ZDM

, Volume 40, Issue 1, pp 3–22

# Early algebra and mathematical generalization

• David W. Carraher
• Mara V. Martinez
• Analúcia D. Schliemann
Original article

## Abstract

We examine issues that arise in students’ making of generalizations about geometrical figures as they are introduced to linear functions. We focus on the concepts of patterns, function, and generalization in mathematics education in examining how 15 third grade students (9 years old) come to produce and represent generalizations during the implementation of two lessons from a longitudinal study of early algebra. Many students scan output values of f(n) as n increases, conceptualizing the function as a recursive sequence. If this instructional route is pursued, educators need to recognize how students’ conceptualizations of functions depart from the closed form expressions ultimately aimed for. Even more fundamentally, it is important to nurture a transition from empirical generalizations, based on conjectures regarding cases at hand, to theoretical generalizations that follow from operations on explicit statements about mathematical relations.

## Keywords

Young Student Algebraic Expression Mathematical Generalization Separate Table Dinner Table
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Notes

### Acknowledgments

This study was developed as part of the National Science Foundation supported project “Algebra in Early Mathematics” (NSF-ROLE grant 0310171). We thank Barbara Brizuela, Anne Goodrow, Darrell Earnest, Susanna Lara-Roth, Camille Burnett, and Gabrielle Cayton for their contributions to the work reported here. We thank also Judah Schwartz for many important insights regarding the role of functions in early mathematics.

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## Authors and Affiliations

• David W. Carraher
• 1
• Mara V. Martinez
• 2
• Analúcia D. Schliemann
• 2
1. 1.TERCCambridgeUSA
2. 2.Department of EducationTufts UniversityMedfordUSA