Table of Contents
Preface
1. Algebra – Setting the scene
1.1 Introduction
1.2 Algebra – Aims, actions and entities
1.2.1 Aims
1.2.2 Actions
1.2.3 Entities
1.3 Why Algebra?
1.4 Chapter Summary
1.5 Thinking further
1.6 References
2. Some lessons from history
2.1 Introduction
2.2 Linear equations in ancient Egypt
2.3 Quadratic equations in ancient Babylonia
2.4 A geometric view of algebra from Arabic mathematics
2.5 Beyond solving equations: The emergence of algebra in Europe
2.5.1 Some algebraic rules and their explanation
2.5.2 Negative numbers
2.6 Chapter summary
2.7 Thinking further
2.8 References
3. Seeing algebra through the eyes of a learner
3.1 Introduction – putting on teachers’ bifocal spectacles
3.2 What do algebraic letters represent?
3.2.1 Fruit salad algebra – misleading teaching with long term consequences
3.3 The process-object duality
3.4 The meaning of the equals sign
3.5 Algebra for recording and revealing mathematical structure
3.6 Transitions from arithmetic to algebra
3.6.1 Contrasting arithmetic and algebraic problem solving
3.6.2 Fundamental ideas of equation solving
3.6.3 Algebra and creativity
3.7 The procedures of equation solving
3.7.1 Process-object distinction in solving simultaneous equations
3.8 Functions as processes and objects
3.9 Chapter Summary
3.10 Thinking Further
3.11 References
Chapter 4: Emphases in algebra teaching
4.1 Introduction
4.2 Teaching algebra in context
4.3 Productive practice
4.4 The reconciliation of routine and insight
4.5 Exploiting student mistakes
4.6 Proofs in algebra teaching
4.7 Chapter summary
4.8 Thinking further
4.9 References
5. Algebra education in the digital era
5.1 Introduction
5.2 Digital tools for algebra
5.2.1 Algebraic functionality of digital tools
5.2.2 Pedagogical roles of digital tools for algebra
5.3 Core algebra entities with digital means
5.3.1 Variable
5.3.2 Equation and equivalence
5.3.3 Function
5.4 Teaching and learning algebra with digital means
5.5 Chapter summary
5.6 Thinking further
5.7 References
Epilogue