Introduction. 1. Approaches to Algebra: Perspectives for Research and Teaching; N. Bednarz, et al. Part I: Historical Perspectives in the Development of Algebra. 2. From Euclid to Descartes: Algebra and its Relation to Geometry; L. Charbonneau. 3. The Roles of Geometry and Arithmetic in the Development of Algebra: Historical Remarks from a Didactic Perspective; L. Radford. 4. The Role of Problems and Problem Solving in the Development of Algebra; T. Rojano. Part II: A Generalization Perspective on the Introduction of Algebra. 5. Expressing Generality and Roots of Algebra; J. Mason. 6. An Initiation into Algebraic Culture Through Generalization Activities; L. Lee. 7. Some Reflections on Teaching Algebra Through Generalization; L. Radford. Part 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; N. Bednarz, B. Janvier. 9. Developing Algebraic Aspects of Problem Solving within a Spreadsheet Environment; T. Rojano. 10. Rough or Smooth? The Transition from Arithmetic to Algebra in Problem Solving; D. Wheeler. 11. Algebraic Thought and the Role of a Manipulable Symbolic Language; A. Bell. 12. Placement and Function of Problems in Algebraic Treatises from Diophantus to Viete; L. Charbonneau, J. Lefebvre. 13. Problem-Solving Approaches to Algebra: Two Aspects; A. Bell. 14. `When is a Problem?': Question from History and Classroom Practice in Algebra; J. Mason. Part IV: AModeling Perspective on the Introduction of Algebra. 15. Mathematical Narratives, Modeling, and Algebra; R. Nemirovsky. 16. Reflections on Mathematical Modeling and the Redefinition of Algebraic Thinking; M.K. Heid. 17. Modeling and the Initiation into Algebra; C. Janvier. Part V: A Functional Perspective on the Introduction of Algebra. 18. A Technology-Intensive Functional Approach to the Emergence of Algebraic Thinking; M.K. Heid. 19. Introducing Algebra by means of a Technology-Supported, Functional Approach; C. Kieran, et al. 20. A Functional Approach to Algebra: Two Issues that Emerge; R. Nemirovsky. Part VI: Synthesis and Directions for Future Research. 21. Backwards and Forwards: Reflections on Different Approaches to Algebra; D. Wheeler. References.