Consecuencias de la reforma

In this study, I focus on learners’ mathematical routines. The empirical data shows learners can express their routines verbally. A challenge to communication however, arose for learners in the ways that they communicated the meaning and their use of specialised

mathematical symbols and keywords. Formal notation which characterises work in function, such as f(x), is a visual mediator. These are visible objects, symbolic artefacts, created and operated on as part of the process of communication (Sfard, 2008). They hold definition attached to them by formal mathematics, and they can symbolise or represent a mathematical object, but they also hold the meanings that learners attribute to them. The broader literature points to how learners conceive of function and what they do within the context of the formal, established structures on function, of which definitions and notation are a part. The formality that learners encounter in school mathematics focuses on literature about notation, keywords and definitions.

One of the purposes of formal education is to prepare learners for the formal thinking that is available to others (Ayalon, et al., 2011). Learners can benefit from explicit effort to promote their understanding of function notation (Carlson, et al., 2008). Translating this to discursive contexts means to prepare learners to participate in the school mathematical discourse. Literate or formal mathematical discourses have among their salient characteristics, a heavy reliance on written symbols and an arsenal of algorithms for making use of the special notation (Sfard, 2008). Different notations can grow different conceptualisations, ranging from the process conception to the object conception (Watson & Harel, 2013). For example, in school

mathematical discourses, f(x) can be seen as representing a process, where, for an input x into a function f, there is an output f(x). This is the meaning attached to the symbols of the visual mediator, 𝑓(𝑥). By contrast, f can also indicate an object; no process being suggested by its symbol. Notation is used interchangeably in texts and curriculum materials, and contributes to the sense of confusion learners communicate about the function object. This duality of meaning

63 was discussed earlier in the way that the words ‘expression’ and ‘equation’ are conflated.

Research shows that function notation is particularly problematic for learners whose previous experience has led to limited sense of what letters symbolise (Watson & Harel, 2013). In such cases, learners can interpret literal symbols as shorthand labels for objects (Kucheman, 1981; McNeil & Weinberg, 2010). The letter f could be taken to stand for ‘function’. This raised a flag for this study: how do learners interpret f or 𝑓(𝑥)?

A weak understanding of functions has been observed in learners’ inability to express function relationships using function notation, where weaknesses showed in not knowing what each symbol in an algebraically defined function means (Carlson, et al., 2008). In highly procedural orientations, similar to classrooms in this study, Carlson (1998) found that learners’ weak understanding of functions could be linked to poor ability to express a function using algebraic symbols and function notation from function values. A discourse that connects symbols with meaning is essential for thinking and learning. Dubinsky & Wilson (2013) mark a transition between iconic and abstract symbols. They say that learners are to begin symbolising iconically, using pictures or diagrams from their real-life experiences. So symbolism links from learners’ informal discourse. These express features which learners see, or are familiar with. These can then be replaced by the abstract symbols used according to convention in formal mathematics. This work resonates with commognition, as it talks of mathematising as moving from ordinary talk (people talk) to the regimented (feature talk) of mathematics. Slavit (1997) has argued that learners should have a proceptual understanding of function notation. Proceptual implies a deeper understanding of function than that which results from just an action or a process orientation. This means that learners should be able to understand notation as cueing both an action and an object, and should work with them flexibly. How do learners interpret and use function notation required in school Mathematics?

Another important component of formal discourse is the mathematical definition or endorsed narrative. Vygotsky states that “the development of the scientific concept begins with a verbal definition. As part of an organised system, the verbal definition descends to the concrete; it descends to the phenomena which the word represents” (Vygotsky, 1987, p. 168). In South Africa, like in most countries, our ‘scientific concepts’ or formal mathematical knowledge, is specified in the NCS. The requirements, order, emphasis and approaches to the formal

64 knowledge on function differ from country to country. The list below is intended to contrast two other approaches with our own.

1. The UK has an informal approach to function. Formal narratives are reserved for the final year of school (year 12) to prepare those intending advanced study in mathematics

(Ayalon, et al., 2011).

2. In the US, textbooks reflect the curriculum and begin almost immediately with the definition of function related concepts, such as relation, function, domain and range (Watson & Harel, 2013).

3. In South Africa, learners encounter the formal notation, as part of pedagogical strategy involving ‘function machines’, where, given a domain, learners substitute into the algebraic expression given, to find the values of the range. Learners’ transition from expressions that initially read as y= ... to f(x) =... (DBE, 2011b). The transition, the significance of different forms of representation, are not emphasised in curriculum, its associated documents, classroom texts and learning programmes in schools.

Not frequently found in research is the importance and the role of objectification in learner communication, as this connects to the formal narrative and explicit approaches to teach formal narratives and notation in learning. From Vygotsky (earlier quote), formal notation and narrative are as important as the automation and embodiment of process for learners to develop a more objectified discourse. Formal mathematical discourse catalyses or bridges the space

between the informal and formal, serving as the link between them. It is noted already that this is particularly difficult in building a discourse on function, as learners have no prior informal experience with the object. As Nachlieli & Tabach have noted, “function has no spontaneously developed precursor-no mathematical predecessor” (Nachlieli & Tabach, 2012, p. 11). More and more research appears to be filling and at once characterising the transitioning space. In South African secondary education, in practice and curriculum documents, it would appear that the learning of a formal narrative coupled with the emphasis on the relationship between variables, is not described explicitly, leaving the teaching of formal narratives at the discretion of the teacher. The context of rampant poor performance could contribute to the formal narrative being deemed too difficult to learn, and hence, will not command attention in teaching.

65 Typical studies over time regarding the learning of formal definitions, show that to

describe learners’ understanding of a mathematical concept, especially to describe initial understanding, focus should be placed on the actions on the concept, its properties and

representations and not so much on the definition (Bloedy-Vinner, 2001; Ronda, 2009; Vinner & Dreyfus, 1989). Surprisingly, within the discursive realm, research shows negligible influence of definitions on learners use of words (Nachlieli & Tabach, 2012). This is probably due to

classroom language not being just a list of technical terms or a recital of definitions, but which involves the use of these terms in relation to each other, across a wide variety of contexts (Lemke, 1990). Venkat & Adler (2012) have suggested the important connection of the initial representation with subsequent transformations and resulting representations. This can be interpreted as the formal definition and the meaning of symbols in the symbolic representation, as having place in what the study calls ‘resulting representations’. The formal definition provides a means for encapsulating the various disparate and compartmentalised discourses of the

different representations of a function. Extending the role of the formal definition as facilitating abstraction is seen in studies which regard defining as responsible for learner beginning to appreciate abstractedness and to stop learners from relating abstract mathematical objects to a specific concrete things (Nachlieli & Tabach, 2012). They suggest combining formal narratives with symbols and examples as an optimal approach. Such research suggests that formal

definitions transition learners from the everyday discourse to the formal mathematical. The same research details this process as the learner participating in the new discourse of the formal

definition, by relying on previously informal use of words. In ‘recycling’ their old uses in the developing formal discourse, learners are not generally aware of inconsistencies with the formal definition (Nachlieli & Tabach, 2012). Consistent ‘recycling’, it appears, produces changes that bring learners closer to the formal. Does this perhaps suggest a means of navigating the

transitioning space between informal and formal?

Work connected to the notion of recycling and reworking words, refers to inconsistencies which arise particularly out of learners’ colloquial use of words to describe what they saw in graphs and how they regarded the object function as well. A keyword, discontinuity, for

example, has been found to be frequently used to describe a ‘gap’ or a ‘hole’, or as a graph that ‘jumped’. Conceptualisations based colloquially in ‘holes’, ‘jumps’ and ‘poles’ have been noted in literature to lead to misconceptions in more complex mathematics, such as the defining of the

66 derivative (Carlson, et al., 2008). Without the formal definition of the object function, and

developed discourse of its meaning, learners regarded discontinuous functions as ‘weird’ or strange. Non-calculable functions are seen as aberrations (Watson & Harel, 2013). The absence of a formal, mathematical definition for the object function showed that learners tend to identify function with one of its representations or realisations, either the graph or algebraic formula (Even, 1992; Leinhardt, Zaslavsky, & Stein, 1990; Sfard, 1992), seldom a table (Dubinsky & Wilson, 2013).

In support of the work above it was found that the word ‘function’ is taken to refer to the algebraic formula in one context and the graph in another, seldom related to or represented by both of them at the same time (Nachlieli & Tabach, 2012). Rather than creating a unified discourse of these representations, learners develop a collection of disparate unrelated

discourses, involving a key word which may be used in different ways on different occasions (op.cit). This study was able to examine the role explicit definitions functions have in connecting and unifying discourses. This is the context of literature in which learners who were able to reproduce the definitions were found to act in ways that contradict the definition (Nachlieli & Tabach, 2012). Could this be a result of ritualised learning of a formal definition, without the reasoning required, or a connection to be established between the object and the definition? The literature cited above suggests a lack of means to connect representations of function to each other, and to the formal defining narrative. Based on this, the question arises as to whether this is a pedagogic imperative. Along with this, could learners without a formal definition of function or formal means to deal with the object be able to make these connections on their own?

A consideration of pedagogic approaches is likewise not so straight forward. The

presentation of an objectified discourse, the idea of function as an object, a ‘thing’, if introduced too early remains beyond the comprehension of many students (Sfard, 1992). I argue that there are ways of thinking in formal mathematics that must be learned together with the mathematics. Paying attention to that transitioning space where learners build connections between objects already existing and new, and between the informal and the formal means of communicating about these, is a highly complex task for both teaching and learning. Commognition calls this meta-level learning. Mostly it develops by participation in the discourse with a knowledgeable other. It is unlikely that learners will develop the narratives that exist in formal mathematics on their own or stumble on these ways of working. Formal mathematics need not be the singular,

67 driving goal of learning mathematics, but rather it can be used to transition from informal to formal discourse, encapsulate related objects by their similarities, and condense and simplify ways of working mathematically, thus providing means for learners to explore mathematics and escape the hold of ritualised practice.

The learning of function is indeed complex. The span of literature covered here and the perspectives examined and synthesised were aimed to develop my understanding of how learners develop a discourse on function, particularly in the back grounding of a formal definition of function as in the SA curriculum. Research in the field has helped focus my investigation into learner discourses, bringing into focus the need to establish discursive connections on the multiple levels established here. Despite this wealth of research, none has yet engaged with explicit pedagogic approaches to do so. Commognition however, is based on learners building onto preceding discursive layers to learn; confirming it as a prudent choice of framework. With certain literature as discussed above having drawn my eye to the necessity of connection, it becomes important to examine further literature on multiple representations as they are relevant to function.