- 1. INTRODUCTION -- 2. SETTING A CONTEXT AND INTRODUCING MODELING TOOLS -- 3. MODELING VS. COUNTING -- 4. POLYGONAL NUMBERS REVISITED -- 5. CHANGE OF MODEL AFFECTS CONTEXT.

6. CHANGE OF CONTEXT REQUIRES NEW MODEL -- 7. INTRODUCING NEW MODELING TOOLS -- 8. QUADRATIC FUNCTIONS AS TOOLS OF SUMMATION -- 9. FINDING THE SUM OF ROOM NUMBERS THAT BELONG TO THE SAME FLOOR -- 10. REFINING OLD MODELS TO MATCH NEW CONTEXT -- 11. INTERPRETING THE RESULTS OF COMPUTATIONAL EXPERIMENTS -- 12. ACTIVITY SET -- Chapter 9 PROGRAMMING DETAILS -- 1. INTRODUCTION -- 2. SPREADSHEETS USED IN CHAPTER 1 -- 2.1. Programming Details for Figure 1.5 -- 2.2. Programming Details for Figure 1.8 -- 2.3. Programming Details for Figure 1.9 -- 2.4. Programming Details for Figures 1.11 and 1.12 -- 2.5. Programming Details for Figures 1.10 and 1.13 -- 3. SPREADSHEETS USED IN CHAPTER 3 -- 3.1. Programming Details for Figures 3.1 and 3.2 -- 3.2. Programming Details for Figures 3.3 and 3.4 -- 3.3. Programming Details for Figure 3.5 -- 3.4. Programming Details for Figure 3.6 -- 3.5. Programming Details for Figure 3.9. -- 4. SPREADSHEETS USED IN CHAPTER 8 -- 4.1. Programming Details for Figure 8.6 -- 4.2. Programming Details for Figure 8.9 -- 4.3. Programming Details for Figure 8.10 -- 4.4. Programming Details for Figure 8.11 -- 4.5. Spreadsheet Programming for Figure 8.12 -- 4.6. Spreadsheet Programming for Figure 8.16 -- 4.7. Spreadsheet Programming for Figure 8.21 -- 4.8. Spreadsheet Programming for Figure 8.22 -- REFERENCES -- INDEX -- Blank Page.