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This book aims at understanding the functioning of algebraic reasoning, its characteristics, the difficulties students encounter in making the transition to algebra, and the situations conducive to its favorable development. Four different perspectives, each related to a corresponding conception of algebra, provide avenues for its introduction: generalization, problem solving, modeling, and functions. The analysis of research on these perspectives is illuminated by a dual focus on epistemological (via the history of the development of algebra) and didactic concerns.
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 Viète; 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: A Modeling 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.