Effective Mathematics Lessons through an Eclectic Singapore Approach : Yearbook 2015, Association of Mathematics Educators

With this seventh volume, as part of the series of yearbooks by the Association of Mathematics Educators in Singapore, we aim to provide a range of learning experiences and teaching strategies that mathematics teachers can judiciously select and adapt in order to deliver effective lessons to their students at the primary to secondary level. Our ultimate goal is to develop successful problem solvers who are able to understand concepts, master fundamental skills, reason logically, apply mathematics, enjoy learning, and strategise their thinking. These qualities will prepare students for life-long learning and careers in the 21st century.The materials covered are derived from psychological theories, education praxis, research findings, and mathematics discourse, mediated by the author's professional experiences in mathematics education in four countries over the past four decades. They are organised into ten chapters aligned with the Singapore mathematics curriculum framework to help teachers and educators from Singapore and other countries deepen their understanding about the so-called'Singapore Maths'.The book strikes a balance between mathematical rigour and pedagogical diversity, without rigid adherence to either. This is relevant to the current discussion about the relative roles of mathematics content knowledge and pedagogical content knowledge in effective teaching. It also encourages teachers to develop their own philosophy and teaching styles so that their lessons are effective, efficient, and enjoyable to teach. Contents 12 Foreword 8 Acknowledgements 10 Chapter 1 Curriculum: Map the Intended, Implemented, and Attained Landscape 16 1 Nature of Mathematics 16 2 Three Types of Curriculum 20 3 Intended Mathematics Curriculum: Why? 25 4 Intended Mathematics Curriculum: What? 28 5 Intended Mathematics Curriculum: Curriculum Framework 29 6 Mathematics Curriculum Development: How to? 31 6.1 Strands of mathematics curriculum development 31 6.2 Situated Socio-Cultural Framework 33 7 Implemented Mathematics Curriculum: How? 34 8 Attained Curriculum: How Well? 37 8.1 Assessment goals 38 8.2 Quality of assessment 38 8.3 Interpretations of assessment data 39 9 Concluding Remarks 42 Chapter 2 Concepts: Build Meanings and Connections 44 1 Hierarchies of Concepts 44 2 Meanings, Examples, Non-examples 45 2.1 Meanings 46 2.2 Examples 49 2.3 Non-examples 51 2.4 Frayer Model 52 3 Modes of Representation 53 3.1 Functions of representations 53 3.2 Concrete Pictorial Abstract (CPA) 55 3.3 Multi-Modal Strategy (MMS) 58 3.4 Modes of representation vs. modes of processing 61 4 Conceptual Connections 63 4.1 Carroll diagram 63 4.2 Venn diagram 64 4.3 Tree diagram 65 4.4 Concept maps 67 5 Concept Questions 69 6 Concluding Remarks 70 Chapter 3 Skills: Use Rules Efficiently 72 1 Nature of Mathematical Skills 72 1.1 Alternative procedures 73 1.2 Conditions for procedures 74 1.3 Hierarchies of mathematical skills 75 2 Skills vs. Concepts 77 2.1 An example from fraction division 77 2.2 Procept 78 2.3 Limit and recurring decimals 79 3 Direct Instruction: An Overview 80 4 Frameworks of Direct Instruction 82 5 Telling and Explaining 83 6 Worked Examples 86 6.1 Correct mathematics and real-world information 87 6.2 I do – We do – You do 88 6.3 Cognitive Load Theory 89 6.4 Problem solving set 91 7 Deliberate Practice 92 7.1 Check seatwork or classwork 93 7.2 Students work on the board 94 7.3 Homework 94 8 Address Student Mistakes 96 9 Skill Questions 98 10 Concluding Remarks 99 Chapter 4 Processes: Sharpen Mathematical Reasoning and Heuristic Use 100 1 Mathematical Processes: Domain-Generic vs. Domain-Specific 100 2 Mathematical Reasoning: Definition 102 3 Intuitive-Experimental Justification 104 3.1 Teaching inductive-experimental justification 104 3.2 Examples of inductive-experimental justification 105 3.3 Caveats about inductive thinking 112 3.4 A brief summary about inductive thinking 118 4 Deductive Proofs 119 4.1 Some proof examples 119 4.2 Logical forms 121 4.3 Converse of Pythagoras’ Theorem 122 4.4 Zero Product Rule (Zero Factor) 123 4.5 Axiomatic system 124 5 Acceptance of Results without Justifications 128 6 Heuristics 129 6.1 Some local studies about teaching of heuristics 130 6.2 Model drawing 133 7 Question-and-Answer (Q&A) 138 8 Mathematics Discussions 142 9 Reasoning Questions 143 10 Concluding Remarks 144 Chapter 5 Applications: View the World Through Mathematical Lenses 146 1 Query about Mathematical Applications 146 2 Context Knowledge 148 3 Direct Applications of Specific Skills 150 4 Applications of Processes 153 5 Applications across School Subjects 156 6 Mathematics-related National Education: A Singapore Initiative 158 7 Mathematical Modelling (MM) 162 7.1 Apply known knowledge 163 7.2 Models based on collected data 165 7.3 From Additional Mathematics 167 7.4 Some teaching issues 168 8 Application Questions 170 9 Concluding Remarks 170 Chapter 6 ICT: Be Its Prudent Master 172 1 Introduction 172 2 Aims of ICT Use in Mathematics Education 173 3 Modes of ICT Use in Education 175 4 Tutor Mode: Learn from the Computer 175 5 Tutee Mode: Learn through Programming the Computer 178 5.1 To teach is to learn twice 179 5.2 A short Logo primer 180 5.3 The Logo approach 184 5.4 Trends in tutee mode 186 6 Tool Mode: Learn with the Computer 187 6.1 Stages of use 188 6.2 Teaching sequence 191 6.3 Teaching and learning issues about tool mode 194 7 Co-Learner Mode 197 8 Computer-based Automatic Assessment (CAA) 198 9 ICT Questions 203 10 Concluding Remarks 203 Chapter 7 Attitudes: Energise Learning with Emotional Power 206 1 Cognitive Domain vs. Affective Domain 206 2 Types and Meanings of Affective Constructs 207 3 Importance of Affective Domain 211 4 Attitudes: Meanings and Measurements 212 4.1 Likert scale 214 4.2 Semantic differential 216 5 Research Issues about Correlations 217 6 Motivation: To Teach is To Sell 220 7 The M_Crest Framework 222 7.1 M = Meaningfulness 222 7.2 C = Confidence 223 7.3 R = Relevance 225 7.4 E = Enjoyment 225 7.5 S = Social Relationships 227 7.6 T = Targets 228 7.7 Combining the Motivators 229 8 Concluding Remarks 229 Chapter 8 Metacognition: Strategic Use of Cognitive Resources 232 1 Metacognition: Meanings and Importance 232 2 Self-Regulation of Learning 234 2.1 A framework about self-regulated learning 235 2.2 Student Question Cards (SQC) 237 2.3 Studies about learning strategies 240 2.4 Promoting alternative learning strategies 247 3 Metacognition during Problem Solving 249 3.1 Cognition vs. metacognition 249 3.2 Non-metacognitive behaviours 251 3.3 Developing metacognition 253 3.4 Investigating metacognition 255 4 Be Mindful 257 5 Concluding Remarks 258 Chapter 9 School Curriculum: Prepare Thoughtful Plans 260 1 Benefits of Preparing Daily Lesson Plans 260 2 Learning or Instructional Objectives 262 2.1 Specific instructional objectives and advance organisers 262 2.2 Informing students about SIOs or AOs 267 3 Assess and Activate Prerequisites 268 4 Motivation for the Lesson 270 5 Development and Consolidation 270 6 Closure 272 7 Overall Organisation 274 8 Review of Lessons 275 9 Worksheets 276 10 Unit Plans and Scheme of Work 279 11 Concluding Remarks 280 Chapter 10 Professional Development: Become Metacognitive Teachers 282 1 Need for Professional Development 282 2 Deepen and Broaden Mathematics Knowledge 284 3 Master New Pedagogy 287 3.1 Teacher standards 287 3.2 Training pedagogy 289 3.3 Research own practices 291 4 Framework of Teacher Professional Development 293 5 Professional Development of Teacher Educators 295 6 Concluding Remarks 296 7 References 297