Mathematics Instructional Practices in Singapore Secondary Schools (Mathematics Education – An Asian Perspective)

"This book offers a detailed look into the how and what of mathematics instruction in Singapore. It presents multiple aspects of mathematics instruction in schools, ranging from the unique instructional core, practices that promote mastery, development of conceptual knowledge through learning experiences, nurturing of positive attitudes, self-regulation of learning and development and use of instructional materials for making connections across mathematical ideas, developing mathematical reasoning, and developing fluency in applying mathematical knowledge in problem solving. The book presents a methodology that is successful in documenting classroom instruction in a comprehensive manner. The research findings illuminate instruction methods that are culturally situated, robust and proven to impact student learning. It demonstrates how a unique data source can be analysed through multiple lenses and provides readers with a rich portrait of how the school mathematics instruction is enacted in Singapore secondary schools"--[Page 4] of cover Mathematics Instructional Practices in Singapore Secondary Schools Series Editors’ Introduction Foreword Preface Acknowledgements Contents Editors and Contributors About the Editors Contributors 1 Overview of the School System and School Mathematics Curriculum in Singapore 1.1 Introduction 1.2 The School System 1.2.1 Primary School 1.2.2 Secondary School 1.3 School Mathematics Curriculum 1.3.1 Framework of the Singapore School Mathematics Curriculum 1.3.2 Primary Mathematics Curriculum 1.3.3 Secondary Mathematics Curriculum 1.4 Concluding Remarks References Part IGlobal Features of Practice 2 A Study of the Enacted School Mathematics Curriculum in Singapore Secondary Schools 2.1 Introduction 2.2 Conceptual Framework 2.2.1 What Did the Findings of CORE 2 Tell Us About Mathematics Teaching and Learning in Singapore Secondary Mathematics Classrooms? 2.2.2 Curriculum Enactment Process 2.2.3 Perspectives of Mathematics Teaching 2.2.4 Summary 2.3 Research Design 2.3.1 Phase 1: Video-Segment of the Project 2.3.2 Phase II: Survey-Segment of the Project Appendix References 3 The Instructional Core That Drives the Enactment of the School Mathematics Curriculum in Singapore Secondary Schools 3.1 Introduction 3.2 An Instructional Core That Drives Mathematics Lessons of Experienced and Competent Teachers 3.3 What Were Mathematics Teachers’ Perceptions About the Instructional Core in General? 3.4 Concluding Remarks References Part IIEnactment of the Intended Curriculum 4 Learning Opportunities to Promote Conceptual Understanding in Singapore Secondary School Mathematics Instruction 4.1 Introduction 4.2 Teaching Mathematics That Promotes Conceptual Understanding 4.2.1 Representations in the Learning of Mathematical Concepts 4.2.2 Attending Explicitly to Mathematical Concepts 4.3 Facilitation of Conceptual Understanding by Two Experienced and Competent Mathematics Teachers 4.4 Analysis of the Lesson for Teacher 8 4.4.1 Properties of Quadratic Equations 4.4.2 Learning Opportunities for Conceptual Understanding 4.5 Analysis of the Lesson for Teacher 26 4.5.1 Distance Between Two Points in a Cartesian Plane 4.5.2 Learning Opportunities for Conceptual Understanding 4.6 Concluding Remarks References 5 Teaching Practices That Promote Mastery in Mathematics Learning in Singapore Secondary School Classrooms 5.1 Introduction 5.2 Mastery in Learning 5.3 Mastery in Learning and the Singapore School Mathematics Curriculum 5.3.1 Phases of Lesson 5.3.2 Worked Example and Class Practice Task Nexus 5.4 Promoting Mastery in Learning in Singapore Secondary Mathematics Classrooms 5.4.1 Phases of Lesson 5.4.2 Worked Example and Class Practice Task Nexus 5.5 Conclusion References 6 Facilitation of Students’ Metacognition: Some Insights Gleaned from Mathematics Classrooms in Singapore Secondary Schools 6.1 Metacognition 6.2 Teachers’ Metacognitive Instructional Strategies in Mathematics Classrooms 6.3 Data Analysis Methods 6.4 Metacognitive Instructional Strategies Used by the 30 Experienced and Competent Mathematics Teachers Gleaned from Video-Recorded Lessons 6.5 Teacher 11 in a Mathematically Challenged Class 6.6 Teacher 17 in a Mathematically Strong Class 6.7 Implications and Future Directions References 7 Cultivation of Positive Attitudes by Experienced and Competent Mathematics Teachers in Singapore Secondary Schools 7.1 Introduction 7.2 Literature Review 7.2.1 Definitions of Attitudes and Affect 7.2.2 Measurements of Attitudes 7.2.3 Research on Attitudes 7.2.4 Nature of Attitudes 7.3 Research Design 7.4 Findings and Discussion 7.4.1 Examples of Different Instructional Strategies in Cultivating Positive Attitudes 7.4.2 LOVE Mathematics Framework to Engage the Hearts of Mathematics Learners 7.4.3 Types of Positive Attitudes Cultivated by the 30 Teachers Across the Four Courses of Study 7.4.4 Possible Factors Influencing Teachers’ Approaches in Imbuing Desired Attitudes 7.5 Conclusion References 8 Balancing an Intuitive-Experimental Approach with Mathematical Rigour: A Case Study of an Experienced and Competent Mathematics Teacher in a Singapore Secondary School 8.1 Introduction 8.2 Teaching of Geometry in Schools 8.3 The Case Study 8.3.1 Method 8.3.2 Data 8.3.3 Analysis of the Data 8.4 Discussion and Conclusion References 9 Meaningful Mathematics Talk That Supports Mathematics Learning in Singapore Secondary Schools 9.1 Introduction 9.2 Meaningful Mathematics Talk 9.3 Mathematics Talk Enacted by an Experienced and Competent Teacher 9.4 Mathematics Talk in the Classrooms of Mathematics Teachers in General 9.5 Conclusion References 10 The Enacted Curriculum—Students’ Perspectives of Good Mathematics Lessons in Singapore Secondary Schools 10.1 The Student Perspective 10.2 Mathematics Instruction in Singapore 10.2.1 Singapore School Mathematics Curriculum Framework 10.3 Singapore Secondary School Students’ Perspectives of Good Mathematics Lessons 10.3.1 Problem-Solving Competencies in the SSMCF 10.3.2 Teacher Approach and Class Activity 10.4 Conclusion References Part IIITasks and Tools 11 Singapore Secondary School Mathematics Teachers’ Selection and Modification of Instructional Materials for Classroom Use 11.1 Introduction 11.2 Instructional Materials 11.2.1 Selection and Modification of Instructional Materials 11.2.2 Characteristics of Instructional Materials 11.3 Teachers’ Reference Materials 11.3.1 Item 1 11.3.2 Item 2 11.3.3 Item 3 11.3.4 Item 4 11.4 Teachers’ Strategies in Selection and Modification for Their Instructional Materials 11.4.1 Research Question 1: How Do Singapore Secondary School Mathematics Teachers Select or Modify Materials for Instructional Practice? 11.4.2 Research Question 2: What Are the Characteristics of Instructional Materials That Will Help Teachers Enact Worthy Instructional Goals of Teaching Materials and Help Students Achieve Desirable Outcomes? 11.5 Discussion References 12 Use of Challenging Items in Instructional Materials by Singapore Secondary School Mathematics Teachers 12.1 Introduction 12.2 Chronologically-Grounded Survey 12.2.1 Challenging Tasks 12.3 Data 12.4 Findings 12.4.1 Task-Affect 12.4.2 Place of Task 12.4.3 Student Ability 12.4.4 Other Supporting Features 12.5 Discussion and Conclusion References 13 Sequencing of Practice Examples for Mathematical Reasoning: A Case of a Singapore Secondary School Teacher’s Practice 13.1 Introduction 13.2 Use of Examples 13.2.1 Task Design 13.2.2 Sequences of Tasks and Variation Theory 13.2.3 Practice Examples 13.3 Method 13.3.1 Context 13.3.2 Teacher 13’s Class and Students 13.3.3 Topic 13.4 Data and Analysis 13.4.1 Data 13.4.2 Analysis 13.5 Results 13.5.1 Analysis of Unit 1: Researchers’ Formation of Conjectures 13.5.2 Analysis of Unit 2: Further Examination of the Conjectures 13.5.3 Advance the Technique 13.5.4 Advance Technique Through Comparing, Inferring and Justifying 13.5.5 Expose and Target Students’ Faulty Reasoning Through Carefully Designed Examples 13.5.6 Consolidate and Formalise Reasoning in Written Form Through Whole Class Instructional Segment 13.6 Discussion 13.7 Conclusion Appendix A References 14 Designing Instructional Materials to Help Students Make Connections: A Case of a Singapore Secondary School Mathematics Teacher’s Practice 14.1 Introduction 14.2 Teaching with a Connectionist Perspective 14.3 Making Connections 14.3.1 Connections Across Multiple Modes of Representations 14.3.2 Conceptual Connections 14.3.3 Temporal Connections 14.3.4 Connections Across Different Methods to Solve Problems 14.4 Method 14.4.1 Data 14.4.2 Analysis of Data 14.5 Findings 14.5.1 Making Connections Within a Unit 14.5.2 Making Connections Between Adjacent Units 14.5.3 Making Connections Across Units 14.6 Discussion 14.7 Conclusion Appendix References 15 Use of Technology by Experienced and Competent Mathematics Teachers in Singapore Secondary Schools 15.1 Introduction 15.2 Literature Review 15.3 Research Design 15.4 Findings and Discussion 15.4.1 Tutor Mode 15.4.2 Tool Mode 15.4.3 Engagement Mode 15.4.4 Assessment Mode 15.4.5 Teachers’ Aids 15.4.6 Students’ Aids 15.5 Conclusion References Part IVConclusion 16 Framing the Portraits of Singapore Secondary Mathematics Pedagogy: An Outsider’s Perspective 16.1 From Machine to Cup, from Plan to Lesson 16.2 Social Cognitive Theory 16.3 Teachers’ Content Knowledge 16.4 Trust in the Leadership 16.5 Students as Disciples 16.6 21st Century Competency Education 16.7 Societal Valuing of Excellence 16.8 Teachers’ Self-Efficacy 16.9 Conclusions References 17 Portraits of the Singapore Secondary School Mathematics Enactment: An Insider’s Perspective 17.1 Bringing the Various Facets Together 17.2 Revisit of the Pentagon Model 17.3 The Singapore “Paradox” 17.4 The Heart of the Paradox 17.5 Conclusion References