Table of Contents
Preface v
Chapter 1 On the Genesis of Problem Posing in Mathematics 1
1.1 Problems from the first printed arithmetic 1
1.1.1 Solving a 15th century problem using the modern-day pedagogy 3
1.1.2 Posing 15th century-like problems through conceptualization 5
1.1.3 Using technology for posing 15th century-like problems 6
1.2 From a classic problem to using the modem spreadsheet 8
1.2.1 The birth of the probability theory through problem posing 8
1.2.2 Using a spreadsheet as a problem-posing tool 11
1.2.3 Duality of the spreadsheet's use 13
1.3 The Problem of the Grand Duke of Tuscany 14
1.4 Conjecturing as posing problems to find proof 17
1.5 Problem posing in a classic context as a springboard into experimental mathematics 19
1.5.1 Triangular numbers with identical digits 20
1.5.2 Triangular number sieves 20
1.6 Problem posing as setting up a research program 22
1.7 Summary 25
Chapter 2 From a Theory of Problem Posing to Classroom Practice of the Digital Era 27
2.1 Problem posing as educational philosophy 27
2.2 Problem posing in the modem educational context 29
2.3 Learning to ask questions about posed/solved problems 32
2.4 Technology as a cultural support of problem posing 35
2.5 Numerical coherence in problem posing 36
2.5.1 Using a spreadsheet to pose a numerically coherent problem 38
2.6 Contextual coherence in problem posing 41
2.7 Pedagogical coherence in problem posing 44
2.8 Didactical coherence in problem posing 48
2.9 Summary 49
Chapter 3 Posing Technology-Immune/Technology-Enabled (TITE) Problems 51
3.1 From teaching machine movement to symbolic computations 51
3.2 Technological advances call for the revision of mathematics curriculum 54
3.3 Definition of a TITE problem and a simple example 58
3.4 Revisiting classic problems in the digital era under the umbrella of the TITE concept 60
3.5 Conceptual bond and arithmetical word problems 67
3.5.1 Looking at the past to develop new teaching ideas 67
3.5.2 Posing similar problems 68
3.6 Revisiting mathematical problems to make them didactically coherent 71
3.6 From numerical to contextual coherence 71
3.6.2 Towards pedagogical coherence 73
3.7 From modeling data to a general formula using technology 74
3.8 Formulating and solving a didactically coherent problem 75
3.9 Maple-based mathematical induction proof 78
3.10 Summary 82
Chapter 4 Linking Algorithmic Thinking and Conceptual Knowledge through Problem Posing 85
4.1 On the hierarchy of two types of knowledge 85
4.2 A simple question leads to revealing hidden creativity 89
4.3 Two levels of conceptual understanding 92
4.4 Solving a problem seeking information 93
4.5 Problem posing leads to conceptual knowledge and collateral learning 95
4.6 Using conceptual bond in posing problems with technology 99
4.7 Summary 102
Chapter 5 Using Graphing Software for Posing Problems in Advanced High School Algebra 105
5.1 Introduction 105
5.2 Location of roots of quadratics about an interval 107
5.3 Digital fabrication 110
5.4 Connecting the coordinate plane with the plane of coefficients 112
5.4.1 The case RREE 112
5.4.2 The case RERE 113
5.4.3 The case REER 113
5.4.4 The case ERER 114
5.4.5 The case EERR 115
5.4.6 The case ERRE 116
5.5 Using Vieta's Theorem 116
5.6 Posing TITE problems in the plane of parameters 119
5.7 Geometric probabilities and the partitioning diagram 123
5.8 Making mathematical connections 125
5.9 Revealing hidden concepts through collateral learning 128
5.10 Summary 130
Chapter 6 Einstellung Effect and Problem Posing 133
6.1 Examples of Einstellung effect 133
6.2 Water jar experiments and Einstellung effect 138
6.3 Posing and solving problems as a remediation of Einstellung effect 141
6.4 Posing problems for water jar experiments using a spreadsheet 143
6.5 Einstellung effect in finding areas on a geoboard 144
6.6 Einstellung effect in solving algebraic equations and inequalities 149
6.7 Einstellung effect in solving trigonometric inequalities 153
6.8 Using technology to pose problems that might lead to Einstellung effect 156
6.9 Einstellung effect in solving logarithmic inequalities 159
6.9.1 Simultaneous extension and contraction of solution set 159
6.9.2 Extension of solution set 162
6.10 Solving logarithmic inequality (6.21) in the general case 166
6.10.1 The case n = 2k 166
6.10.2 The case 77 = 2k + 1 172
6.11 Summary 176
Chapter 7 Explorations with Integer Sequences as TITE Problem Posing 179
7.1 Introduction 179
7.2 Exploring patterns formed by the last digits of the sums of powers of integers 180
7.3 Discovering patters in the last digits of the polygonal numbers 186
7.3.1 The triangular number sieves 186
7.3.2 Triangular number sieves and the last digits of their terms 188
7.3.3 Rises and falls in permutations 189
7.3.4 Connecting triangular and square numbers within the multiplication table 189
7.3.5 The square number sieves 191
7.3.6 The pentagonal number sieves 193
7.3.7 The general case of the m-gonal number sieves 195
7.4 Patterns in the behavior of the greatest common divisors of two polygonal numbers 200
7.5 Exploring sequences formed by the sums of powers of integers 202
7.6 Exploring sieves developed from the sums of powers of integers 204
7.7 Summary 207
Appendix 209
8.1 Spreadsheets included in Chapter 1 209
8.2 Spreadsheets included in Chapter 2 210
8.3 Spreadsheets included in Chapter 3 211
8.4 Spreadsheets included in Chapter 4 212
8.5 Spreadsheets included in Chapter 6 212
8.6 Spreadsheets included in Chapter 7 213
Bibliography 217
Index 233
