Early Introduction to Algebraic Thinking: the Role of Logo as link between Algebra and Geometry

Cristianne Butto

CINVESTAV-IPN Mexico

Teresa Rojano

CINVESTAV-IPN Mexico

Mrojanoa@mailer.main.conacyt.mx

Olimpia Figueras

CINVESTAV-IPN Mexico

Transition from Arithmetic to Algebra has been studied by diverse authors from the perspective of general arithmetic, break evolution, reification, the sense of operations, symbol interpretation, and methods. The present article reports on a study about transition to algebra incorporating a teaching model that takes into account geometric background and proportional reasoning of the appearance of algebraic language, as well as generalization processes that allow access to more abstract algebraic thinking. In the history of the development of algebraic ideas, geometric thinking appears as a relevant background regarding the meanings underlying the symbols of early algebraic expressions (Witmer 1983) as well as ideas of geometric proportionality (Radford, 1996).

In light of these considerations, a path to Algebra that has been conceived incorporates meaning sources such as proportional reasoning (numerical and geometric), aspects of proportional variation and generalization processes (Mason, 1985) toward the construction of a teaching model (Filloy, 1999) in which students may build meaning sources beyond numbers and its operations. The teaching model thus proposed (Filloy,1999) intends to provide students with early mathematical experience as meaningful initiation to the learning of algebra. In this direction, the Logo environment is incorporated for the relationship among proportional numerical thinking, proportional geometric thinking and generalization processes to become explicit. The study includes 10-11 year olds and it involves a pre-questionnaire to analyze notions of geometric proportionality and the arithmetic handled by students at the beginning of the study; experimental work involving pairs of students working in a Logo environment; and a post questionnaire to study influences of the three components (numerical, geometric and generalization) in the construction of early algebraic notions. Another aspect of the study focuses on student-student, student-interviewer, student-environment interactions, through the study of meaning negotiation while discourse takes place. Structural analysis of the text by Clarke (1998) is included where social interaction levels proposed by Kieran and Dreyfus (1999) are identified.

References

Clarke, D. J. (1998). Studying in classroom negotiation of meaning: Complementary accounts methodology, chapter 7. In A Teppo (Ed.), Qualitative Reserach methods in Mathematics Education. Reston, VA: National Council of Teachers of Mathematics.

Filloy, E. (1999). Modelos Teóricos locales (MTL): Un marco teórico y metodológico para la observación experimental en matemática educativa. In Aspectos teóricos del álgebra educativa. Grupo Editorial Iberoamerica S.A de C.V

Kieran, C (1999). Collaborative problem solving by 13 year-old Algebra students: Didactic productivity, patterns of interaction and the role of the computer. (School Algebra Theory and Practice Working Group, E. Filloy & T. Rojano). In f. Hitt & M. Santos (Eds.), Proceedings of the XXI Annual Conference of the Psychology of Mathematics Education, North American Chapter, Vol. 1 (pp. 158-160). Columbus, Ohio: ERIC.

Mason, J.; Graham, A.; Pimm, D., & Gower, N. (1985). Routes and Roots of Algebra. Great Britian: The Open University Press.

Radford, L. (1996). The role of Geometry and Arithmetic in the development of Algebra: Historical remarks from a didactic perspective. In N. Bernardz, C. Kieran and I. Lee (Eds), Approaches to Algebra. Perspectives for Reserach and Teaching. The Netherlands: Kluwer Academic Publishers.

Viète, F. (1983). The Analytic Art (T.R. Witmer, trans.). Kent, OH: Kent State University Press.