section now examines the discourse of these routines so as to classify them as part of a
ritualised or exploratory practice. This is the second layer of description. The pie charts below show the accumulated frequencies of ritual and exploration codes embedded in learner
utterances involving the algebraic representation, in the three performance groups:
Figure 13 Exploration and ritual per performance group.
The pie charts above were not surprising. It was expected that instances of ritualised discourse would far outweigh that of exploratory discourse in the data. As stated, new routines begin as rituals and gradually turn into explorations through a process of individualisation.
127 Individualisation is a transitory phase in ritualisation, where a learner participates in routines as part of a collective, most likely with a teacher and other learners in a classroom. Ritual is the form that routines take in Vygotskys’ zone of proximal development (ZPD), where this process of individualisation, mentioned above, is termed rationalisation. This asserts the claim made by commognitive frameworks that rituals are a necessary “developmental precursor to explorations” (Sfard, 2008, p. 223). Discursive theory from Vygotsky and Sfard are additionally supported by this study’s initial aggregation of codes. At the outset, this study acknowledges the importance of the contexts of learning in the six schools involved. Factors exist in each of the schools and in the cohort which contribute to this overwhelmingly ritualised aggregate. This was discussed in Chapters 1 and 4. They speak to learning contexts that are typical of most South African schools: the discourses of teaching and learning in school Mathematics, the nature of the school
Mathematical discourse, which has as emphasis the ‘how’ of a routine at the expense of the ‘when’, and the resources that are available to learners are some typical contributors.
Routines begin initially as a loose collection of realisations, which are individualised into an integrated discourse. Exploratory talk will often show learners are able to group realisations and routines which are related to each other. What was interesting in this study was that learners from all performance groups showed instances of exploratory discourse, despite the antagonistic factors to learning and the dominance of the ritualised practice of school Mathematics. How could these exploratory utterances be characterised across performance groups? A focus on PG3 communication helped this characterisation, by providing the least complicated view into learner thinking. The incidence of exploration occurred less frequently in this group, and these
incidences could be reflected on to distil the most basic of exploration activity. PG3 showed exploratory talk in identification of objects and symbols, in objectified ways, as opposed to the relations of routines pertaining to objects. This was indication that certain rituals related to naming and a limited number of procedures were being reified. While this exploratory portion was small, and starkly depicted on the pie charts above, its occurrence was interesting in provoking investigation into the nature of this participation as it pertains to our poorly
performing learners. Certainly, it can mark a significant point on which to build successive layers of discourse for these learners. Where, and around which aspects of the mathematical discourse do these instances arise? The significance of this for the broader study permits a view of the nature of the exploratory discourse in each performance group, and how it thus compares across
128 performance groups. A broad description of the ritualised discourse is discussed first as it provides a window into the exploratory discourse.
Table 3 Ritual codes in each performance group for the algebraic representation.
Ritual: PG1 PG2 PG3 Total of codes
classified ritual Total number ritual codes for the algebraic
representation
701 1096 1122 2919
Total percentage 23.43% 36.63% 37.50% 97.56%
PG1 show the lowest number and proportion of ritualised codes on the algebraic representation compared to the other two groups (as shown on the table and pie charts above). This is because they were able to say more with fewer words through compression of their talk with symbolisation and reification. With a wider repertoire of routines, they could thus pick the most economical discursive means for the task required without being ambiguous. Generally, PG1 talk showed they were able to realise far more from the signifier than the other groups. Their utterances showed greater frequency of objectified talk. It ought to be recalled here that an object is a discursive construct which permits the connecting of processes, properties, routines, ‘things’ which are all linked to each other by a specific criteria. The word ‘function’, as a noun, or one such ‘thing’, subsumes all communication on algebraic expressions, graphs and tables representing a one-to-one or many-to-one relationship between the variables involved. Using key mathematical words in such objectified ways enabled PG1 to say more with less.
The objectifying potential of the discourse of PG1 is further supported by fewer
subjectifying utterances noted in this group. Subjectifying utterances places the learner as central to the mathematics, as opposed to having the mathematics stand alone as talk of objects. School M’s, PG1 and PG2 illustrate this, when responding to the question: “do you see a function in 𝑓(𝑥) = 𝑥2 ?” Both PG1 and PG2 could identify the symbolic expression as a function, named the expression, as representing a parabolic graph or being a quadratic, in five or fewer utterances, between the pair of learners involved. PG3, in contrast, took a 23-utterance exchange.12 This suggested early on in the analysis that better-performing learners were more adept at using the
12 Appendix 6 - Transcript for school M, showing utterances across performance groups, particularly PG3 M104- M127.
129 compressing apparatus offered by formal mathematics. The summary of the exchange of 23 utterances can best be articulated in the M-PG3 learners’ written response: ‘using a table method, a graph can be formed from x and y’ (Appendix 6). Transformation of the algebraic expression to table of values to the graphical representation was identified as a dominant ritual of PG3, and showed an entirely procedural orientation to the algebraic expression. Learners
generally took any algebraic expression provided to them and made y the subject of the formula, resorting to the completion of a table of values, which was then graphed. Once graphed, and solely contingent on the possibility of graphing, the equation was classified as either a function, or not. Confirmed by Learner S in performance Group 3 of school M, utterance S114 (denoted in short by M-PG3-S114): ‘...it has to be a function because a function has to be a graph’. Subsection 5.3 of this chapter examines learners routines for discernment of a function.
In summary, the most frequent ritual across particularly the poorly performing groups related to being able to sketch the graph from the algebraic expression. They showed a strong processual orientation to the object. The graph was the critical condition defining a function relationship for these learners. The next section presents a broad description of what exploration codes show. The nature of the exploration routines of PG3 is again broadly discussed as it mimics the foundational exploratory routines of the better-performing groups. Table 4, which follows, shows the aggregation of exploration codes characterising the utterances across performance groups. It provides detail and supports the pie charts presented earlier. Table 4 Exploration codes in each performance group for the algebraic expression.
Exploration PG1 PG2 PG3 Total Exploration
Codes Total number codes indicating exploration for
the algebraic representation
35 28 10 73
Total percentage 1.17% 0.94% 0.33% 2.44%
PG3 held a portion of the exploratory codes. Earlier in this chapter, the question arose about ‘where and around what aspects of mathematical content these instances arise’. The coded utterances show exploration routines in PG3 revolving around talk:
130
that connected different representations-equivalence of the algebraic expression and the graph (4 codes) ;
of specific features of functions (4 codes);
giving meaning to what symbolic representations signify (2 codes).
To illustrate this, two instances from the data have been chosen below:
1. T-PG3- B274: (points to equation on Card D: 𝑦 = −2𝑥 + 3): ‘...the equation of a straight line graph, its y equals mx plus c. And then here, ja here [sic], this is c, this is m, which is the gradient.’
This utterance B274 involved Learner B seeing equivalence in the graph and the equation (E4). The specific arrangement of algebraic symbols on Card D signified the graph of a straight line to the learner. The utterance showed further that the algebraic symbols m and c signified independent entities, where m was decribed as the ‘gradient’. This was coded exploration code (E5) as ‘gradient’ was referred to as a noun, an entity. The symbol held meaning for the learner. Identification of objects is regarded as low-level objectified talk within the commognitive framework (Sfard, 2008).
2. T-PG3- B313: (refers to Card P: (𝑥) = 7) :‘ ... the gradient, its zero because... the graph is now in line with, with, with, with x’[sic] . Then B315: ‘it’s, it’s, it’s parallel to, to x... ja [sic]. It’s parallel to x. [The] x-axis.’
Here we see learner B, realising relevant and key features of the graph from the equation. The algebraic expression on Card P signifies a line of zero gradient, parallel to the x-axis. This was coded E3.
PG3 showed restricted exploration presence in codes in the categories of mathematical objects, and in flexibility in working with the linear function specifically. It is worth noting that the codes present in PG3 were common to the other groups. PG3 codes relating to their goals in mathematics, the applicability of routines, who the mathematics is addressed to, and reasons for accepting narratives given or derived- remain ritualised for this group (discussed in 5.3.2). The zoom out sets the background of the study with broad discussion of the rituals and explorations which exist in the dynamic interaction between pairs of learners when they
131 communicate about the object. Finer details of routines emerge as they are zoomed in on, and as the chapter progresses. The broad interpretation so far is that discourse around the algebraic representation is, in the main, ritualised across groups. Learners in all performance groups tend in specific and similar instances towards a more objectified and exploratory discourse. These instances are located most frequently when learners worked with the linear function. The objectified utterances pertained broadly to three areas:
identification of the linear function from the arrangement of symbols in the expression;
identification of the features of the function; and
showing equivalence between the algebraic expression and the graphical representation. Despite these limited occurrences, they still enable a starting point for a more nuanced description of learner participation in the discourse on function. Importantly, they identify, particularly for poorer-performing learners, a starting point on which teaching can capitalise. The presence of exploratory codes based on how learners at different performance groups were communicating, opened the analytic lens for classification of the exploratory codes. The codes for ritual and exploration as they occurred in learner utterances at different levels of performance can now be examined.
5.3 Frequent routine codes related to performance across schools: ‘zoom out’