Early Algebraization

1. Preface to Part I
Jinfa Cai, Eric Knuth

2. Functional Thinking as a Route Into Algebra intheElementary Grades
Maria L. Blanton, James J. Kaput

3. Developing Students’ Algebraic Thinking inEarlier Grades: Lessons from China andSingapore
Jinfa Cai, Swee Fong Ng, John C. Moyer

4. Developing Algebraic Thinking in the Context ofArithmetic
Susan Jo Russell, Deborah Schifter, Virginia Bastable

5. The Role of Theoretical Analysis in Developing Algebraic Thinking: A Vygotskian Perspective
Jean Schmittau

6. The Arithmetic-Algebra Connection: AHistorical-Pedagogical Perspective
K. Subramaniam, Rakhi Banerjee

7.
Shiki: A Critical Foundation for School Algebra in Japanese Elementary School Mathematics
Tad Watanabe

8. Commentary on Part I
Jeremy Kilpatrick

9. Preface to PartII
Eric Knuth, Jinfa Cai

10. Algebraic Thinking with and without Algebraic Representation: A Pathway for Learning
Murray S. Britt, Kathryn C. Irwin

11. Examining Students’ Algebraic Thinking inaCurricular Context: ALongitudinal Study
Jinfa Cai, John C. Moyer, Ning Wang, Bikai Nie

12. Years 2 to 6 Students’ Ability to Generalise: Models, Representations and Theory forTeaching and Learning
Tom J. Cooper, Elizabeth Warren

13. Algebra in the Middle School: Developing Functional Relationships Through Quantitative Reasoning
Amy B. Ellis

14. Representational Competence and Algebraic Modeling
Andrew Izsák

15. Middle School Students’ Understanding of Core Algebraic Concepts: Equivalence & Variable
Eric J. Knuth, Martha W. Alibali, Nicole M. McNeil, Aaron Weinberg, Ana C. Stephens

16. An Approach to Geometric and Numeric Patterning that Fosters Second Grade Students’ Reasoning and Generalizing about Functions and Co-variation
Joan Moss, Susan LondonMcNab

17. Grade 2 Students’ Non-Symbolic Algebraic Thinking
Luis Radford

18. Formation of Pattern Generalization Involving Linear Figural Patterns Among Middle School Students: Results of a Three-Year Study
F. D. Rivera, Joanne Rossi Becker

19. Commentary on PartII
Bharath Sriraman, Kyeong-Hwa Lee

20. Preface to PartIII
Eric Knuth, Jinfa Cai

21. Prospective Middle-School Mathematics Teachers’ Knowledge of Equations andInequalities
Nerida F. Ellerton, M. A. (Ken) Clements

22. The Algebraic Nature of Fractions: Developing Relational Thinking in Elementary School
Susan B. Empson, Linda Levi, Thomas P. Carpenter

23. Professional Development to Support Students’ Algebraic Reasoning: An Example fromtheProblem-Solving Cycle Model
Karen Koellner, Jennifer Jacobs, Hilda Borko, Sarah Roberts, Craig Schneider

24. Using Habermas’ Theory ofRationality toGain Insight intoStudents’ Understanding ofAlgebraic Language
Francesca Morselli, Paolo Boero

25. Theoretical Issues and Educational Strategies for Encouraging Teachers to Promote aLinguistic and Metacognitive Approach toEarly Algebra
Annalisa Cusi, Nicolina A. Malara, Giancarlo Navarra

26. A Procedural Focus and a Relationship Focus toAlgebra: How U.S. Teachers and Japanese Teachers Treat Systems of Equations
Margaret Smith

27. Teaching Algebraic Equations with Variation inChinese Classroom
Jing Li, Aihui Peng, Naiqing Song

28. Commentary on Part III
John Mason

29. Overall Commentary on Early Algebraization: Perspectives for Research and Teaching
Carolyn Kieran

Keywords: Education, Mathematics Education, Teaching and Teacher Education, Learning & Instruction, International and Comparative Education, Curriculum Studies