Generalization in fifth graders within a functional approach

“According to some researchers, generalization is the key element in algebra (e.g., Mason, 1996).”

“For example, Mason and Pimm (1984) describe the use of everyday language as a fundamental resource to express generalization and its use can have influence in the use of algebraic symbolism to express the generalization.”

“Different researchers show types or levels of generalization (e.g., Harel & Tall, 1991; Kaput, 1999; Kruteskii, 1976; Stacey, 1989).”

“Representations used by the students to generalize: pictorial, verbal, numerical or with algebraic notation, as well as combinations of one or the other or both with other representations (Blanton et al., 2011; Carraher et al., 2008).”

“…in mathematics education is growing around elementary school students’ understanding and expression of notions about algebraic concepts, particularly these students’ generalization and how they express general relationships when solving different problems (Carpenter & Levi, 2000; Kaput, 2008).”

“Although such functional thinking appears to be beneficial for students, its application in the elementary grades has received scant attention (Blanton & Kaput, 2011).”

“…reorganization process in which PNA 12 (3) a person identifies the essence of one idea, which implies deliberate reasoning that builds on specific cases to identify inter-model, inter-procedural or inter-structural relationships (Carraher & Schliemann, 2002; Kaput, 1999; Mitchelmore & White, 2007).”

“In line with Blanton et al. (2015), the present authors venture that using algebraic notation would enable students to visualize generalization in fuller detail.”

“…reorganization process in which PNA 12 (3) a person identifies the essence of one idea, which implies deliberate reasoning that builds on specific cases to identify inter-model, inter-procedural or inter-structural relationships (Carraher & Schliemann, 2002; Kaput, 1999; Mitchelmore & White, 2007).”

“Traditionally, generalization process has been linked to the use of algebraic symbolism and much of the previous research has focused on the students’ difficulties in generalizing (Dienes, 1961; Ellis, 2007).”

“Radford (2010) recognizes the importance of gestures as a way to express generalization.”

“Smith (2008) defined the functional relationships involving two quantities that co-vary to be: (a) correspondence, or the relationship between the pairs of values for the two variables ; and (b) covariation, or the relationship that describes how changes in one variable affect the other.”

“Generalization and representation notions are very close in Elementary education (Kaput, Blanton, & Moreno, 2008).”

“…reorganization process in which PNA 12 (3) a person identifies the essence of one idea, which implies deliberate reasoning that builds on specific cases to identify inter-model, inter-procedural or inter-structural relationships (Carraher & Schliemann, 2002; Kaput, 1999; Mitchelmore & White, 2007).”

“…in quantities, (b) use different representations, (c) expand numerical contents working with domain and codomain of variables, (d) help to increase students’ ability to get generalization, among others (Blanton et al., 2011; Carraher & Schliemann, 2018; Romberg, Fennema, & Carpenter, 1993).”

“Students at Elementary, or even at Kindergarten, are able to identify regularities that serve as first approach to generalization with no knowledge about algebraic symbolism (e.g., Castro, Cañadas, & Molina, 2017; Schifter, Monk, Russell, & Bastable, 2008).”

“Within this project, previous studies have described fifth graders’ (10 to 11 years old) ability to generalise, identifying the functional relationships they applied and the types of representation they used to express them (Pinto & Cañadas, 2018a; b).”

“An increasing number of studies have reported the work of elementary students with functional thinking problems (e.g., Pinto & Cañadas, 2018; Blanton et al., 2015; Carraher & Schliemann, 2007).”

“There are publications that are part of research projects in various countries, which emphasize the evidence that children in early grades can think in a rather more sophisticated manner than previously assumed (e.g. Kaput et al., 2008; Pinto & Cañadas, 2017; Radford, 2018; Torres et al., 2018).”

“They have highlighted students’ preference for expressing generalization verbally (Merino et al., 2013; Torres et al., 2022), and there is evidence that fifth-graders (10- to 11-year-olds) have the potential to use algebraic notation and verbal representation to generalize (Pinto & Cañadas, 2018).”

“Structures are the regularities observed in relationships between variables within a function (Pinto & Cañadas, 2017a).”

“Recent research shows that students from early educational ages can solve tasks involving algebraic notions, such as those relating to functional thinking (e.g., Blanton et al., 2015; Cañadas et al., 2016; Morales et al., 2016; Pinto & Cañadas, 2018).”

“In the same line, Pinto et al. (2018) showed that students aged nine to 11 used covariation and correspondence in a task whose function was f(x)=2x+6 and generalized it using algebraic representation and verbal representation.”

“Research indicates that functional thinking capabilities can be present from early educational stages and can be further developed through targeted instructional activities (Blanton & Kaput, 2004; Cañadas et al., 2016; Pinto & Cañadas, 2018; Türkmen & Tanışlı, 2019).”