5 Representational Thinking as a Framework for Introducing Functions in the Elementary Curriculum

“A correspondence relationship is a correlation between corresponding pairs of independent and dependent variables, typically expressed as a function rule (Smith, 2008).”

“Smith (2008) related these two aspects of generalisation with different types of thinking—Brepresentational thinking^ and Bsymbolic thinking^, respectively (p. 133).”

“…their research on the functional thinking found in classroom data is based on the three modes of analyzing patterns and relationships outlined by Smith (2008): (i) recursive thinking, which involves finding variation within a single sequence of values; (ii) covariational thinking, which is…”

“Smith (2008) proposed a framework for functional thinking in which algebraic thinking occurs as the learner invents or appropriates representational systems to represent generalizations of a relationship between variables.”

“T13 answered both questions following the same “closed form rule” to describe a relation between quantities (Confrey and Smith 1994), adding the number of top and bottom gray tiles to the number of tiles on the left and right sides (3 + 3).”

“An understanding of linear functional relationships is considered an appropriate starting point for developing functional thinking, and this can extend naturally from counting experiences involving repeated addition (Smith 2008).”

“Questions addressed the correspondence relationship (primarily) and covariation (Smith, 2008) and revolved around both direct (how the dependent was related to the independent variable) and inverse (how the independent was related to the dependent variable) functional relationships.”

“For Smith (2008), the genesis of functional thinking implies paying attention to the varying quantities and starting to focus on the relationship between these quantities. For Mestre (2014), young children face a variety of causal and dependence events in their daily lives, enabling this type of thinking to develop lo…”

“The two key intuitions for developing functional thinking are correspondence and covariation (Smith, 2008).”

“Recursive patterns are centered on the values of a single variable and correspondence/covariation are part of functional thinking given “the emphasis on constructing a relationship between variables” (Smith, 2008, p. 146).”

“In fact, beyond recursive patterning , Smith (2008) outlines two other modes of analysing patterns and relationships: correspondence relationship and covariational thinking .”

“Students seemed to have made the shift from term-to-term reasoning to understanding covariation – not a trivial step (Blanton & Kaput, 2011; Smith, 2008).”

“These two meanings are foundational for generalizing figural growing patterns (e.g., Smith, 2008) and arguably for creating them: to find accurate quantities for each figure using correspondence and to create a coherent structural arrangement evidencing a particular type of growth (linear,…”

“Functional thinking is present when students establish covariation or correspondence relationships between the variables involved in problems (Smith, 2008).”

“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.”

“…defined as a type of Brepresentational thinking that focuses on the relationship between two (or more) varying quantities, specifically the kinds of thinking that lead from specific relationships (individual incidences) to generalizations of that relationship across instances^ (Smith 2008, p. 143).”

“Nonetheless, this finding is consistent with past literature and supports the notion that the ability to work with patterns is strongly associated with algebraic thinking (Blanton & Kaput, 2003, 2004; Childs, 1995; Ferrini-Mundy et al., 1997; Smith, 2008).”

“…as a type of “representational thinking that focuses on the relationship between two (or more) varying quantities, specifically the kinds of thinking that lead from specific relationships (individual incidences) to generalisations of that relationship across instances” (Smith, 2008, p. 143).”

“Functional thinking abilities involve (a) identifying and expressing relationships between variables presented through numerical and figural patterns (Smith, 2008; Pittalis et al., 2020) and (b) linking various types of function representations (i.”

“…relationships, use a variety of representations – including algebraic symbolism – and understand, represent and progress in the expression of functional relationships, after receiving instruction (e.g. Blanton et al., 2015; Blanton & Kaput, 2004; Carraher et al., 2008; Warren & Cooper, 2005, 2008).”

“Uno de los factores analizados por la investigación sobre pensamiento funcional es la generalización de las relaciones (Blanton et al., 2011; Smith, 2008) mediante la descripción de la regla general según la cual las cantidades covarían y que las representa (Blanton et al., 2011; Hitt &…”

“…pattern generalisation: a concrete context intended to elicit visual reasoning, different views of function such as correspondence and co-variation (Smith, 2008), and conceptual connections between functional relationships and their symbolisa-tion (e.g. Mason, 2017; Radford, 2003; Rivera, 2010).”

“When students express the generalization of the functional relationship, they may represent it through spoken language, graphs, symbolic representations, or a combination of different representations (Carraher et al. 2008; Smith 2008).”

“Covariational thinking involves stating the rule based on which two quantities covary, while emphasizing coordinating the underlying rate of change and conceptualizing the dependency between quantities as invariant across quantity values (Smith, 2008; Thompson & Carlson, 2017).”

“x-value, thus creating a correspondence between x and y (Smith, 2008).”

“Smith (2008) defines functional thinking as “representational thinking that focuses on the relationship between two (or more) varying quantities, specifically the kinds of thinking that lead from specific relationships (individual incidences) to generalizations of that relationship across…”

“Functional thinking lessons are expected to facilitate students to identify relationships between variables that are presented through numerical and figural patterns, such as re-cursive, co-variational, and correspondence relationships (Smith, 2008; Stephens et al., 2017).”

“Smith (2008) proposed three types of relationships: (a) recursive patterns, which focus on the variation of just one of the variables and finding its values from others previously obtained; (b) correspondence, in which the values of pairs of independent/dependent variables are correlated; and (c)…”

“Instead of emphasizing the simultaneous change, as in covariation, correspondence thinking underscores the direct association or pairing of values between the two variables (Pittalis et al., 2020; Smith, 2008).”

“Many authors stress the importance of hierarchy in the level of meaning of used representations in the process of developing adequate mental images (Hiebert and Carpenter 1992; Sfard 2000; Smith 2006; Cooper and Warren 2011).”

“Individuals reason covariationally when they envision that the values of two quantities vary simultaneously (Johnson, 2023; Smith,2008).”

“These findings suggest that non-ASD students can apply broader contextual insights to new situations and integrate them with abstract reasoning to tackle pattern tasks and ultimately identify correspondence relationships (Smith, 2008).”

“In this case, they might demonstrate the covariation approach, i.e., how the change in one (the independent variable) produces a change in the other (the dependent variable) or the general relation between variables as in the correspondence (Confrey & Smith, 1994; Smith, 2008).”

“After coding the identified episodes, (ir)regularities between the essential practices of generalizations (Blanton et al., 2019) and the modes of functional thinking (Smith, 2008) used in the interaction were searched for.”

“In addition, as different representations highlight different aspects of concepts (Smith, 2008), having a range of representations and being able to move both within and between them can lead students a conceptual understanding of the distributive property (Barmby et al., 2009; Goldin & Shteingold,…”

“Por ejemplo, Smith (2008), y Warren (2005)”

“Moreover, other studies have focused on the pattern generalization related to the development of algebraic thinking and functional thinking (Blanton & Kaput, 2011; Kaput et al., 2007; Markworth, 2010; Oliveira et al., 2021; Smith, 2008; Wilkie & Clarke, 2016).”

“En este sentido, Smith (2008) menciona que, en problemas relacionados con las funciones, los estudiantes se orientan creando una tabla para registrar las entradas correspondientes para los valores de las variables de interés, situación que también sucede con los futuros profesores.”

“Smith (2008), based on a paper by Confrey and Smith (1991), defined three types of relationships: (a) recursion, which entails identifying the pattern of change in a series of values, i.e., that describes the variation in the values adopted by a single variable; (b) correspondence, in which a…”

“For their part, the relationship of covariation and correspondence does refer to a relationship between variables, commonly called functional relationships (Blanton et al., 2011; Confrey & Smith, 1991; Smith, 2008).”

“In understanding the characteristics of the square root relationship, these two forms of reasoning supported one other and were both significant for engendering functional thinking (Smith, 2008).”

“FT can be understood in different ways (e.g., Doorman & Drijvers, 2011; Freudenthal, 1983; Smith, 2008).”

“Thus, a student demonstrates functional thinking when focused on describing the relationship between two or more covariate quantities (Confrey & Smith, 1991; Smith, 2008).”

“Se trata de un proceso cognitivo que se centra en establecer relaciones entre dos o más cantidades que varían simultáneamente (Smith, 2008).”

“Functional thinking – which emphasises the relationships between two or more variable quantities – is regarded as a fundamental component of mathematics and has applications in numerous other fields as well as in real-world settings (Bayazit & Aksoy, 2013; Frey et al., 2022; Smith, 2003).”

“They are using the definition from Smith (2008): Representational thinking that focuses on the relationship between two (or more) varying quantities, specifically the kinds of thinking that lead from specific relationships (individual incidences) to generalizations of that relationship across…”

“Correspondence is the rule established between pairs of values for the two variables (Confrey & Smith, 1991; Smith, 2008).”

“The development of functional thinking can emerge in fundamentally different strategies (e.g. Smith 2008), which can be labelled as recursive, co-variation, and explicit or correspondence: recursive : Consecutive values of the dependent variables are calculated by attending to the difference of the…”