Abstract
This study describes how 24 third graders (8–9 years old) relate and represent the relationships between variables when working with a functional thinking problem. This aspect contributes to providing insights about how elementary school students attend properties and relationships between covarying quantities rather than isolated computations. From a functional approach to early algebra, we describe written students’ answers when working with a problem that involves a function, which includes questions for specific values and to generalize. Design research guidelines, specifically those set out for Classroom Teaching Experiment were followed. This study addresses the fourth and last Classroom Teaching Experiment session, which involved a function of the type y = ax + b and students had not previously worked it. Students primarily evidenced correspondence relationship, using natural language and numerical representation to express this functional relationship. Our findings let us to state that (a) although students were not used to working with these types of problems, eleven of them go beyond arithmetic computations, finding relationships that relate the variables; and (b) three students generalized using natural language as a useful vehicle, while there are other students who perceived the same regularity for different specific values but they are unable to represent generalization clearly.
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Notes
In this study, we focus on external representations to distinguish them from mental or internal ones. Therefore, every time we mention the term “representation” or “representations,” we refer to external representations, made with pencil or paper, which are intentional, permanent, and have a spatial nature.
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Acknowledgements
This study was conducted under National R&D Project EDU2016-75771-P funded by the Spanish Ministry of Economy and Competitiveness; the first author benefited from a PhD. grant awarded the by Chilean Government through the Comisión Nacional de Investigación Científica y Tecnológica (CONICYT), folio 72160307-2015.
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Pinto, E., Cañadas, M.C. & Moreno, A. Functional Relationships Evidenced and Representations Used by Third Graders Within a Functional Approach to Early Algebra. Int J of Sci and Math Educ 20, 1183–1202 (2022). https://doi.org/10.1007/s10763-021-10183-0
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DOI: https://doi.org/10.1007/s10763-021-10183-0