1. Approaches to Algebra: Perspectives for Research and Teaching.- I. Historical Perspectives in the Development of Algebra.- 2. From Euclid to Descartes: Algebra and its Relation to Geometry.- 3. The Roles of Geometry and Arithmetic in the Development of Algebra: Historical Remarks from a Didactic Perspective.- 4. The Role of Problems and Problem Solving in the Development of Algebra.- II. A Generalization Perspective on the Introduction of Algebra.- 5. Expressing Generality and Roots of Algebra.- 6. An Initiation into Algebraic Culture through Generalization Activities.- 7. Some Reflections on Teaching Algebra through Generalization.- III. A Problem-Solving Perspective on the Introduction of Algebra.- 8. Emergence and Development of Algebra as a Problem-Solving Tool: Continuities and Discontinuities with Arithmetic.- 9. Developing Algebraic Aspects of Problem Solving within a Spreadsheet Environment.- 10. Rough or Smooth? The Transition from Arithmetic to Algebra in Problem Solving.- 11. Algebraic thought and the Role of a Manipulable Symbolic Language.- 12. Placement and Function of Problems in Algebraic Treatises from Diophantus to Viète.- 13. Problem-Solving Approaches to Algebra: Two Aspects.- 14. "When is a Problem?": Questions from History and Classroom Practice in Algebra.- IV. A Modeling Perspective on the Introduction of Algebra.- 15. Mathematical Narratives, Modeling, and Algebra.- 16. Reflections on Mathematical Modeling and the Redefinition of Algebraic Thinking.- 17. Modeling and the Initiation into Algebra.- V. A Functional Perspective on the Introduction of Algebra.- 18. A Technology-Intensive Functional Approach to the Emergence of Algebraic Thinking.- 19. Introducing Algebra by Means of a Technology-Supported, Functional Approach.- 20. AFunctional Approach to Algebra: Two Issues that Emerge.- VI. Synthesis and Directions for Future Research.- 21. Backwards and Forwards: Reflections on Different Approaches to Algebra.- References.- Author Affiliations.