Contents -- Part I Introduction -- Chapter 1 Mathematical Problem Solving in Singapore Schools -- 1 Introduction -- 2 Mathematical Problem Solving -- 3 Pedagogy and Practice in Mathematical Problem Solving -- 4 Mathematical Problem-Solving Tasks -- 5 Mathematical Problem Solving and the Education System in Singapore -- 6 Concluding Remarks -- References -- Part II The Processes and Pedagogies -- Chapter 2 Tasks and Pedagogies that Facilitate Mathematical Problem Solving -- 1 Introduction -- 2 Assumptions About Problem Solving and Classroom Activity -- 3 Fostering Problem Solving by Posing Open-Ended Tasks -- 4 Content-Specific Open-Ended Mathematical Tasks -- 5 Mathematical Problem Solving and our Planning and Teaching Model -- 5.1 The tasks and their sequence -- 5.2 Enabling prompts -- 5.3 Extending prompts -- 5.4 Explicit pedagogies -- 5.5 Learning community -- 6 The Next Phase of the Research -- 6.1 Mr Smith's context and goals -- 6.1.1 Pre- and post-test results -- 6.1.2 Nature of the teaching -- 6.1.3 Analysis of students' responses to various tasks -- 6.1.4 The delayed post test -- 7 Summary and Conclusion -- References -- Chapter 3 Learning through Productive Failure in Mathematical Problem Solving -- 1 Introduction -- 2 Arguments Supporting the Case for Productive Failure -- 3 Exploring Productive Failure in a Singapore Math Classroom -- 3.1 Findings -- 4 Extending Productive Failure in a Singapore Math Classroom -- 4.1 Findings -- 5 General Discussion -- 5.1 Explaining productive failure -- 5.2 Implications for teaching and learning design -- 6 Conclusion -- Acknowledgements -- References -- Appendix A -- Appendix B -- Appendix C -- Chapter 4 Note Taking as Deliberate Pedagogy: Scaffolding Problem Solving Learning -- 1 Introduction -- 2 The Deliberate Nature of Note Taking: Bruner and Vygostsky.

3 Practical Applications of Deliberate Note Taking -- 3.1 Scaffolding in the zone of proximal development -- 3.2 Episode one: Mr. Orland -- 3.3 Episode two: Miss Lipan -- 3.4 Lesson debriefing with the teacher educator -- 3.5 Why does this model work? -- 4 Conclusion -- References -- Chapter 5 Japanese Approach to Teaching Mathematics via Problem Solving -- 1 Introduction -- 2 Mathematics Lesson as Structured Problem-Solving -- 2.1 A typical organization of a lesson -- 2.2 The Japanese lesson pattern -- 2.3 Beyond the pattern -- 2.4 A story or a drama as a metaphor for an excellent lesson -- 3 Preparing a Lesson by Focusing on Students' Problem Solving -- 4 Some Practical Ideas Shared by Japanese Teachers -- 5 Final Remarks -- References -- Chapter 6 Mathematical Problem Posing in Singapore Primary Schools -- 1 Introduction -- 2 Mathematical Problem Solving and Problem Posing -- 3 Mathematical Problem-Posing Processes -- 3.1 Posing primitives -- 3.2 Posing related problems -- 3.3 Constructing meaning for mathematical symbols -- 3.4 Engaging in metacognition -- 3.5 Connecting to one's experiences -- 4 Mathematical Problem Posing in the Classroom -- 4.1 Developing a concept -- 4.2 Providing drill-and-practice -- 4.3 Problem solving -- 4.4 Assessing understanding -- 4.5 Differentiating instruction -- 5 Conclusion -- References -- Chapter 7 Solving Mathematical Problems by Investigation -- 1 Introduction -- 2 Relationship between Problem Solving and Investigation -- 3 Solving Mathematical Problems by Investigation -- 4 Conclusion and Implications -- References -- Chapter 8 Generative Activities in Singapore (GenSing): Pedagogy and Practice in Mathematics Classrooms -- 1 Introduction -- 2 What is a Generative Activity? -- 3 Narrative -- 4 GenSing Theoretical Foundations -- 5 Generative Design Theory and Practice -- 5.1 Breadth -- 5.2 Pedagogy.

6 Space Creating Play and Dynamic Structure -- 7 Agency and Participation -- 8 Examples from Singapore Classrooms -- 8.1 Student learning: Powerful means of learning mathematics -- 8.2 Teacher pedagogy: Model of teacher professional development -- 8.3 Technology innovation -- 9 Conclusion -- References -- Chapter 9 Mathematical Modelling and Real Life Problem Solving -- 1 Introduction -- 1.1 What is mathematical modelling -- 1.2 Problem solving examples from textbooks -- 2 Approaches to Mathematical Modelling -- 2.1 Empirical models -- 2.2 Simulation models -- 2.3 Deterministic models -- 2.4 Stochastic models -- 3 Examples -- 4 Pedagogical Implications -- 5 Concluding Remarks -- References -- Part III Mathematical Problems and Tasks -- Chapter 10 Using Innovation Techniques to Generate 'New' Problems -- 1 Introduction -- 2 Innovation: A Borrowed Idea from Literature -- 3 Innovation on Problems in Mathematics -- 4 Using Innovation to Generate Problems -- 4.1 Example 1 -- 4.1.1 Innovation by replacement -- 4.1.2 Innovation by addition -- 4.1.3 Innovation by modification -- 4.1.4 Innovation by contextualizing the problem -- 4.1.5 Innovation by turning the problem around -- 4.1.6 Innovation by reformulation -- 4.2 Example 2 -- 4.2.1 Innovation by replacement -- 4.2.2 Innovation by addition -- 4.2.3 Innovation by modification -- 4.2.4 Innovation by contextualizing the problem -- 4.2.5 Innovation by turning the problem around -- 4.2.6 Innovation by reformulation -- 5 Cognitive Value of Innovation in Problem Solving -- 6 Benefits of Innovation on Existing Problems -- 7 Practical Aspects of Innovation in Generating Mathematical Problems -- 8 Concluding Remarks -- Acknowledgement -- References -- Appendix -- Chapter 11 Mathematical Problems for the Secondary Classroom -- 1 Introduction -- 2 Problem and Problem Solving -- 2.1 Using generalisation and extension.

2.2 Using different representations -- 2.3 Making connections -- 2.4 Finding multiple methods of solution -- 2.5 Using technology -- 2.6 Drawing or constructing -- 2.7 Proving -- 2.8 Carrying out simple investigations -- 2.9 Solving open-ended problems -- 3 Conclusion -- References -- Chapter 12 Integrating Open-Ended Problems in the Lower Secondary Mathematics Lesson -- 1 Introduction -- 2 Characteristics of Open-Ended Problems -- 3 Process of Solving Open-Ended Problems -- 4 Sample Open-Ended Problems for Lower Secondary Students -- 5 Conclusion -- References -- Chapter 13 Arousing Students' Curiosity and Mathematical Problem Solving -- 1 Introduction -- 2 Arousing Students' Curiosity in Mathematics and Problem Solving -- 2.1 Introducing mathematics through daily activities -- 2.1.1 Calendar -- 2.1.2 Page numbers of books -- 2.2 Modifying the usual approach of classroom teaching -- 2.2.1 An example from teaching mensuration -- 2.2.2 Another example from teaching algebraic manipulation -- 2.2.3 Extension from pattern gazing -- 2.3 Making classroom mathematics relevant in real-life -- 2.3.1 Two examples from mensuration: Area of trapezium and volume of frustum -- 2.3.2 Two more examples from the use of the common logarithms -- 2.4 Elaborating on the less prominent results -- 2.4.1 Volume and surface area of a sphere -- 3 Conclusion -- References -- Part IV Future Directions -- Chapter 14 Moving beyond the Pedagogy of Mathematics: Foregrounding Epistemological Concerns -- 1 Introduction -- 2 What does it mean to learn Mathematics? -- 3 Implications for mathematics education research and practice -- 3.1 Understanding children's inventive and constructive resources -- 3.2 Formal and informal learning designs that help build upon children's constructive resources -- 3.3 Developing teacher capacity -- 4 Conclusion -- References -- Contributing Authors.