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Early algebra and mathematical generalization

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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.

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Notes

  1. The hypergeometric sense of closed form (Weisstein, 1999, viz., Closed form, Generalized hypergeometric function) is not related to the present usage.

  2. Recursion and iteration refer to a process that “runs again and again.” We are not using recursive in the logic programming sense of recursion, according to which algorithm execution begins with the repeating condition.

  3. (Weisstein, 1999, viz., Sequence, Arithmetic progression) defines a sequence as an ordered set of mathematical objects, {o 1, o 2, o 3, … o n}.

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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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  1. TERC, Cambridge, MA, USA

    David W. Carraher

  2. Department of Education, Tufts University, Paige Hall, Medford, MA, 02155, USA

    Mara V. Martinez & Analúcia D. Schliemann

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  1. David W. Carraher
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Correspondence to Analúcia D. Schliemann.

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Carraher, D.W., Martinez, M.V. & Schliemann, A.D. Early algebra and mathematical generalization. ZDM Mathematics Education 40, 3–22 (2008). https://doi.org/10.1007/s11858-007-0067-7

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