Word problems associated with the use of functional strategies among grade 4 students
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Word problems associated with the use of functional strategies among grade 4 students Rafael Ramírez 1
& Bárbara
M. Brizuela 2
& Cristina
Ayala-Altamirano 1
Received: 31 October 2019 / Revised: 29 June 2020 / Accepted: 10 July 2020 # Mathematics Education Research Group of Australasia, Inc. 2020
Abstract This article discusses the characteristics of word problems that are associated with students’ use of functional strategies and their ability to represent the generalization of functions. In the context of a broader research project designed to explore and foster functional thinking among elementary school students, twenty-five grade 4 (9- to 10-year-old) students were asked to identify functional relationships in five problems involving specific or indeterminate quantities. Their responses to a number of questions involving the generalization of the relationships in the problems were analyzed and associated to the characteristics of the problems. The type of representation of generalization used (verbal, generic, or symbolic) was also identified. Our findings indicate that grade 4 students showed potential for functional thinking prior to receiving instruction on variables and their notation. Such thinking was most effectively prompted when they worked with word problems that explicitly involved an additive function. When students generalized functional relationships, they represented them verbally or with generic examples. None of the students used symbolic representation. The originality of this study lies in the description of the specific characteristics of word problems that are associated with functional thinking; this information will prove useful to both teachers and curriculum designers. Identifying these characteristics could help build and propose tasks that encourage students to use more than one and more sophisticated strategies. Keywords Early algebra . Functional thinking . Generalization . Representations . Word
problems * Rafael Ramírez [email protected] Bárbara M. Brizuela [email protected] Cristina Ayala-Altamirano [email protected]
1
University of Granada, Campus de Cartuja s/n, 18071 Granada, Spain
2
Department of Education, Tufts University, 12 Upper Campus Rd., Paige Hall, Medford, MA 02155, USA
R. Ramirez-Úcles et al.
Introduction One of the goals pursued in today’s mathematics education research is to establish the connection between arithmetic and algebraic thinking in the early grades (Warren et al. 2016). According to the literature, understanding patterns, relationships, and functions in different contexts is critical to algebraic thinking (Blanton and Kaput 2011; Kieran et al. 2016; Lee et al. 2018). The study reported in this paper was designed to explore functional thinking among grade 4 students who had no prior instruction on the use of variable notation and in early algebra (specifically, early algebra that included problems focused on everyday contexts). Functional thinking includes generalizing relationships between covariant quantities: expressin
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