Concept Mapping in Mathematics (eBook)

Concept Mapping in Mathematics 2
Foreword 5
Contents 7
Contributors 8
Introduction 10
Part I A Historical Overview of Concept Mapping 14
1 The Development and Evolution of the Concept Mapping Tool Leading to a New Model for Mathematics Education 15
Introduction: The Invention of Concept Mapping 16
The Use of Concept Maps in Mathematics 19
CmapTools and the Internet 19
A New Model for Education 20
Expert Skeleton Concept Maps 22
Adding Concepts and Resources Using CmapTools 22
Collaboration Among Students 23
Exploration with Real World Problems 24
Written, Oral, and Video Reports and Developing Knowledge Models 24
Sharing and Assessing Team Knowledge Models 26
In Conclusion 26
Part II Primary Mathematics Teaching and Learning 29
2 Analysing the 0Measurement0 Strand Using Concept Maps and Vee Diagrams 30
Introduction 30
Literature Review of Concept Mapping and Vee Diagrams 31
Methodology 32
Data Collected and Analysis 33
"Length" Concept Maps 33
Volume "Concept Maps" 38
Vee Diagrams of Mathematics Problems 46
Journal of Reflections 52
Critical Ability to Analyse Topics and Problem 52
Solve Mathematics Problems 53
Communicate Effectively 53
Develop a Deep Conceptual Understanding of a Topic 53
Discussion 53
Implications 55
3 Concept Mapping as a Means to Develop and Assess Conceptual Understanding in Primary Mathematics Teacher Education 58
Introduction 58
Developing the Concept of a Positional System in Teacher Education 59
Maryannes Map 60
Death by Decimal 66
Summary 67
4 Using Concept Maps and Vee Diagrams to Analyse the 0Fractions0 Strand in Primary Mathematics 69
Introduction 69
Definitions of Concept Maps and Vee Diagrams 70
Case Study 71
Context 71
Data Collected and Analysis 72
Task 1 Data and Analysis 72
Early Stage 1 and Stage 1 Concept Maps 72
Stage 2 Concept Map 74
Stage 3 Concept Map 75
Stage 4 Concept Map 77
Task 2 Data and Analysis 78
Overview "Fractions" Concept Map 78
Concept Maps and Vee Diagrams of "Fraction" Problems 85
Discussion and Implications 91
5 Concept Maps as Innovative Learning and Assessment Tools in Primary Schools 97
Introduction 97
Methodology 98
Results 99
Professional Development Workshops and Reflection Sessions 99
Teachers' Professional Development Workshop 99
Reflection Sessions 100
On-Going Professional Support for the Teachers 101
Preparation of Teaching and Learning Resources 101
Initial Site Visits 101
Classroom Observation Visit 102
Portfolio of Teacher Resources 102
More Site Visits and Reflection Sessions 103
Concept Mapping Activities 105
First Mapping Activities 105
Second Mapping Activities 107
Additional Mapping Activities 110
Teachers' Self-Designed Final Mapping Activities 113
Post-Activity Reflective Session 117
Discussion 117
Reflective Sessions 118
School Realities 118
On-going Professional Support 119
Summary of Class Contributions 119
Innovation and Professional Practice 120
Implications 121
Part III Secondary Mathematics Teaching and Learning 124
6 Evidence of Meaningful Learning in the Topic of 0Proportionality0 in Second Grade Secondary Education 125
Theoretical Background 125
The Topic of the Proportionality 127
Research Planning and Design 129
The Setting in Which the Concept Maps were Used 130
Description and Discussion of the Findings 132
The Way in Which the Pupils Differentiate Concepts from Links 133
Utilisation of Concepts 133
Propositions Formed from Links Between Concepts 133
Levels of Hierarchy 135
Crossed Links 136
Conclusion 140
7 Concept Mapping as a Means to Develop and Assess Conceptual Understanding in Secondary Mathematics Teacher Education 144
Case Study 144
Epistemological Value 152
8 Concept Mapping a Teaching Sequence and Lesson Plan for 0Derivatives0 155
Introduction 155
Methodology 156
Data Collected 157
Data Analysis 157
Learning to Concept Map 158
Overview Concept Maps 160
Teaching Sequence Concept Map 162
Lesson Plan Concept Map 165
Discussion 170
Concept Maps of Critical Analysis 171
Workshop Discourse 172
Socio-Mathematical Norms 172
Practical Management of the Learning Ecology 173
Main Insights and Implications 173
9 Curricular Implications of Concept Mapping in Secondary Mathematics Education 176
Introduction 177
Concept Mapping and Historical Research as a Combined Epistemological Tool 177
Conceptual Analysis From a Cultural Historical Perspective 178
Historical Foundation 178
Conceptual Essence of the Logarithm Concept from a Cultural-Historical Perspective 180
More Fully Developed Concept Map of a Logarithm 180
The Problem of Generative Metonymy 183
Conceptual Representation the Teachers View 183
Conceptual Representation The Students View 184
Curriculum Proposal 186
Lesson 1: "Introducing the Logarithm" 187
Lesson 2: "The Logarithmic Graph" 188
Lesson 3: "Logarithms, So What?" 189
Lesson 4: "Logarithmic Scales and Logarithmic Graph Paper" 190
Curriculum Design Questions to Use in Conjunction with Concept Mapping 191
Conclusions and Implications 192
10 Using Concept Maps and Gowin0s Vee to Understand Mathematical Models of Physical Phenomena 194
Introduction 194
Theoretical background: Concept Maps and Gowins Vee 195
What is a Concept Map? 197
Gowins Vee 203
Phases of the Implementation of the Strategy 204
Phase I: Learning About Concept Maps and Gowin's Vee 205
Phase II: Eliciting Basic Conceptual Knowledge of Mathematical Functions 206
Phase III: Acquisition of the Concept "model" in Physics 206
Phase IV: Students Training in the Application of the Concept "Model" to Physical Phenomena 206
The Evaluation of the Strategy 206
The Strategy -- Trial and Results 207
Phase I: Learning About Concept Maps and Gowin's Vee. 207
Phase II: Eliciting Basic Conceptual Knowledge of Mathematical Functions 207
Phase III: Acquisition of the Concept "Model" in Physics 209
Phase IV: Modelling Physical Phenomena 219
Conclusions 219
11 Applying Concept Mapping to Algebra I 222
Introduction 222
Developing an Algebra I Course Through Concept Mapping 224
Course Prerequisites 224
Narrative on the Development of Polynomials 225
Working With Monomials 226
Polynomials are Developed From Monomials 228
Operations on Polynomials 229
Degree of a Polynomial 230
Evaluating Polynomials and Solving Polynomial Equations 231
Quadratic Polynomials 232
Part IV University Mathematics Teaching and Learning 240
12 Enhancing Undergraduate Mathematics Learning Using Concept Maps and Vee Diagrams 241
Introduction 241
Theoretical Framework 243
Methodology 245
Data Analysis 246
Concept Maps 246
Vee Diagrams 247
Data Collected and Analysis 247
Concept Map Data 248
Student 1's Topic ' Laplace's Transform (LT) 248
Student 2--s Topic -- Trigonometric Approximations 249
Student 3--s Topic -- Least Squares Polynomial Approximations (LSPA) 250
Student 4--s Topic -- Multivariable Functions 251
Student 5--s topic -- Numerical methods 252
Student 6--s Topic -- Partial Differential Equations (pdes) 254
Vee Map Data 254
Overall Criteria 254
Specific Criteria 257
Discussion 257
Conclusions and Implications 258
13 Concept Mapping: An Important Guide for the Mathematics Teaching Process 262
Introduction and Antecedents 262
Concept Maps and The Learning Process 265
Basic Processes and The Learning of Mathematics 267
Concept Mapping for Mathematics Teaching Process: Some Examples 268
"Critical Point of a Function" Concept Map 269
"Extreme Point" Concept Map 271
"Real Numbers" Concept Map 272
"Displacement of Functions" Concept Map 273
Mathematics Problem Concept Maps 273
"Solution of Certain Inequalities" Concept Map 274
"Elements of a Function" Concept Map 274
Cognitive Development of the Learner 276
Concept Map and Students: Important Considerations 276
The Intervention and Its Methodology 277
Results 277
Discussion and Implications 278
14 Concept Mapping and Vee Diagramming 0Differential Equations0 281
Introduction 281
Concept Mapping and Vee Diagram Studies 282
Theoretical Perspectives 283
Nats Case Study 285
Nats Data and Analysis 286
Concept Map Data Analysis 286
Concept Map Criteria 286
Concept Map Data 288
Vee Diagram Data Analysis 291

Discussion 295
Implications 297
15 Using Concept Maps to Mediate Meaning in Undergraduate Mathematics 300
Introduction 300
Mathematics Education in Samoa 301
Literature Review 302
Theoretical Framework 302
Relevant Studies 304
Concept Mapping and Vee Diagram Studies 304
Methodology of the Study 305
Analysis of Concept Maps 305
Concept Maps Collected 306
Examples of "Good" Concept Maps 309
Student 3: Fia -- Numerical Methods 309
Student 4: Vae -- Limits and Continuity 310
Student 9: Toa -- Normal Distributions (ND) 313
Example of an "Above Average" Final Concept Map 315
Student 5: Heku -- Motion 315
Example of an "Average" Final Concept Map 315
Student 2: Loke -- Differentiation 315
Example of a "Below Average" Final Concept Map 317
Student 8: Pasi -- Integration 317
Example of a "Poor" Final Concept Map 318
Student 1: Pene -- Indeterminate Forms 318
Discussion 319
Implications 325
Part V Future Directions 328
16 Implications and Future Research Directions 329
Index 335