Multiple representations

Much of what is written emphasises the separateness of the discourses around the representations studied in secondary school, namely: the algebraic representation, the table and the graph. It has been found that learners have difficulty making transitions from one

meditational mode to another or from one representation to another. Tables, graphs and expressions might be multiple representations to us, but there is no evidence they are multiple representations of anything to learners (Thompson, 1994b). As emphasised in prior sections, there are two necessary discursive connections that have to be sought: first, the connection and transformation across the different representations themselves, and second, the connectedness of the representations to the new subsuming discursive object. The dominance of literature from an acquisitionist/Piagetian paradigm mostly examines the flexibility of moving between

representations by transforming from one representation to another. It flags, for the

participationist/discursive researcher, useful points of contention in learning, where dominant research paradigms are unable to adequately account for learners’ poor performance. A discourse connecting representations to subsuming discourse of function is scarce in extant literature.

68 Regarding flexibility in working with the different representations, Ronda (2009) found that a full understanding of the concept function necessitates the understanding of, and the ability to work with each of the representations. This suggests conceptually and discursively, seeing the equivalence of the representations, and hints at the larger object. Other similar work suggests that the ability to identify and represent the same thing in different ways, and flexibility in moving from one representation to another, allows learners to see rich relationships and develop a better conceptual understanding, which broadens and strengthens one’s ability to solve problems (Even, 1998; Slavit, 1997). This extends and links flexibility to problem-solving. The ability to solve problems and develop in abstraction is a goal of mathematics. However, the discrete packaging of the different representations, which best describes the pedagogy of school mathematics, does not necessarily enable the notion of equivalence among representations to develop. Couple this with the absence of the relationship notion, could account for function being seen in a single representation only. This severely curtails the opportunity for learners to solve problems as complexity increases and may account for the poor performance in function described in Chapter 1.

Learning functions is not simple; there are multiple layers for learners to connect when dealing with this abstract mathematical object. Sierpinska (1992) found learners have difficulty making sense of covariation, that is, seeing function as the rate at which one quantity changes with respect to another. Multiple meaning attached to symbols, in an expression or equation, adds to the complexity. For example, learners’ early experience with equations involve equations not as a function, but as a statement where one quantity equals another involving a single

variable. The equals sign is interpreted as a signal to ‘do something’, or ‘perform an operation’, rather than denoting a relationship of equality between the expressions on either side of the equal sign (Kieran, 2007). When a function is represented by an equation, it shows a relationship between two quantities or an arrangement of algebraic symbols, which can be manipulated and transformed. The dichotomy of process-object at this point becomes critical, depending on which receives emphasis. The arrangement of symbols in an equation conveys conceptual knowledge and possibly an object conception (Ronda, 2009). The parameters of the equation themselves signify entities which can be used to reason (Kaput, 1989). However it had been shown that learners are not acquainted with the roles of parameters in different representations (Even, 1998). Understanding function from an equation is considered a major conceptual node

69 for learners (Ronda, 2009). An equation would require that a learner develops a discourse of the object in its entirety, but also a discourse of its individual component symbolic parts.

Literature on how to achieve flexibility and develop an objectified notion of function was

difficult to find. To focus on flexibility, Venkat & Adler (2012) talk of producing transformation sequences that connect across representations. Now this goes beyond the emphasis of

transformation sequences within a particular representation. It is wider-arching, in seeking to connect sequences between representations. With transformation in emphasis, particularly in school Mathematics, learners concentrate on moving symbols around, as opposed to connecting the symbols and process to the object represented by those symbols. For this reason, pure process orientations were found to be absent of meaning for learners, who appear unable to offer

interpretations or use function in and across varied representational instances (Carlson, et al., 2008). A process orientation is defined as an understanding of the transformational activity performed on a function (Slavit, 1997).

The literature of multiple representations was scoured for notions of the ways that learners see equivalence. From any view, acquisitionist or participationist, the research largely seems to suggest that working with two representations, particularly the transformation of one to the other, is taken as a sign that learners see equivalence. It is thus important to ask learners to substantiate their thinking, and thus gain access to whether and how the representations remain separate or are related.