Early literature attributes the development of the object function to the need to show variation (Ayalon, Lerman, & Watson, 2011; Bakar & Tall, 1991; Bloedy-Vinner, 2001; Carlson, 1998; Carlson, Jacobs, Coe, Larsen, & Hsu, 2002). The early work of Sfard saw function born as a result of a long search after a mathematical description for physical phenomena involving variable quantities (Sfard, 1991). This seemed to be the common discourse of the time, e.g. where function was seen as being the first tool for dealing with changing, rather than with constant magnitudes (Dubinsky & Harel, 1992). More recently, discursive literature marks the complexity of the transition from algebraic discourse, with its own objects and rules, to the notion of function that “is new to learners and for which much meta-level learning has to occur before learners gain reasonable command of the commognitive activity” (Ben-Zvi & Sfard, 2007, p. 135).
Unlike algebra with its links to arithmetic, in function, learners have little informal preparation on which to build a new discourse, no preceding discursive layer. Possible previous experience could have involved predicting terms of a sequence, where the relationship between variables is backgrounded. Learners essentially work with a single set of numbers. Thus I would agree that, the new discursive object, function, has little or no spontaneous mathematical experience from which to be developed (Nachlieli & Tabach, 2012). In addition, because of the circularity of discourse development and object construction, effort from both the teacher (knowledgeable other) and the learner, must be invested in coming to grips with an unfamiliar,
57 abstract mathematical object (Ben-Zvi & Sfard, 2007) called a function. My teaching experience shows that learners come to the mathematical classroom with ‘everyday’ spontaneous, amalgam meanings of the word derived from English and Natural Science classrooms. The extension of these meanings to a formal mathematical narrative is not a direct translation from a purely concrete meaning to an abstract mathematical one. This is especially difficult in multilingual contexts, where English as the language of instruction is not the first or second language of many our mathematical learners. Learners are expected to engage a conversation on functions as a precondition for objectification, before they build realisations of the object or are able to relate the object to subsuming discourses (Ben-Zvi & Sfard, 2007). The discourse on functions subsumes the discourse on algebra.
To reiterate, this study examines how the discourse on function has developed for a group of Grade 11 learners in terms of their discursive routines. Research into teaching and learning in function reveal two main themes: theory to analyse how learners understand function as a concept, and explanations for the difficulties learners have in learning function. There is much research rooted in an acquisitionist perspective. Many learners believe that all functions should be definable by a single algebraic formula (Carlson, Oehrtman, & Thompson, 2008). Learners are found to have difficulty distinguishing between an algebraically-defined function, and an equation (Carlson, 1998). This is attributed to the many uses for the equal sign and that teachers and the curriculum refer to a formula as an equation (Carlson, et al., 2008). Students are shown to benefit from explicit efforts to help them distinguish between functions and equations (Carlson, et al., 2008). Even (1990, p. 530) looked at the definition of a function as “every element in the domain corresponding to exactly one element in the range”, and has observed that learners do not understand the meaning of this definition. They can recite the definition correctly, but they could not explain what it means, nor why function is defined the way it is. Ronda (2009, p. 34) has called definitions an ‘abstraction of a concept’. She has further noted that “knowing the definition of a concept does not necessarily translate into an object conception” (op cit, p. 34). Reciting a formal definition, therefore does not guarantee that the learner has objectified. This is only a partial development of the discursive object. The current study examines learner discourse on function and asks, if through related processes on the object or through the recognition of its features, have learners objectified function?
58 Earlier constructivist research shows that a primarily procedural orientation to using functions to solve specific problems lacks meaning and coherence for learners (Carlson, 1998). In discursive terms, this relates to learners knowing the how of a mathematical routine, and not the when. Learners who think about functions only in terms of symbolic manipulations and procedural techniques are unable to comprehend a general mapping of a set of input values to a set output values; they also lack the conceptual structures for modelling function relationships in which the function value (output variable) changes simultaneously with continuous changes in the input variable (Carlson, 1998; Carlson, et al., 2008; Monk & Nemirovsky, 1994; Thompson, 1994a). This research showed that the strong emphasis on procedures without accompanying tasks to develop deep understanding of the concept has not been effective for building
foundational function conceptions. Tasks should allow for meaningful interpretation and use of function in various contexts. Carlson et al. (2008) showed that learners, when probed to explain their thinking, typically providing some memorised rule or procedure to support their ideas. They appear to resort to memorised facts to guide their explanations (op.cit). Such literature raised questions as to how learners work with variables in the current study.
To describe this object or function that learners find problematic, other cognitive-based research attempts to develop a conceptual map for functions (Ayalon, et al., 2011) and
emphasises the function concept as foundational in advanced study of mathematics. Notions of variation and the particular relationship between variables theme such research. The importance of understanding function through concept maps, was an attempt to understand how learners’ see function at various stages of its development. To this end, Ronda (2009), developed growth points, which are ‘big ideas’ of learners understanding of function. These provide a way to monitor and analyse learner understanding of functions in equation form. Growth points describe learners’ big ideas in terms of strategies, knowledge and procedures that learners apply in
working with tasks and problem situations (Ronda, 2009). While cognitively based as the development of schema, growth points (GP) deviate from the order of knowledge and skills learners are to acquire as specified in the NCS. Primarily, this occurs as a result of the process orientation of the South African curriculum. Growth points are helpful, as they flag transitions in learning about functions. Growth points are evident when learners’ progress:
1. from generating values from the equation (GP1);
59 3. to equations describing properties of relationships (GP3); and
4. to functions as objects that can be manipulated and transformed (GP4).
Growth points 1 to 4 listed above are restricted to working with algebraic expressions for functions only. I have restricted myself to growth points of this representation as it was the representation where learners in the current study had the most to say. It has been interpreted to mark transitions in learning that the broader literature had already uncovered. It made it possible to see if the learners in my study progressed through these growth points in the order suggested, and how these transitions showed in an objectified discourse.
Certainly, growth points have an advantage in a curriculum with an orientation to process and assessments such as the NCS. It is the context of continued of poor performance and poorly prepared teachers that mathematics is recontextualised in our classrooms. Under these (and other) sinister constraints, we have to question whether it offers significant difference from what we have already been doing. Growth points offer transitions for objectification through process or properties. How is it possible to tell that a learner has objectified or shown a mastery of process at each of these levels? This challenge to operationalisation brought on by the many acquisitionist studies reviewed, became a research imperative of my study. Would learners in my study cohere to the order specified by growth points in their knowledge of functions? If not, what did this show about our transitions in developing discourse on function? Would an examination of discourse for objectification show that learners could ‘unpack and repack’ through the levels flexibly?
Broader conceptual approaches examined function across multiple representations and developed constructs to describe and view learning. The earliest acquisitionist based work I could find that described learning in ‘layers’, to contrast the discursive approach used here in describing layers of formal and informal discourse, was that of Freudenthal (1978). Coming from a process-object conception of learning, learning through engaging process at one level produces what learners will observe at the next level (op cit). Similarly, other acquisitionist approaches classified mathematical concepts as operational and structural. ‘Operational’ suggests
approaching a concept as a process, whereas ‘structural’ suggests approaching a concept as an object (Dubinsky & Harel, 1992; Sfard, 1991). Computation leads to a process conception which can later be encapsulated as a mental object (Sfard, 1991). By process-object theory, an object
60 conception is attained, generally after experience has been gained performing actions on the object (Ronda, 2009). Another route to objectification, lists understanding the properties of functions (Ronda, 2009; Slavit, 1997). Here, learners conceive of a function through the properties functions have or do not have. This characteristic was described in the section on algebra. It recurs under function as a result of the subsuming nature of the discourses. Slavit (1997) zooms further in on properties to classify them as global or local properties. Global properties pertain to class of functions, for example, in the way that symmetry can similarly be described for all quadratic functions; local properties result from properties of individual or a selection of ordered pairs of the function (op cit). Perceiving a function as an object, in school Mathematics, brings function closer to learners’ experience, when we consider the abstract definitions of functions, like the set-theory definition, which is removed from learners’ experiences (Sfard, 1992).
Another dichotomy noted in literature is that between a global approach on the one hand, and a pointwise approach on the other, when it comes to working with functions (Even, 1998; Monk, 1988). Pointwise means to plot, read or deal with discrete points, as distinct from the process, operational and local conceptions described above. A global approach looks at the behaviour of a function. It seems, similarly, to mirror the structural, object, and global approach to function described in literature. Both Slavit (1997) and Even (1998) use the term ‘global’ in similar ways. This entails examining a function as an object, with properties or features, and which behaves in ways that are describable and predictable. While both these approaches have their strengths, the global approach appears to give a better and more powerful understanding of the relationship between the graphical and symbolic representations (Even, 1998). It is also a stated goal in the curriculum. Learners would need to reverse algebraic processes and have flexibility across all representations (Ronda, 2009). This is synonymous with the packing and repacking, or working forwards and backwards, as previously described. Ronda’s growth points 3 and 4, mentioned above, seem to develop from a global understanding of a particular
representation. In doing this, learners begin to explore dynamic function relationships, with regards to how one variable changes while imagining change in the other (Carlson, et al., 2008). Individualising such changes means to be able to discuss and interpret the changes to important features of the graph and its shape (op cit). The South African Mathematics curriculum includes the global conception of functions as part of its topics in Grade 11, relating to the
61 transformations of functions, where changes of critical features occur under transformation. The questions that lead from this discussion are: from learners discourse on function, how can I describe their notions of function? Are learners oriented to process or object? Do they have a global or pointwise view of functions from school Mathematics? And, what implications do these orientations have for learning function?
The discussion of the South African curriculum earlier, showed an orientation to process, through emphasis on symbolic manipulation and procedural techniques. Research below shows that this can account for learner difficulties with constant functions (e.g., y=5), also known as special or univariate functions. Research over two decades reports that these are not considered as functions because they do not vary (Bakar & Tall, 1991; Carlson, et al., 2008; Dubinsky & Wilson, 2013). Carlson et al. (2008, p. 12) have shown that ‘learners carry out rote procedures’ when asked to solve these special functions. Earlier Carlson (1998) found 7% of A-students could produce a correct example of a function where all output values were equal to one another. This may indicate learners still stuck in a pointwise approach. Venkat & Adler (2012) showed a ‘block’ in flexibility, when learners have access to a particular method, which did not provide ways of dealing with the special cases of horizontal and vertical lines, as it does not allow learners to focus on the features of the input or other representations. Explanations supporting this idea of reliance on one route or lack of connection between routes is supported (Ayalon, et al., 2011). Within a rigid procedural orientation, Dubinsky & Wilson (2013) explain that learners expect a one-to-one correspondence between variables before they are able to see a functional relationship. The criterion of a change in one variable resulting in a change of another is a necessary condition for recognition of a functional relationship for learners. This notion appears as problematic for learners to individualise across all the research surveyed.
It is clear from the review of literature that common problems persist in learning function despite great theoretical strides in pedagogy. Given the work, describing the difficulties learners have with the idea of variation, it became important for me to locate my enquiry in learners’ mathematical routines with functions required in school mathematics. The connections between the function as an object and the processes involved appear to reside in a transitioning gap. Literature suggests that learners could objectify through two routes, namely through reifying processes, or through the features and properties of functions. When we understand as teachers and researchers what learners know through the way in which they communicate what they
62 know, we can work towards developing an approach to function discourse that makes
objectification, hence abstraction and complexity, possible. It is worthwhile asking whether a focus on learner routines in function would illuminate their thinking around the transitioning space from informal to formal function discourse; from algebra to function; from the various representations of function to the object itself; and from the individual features and properties of various functions to a class of function.